It is mandatory for the students to go through all the topics included in this chapter to kickstart their preparation. The list of topics is mentioned below along with the quick links. Understand each topic by referring to every link and solving every problem available below. Find the various methods to solve the questions and choose the best one to practice well for the exam. All the topics such as Simplest form, Subtract Fractions with Unlike Denominators, Find Common Denominators, Estimate Sums, and Differences of Fractions, Add Fractions with Unlike Denominators are prepared with a clear explanation in BIM Grade 5 Ch 8 Add and Subtract Fractions Solution key.

Lesson: 1 Simplest Form

Lesson: 2 Estimate Sums and Differences of Fractions

Lesson: 3 Find Common Denominators

Lesson: 4 Add Fractions with Unlike Denominators

Lesson: 5 Subtract Fractions with Unlike Denominators

Lesson: 7 Subtract Mixed Numbers

Lesson: 8 Problem Solving: Fractions

Chapter 8 – Add and Subtract Fractions

### Lesson 8.1 Simplest Form

Explore and Grow

Use the model to write as many fractions as possible that are equivalent to $$\frac{36}{72}$$ but have numerators less than 36 and denominators less than 72.

Which of your fractions has the fewest equal parts? Explain.

Construct Arguments
When might it be helpful to write $$\frac{48}{72}$$ as $$\frac{2}{3}$$ in a math problem?

Think and Grow: Simplest Form

Key Idea
When the numerator and denominator of a fraction have no common factors other than 1, the fraction is in simplest form. To write a fraction in simplest form, divide the numerator and the denominator by the greatest of their common factors.
Example
simplify $$\frac{6}{8}$$ in the simplest form.

Step 1: Find the common factors of 6 and 8.

The common factors of 6 and 8 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

Show and Grow

Question 1.
Use the model to write $$\frac{2}{4}$$ in simplest form.

Step 1: Find the common factors of 2 and 4.
Factors of 2:   1, 2
Factors of 4:   1, 2, 4
The common factors of 2 and 4 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

Because 1 and 2 have no common factors other than 1, $$\frac{2}{4}$$ is in simplest form.

Question 2.
Write $$\frac{8}{12}$$ in simplest form
Step 1: Find the common factors of 8 and 12.
Factors of 8:    1, 2, 4, 8
Factors of 12:  1, 2, 3, 4, 6, 12
The common factors of 8 and 12 are 1, 2 and 4.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

Because 2 and 3 have no common factors other than 1, $$\frac{8}{12}$$ is in simplest form.

Apply and Grow: Practice

Use the model to write the fraction in simplest form.

Question 3.

Step 1: Find the common factors of 8 and 10.
Factors of 8:    1, 2, 4, 8
Factors of 10:  1, 2, 5, 10
The common factors of 8 and 10 are 1 and 2.

Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

$\dpi{100} \small \frac{8}{10}= \frac{8 \div 2}{10 \div 2} = \frac{4}{5}$

Because 4 and 5 have no common factors other than 1, $$\frac{8}{10}$$ is in simplest form.

Question 4.

Step 1: Find the common factors of 5 and 15.
Factors of 5: 1, 5
Factors of 15: 1, 3, 5,15
The common factors of 5 and 15 are 1 and 5.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{5}{15}= \frac{5 \div 5}{15 \div 5} = \frac{1}{3}$
Because 1 and 3 have no common factors other than 1, $$\frac{5}{15}$$ is in simplest form.

Question 5.

Write the fraction in simplest form.
Step 1: Find the common factors of 10 and 12.
Factors of 10:  1, 2, 5, 10
Factors of 12:  1, 2, 3, 4, 6, 12
The common factors of 10 and 12 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{10}{12}= \frac{10 \div 2}{12 \div 2} = \frac{5}{6}$
Because 5 and 6 have no common factors other than 1, $$\frac{10}{12}$$ is in simplest form.

Question 6.
$$\frac{3}{6}$$
Step 1: Find the common factors of 3 and 6.
Factors of 3:  1, 3
Factors of 6:  1, 2, 3, 6
The common factors of 3 and 6 are 1 and 3.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{3}{6}= \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1, $$\frac{3}{6}$$ is in simplest form.

Question 7.
$$\frac{2}{10}$$
Step 1: Find the common factors of 2 and 10.
Factors of 2:  1, 2
Factors of 10:  1, 2, 5, 10
The common factors of 2 and 10 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{2}{10}= \frac{2 \div 2}{10 \div 2} = \frac{1}{5}$
Because 1 and 5 have no common factors other than 1, $$\frac{2}{10}$$ is in simplest form.

Question 8.
$$\frac{6}{8}$$
Step 1: Find the common factors of 6 and 8.
Factors of 6:  1, 2, 3, 6
Factors of 8:  1, 2, 4, 8
The common factors of 6 and 8 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{6}{8}= \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$
Because 3 and 4 have no common factors other than 1, $$\frac{6}{8}$$ is in simplest form.

Question 9.
$$\frac{7}{14}$$
Step 1: Find the common factors of 7 and 14.
Factors of 7:  1, 7
Factors of 14:  1, 2, 7, 14
The common factors of 7 and 14 are 1 and 7.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{7}{14}= \frac{7 \div 7}{14 \div 7} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1, $$\frac{7}{14}$$ is in simplest form.

Question 10.
$$\frac{10}{100}$$
Step 1: Find the common factors of 10 and 100.
Factors of 10: 1, 2, 5, 10
Factors of 100:  1, 2, 4, 5, 10, 20, 25, 50, 100
The common factors of 10 and 100 are 1, 2, 5 and 10.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{10}{100}= \frac{10 \div 10}{100 \div 10} = \frac{1}{10}$
Because 1 and 10 have no common factors other than 1, $$\frac{10}{100}$$ is in simplest form.

Question 11.
$$\frac{12}{4}$$
Step 1: Find the common factors of 12 and 4.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 4:  1, 2, 4
The common factors of 12 and 4 are 1, 2 and 4.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{12}{4}= \frac{12 \div 4}{4 \div 4} = \frac{3}{1}$
Because 3 and 1 have no common factors other than 1, $$\frac{12}{4}$$ is in simplest form.

Question 12.
Three out of nine baseball players are in the outfield. In simplest form, what fraction of the players are in the outfield?

Step 1: Find the common factors of 3 and 9.
Factors of 3: 1, 3
Factors of 9:  1, 3, 9
The common factors of 3 and 9 are 1 and 3
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{3}{9}= \frac{3 \div 3}{9 \div 3} = \frac{1}{3}$
Because 1 and 3 have no common factors other than 1.
Therefore, players are in the outfield.

Question 13.
YOU BE THE TEACHER
Your friend writes $$\frac{2}{6}$$ in simplest form. Is your friend correct? Explain

The numerator and the denominator has to divide by the greatest of the common factors. You have divided only the denominator.

Explanation for $$\frac{2}{6}$$ in simplest form.
Step 1: Find the common factors of 2 and 6.
Factors of 2: 1, 2
Factors of 6:  1, 2, 3, 6
The common factors of 2 and 6 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{2}{6}= \frac{2 \div 2}{6 \div 2} = \frac{1}{3}$
Because 1 and 3 have no common factors other than 1, $$\frac{2}{6}$$ is in simplest form.

Question 14.
Reasoning
The numerator and denominator of a fraction have 1, 2, and 4 as common factors. After you divide the numerator and denominator by 2, the fraction is still not in simplest form. Why?
Given that, common factors are 1, 2 and 4
For the fraction to be in the simplest form, the numerator and denominator has to divide by the greatest of the common factors.
Here 4 is the greatest common factor. So, divide both the numerator and denominator by 4 to get the simplest form.
For example 4 and 8
Factors for 4: 1, 2, 4
Factors for 8: 1, 2, 4, 8
common factors: 1, 2 and 4
Simplest form:    $\dpi{100} \small \frac{4}{8}= \frac{4 \div 4}{8 \div 4} = \frac{1}{2}$

Think and Grow: Modeling Real Life

Example
A quarterback passes the ball 45 times during a game. The quarterback completes 35 passes. What fraction of the passes, in simplest form, does the quarterback complete?

Find the number of passes that are not completed by subtracting the pass completions from the total number of passes.
45 – 35 = 10
Write a fraction for the passes the quarterback does not complete.

Find common factors of 10 and 45. Then write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

The quarterback does not complete __ of the passes.

The quarterback does not complete 2/9 of the passes.

Show and Grow

Question 15.
There are 24 students in your class. Four of the students have blue eyes. What fraction of the class, in simplest form, do not have blue eyes?
Given that,
Total no. of students in the class = 24
Students have blue eyes = 4
Students do not have blue eyes = 24 – 4 = 20
Step 1: Find the common factors of 24 and 20.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 20:  1, 2, 4, 5, 10, 20
The common factors of 24 and 20 are 1, 2 and 4.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

$\dpi{100} \small \frac{20}{24}= \frac{20 \div 4}{24 \div 4} = \frac{5}{6}$

Because 5 and 6 have no common factors other than 1.
Therefore, $\dpi{100} \small \frac{5}{6}$th of the class do not have blue eyes.

Question 16.
DIG DEEPER!
A student answers 4 out of 12 questions on a test incorrectly. What fraction of the questions, in simplest form, does the student answer incorrectly? Interpret the fraction.
Given that,
Total no. of questions = 12
No. of incorrect answers = 4
Step 1: Find the common factors of 12 and 4.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 4:  1, 2, 4
The common factors of 12 and 4 are 1, 2 and 4.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{4}{12} = \frac{4 \div 4}{12 \div 4} = \frac{1}{3}$
Because 1 and 3 have no common factors other than 1.
Therefore, $\dpi{100} \small \frac{1}{3}$ of the questions student answered incorrectly.

### Lesson 8.1 Simplest Form Homework & Practice 8.1

Use the model to write the fraction in simplest form.

Question 1.

Step 1: Find the common factors of 6 and 9.
Factors of 6: 1, 2, 3, 6
Factors of 9:  1, 3, 9
The common factors of 6 and 9 are 1 and 3.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

$\dpi{100} \small \frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$
Because 2 and 3 have no common factors other than 1, $\dpi{100} \small \frac{6}{9}$ is in simplest form.

Question 2.

Step 1: Find the common factors of 3 and 12.
Factors of 3: 1, 3
Factors of 12:  1, 3, 4, 6, 12
The common factors of 3 and 12 are 1 and 3.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4}$
Because 1 and 4 have no common factors other than 1, $\dpi{100} \small \frac{3}{12}$ is in simplest form.

Question 3.

Write the fraction in simplest form
Step 1: Find the common factors of 5 and 10.
Factors of 5: 1, 5
Factors of 10:  1, 2, 5, 10
The common factors of 5 and 10 are 1 and 5.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{5}{10} = \frac{5 \div 5}{10 \div 5} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1, $\dpi{100} \small \frac{5}{10}$ is in the simplest form.

Question 4.
$$\frac{4}{8}$$
Step 1: Find the common factors of 4 and 8.
Factors of 4: 1, 2, 4
Factors of 8:  1, 2, 4, 8
The common factors of 4 and 8 are 1, 2 and 4.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1, $$\frac{4}{8}$$ is in simplest form.

Question 5.
$$\frac{5}{100}$$
Step 1: Find the common factors of 5 and 100.
Factors of 5: 1, 5
Factors of 100:  1, 2, 4, 5, 10, 20, 25, 50, 100
The common factors of 5 and 100 are 1 and 5.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{5}{100} = \frac{5 \div 5}{100 \div 5} = \frac{1}{20}$
Because 1 and 20 have no common factors other than 1, $$\frac{5}{100}$$ is in simplest form.

Question 6.
$$\frac{20}{15}$$
Step 1: Find the common factors of 20 and 15.
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 15:  1, 3, 5, 15
The common factors of 20 and 15 are 1 and 5.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{20}{15} = \frac{20 \div 5}{15 \div 5} = \frac{4}{3}$
Because 4 and 3 have no common factors other than 1, $$\frac{20}{15}$$ is in simplest form.

Question 7.
There are 18 students in your class. Six of the students pack their lunch. In simplest form, what fraction of the students in your class pack their lunch?

Total students in the class = 18
No. of students pack their lunch = 6
Step 1: Find the common factors of 6 and 18.
Factors of 6: 1, 2, 3, 6
Factors of 18:  1, 2, 3, 6, 9, 18
The common factors of 6 and 18 are 1, 2, 3 and 6
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{6}{18} = \frac{6 \div 6}{18 \div 6} = \frac{1}{3}$
Because 1 and 3 have no common factors other than 1.
Therefore, $\dpi{100} \small \frac{1}{3}$ of the students pack their lunch.

Question 8.
Reasoning
Why do you have to divide a numerator and a denominator by the greatest of their common factors to write a fraction in simplest form?
To simplify a fraction to lowest terms, divide both the numerator and the denominator by their common factors. Repeat as needed until the only common factor is 1.

Question 9.
Writing
Explain how you know when a fraction is in simplest form.
If the fraction has no common factors other than 1, then it is said to be the simplest form of the fraction.

Question 10.
Open-Ended
Write a fraction in which the numerator and the denominator have 1, 2, 4, and 8 as common factors. Then write the fraction in the simplest form.
The fraction in which the numerator and the denominator is $\dpi{100} \small \frac{8}{16}$.
Step 1: Find the common factors of 8 and 16.
Factors of 8: 1, 2, 4, 8
Factors of 16:  1, 2, 4, 8, 16
The common factors of 8 and 16 are 1, 2, 4 and 8.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{8}{16} = \frac{8 \div 8}{16 \div 8} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1.

Question 11.
Modeling Real Life
A flight attendant has visited 30 of the 50 states. What fraction of the states, in simplest form, has he not visited?
Given that,
No. of states = 50
A flight attendant has visited 30 states.
The no. of states he has not visited = 50 – 30 = 20
Step 1: Find the common factors of 20 and 50.
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 50: 1, 2, 5, 10, 25, 50
The common factors of 20 and 50 are 1, 2, 5 and 10.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{20}{50} = \frac{20 \div 10}{50 \div 10} = \frac{2}{5}$
Because 2 and 5 have no common factors other than 1.
So the flight attendant has not visited $\dpi{100} \small \frac{2}{5}$ of the states.

Question 12.
DIG DEEPER!
A bin has red, orange, yellow, green, blue, and purple crayons. There are 4 of each color in the bin. In simplest form, what fraction of the crayons are red, orange, yellow, or green?
Given that, a bin has 6 colors(red, orange, yellow, green, blue, and purple) of crayons.
There are 4 crayons in the each color = 4 x 6 = 24
If the bin having only 3 colors(red, orange, yellow or green) = 4 x 3 = 12
Step 1: Find the common factors of 12 and 24.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The common factors of 12 and 24 are 1, 2, 3, 4, 6 and 12.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{12}{24} = \frac{12 \div 12}{24 \div 12} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1.
$\dpi{100} \small \frac{1}{2}$ of the crayons are red, orange, yellow or green.

Review & Refresh

Estimate the sum or difference.

Question 13.
598.44 – 45.61 =
The number round to 598.44 is 600.
The number round to 45.61 is 50.
600 – 50 = 550
Thus the estimated difference is 550.

Question 14.
93.8 + 4.3 =
The number round to 93.8 is 94.
The number round to 4.3 is 4.
94 + 4 = 98
Thus the estimated addition is 98.

### Lesson 8.2 Estimate Sums and Differences of Fractions

Explore and Grow

Plot $$\frac{7}{12}$$, $$\frac{5}{6}$$ and $$\frac{1}{10}$$ on the number line.

How can you estimate $$\frac{7}{12}$$ + $$\frac{5}{6}$$ ?
How can you estimate $$\frac{2}{3}$$ – $$\frac{1}{10}$$?

Reasoning
Write two fractions that have a sum of about $$\frac{1}{2}$$. Then write two fractions that have a difference of about $$\frac{1}{2}$$. Explain your reasoning.

Think and Grow: Estimate Sums and Differences

You have used the benchmarks $$\frac{1}{2}$$ and 1 to compare fractions. You can use the benchmarks 0, $$\frac{1}{2}$$, and 1 to estimate sums and differences of fractions.
Example
Estimate $$\frac{1}{6}$$ + $$\frac{5}{8}$$
Step 1: Use a number line to estimate each fraction.

Step 2: Estimate the sum.
An estimate of $$\frac{1}{6}$$ + $$\frac{5}{8}$$ is __ + __ = ___
Example
Estimate $$\frac{9}{10}$$ – $$\frac{2}{5}$$.

Show and Grow

Estimate the sum or difference

Question 1.
$$\frac{1}{3}$$ + $$\frac{11}{12}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{1}{3}$ is between 0 and $\dpi{100} \small \frac{1}{2}$ , but is closer to $\dpi{100} \small \frac{1}{2}$
$\dpi{100} \small \frac{11}{12}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1
Step 2: Estimate the sum.
An estimate of $$\frac{1}{3}$$ + $$\frac{11}{12}$$ is   $\dpi{100} \small \frac{1}{2}$ +  1 =  $\dpi{100} \small \frac{3}{2}$

Question 2.
$$\frac{3}{5}$$ + $$\frac{5}{6}$$
Step 1: Estimate each fraction
$\dpi{100} \small \frac{3}{5}$ is between  $\dpi{100} \small \frac{1}{2}$ and 1 , but is closer to $\dpi{100} \small \frac{1}{2}$
$\dpi{100} \small \frac{5}{6}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1
Step 2: Estimate the sum.
An estimate of $$\frac{3}{5}$$ + $$\frac{5}{6}$$ is $\dpi{100} \small \frac{1}{2}$ +  1 =  $\dpi{100} \small \frac{3}{2}$

Question 3.
$$\frac{15}{16}$$ – $$\frac{7}{8}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{15}{16}$ is about
Think : The numerator is about the same as the denominator.
$\dpi{100} \small \frac{7}{8}$ is about
Think : The numerator is about the same as the denominator.
Step 2: Estimate the difference.
An estimate of $\dpi{100} \small \frac{15}{16}$$\dpi{100} \small \frac{7}{8}$  is 1 – 1 = 0.

Apply and Grow: Practice

Estimate the sum or difference.

Question 4.
$$\frac{1}{6}$$ + $$\frac{3}{5}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{1}{6}$ is between 0 and $\dpi{100} \small \frac{1}{2}$, but is closer to 0.
$\dpi{100} \small \frac{3}{5}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to $\dpi{100} \small \frac{1}{2}$.
Step 2: Estimate the sum.
An estimate of $$\frac{1}{6}$$ + $$\frac{3}{5}$$ = 0 + $\dpi{100} \small \frac{1}{2}$ = $\dpi{100} \small \frac{1}{2}$

Question 5.
$$\frac{4}{5}$$ – $$\frac{5}{12}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{4}{5}$ is about
Think: The numerator is about the same as the denominator.
$\dpi{100} \small \frac{5}{12}$ is about
Think: The numerator is about half of the denominator.
Step 2: Estimate the difference.
An estimate of $\dpi{100} \small \frac{15}{16}$$\dpi{100} \small \frac{7}{8}$  is 1 – $\dpi{100} \small \frac{1}{2}$  = $\dpi{100} \small \frac{1}{2}$

Question 6.
$$\frac{13}{16}$$ + $$\frac{5}{6}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{13}{16}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1.
$\dpi{100} \small \frac{5}{6}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1.
Step 2: Estimate the sum.
An estimate of $$\frac{13}{16}$$ + $$\frac{5}{6}$$ = 1+1 = 2

Question 7.
$$\frac{3}{6}$$ – $$\frac{1}{8}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{3}{6}$ is about
Think: The numerator is about half of the denominator.
$\dpi{100} \small \frac{1}{8}$ is about
Think: The numerator is near to zero.
Step 2: Estimate the difference.
An estimate of $\dpi{100} \small \frac{3}{6}$$\dpi{100} \small \frac{1}{8}$ is  $\dpi{100} \small \frac{1}{2}$  – 0 = $\dpi{100} \small \frac{1}{2}$ .

Question 8.
$$\frac{1}{14}$$ + $$\frac{98}{100}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{1}{14}$ is closer to 0
$\dpi{100} \small \frac{98}{100}$ is closer to 1.
Step 2: Estimate the sum.
An estimate of $$\frac{1}{14}$$ + $$\frac{98}{100}$$ = 0 +1 = 1

Question 9.
$$\frac{11}{12}$$ – $$\frac{1}{8}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{11}{12}$ is about
Think: The numerator is about the same as the denominator.
$\dpi{100} \small \frac{1}{8}$ is about
Think: The numerator is near to zero.
Step 2: Estimate the difference.
An estimate of  $\dpi{100} \small \frac{11}{12}$ – $\dpi{100} \small \frac{1}{8}$ is  1  – 0 = 1.

Question 10.
You walk $$\frac{1}{10}$$ mile to your friend’s house and then you both walk $$\frac{2}{5}$$ mile. Estimate how much farther you walk with your friend than you walk alone.

To find how much farther I walk with my friend than I walk alone, subtract the distance that I walk alone from we both walk.
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{1}{10}$ is about____
Think: The numerator is near to zero.
$\dpi{100} \small \frac{2}{5}$ is about____
Think: The numerator is about half of the denominator.
Step 2: Estimate the difference.
An estimate of  $\dpi{100} \small \frac{2}{5}$$\dpi{100} \small \frac{1}{10}$ = $\dpi{100} \small \frac{1}{2}$ – 0 = $\dpi{100} \small \frac{1}{2}$
So the distance I walk with my friend than I walk alone is $\dpi{100} \small \frac{1}{2}$ mile.

Question 11.
A carpenter has two wooden boards. One board is $$\frac{3}{4}$$ foot long and the other board is $$\frac{1}{6}$$ foot long. To determine whether the total length of the boards is 1 foot, should the carpenter use an estimate, or is an exact answer required? Explain.
Given,
A carpenter has two wooden boards. One board is $$\frac{3}{4}$$ foot long and the other board is $$\frac{1}{6}$$ foot long.
$$\frac{3}{4}$$ + $$\frac{1}{6}$$
The fractions have unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCM = 12
$$\frac{3}{4}$$ × $$\frac{3}{3}$$ = $$\frac{9}{12}$$
$$\frac{1}{6}$$ × $$\frac{2}{2}$$ = $$\frac{2}{12}$$
$$\frac{9}{12}$$ + $$\frac{2}{12}$$ = $$\frac{11}{12}$$
$$\frac{11}{12}$$ is approximately equal to 1 foot.

Question 12.
Number Sense
A fraction has a numerator of 1 and a denominator greater than 4. Is the fraction closer to 0, $$\frac{1}{2}$$, or 1? Explain.
Given,
A fraction has a numerator of 1 and a denominator greater than 4.
$$\frac{1}{4}$$ = 0.25
If the denominator is greater than 4. Let’s consider 8.
$$\frac{1}{8}$$ = 0.125
The fraction will be close to 0.

Think and Grow: Modeling Real Life

Example
In the human body, the small intestine is about 20$$\frac{1}{12}$$ feet long. The large intestine is about 4$$\frac{5}{6}$$ feet long. About how long are the intestines in the human body?

To ﬁnd the total length of the intestines, estimate 20$$\frac{1}{12}$$ + 4$$\frac{5}{6}$$.
Step 1: Use mental math to round each mixed number to the nearest whole number.

Show and Grow

Question 13.
A bullfrog jumps 5$$\frac{11}{12}$$ feet. A leopard frog jumps 4$$\frac{1}{3}$$ feet. About how much farther does the bullfrog jump than the leopard frog?
Step 1: Use mental math to round each mixed number to the nearest whole number.
5$\dpi{100} \small \frac{11}{12}$ is about,  $\dpi{100} \small \frac{11}{12}$ is closer to 1 than 0.
4$\dpi{100} \small \frac{1}{3}$ is about, $\dpi{100} \small \frac{1}{3}$ is closer to 0 than 1.
Step 2: Estimate the difference
An estimate of 5$\dpi{100} \small \frac{11}{12}$  – 4$\dpi{100} \small \frac{1}{3}$ = 1 – 0 = 1
So, bullfrog jumps 1 feet farther than the leopard frog.
Question 14.
DIG DEEPER!
A cell phone has 32 gigabytes of storage. The amounts of storage used by photos, songs, and apps are shown. About how many gigabytes of storage are left?

Given that,
No. of gigabytes of storage in cellphone = 32
Step 1: Use mental math to round each mixed number to the nearest whole number.
Photos —>  8$\dpi{100} \small \frac{4}{5}$ is about,  $\dpi{100} \small \frac{4}{5}$ is closer to 1 than 0
Songs —>   2$\dpi{100} \small \frac{3}{100}$ is about,  $\dpi{100} \small \frac{3}{100}$ is closer to 0 than 1
Apps —>     6$\dpi{100} \small \frac{7}{10}$  is about,  $\dpi{100} \small \frac{7}{10}$ is closer to 1 than 0
Step 2: Storage left in the phone = Total storage – storage(photos + songs + apps)
= 32 – (1 + 0 + 1)
Therefore, storage left = 30 gigabytes.

Question 15.
DIG DEEPER!
Use two different methods to estimate how many cups of nut medley the recipe makes. Which estimate do you think is closer to the actual answer? Explain.

1 $$\frac{3}{8}$$ + $$\frac{5}{8}$$ + 2 $$\frac{1}{3}$$
1 + 2 = 3
$$\frac{3}{8}$$ + $$\frac{5}{8}$$ + $$\frac{1}{3}$$
1 $$\frac{1}{3}$$
3 + 1 $$\frac{1}{3}$$ = 4 $$\frac{1}{3}$$
The fraction is 4 $$\frac{1}{3}$$
4 is equal to the actual answer.

### Estimate Sums and Differences of Fractions Homework & Practice 8.2

Estimate the sum or difference

Question 1.
$$\frac{11}{12}$$ – $$\frac{5}{6}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{11}{12}$ is about
Think : The numerator is about the same as the denominator.
$\dpi{100} \small \frac{5}{6}$ is about
Think : The numerator is about same as the denominator.
Step 2: Estimate the difference.
An estimate of  $\dpi{100} \small \frac{11}{12}$$\dpi{100} \small \frac{5}{6}$  is  1  – 1 = 0.

Question 2.
$$\frac{17}{20}$$ + $$\frac{13}{20}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{17}{20}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1.
$\dpi{100} \small \frac{13}{20}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to $\dpi{100} \small \frac{1}{2}$.
Step 2: Estimate the sum.
An estimate of $$\frac{17}{20}$$ + $$\frac{13}{20}$$ = 1 + $\dpi{100} \small \frac{1}{2}$ = $\dpi{100} \small \frac{3}{2}$.

Question 3.
$$\frac{3}{8}$$ – $$\frac{1}{6}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{3}{8}$ is about
Think : The numerator is about half of the denominator.
$\dpi{100} \small \frac{1}{6}$ is about
Think : The numerator is near to zero.
Step 2: Estimate the difference.
An estimate of  $\dpi{100} \small \frac{3}{8}$  – $\dpi{100} \small \frac{1}{6}$   is  $\dpi{100} \small \frac{1}{2}$  – 0 = $\dpi{100} \small \frac{1}{2}$.

Question 4.
$$\frac{7}{12}$$ + $$\frac{2}{5}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{7}{12}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to $\dpi{100} \small \frac{1}{2}$.
$\dpi{100} \small \frac{2}{5}$ is between 0 and $\dpi{100} \small \frac{1}{2}$, but is closer to $\dpi{100} \small \frac{1}{2}$.
Step 2: Estimate the sum.
An estimate of $$\frac{7}{12}$$ + $$\frac{2}{5}$$ = $\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{1}{2}$ = 1.

Question 5.
$$\frac{4}{5}$$ – $$\frac{7}{12}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{4}{5}$ is about
Think: The numerator is about the same as the denominator.
$\dpi{100} \small \frac{7}{12}$ is about
Think: The numerator is about half of the denominator.
Step 2: Estimate the difference.
An estimate of  $\dpi{100} \small \frac{4}{5}$  – $\dpi{100} \small \frac{7}{12}$   is  1 –  $\dpi{100} \small \frac{1}{2}$  = $\dpi{100} \small \frac{1}{2}$

Question 6.
$$\frac{1}{5}$$ + 1$$\frac{10}{21}$$
Step 1: Use mental math to round each mixed number to the nearest whole number.
$$\frac{1}{5}$$ is closer to 0
1$$\frac{10}{21}$$ is close to 1.
Step 2: Estimate the sum.
An estimate of $$\frac{1}{5}$$ + 1$$\frac{10}{21}$$
= 0 + 1 = 1

Question 7.
3$$\frac{5}{8}$$ – $$\frac{1}{10}$$
Step 1: Use mental math to round each mixed number to the nearest whole number.
3$\dpi{100} \small \frac{5}{8}$  is about, $\dpi{100} \small \frac{5}{8}$ is closer to 1 than 0
$\dpi{100} \small \frac{1}{10}$ is near to 0
Step 2: Estimate the difference
An estimate of 3$\dpi{100} \small \frac{5}{8}$  – $\dpi{100} \small \frac{1}{10}$ = 1 – 0 = 1.

Question 8.
6$$\frac{1}{3}$$ + 2$$\frac{4}{6}$$
Step 1: Use mental math to round each mixed number to the nearest whole number.
$$\frac{1}{3}$$ is close to 0 than 1.
$$\frac{4}{6}$$ is to 1 than 0.
Step 2: Estimate the sum
An estimate of 6$$\frac{1}{3}$$ is 6.
An estimate of 2$$\frac{4}{6}$$ is 3.
6 + 3 = 9

Question 9.
5$$\frac{7}{8}$$ – 4$$\frac{49}{100}$$
Step 1: Use mental math to round each mixed number to the nearest whole number.
5$\dpi{100} \small \frac{7}{8}$  is about, $\dpi{100} \small \frac{7}{8}$ is closer to 1 than 0
4$\dpi{100} \small \frac{49}{100}$  is about, $\dpi{100} \small \frac{49}{100}$ is closer to 0 than 1
Step 2: Estimate the difference
An estimate of 5$\dpi{100} \small \frac{7}{8}$  – 4$\dpi{100} \small \frac{49}{100}$  = 1 – 0 =  1.

Question 10.
You make a bag of trail mix with $$\frac{2}{3}$$ cup of raisins and $$\frac{9}{8}$$ cups of peanuts. About how much trail mix do you make?
Raisins = $\dpi{100} \small \frac{2}{3}$ cups
Peanuts = $\dpi{100} \small \frac{9}{8}$ cups
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{2}{3}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to $\dpi{100} \small \frac{1}{2}$.
$\dpi{100} \small \frac{9}{8}$ is closer to 1.
Step 2: Estimate the sum.
An estimate of $$\frac{2}{3}$$ + $$\frac{9}{8}$$ = $\dpi{100} \small \frac{1}{2}$ + 1 = $\dpi{100} \small \frac{3}{2}$.
So trail mix = $\dpi{100} \small \frac{3}{2}$.

Question 11.
You have $$\frac{2}{3}$$ cup of ﬂour in a bin and $$\frac{7}{8}$$ cup of ﬂour in a bag. To determine whether you have enough ﬂour for a recipe that needs 1$$\frac{3}{4}$$ cups of ﬂour, should you use an estimate, or is an exact answer required? Explain.
Given,
You have $$\frac{2}{3}$$ cup of ﬂour in a bin and $$\frac{7}{8}$$ cup of ﬂour in a bag.
$$\frac{2}{3}$$ + $$\frac{7}{8}$$ = 1 $$\frac{13}{24}$$
Th estimated fraction of 1 $$\frac{13}{24}$$ is 1$$\frac{3}{4}$$

Question 12.
Writing
Explain how you know $$\frac{9}{10}$$ – $$\frac{3}{5}$$ is about $$\frac{1}{2}$$.
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{9}{10}$ is about
Think: The numerator is about the same as the denominator.
$\dpi{100} \small \frac{3}{5}$ is about
Think : The numerator is about half of the denominator.
Step 2: Estimate the difference.
An estimate of $\dpi{100} \small \frac{9}{10}$$\dpi{100} \small \frac{3}{5}$  is 1 – $\dpi{100} \small \frac{1}{2}$  = $\dpi{100} \small \frac{1}{2}$.

Question 13.
Precision
Your friend says $$\frac{5}{8}$$ + $$\frac{7}{12}$$ is about 2. Find a closer estimate. Explain why your estimate is closer.
$$\frac{5}{8}$$ + $$\frac{7}{12}$$ = 1 $$\frac{5}{24}$$
$$\frac{5}{24}$$ is closer to 1 than 0.
1 $$\frac{5}{24}$$ is about 2.

Question 14.
Modeling Real Life
About how much taller is Robot A than Robot B?

Step 1: Use mental math to round each mixed number to the nearest whole number.
1 $\dpi{100} \small \frac{6}{10}$ is about, $\dpi{100} \small \frac{6}{10}$ is closer to 1 than 0.
1 $\dpi{100} \small \frac{1}{5}$  is about, $\dpi{100} \small \frac{1}{5}$ is closer to 0 than 1.
Step 2: Estimate the difference
An estimate of 1 $\dpi{100} \small \frac{6}{10}$  – 1 $\dpi{100} \small \frac{1}{5}$   = 1 – 0 =  1.
Robot A is 1 meter taller than Robot B.

Question 15.
Modeling Real Life
A class makes a paper chain that is 5$$\frac{7}{12}$$ feet long. The class adds another 3$$\frac{5}{6}$$ feet to the chain. About how long is the chain now?

Step 1: Use mental math to round each mixed number to the nearest whole number.
5$\dpi{100} \small \frac{7}{12}$ is about,  $\dpi{100} \small \frac{7}{12}$ is closer to 1 than 0.
3$\dpi{100} \small \frac{5}{6}$ is about, $\dpi{100} \small \frac{5}{6}$ is closer to 1 than 0.
Step 2: Estimate the sum
An estimate of 5$\dpi{100} \small \frac{7}{12}$  + 3$\dpi{100} \small \frac{5}{6}$ = 1 + 1 = 2
Therefore, the chain is now 2 feet long.

Review & Refresh

Question 16.
509 × 5 = ___
The number 509 round to hundred is 500.
500 × 5 = 2500
509 × 5 = 2545

Question 17.
7,692 × 6 = ___
The number 7692 round to hundred is 7700.
7700 × 6 = 46200
7,692 × 6 = 46152

Question 18.
31,435 × 7 = ___
The number 31435 round to hundred is 31,400.
31,400 × 7 = 219800
31,435 × 7 = 220045

### Lesson 8.3 Find Common Denominators

Explore and Grow

You cut a rectangular pan of vegetable lasagna into equal-sized pieces. You serve $$\frac{1}{2}$$ of the lasagna to a large table and $$\frac{1}{3}$$ of the lasagna to a small table. Draw a diagram that shows how you cut the lasagna.
What fraction of the lasagna does each piece represent? How does the denominator of the fraction compare to the denominators of $$\frac{1}{2}$$ and $$\frac{1}{3}$$ ?

Reasoning
Is there another way you can cut the lasagna? Explain your reasoning.

Think and Grow: Find Common Denominators

Key Idea
Fractions that have the same denominator are said to have a common denominator. You can find a common denominator either by finding a common multiple of the denominators or by finding the product of the denominators.
Example
Use a common denominator to write equivalent fractions for $$\frac{1}{2}$$ and $$\frac{5}{8}$$.
List multiples of the denominators.

Example
Use a common denominator to write equivalent fractions for $$\frac{2}{3}$$ and $$\frac{1}{4}$$. Use the product of the denominators: 3 × 4 = __.
Write equivalent fractions with denominators of 12.

Show and Grow

Use a common denominator to write an equivalent fraction for each fraction.

Question 1.
$$\frac{2}{3}$$ and $$\frac{1}{6}$$
Use the product of the denominators : 3 $\dpi{100} \small \times$ 6 = 18
Write equivalent fractions with denominators of 18
$\dpi{100} \small \frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}$
$\dpi{100} \small \frac{1}{6} = \frac{1 \times 3}{6 \times 3} = \frac{3}{18}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{12}{18}$  and $\dpi{100} \small \frac{3}{18}$.

Question 2.
$$\frac{5}{6}$$ and $$\frac{3}{4}$$
Step 1: Use the product of the denominators : 6 $\dpi{100} \small \times$ 4 = 24
Step 2: Write equivalent fractions with denominators of 24
$\dpi{100} \small \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$
$\dpi{100} \small \frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{20}{24}$  and $\dpi{100} \small \frac{18}{24}$.

Apply and Grow: Practice

Use a common denominator to write an equivalent fraction for each fraction.

Question 3.
$$\frac{2}{3}$$ and $$\frac{5}{6}$$
Step 1: Use the product of the denominators : 3 $\dpi{100} \small \times$ 6 = 18
Step 2: Write equivalent fractions with denominators of 18

$\dpi{100} \small \frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}$

$\dpi{100} \small \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{12}{18}$  and $\dpi{100} \small \frac{15}{18}$.

Question 4.
$$\frac{3}{4}$$ and $$\frac{1}{2}$$
Step 1: Use the product of the denominators : 4 $\dpi{100} \small \times$ 2 = 8
Step 2: Write equivalent fractions with denominators of 8
$\dpi{100} \small \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$
$\dpi{100} \small \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{6}{8}$  and $\dpi{100} \small \frac{4}{8}$.

Question 5.
$$\frac{5}{9}$$ and $$\frac{2}{3}$$
Step 1: Use the product of the denominators: 9 $\dpi{100} \small \times$ 3 = 27
Step 2: Write equivalent fractions with denominators of 27
$\dpi{100} \small \frac{5}{9} = \frac{5 \times 3}{9 \times 3} = \frac{15}{27}$
$\dpi{100} \small \frac{2}{3} = \frac{2 \times 9}{3 \times 9} = \frac{18}{27}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{15}{27}$  and $\dpi{100} \small \frac{18}{27}$.

Question 6.
$$\frac{8}{21}$$ and $$\frac{3}{7}$$
Step 1: Use the product of the denominators: 21 $\dpi{100} \small \times$ 7 = 147
Step 2: Write equivalent fractions with denominators of 147
$\dpi{100} \small \frac{8}{21} = \frac{8 \times 7}{21 \times 7} = \frac{56}{147}$
$\dpi{100} \small \frac{3}{7} = \frac{3 \times 21}{7 \times 21} = \frac{63}{147}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{56}{147}$  and $\dpi{100} \small \frac{63}{147}$.

Question 7.
$$\frac{1}{5}$$ and $$\frac{1}{2}$$
Step 1: Use the product of the denominators: 5 $\dpi{100} \small \times$ 2 = 10
Step 2: Write equivalent fractions with denominators of 10
$\dpi{100} \small \frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10}$
$\dpi{100} \small \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{2}{10}$  and $\dpi{100} \small \frac{5}{10}$.

Question 8.
$$\frac{3}{4}$$ and $$\frac{1}{6}$$
Step 1: Use the product of the denominators: 4 $\dpi{100} \small \times$ 6 = 24
Step 2: Write equivalent fractions with denominators of 24
$\dpi{100} \small \frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}$
$\dpi{100} \small \frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{18}{24}$  and $\dpi{100} \small \frac{4}{24}$.

Question 9.
$$\frac{3}{7}$$ and $$\frac{2}{9}$$
Step 1: Use the product of the denominators: 7 $\dpi{100} \small \times$ 9 = 63
Step 2: Write equivalent fractions with denominators of 63
$\dpi{100} \small \frac{3}{7} = \frac{3 \times 9}{7 \times 9} = \frac{27}{63}$
$\dpi{100} \small \frac{2}{9} = \frac{2 \times 7}{9 \times 7} = \frac{14}{63}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{27}{63}$  and $\dpi{100} \small \frac{14}{63}$.

Question 10.
$$\frac{3}{8}$$ and $$\frac{5}{11}$$
Step 1: Use the product of the denominators: 8 $\dpi{100} \small \times$ 11 = 88
Step 2: Write equivalent fractions with denominators of 88
$\dpi{100} \small \frac{3}{8} = \frac{3 \times 11}{8 \times 11} = \frac{33}{88}$
$\dpi{100} \small \frac{5}{11} = \frac{5 \times 8}{11 \times 8} = \frac{40}{88}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{33}{88}$ and $\dpi{100} \small \frac{40}{88}$.

Question 11.
You walk your dog $$\frac{3}{4}$$ mile on Saturday and $$\frac{5}{8}$$ mile on Sunday. Use a common denominator to write an equivalent fraction for each fraction.

Step 1: Use the product of the denominators: 4 $\dpi{100} \small \times$ 8 = 32
Step 2: Write equivalent fractions with denominators of 32
$\dpi{100} \small \frac{3}{4} = \frac{3 \times 8}{4 \times 8} = \frac{24}{32}$
$\dpi{100} \small \frac{5}{8} = \frac{5 \times 4}{8 \times 4} = \frac{20}{32}$
Dog walk on Saturday, equivalent fraction = $\dpi{100} \small \frac{24}{32}$
Dog walk on Sunday, equivalent fraction = $\dpi{100} \small \frac{20}{32}$

Question 12.
Writing
Explain how to use the models to ﬁnd a common denominator for $$\frac{1}{2}$$ and $$\frac{3}{5}$$. Then write an equivalent fraction for each fraction.

Initially find the product of the denominators and that product value is the denominator for both fractions.
Step 1: Use the product of the denominators: 2 $\dpi{100} \small \times$ 5 = 10
Step 2: Write equivalent fractions with denominators of 10
$\dpi{100} \small \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$
$\dpi{100} \small \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{5}{10}$ and $\dpi{100} \small \frac{6}{10}$.

Question 13.
Number Sense
Which pairs of fractions are equivalent to $$\frac{1}{2}$$ and $$\frac{2}{3}$$ ?

$\dpi{100} \small \frac{6}{12}$ and $\dpi{100} \small \frac{8}{12}$

$\dpi{100} \small \frac{3}{6}$ and $\dpi{100} \small \frac{4}{6}$
These two fractions are equivalent to $$\frac{1}{2}$$ and $$\frac{2}{3}$$.

Think and Grow: Modeling Real Life

Example
You and your friend make woven key chains. Your key chain is $$\frac{2}{4}$$ foot long. Your friend’s is $$\frac{3}{6}$$ foot long. Are the key chains the same length?

Use a common denominator to write equivalent fractions for the lengths of the key chains. Use the product of the denominators.

Write equivalent fractions with denominators of 24.

Compare the lengths of the key chains.
So, the key chains __ the same length.

4 × 6 = 24
Write equivalent fractions with denominators of 24.

So, the key chains has the same length.

Show and Grow

Question 14.
Your hamster weighs $$\frac{13}{16}$$ ounce. Your friend’s hamster weighs $$\frac{6}{8}$$ ounce. Do the hamsters weigh the same amount?

Use a common denominator to write equivalent fractions for the hamsters weight.
Step 1: Use the product of the denominators: 16 $\dpi{100} \small \times$ 8 = 128
Step 2: Write equivalent fractions with denominators of 128

$\dpi{100} \small \frac{13}{16} = \frac{13 \times 8}{16 \times 8} = \frac{104}{128}$

$\dpi{100} \small \frac{6}{8} = \frac{6 \times 16}{8 \times 16} = \frac{96}{128}$
Hamsters weigh the different amount. One is $\dpi{100} \small \frac{104}{128}$ ounce and the other one is $\dpi{100} \small \frac{96}{128}$ ounce.

Question 15.
DIG DEEPER!
You have three vegetable pizzas of the same size. One has 4 equal slices. The second has 8 equal slices. The third has 6 equal slices. You cut the pizzas until all of them have the same number of slices. How many slices does each pizza have?
Given,
You have three vegetable pizzas of the same size.
One has 4 equal slices. The second has 8 equal slices.
The third has 6 equal slices. You cut the pizzas until all of them have the same number of slices.
4 + 6 + 8 = 18
one has $$\frac{4}{18}$$
second has $$\frac{6}{18}$$
Third has $$\frac{8}{18}$$
Total there are 18 slices.

### Find Common Denominators Homework & Practice 8.3

Use a common denominator to write an equivalent fraction for each fraction.

Question 1.
$$\frac{1}{2}$$ and $$\frac{3}{8}$$
Step 1: Use the product of the denominators: 2 $\dpi{100} \small \times$ 8 = 16
Step 2: Write equivalent fractions with denominators of 16

$\dpi{100} \small \frac{1}{2} = \frac{1 \times 8}{2 \times 8} = \frac{8}{16}$

$\dpi{100} \small \frac{3}{8} = \frac{3 \times 2}{8 \times 2} = \frac{6}{16}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{8}{16}$  and $\dpi{100} \small \frac{6}{16}$.

Question 2.
$$\frac{7}{9}$$ and $$\frac{2}{3}$$
Step 1: Use the product of the denominators: 9 $\dpi{100} \small \times$ 3 = 27
Step 2: Write equivalent fractions with denominators of 27.

$\dpi{100} \small \frac{7}{9} = \frac{7 \times 3}{9 \times 3} = \frac{21}{27}$

$\dpi{100} \small \frac{2}{3} = \frac{2 \times 9}{3 \times 9} = \frac{18}{27}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{21}{27}$  and $\dpi{100} \small \frac{18}{27}$.

Question 3.
$$\frac{5}{6}$$ and $$\frac{1}{2}$$
Step 1: Use the product of the denominators: 6 $\dpi{100} \small \times$ 2 = 12
Step 2: Write equivalent fractions with denominators of 12.

$\dpi{100} \small \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

$\dpi{100} \small \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{10}{12}$  and $\dpi{100} \small \frac{6}{12}$.

Question 4.
$$\frac{3}{4}$$ and $$\frac{5}{16}$$
Step 1: Use the product of the denominators: 4 $\dpi{100} \small \times$ 16 = 64
Step 2: Write equivalent fractions with denominators of 64.

$\dpi{100} \small \frac{3}{4} = \frac{3 \times 16}{4 \times 16} = \frac{48}{64}$

$\dpi{100} \small \frac{5}{16} = \frac{5 \times 4}{16 \times 4} = \frac{20}{64}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{48}{64}$  and $\dpi{100} \small \frac{20}{64}$.

Question 5.
$$\frac{18}{24}$$ and $$\frac{5}{6}$$
Step 1: Use the product of the denominators: 24 $\dpi{100} \small \times$ 6 = 144
Step 2: Write equivalent fractions with denominators of 144.

$\dpi{100} \small \frac{18}{24} = \frac{18 \times 6}{24 \times 6} = \frac{108}{144}$

$\dpi{100} \small \frac{5}{6} = \frac{5 \times 24}{6 \times 24} = \frac{120}{144}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{108}{144}$  and $\dpi{100} \small \frac{120}{144}$.

Question 6.
$$\frac{1}{3}$$ and $$\frac{1}{5}$$
Step 1: Use the product of the denominators: 3 $\dpi{100} \small \times$ 5 = 15
Step 2: Write equivalent fractions with denominators of 15.

$\dpi{100} \small \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$

$\dpi{100} \small \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{5}{15}$  and $\dpi{100} \small \frac{3}{15}$.

Question 7.
$$\frac{3}{5}$$ and $$\frac{4}{7}$$
Step 1: Use the product of the denominators: 5 $\dpi{100} \small \times$ 7= 35
Step 2: Write equivalent fractions with denominators of 35.

$\dpi{100} \small \frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$

$\dpi{100} \small \frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{21}{35}$  and $\dpi{100} \small \frac{20}{35}$.

Question 8.
$$\frac{5}{8}$$ and $$\frac{2}{9}$$
Step 1: Use the product of the denominators: 8 $\dpi{100} \small \times$ 9 = 72
Step 2: Write equivalent fractions with denominators of 72.

$\dpi{100} \small \frac{5}{8} = \frac{5 \times 9}{8 \times 9} = \frac{45}{72}$

$\dpi{100} \small \frac{2}{9} = \frac{2 \times 8}{9 \times 8} = \frac{16}{72}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{45}{72}$  and $\dpi{100} \small \frac{16}{72}$.

Question 9.
A mint plant grows $$\frac{7}{8}$$ inch in 1 week and $$\frac{13}{16}$$ inch the next week. Use a common denominator to write an equivalent fraction for each fraction.
Step 1: Use the product of the denominators : 8 $\dpi{100} \small \times$ 16 = 128
Step 2: Write equivalent fractions with denominators of 128.

$\dpi{100} \small \frac{7}{8} = \frac{7 \times 16}{8 \times 16} = \frac{112}{128}$

$\dpi{100} \small \frac{13}{16} = \frac{13 \times 8}{16 \times 8} = \frac{104}{128}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{112}{128}$  and $\dpi{100} \small \frac{104}{128}$.

Question 10.
Which One Doesn’t Belong? Which pair of fractions is not equivalent to $$\frac{2}{5}$$ and $$\frac{1}{10}$$?

So, $\dpi{100} \small \frac{6}{15}$ and $\dpi{100} \small \frac{5}{15}$ is not equivalent to $$\frac{2}{5}$$ and $$\frac{1}{10}$$ and remaining all the pairs are equivalent.

Question 11.
YOU BE THE TEACHER
Your friend says she used a common denominator to ﬁnd fractions equivalent to $$\frac{2}{3}$$ and $$\frac{8}{9}$$. Is your friend correct? Explain.

No, she is wrong because 9 and 12 are not common denominators.
Step 1: Use the product of the denominators : 3 $\dpi{100} \small \times$ 9 = 27
Step 2: Write equivalent fractions with common denominator of 27.

$\dpi{100} \small \frac{2}{3} = \frac{2 \times 9}{3 \times 9} = \frac{18}{27}$

$\dpi{100} \small \frac{8}{9} = \frac{8 \times 3}{9 \times 3} = \frac{24}{27}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{18}{27}$  and $\dpi{100} \small \frac{24}{27}$.

Question 12.
Modeling Real Life
Some friends spend $$\frac{1}{3}$$ hour collecting sticks and $$\frac{5}{6}$$ hour building a fort. Do they spend the same amount of time on each? Explain.

No, they can not spend the same time on each because $\dpi{100} \small \frac{1}{3}$ is not equivalent to $\dpi{100} \small \frac{5}{6}$.
$\dpi{100} \small \frac{1}{3}$ is equivalent to $\dpi{100} \small \frac{2}{6}$.

Question 13.
DIG DEEPER!
Use a common denominator to write an equivalent fraction for each fraction. Which two students are the same distance from the school? Are they closer to or farther from the school than the other student?

Step 1: Use the LCM of the denominators. LCM of 12, 8 and 6 = 24
Step 2: Write equivalent fractions with common denominator of 24.
Student A –>  $\dpi{100} \small \frac{10}{12} = \frac{10 \times 2}{12 \times 2} = \frac{20}{24}$
Student B –>  $\dpi{100} \small \frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}$
Student C –>  $\dpi{100} \small \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$
Therefore, student A and student C are the same distance from the school i.e. $\dpi{100} \small \frac{20}{24}$ mile.
They are closer to student B.

Review & Refresh

Find the value of the expression.

Question 14.
102

Question 15.
8 × 104

Question 16.
6 × 103

Question 17.
9 × 105

### Lesson 8.4 Add Fractions with Unlike Denominators

Use a model to find the sum.

Explain how you can use a model to add fifths and tenths.

Construct Arguments
How can you add two fractions with unlike denominators without using a model? Explain why your method makes sense.

Think and Grow: Add Fractions with Unlike Denominators

You can use equivalent fractions to add fractions that have unlike denominators.
Example
Find $$\frac{1}{4}$$ + $$\frac{3}{8}$$
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is a multiple of 4, so rewrite $$\frac{1}{4}$$ with a denominator of 8.

Example
Find $$\frac{7}{8}$$ + $$\frac{1}{6}$$ Estimate __
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is not a multiple of 6, so rewrite each fraction with a denominator of 8 × 6 = 48.

Show and Grow

Question 1.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 6 is a multiple of 3, so rewrite it with a denominator of 6.
Rewrite $\dpi{100} \small \frac{2}{3}$ as $\dpi{100} \small \frac{2 \times 2}{3 \times 2}$ = $\dpi{100} \small \frac{4}{6}$

$\dpi{100} \small \frac{5}{6}$ + $\dpi{100} \small \frac{2}{3}$  =  $\dpi{100} \small \frac{5}{6}$ + $\dpi{100} \small \frac{4}{6}$

= $\dpi{100} \small \frac{9}{6}$ or $\dpi{100} \small \frac{3}{2}$

Question 2.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 5 is not a multiple of 4, so rewrite each fraction with a denominator of 5 $\dpi{100} \small \times$ 4 = 20

Rewrite $\dpi{100} \small \frac{1}{5}$  as $\dpi{100} \small \frac{1 \times 4}{5 \times 4}$ = $\dpi{100} \small \frac{4}{20}$  and  $\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 5}{4 \times 5}$ = $\dpi{100} \small \frac{15}{20}$

$\dpi{100} \small \frac{1}{5}$ + $\dpi{100} \small \frac{3}{4}$ = $\dpi{100} \small \frac{4}{20}$ + $\dpi{100} \small \frac{15}{20}$

= $\dpi{100} \small \frac{19}{20}$

Question 3.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 6 is not a multiple of 4, so rewrite each fraction with a denominator of 6 $\dpi{100} \small \times$ 4 = 24
Rewrite $\dpi{100} \small \frac{1}{6}$  as $\dpi{100} \small \frac{1 \times 4}{6 \times 4}$ = $\dpi{100} \small \frac{4}{24}$  and  $\dpi{100} \small \frac{1}{4}$ as $\dpi{100} \small \frac{1 \times 6}{4 \times 6}$ = $\dpi{100} \small \frac{6}{24}$

$\dpi{100} \small \frac{1}{6}$ + $\dpi{100} \small \frac{1}{4}$ = $\dpi{100} \small \frac{4}{24}$ + $\dpi{100} \small \frac{6}{24}$

= $\dpi{100} \small \frac{10}{24}$ or $\dpi{100} \small \frac{5}{12}$

Apply and Grow: Practice

Question 4.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is a multiple of 4, so rewrite it with a denominator of 8.
Rewrite $\dpi{100} \small \frac{1}{4}$ as $\dpi{100} \small \frac{1 \times 2}{4 \times 2}$ = $\dpi{100} \small \frac{2}{8}$

$\dpi{100} \small \frac{5}{8}$ + $\dpi{100} \small \frac{1}{4}$ = $\dpi{100} \small \frac{5}{8}$ + $\dpi{100} \small \frac{2}{8}$

= $\dpi{100} \small \frac{7}{8}$

Question 5.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 12 is a multiple of 3, so rewrite it with a denominator of 12.
Rewrite $\dpi{100} \small \frac{2}{3}$ as $\dpi{100} \small \frac{2 \times 4}{3 \times 4}$ = $\dpi{100} \small \frac{8}{12}$

$\dpi{100} \small \frac{2}{3}$ + $\dpi{100} \small \frac{7}{12}$$\dpi{100} \small \frac{8}{12}$ + $\dpi{100} \small \frac{7}{12}$

= $\dpi{100} \small \frac{15}{12}$ or $\dpi{100} \small \frac{5}{4}$

Question 6.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 15 is a multiple of 5 , so rewrite it with a denominator of 15.
Rewrite $\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 3}{5 \times 3}$ = $\dpi{100} \small \frac{6}{15}$

$\dpi{100} \small \frac{2}{5}$ + $\dpi{100} \small \frac{10}{15}$ = $\dpi{100} \small \frac{6}{15}$ + $\dpi{100} \small \frac{10}{15}$

= $\dpi{100} \small \frac{16}{15}$

Question 7.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is not a multiple of 6, so rewrite each fraction with a denominator of 8 × 6 = 48.
Rewrite $\dpi{100} \small \frac{1}{6}$ as $\dpi{100} \small \frac{1 \times 8}{6 \times 8}$ = $\dpi{100} \small \frac{8}{48}$  and  $\dpi{100} \small \frac{4}{8}$ as $\dpi{100} \small \frac{4 \times 6}{8 \times 6}$ = $\dpi{100} \small \frac{24}{48}$

$\dpi{100} \small \frac{1}{6}$ + $\dpi{100} \small \frac{4}{8}$ = $\dpi{100} \small \frac{8}{48}$ + $\dpi{100} \small \frac{24}{48}$

= $\dpi{100} \small \frac{32}{48}$ or $\dpi{100} \small \frac{2}{3}$

Question 8.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 12 is not a multiple of 5, so rewrite each fraction with a denominator of 12 × 5 = 60.
Rewrite $\dpi{100} \small \frac{11}{12}$ as $\dpi{100} \small \frac{11 \times 5}{12 \times 5}$ = $\dpi{100} \small \frac{55}{60}$  and  $\dpi{100} \small \frac{3}{5}$ as $\dpi{100} \small \frac{3 \times 12}{5 \times 12}$ = $\dpi{100} \small \frac{36}{60}$

$\dpi{100} \small \frac{11}{12}$ + $\dpi{100} \small \frac{3}{5}$ = $\dpi{100} \small \frac{55}{60}$ + $\dpi{100} \small \frac{36}{60}$

= $\dpi{100} \small \frac{91}{60}$

Question 9.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 9 is a multiple of 3 , so rewrite it with a denominator of 9.
Rewrite $\dpi{100} \small \frac{4}{3}$ as $\dpi{100} \small \frac{4 \times 3}{3 \times 3}$ = $\dpi{100} \small \frac{12}{9}$

$\dpi{100} \small \frac{2}{9}$ + $\dpi{100} \small \frac{4}{3}$ + $\dpi{100} \small \frac{5}{9}$  =  $\dpi{100} \small \frac{2}{9}$ + $\dpi{100} \small \frac{12}{9}$ + $\dpi{100} \small \frac{5}{9}$

= $\dpi{100} \small \frac{19}{9}$

Question 10.
Your friend buys $$\frac{1}{8}$$ pound of green lentils and $$\frac{3}{4}$$ pound of brown lentils. What fraction of a pound of lentils does she buy?

Given that,
pound of green lentils = $\dpi{100} \small \frac{1}{8}$
Pound of brown lentils = $\dpi{100} \small \frac{3}{4}$
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is a multiple of 4, so rewrite $\dpi{100} \small \frac{3}{4}$ with a denominator of 8.
Rewrite $\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 2}{4 \times 2}$ = $\dpi{100} \small \frac{6}{8}$
Pound of lentils = $\dpi{100} \small \frac{1}{8}$ + $\dpi{100} \small \frac{3}{4}$
= $\dpi{100} \small \frac{1}{8}$ + $\dpi{100} \small \frac{6}{8}$
Fraction of pound of lentils = $\dpi{100} \small \frac{7}{8}$

Question 11.
Reasoning
Newton and Descartes find $$\frac{1}{2}$$ + $$\frac{1}{6}$$. Newton says the sum is $$\frac{4}{6}$$. Descartes says the sum is $$\frac{2}{3}$$. Who is correct? Explain.
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
$\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{1}{6}$
Think: 6 is a multiple of 2, so rewrite $\dpi{100} \small \frac{1}{2}$ with a denominator of 6.
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 3}{2 \times 3}$ = $\dpi{100} \small \frac{3}{6}$
$\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{1}{6}$ = $\dpi{100} \small \frac{3}{6}$ + $\dpi{100} \small \frac{1}{6}$
= $\dpi{100} \small \frac{4}{6}$ or $\dpi{100} \small \frac{2}{3}$
Therefore, both Newton and Descartes answers are correct.

Question 12.
DIG DEEPER!
Write two fractions that have a sum of 1 and have different denominators.
$$\frac{1}{2}$$ + $$\frac{3}{6}$$
= $$\frac{1}{2}$$ × $$\frac{3}{3}$$ + $$\frac{3}{6}$$
= $$\frac{3}{6}$$ + $$\frac{3}{6}$$
= $$\frac{6}{6}$$
= 1

Think and Grow: Modeling Real Life

Example
About $$\frac{17}{15}$$ of Earth’s surface is covered by ocean water.
About $$\frac{3}{100}$$ of Earth’s surface is covered by other water resources.

About how much of Earth’s surface is covered by water?

Show and Grow

Question 13.
The George Washington Bridge links Manhattan, NY, to FortLee, NJ. The part of the bridge in New Jersey is about $$\frac{1}{2}$$ mile long. The part in New York is about $$\frac{2}{5}$$ mile long. About how long is the George Washington Bridge?

Given that,
New Jersey bridge = $$\frac{1}{2}$$ mile long
New York bridge = $$\frac{2}{5}$$ mile long
Add $\dpi{100} \small \frac{1}{2}$ and $\dpi{100} \small \frac{2}{5}$ to find how long is the George Washington Bridge
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 5 is not a multiple of 2, so rewrite each fraction with a denominator of 5 $\dpi{100} \small \times$ 2 = 10
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 5}{2 \times 5}$ = $\dpi{100} \small \frac{5}{10}$  and $\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 2}{5 \times 2}$ = $\dpi{100} \small \frac{4}{10}$
$\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{2}{5}$  = $\dpi{100} \small \frac{5}{10}$ + $\dpi{100} \small \frac{4}{10}$
= $\dpi{100} \small \frac{9}{10}$
So, George Washington Bridge is about $\dpi{100} \small \frac{9}{10}$ mile long.

Question 14.
DIG DEEPER!
Your goal is to practice playing the saxophone for at least 2 hours in 1 week. Do you reach your goal? Explain.

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
$\dpi{100} \small \frac{3}{4}$ + $\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{2}{3}$
Think: Rewrite the denominators as 4 $\dpi{100} \small \times$ 2 $\dpi{100} \small \times$ 3 = 24
Rewrite $\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 6}{4 \times 6}$ = $\dpi{100} \small \frac{18}{24}$
$\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 12}{2 \times 12}$ = $\dpi{100} \small \frac{12}{24}$
$\dpi{100} \small \frac{2}{3}$ as $\dpi{100} \small \frac{2 \times 8}{3 \times 8}$ = $\dpi{100} \small \frac{16}{24}$
$\dpi{100} \small \frac{3}{4}$ + $\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{2}{3}$ = $\dpi{100} \small \frac{18}{24}$ + $\dpi{100} \small \frac{12}{24}$ + $\dpi{100} \small \frac{16}{24}$
= $\dpi{100} \small \frac{46}{24}$ or $\dpi{100} \small \frac{23}{12}$
Total practice time in a week = $\dpi{100} \small \frac{23}{12}$ = 1.91 hours
So, goal does not reached.

### Add Fractions with Unlike Denominators Homework & Practice 8.4

Question 1.
$$\frac{1}{9}$$ + $$\frac{2}{3}$$ = ___
$\dpi{100} \small \frac{1}{9}$ + $\dpi{100} \small \frac{2}{3}$
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 9 is a multiple of 3, so rewrite it with a denominator of 9
Rewrite $\dpi{100} \small \frac{2}{3}$ as $\dpi{100} \small \frac{2 \times 3}{3 \times 3}$ = $\dpi{100} \small \frac{6}{9}$
$\dpi{100} \small \frac{1}{9}$ + $\dpi{100} \small \frac{2}{3}$ = $\dpi{100} \small \frac{1}{9}$ + $\dpi{100} \small \frac{6}{9}$
= $\dpi{100} \small \frac{7}{9}$
$$\frac{1}{9}$$ + $$\frac{2}{3}$$ = $\dpi{100} \small \frac{7}{9}$

Question 2.
$$\frac{1}{2}$$ + $$\frac{3}{4}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 4 is a multiple of 2, so rewrite it with a denominator of 4
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$
$\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{3}{4}$ = $\dpi{100} \small \frac{2}{4}$ + $\dpi{100} \small \frac{3}{4}$
= $\dpi{100} \small \frac{5}{4}$
$$\frac{1}{2}$$ + $$\frac{3}{4}$$ = $\dpi{100} \small \frac{5}{4}$

Question 3.
$$\frac{4}{6}$$ + $$\frac{5}{12}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 12 is a multiple of 6, so rewrite it with a denominator of 12
Rewrite $\dpi{100} \small \frac{4}{6}$ as $\dpi{100} \small \frac{4 \times 2}{6 \times 2}$ = $\dpi{100} \small \frac{8}{12}$
$\dpi{100} \small \frac{4}{6}$ + $\dpi{100} \small \frac{5}{12}$ = $\dpi{100} \small \frac{8}{12}$ + $\dpi{100} \small \frac{5}{12}$
= $\dpi{100} \small \frac{13}{12}$
$$\frac{4}{6}$$ + $$\frac{5}{12}$$ = $\dpi{100} \small \frac{13}{12}$

Question 4.
$$\frac{1}{3}$$ + $$\frac{1}{4}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 4 is not a multiple of 3, so rewrite each fraction with a denominator of 4 $\dpi{100} \small \times$ 3 = 12
Rewrite $\dpi{100} \small \frac{1}{3}$ as $\dpi{100} \small \frac{1 \times 4}{3 \times 4}$ = $\dpi{100} \small \frac{4}{12}$  and $\dpi{100} \small \frac{1}{4}$ as $\dpi{100} \small \frac{1 \times 3}{4 \times 3}$ = $\dpi{100} \small \frac{3}{12}$
$\dpi{100} \small \frac{1}{3}$ + $\dpi{100} \small \frac{1}{4}$  = $\dpi{100} \small \frac{4}{12}$ + $\dpi{100} \small \frac{3}{12}$
= $\dpi{100} \small \frac{7}{12}$
$$\frac{1}{3}$$ + $$\frac{1}{4}$$ = $\dpi{100} \small \frac{7}{12}$

Question 5.
$$\frac{3}{2}$$ + $$\frac{4}{5}$$ = __
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 5 is not a multiple of 2, so rewrite each fraction with a denominator of 5 $\dpi{100} \small \times$ 2 = 10
Rewrite $\dpi{100} \small \frac{3}{2}$ as $\dpi{100} \small \frac{3 \times 5}{2 \times 5}$ = $\dpi{100} \small \frac{15}{10}$  and $\dpi{100} \small \frac{4}{5}$ as $\dpi{100} \small \frac{4 \times 2}{5 \times 2}$ = $\dpi{100} \small \frac{8}{10}$
$\dpi{100} \small \frac{3}{2}$ + $\dpi{100} \small \frac{4}{5}$  = $\dpi{100} \small \frac{15}{10}$ + $\dpi{100} \small \frac{8}{10}$
= $\dpi{100} \small \frac{23}{10}$
$$\frac{3}{2}$$ + $$\frac{4}{5}$$ = $\dpi{100} \small \frac{23}{10}$

Question 6.
$$\frac{6}{8}$$ + $$\frac{9}{10}$$ + $$\frac{1}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 10 is not a multiple of 8, so rewrite each fraction with a denominator of 10 $\dpi{100} \small \times$ 8 = 80
Rewrite $\dpi{100} \small \frac{6}{8}$ as $\dpi{100} \small \frac{6 \times 10}{8 \times 10}$ = $\dpi{100} \small \frac{60}{80}$
$\dpi{100} \small \frac{9}{10}$ as $\dpi{100} \small \frac{9 \times 8}{10 \times 8}$ = $\dpi{100} \small \frac{72}{80}$
$\dpi{100} \small \frac{1}{8}$ as $\dpi{100} \small \frac{1 \times 10}{8 \times 10}$ = $\dpi{100} \small \frac{10}{80}$
$\dpi{100} \small \frac{6}{8}$ + $\dpi{100} \small \frac{9}{10}$ + $\dpi{100} \small \frac{1}{8}$ = $\dpi{100} \small \frac{60}{80}$ + $\dpi{100} \small \frac{72}{80}$ + $\dpi{100} \small \frac{10}{80}$ = $\dpi{100} \small \frac{142}{80}$
$$\frac{6}{8}$$ + $$\frac{9}{10}$$ + $$\frac{1}{8}$$ = $\dpi{100} \small \frac{142}{80}$

Question 7.
You use beads to make a design. Of the beads, $$\frac{1}{3}$$ are red and $$\frac{1}{6}$$ are blue. The rest are white.What fraction of the beads are red or blue?
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 6 is a multiple of 3, so rewrite it with a denominator of 6
Rewrite $\dpi{100} \small \frac{1}{3}$ as $\dpi{100} \small \frac{1 \times 2}{3 \times 2}$ = $\dpi{100} \small \frac{2}{6}$
$\dpi{100} \small \frac{1}{3}$ + $\dpi{100} \small \frac{1}{6}$ = $\dpi{100} \small \frac{2}{6}$ + $\dpi{100} \small \frac{1}{6}$
= $\dpi{100} \small \frac{3}{6}$
The fraction of beads are red or blue = $\dpi{100} \small \frac{3}{6}$ = $\dpi{100} \small \frac{1}{2}$
Rest are white = $\dpi{100} \small \frac{1}{2}$

Question 8.
YOU BE THE TEACHER
Your friend says the sum of $$\frac{1}{5}$$ and $$\frac{9}{10}$$ is $$\frac{10}{15}$$. Is your friend correct? Explain.
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 10 is a multiple of 5, so rewrite it with a denominator of 10
Rewrite $\dpi{100} \small \frac{1}{5}$ as $\dpi{100} \small \frac{1 \times 2}{5 \times 2}$ = $\dpi{100} \small \frac{2}{10}$
$\dpi{100} \small \frac{1}{5}$ + $\dpi{100} \small \frac{9}{10}$ = $\dpi{100} \small \frac{2}{10}$ + $\dpi{100} \small \frac{9}{10}$
= $\dpi{100} \small \frac{11}{10}$
Therefore, the sum of $$\frac{1}{5}$$ and $$\frac{9}{10}$$ is $\dpi{100} \small \frac{11}{10}$.
So, my friend answer is wrong.

Question 9.
Reasoning
Which expressions are equal to $$\frac{14}{15}$$?

$\dpi{100} \small \frac{3}{5}$ + $\dpi{100} \small \frac{1}{3}$ and $\dpi{100} \small \frac{1}{5}$ + $\dpi{100} \small \frac{11}{15}$ are equal to $$\frac{14}{15}$$ and these two expressions only having denominator of 15.

Question 10.
Modeling Real Life
There are 100 senators in the 115th Congress. Democrats make up of the senators, and Republicans make up $$\frac{13}{25}$$ of the 25 senators. The rest are Independents. What fraction of the senators are Democrat or Republican?
Given,
There are 100 senators in the 115th Congress.
Democrats make up of the senators, and Republicans make up $$\frac{13}{25}$$ of the 25 senators.
$$\frac{100}{115}$$ + $$\frac{13}{25}$$
= $$\frac{100}{115}$$ × $$\frac{5}{5}$$ + $$\frac{13}{25}$$ × $$\frac{23}{23}$$
= $$\frac{799}{575}$$
= 1 $$\frac{224}{575}$$

Question 11.
Modeling Real Life
Your friend needs 1 cup of homemade orange juice. He squeezes $$\frac{1}{2}$$ cup of orange juice from one orange and $$\frac{3}{8}$$ cup from another orange. Does your friend need to squeeze another orange? Explain.
$\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{3}{8}$
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is a multiple of 2, so rewrite it with a denominator of 8
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 4}{2 \times 4}$ = $\dpi{100} \small \frac{4}{8}$
$\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{3}{8}$ = $\dpi{100} \small \frac{4}{8}$ + $\dpi{100} \small \frac{3}{8}$ = $\dpi{100} \small \frac{7}{8}$
Juice from 2 oranges = $\dpi{100} \small \frac{7}{8}$
My friend needs 1 cup of orange juice = 1 – $\dpi{100} \small \frac{7}{8}$ = $\dpi{100} \small \frac{1}{8}$
So, my friend needs to squeeze $\dpi{100} \small \frac{1}{8}$ cup from another orange.

Question 12.
DIG DEEPER!
Of all the atoms in caffeine, $$\frac{1}{12}$$ are oxygen atoms, $$\frac{1}{6}$$ are nitrogen atoms, and $$\frac{1}{3}$$ are carbon atoms. The rest of the atoms are hydrogen. What fraction of the atoms in caffeine are oxygen, nitrogen, or hydrogen?
From the given information, hydrogen atoms = 1- ($\dpi{100} \small \frac{1}{12}$ + $\dpi{100} \small \frac{1}{6}$ + $\dpi{100} \small \frac{1}{3}$) = $\dpi{100} \small \frac{5}{12}$
$\dpi{100} \small \frac{1}{12}$ + $\dpi{100} \small \frac{1}{6}$ + $\dpi{100} \small \frac{5}{12}$ = ?
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 12 is a multiple of 6, so rewrite it with a denominator of 12
Rewrite $\dpi{100} \small \frac{1}{6}$ as $\dpi{100} \small \frac{1 \times 2}{6 \times 2}$ = $\dpi{100} \small \frac{2}{12}$
$\dpi{100} \small \frac{1}{12}$ + $\dpi{100} \small \frac{1}{6}$ + $\dpi{100} \small \frac{5}{12}$ = $\dpi{100} \small \frac{1}{12}$ + $\dpi{100} \small \frac{2}{12}$ + $\dpi{100} \small \frac{5}{12}$
= $\dpi{100} \small \frac{8}{12}$
= $\dpi{100} \small \frac{2}{3}$
So $\dpi{100} \small \frac{2}{3}$ of the atoms in caffeine are oxygen, nitrogen, or hydrogen.

Review & Refresh

Use properties to find the sum or product.

Question 13.
5 × 84

Explanation:
We can find the product by using the distributive property.
5 × 84 = 5 × (80 + 4)
= (5 × 80) + (5 × 4)
= 400 + 20
= 420

Question 14.
521 + 0 + 67

Explanation:
We can find the sum of the given expression using the additive identity.
521 + 0 + 67 = 521 + 67
= 588

Question 15.
25 × 8 × 4

Explanation:
25 × 8 × 4
= 25 × 32
= 800

### Lesson 8.5 Subtract Fractions with Unlike Denominators

Explore and Grow

Use a model to find the difference.

Explain how you can use a model to subtract fourths from twelfths.

Construct Arguments
How can you subtract two fractions with unlike denominators without using a model? Explain why your method makes sense.

Think and Grow: Subtract Fractions with Unlike Denominators

You can use equivalent fractions to subtract fractions that have unlike denominators.
Example
Find $$\frac{9}{10}$$ – $$\frac{1}{2}$$.
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 10 is a multiple of 2, so rewrite $$\frac{1}{2}$$ with a denominator of 10.

Example
Find $$\frac{4}{3}$$ – $$\frac{1}{4}$$. Estimate _____
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 4 is not a multiple of 3, so rewrite each fraction with a denominator of 3 × 4 = 12.

Show and Grow

Subtract.

Question 1.
$$\frac{1}{2}$$ – $$\frac{1}{4}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 4 is a multiple of 2, so rewrite $$\frac{1}{2}$$ with a denominator of 4.
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 2}{2 \times 2}$ = $\dpi{100} \small \frac{2}{4}$
$\dpi{100} \small \frac{1}{2}$$\dpi{100} \small \frac{1}{4}$ = $\dpi{100} \small \frac{2}{4}$$\dpi{100} \small \frac{1}{4}$
= $$\frac{1}{4}$$
$$\frac{1}{2}$$ – $$\frac{1}{4}$$ = $$\frac{1}{4}$$

Question 2.
$$\frac{7}{9}$$ – $$\frac{2}{3}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 9 is a multiple of 3, so rewrite $$\frac{2}{3}$$ with a denominator of 9.
Rewrite $\dpi{100} \small \frac{2}{3}$ as $\dpi{100} \small \frac{2 \times 3}{3 \times 3}$ = $\dpi{100} \small \frac{6}{9}$
$\dpi{100} \small \frac{7}{9}$$\dpi{100} \small \frac{2}{3}$ = $\dpi{100} \small \frac{7}{9}$$\dpi{100} \small \frac{6}{9}$
= $\dpi{100} \small \frac{1}{9}$
$$\frac{7}{9}$$ – $$\frac{2}{3}$$ = $$\frac{1}{9}$$

Question 3.
$$\frac{6}{5}$$ – $$\frac{3}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 8 is not a multiple of 5, so rewrite each fraction with a denominator of 5 $\dpi{100} \small \times$ 8 = 40
Rewrite $\dpi{100} \small \frac{6}{5}$ as $\dpi{100} \small \frac{6 \times 8}{5 \times 8}$ = $\dpi{100} \small \frac{48}{40}$
$\dpi{100} \small \frac{3}{8}$ as $\dpi{100} \small \frac{3 \times 5}{8 \times 5}$ = $\dpi{100} \small \frac{15}{40}$
$\dpi{100} \small \frac{6}{5}$$\dpi{100} \small \frac{3}{8}$ = $\dpi{100} \small \frac{48}{40}$ – $\dpi{100} \small \frac{15}{40}$
= $\dpi{100} \small \frac{33}{40}$
$$\frac{6}{5}$$ – $$\frac{3}{8}$$ = $\dpi{100} \small \frac{33}{40}$

Apply and Grow: Practice

Subtract.

Question 4.
$$\frac{10}{12}$$ – $$\frac{3}{4}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 12 is a multiple of 4, so rewrite $$\frac{3}{4}$$ with a denominator of 12.
Rewrite $\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 3}{4 \times 3}$ = $\dpi{100} \small \frac{9}{12}$
$\dpi{100} \small \frac{10}{12}$$\dpi{100} \small \frac{3}{4}$ = $\dpi{100} \small \frac{10}{12}$$\dpi{100} \small \frac{9}{12}$
=$\dpi{100} \small \frac{1}{12}$
$$\frac{10}{12}$$ – $$\frac{3}{4}$$ = $$\frac{1}{12}$$

Question 5.
$$\frac{1}{3}$$ – $$\frac{1}{6}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 6 is a multiple of 3, so rewrite $$\frac{1}{3}$$ with a denominator of 6.
Rewrite $\dpi{100} \small \frac{1}{3}$ as $\dpi{100} \small \frac{1 \times 2}{3 \times 2}$ = $\dpi{100} \small \frac{2}{6}$
$\dpi{100} \small \frac{1}{3}$$\dpi{100} \small \frac{1}{6}$ = $\dpi{100} \small \frac{2}{6}$$\dpi{100} \small \frac{1}{6}$
= $\dpi{100} \small \frac{1}{6}$
$$\frac{1}{3}$$ – $$\frac{1}{6}$$ = $$\frac{1}{6}$$

Question 6.
$$\frac{9}{10}$$ – $$\frac{2}{5}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 10 is a multiple of 5, so rewrite $$\frac{2}{5}$$ with a denominator of 10.
Rewrite $\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 2}{5 \times 2}$ = $\dpi{100} \small \frac{4}{10}$
$\dpi{100} \small \frac{9}{10}$$\dpi{100} \small \frac{2}{5}$ = $\dpi{100} \small \frac{9}{10}$$\dpi{100} \small \frac{4}{10}$
= $\dpi{100} \small \frac{5}{10}$ or $\dpi{100} \small \frac{1}{2}$
$$\frac{9}{10}$$ – $$\frac{2}{5}$$ = $$\frac{1}{2}$$

Question 7.
$$\frac{5}{4}$$ – $$\frac{2}{5}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 5 is not a multiple of 4, so rewrite each fraction with a denominator of 4 $\dpi{100} \small \times$ 5 = 20
Rewrite $\dpi{100} \small \frac{5}{4}$ as $\dpi{100} \small \frac{5 \times 5}{4 \times 5}$ = $\dpi{100} \small \frac{25}{20}$
$\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 4}{5 \times 4}$ = $\dpi{100} \small \frac{8}{20}$
$\dpi{100} \small \frac{5}{4}$$\dpi{100} \small \frac{2}{5}$ = $\dpi{100} \small \frac{25}{20}$$\dpi{100} \small \frac{8}{20}$
= $\dpi{100} \small \frac{17}{20}$
$$\frac{5}{4}$$ – $$\frac{2}{5}$$ = $$\frac{17}{20}$$

Question 8.
$$\frac{13}{16}$$ – $$\frac{3}{16}$$ – $$\frac{5}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 16 is a multiple of 8, so rewrite $$\frac{5}{8}$$ with a denominator of 16.
Rewrite $\dpi{100} \small \frac{5}{8}$ as $\dpi{100} \small \frac{5 \times 2}{8 \times 2}$ = $\dpi{100} \small \frac{10}{16}$
$\dpi{100} \small \frac{13}{16}$$\dpi{100} \small \frac{3}{16}$$\dpi{100} \small \frac{10}{16}$ = 0
$$\frac{13}{16}$$ – $$\frac{3}{16}$$ – $$\frac{5}{8}$$ = 0

Question 9.
$$\frac{8}{9}$$ – ($$\frac{2}{3}$$ + $$\frac{1}{6}$$) = ___
Use equivalent fractions to write the fractions with a common denominator.
common denominator for 9, 3 and 6 = 18
Rewrite $\dpi{100} \small \frac{8}{9}$ as $\dpi{100} \small \frac{16}{18}$
$\dpi{100} \small \frac{2}{3}$ = $\dpi{100} \small \frac{12}{18}$
$\dpi{100} \small \frac{1}{6}$ = $\dpi{100} \small \frac{3}{18}$
$\dpi{100} \small \frac{8}{9}$$\dpi{100} \small \frac{2}{3}$ + $\dpi{100} \small \frac{1}{6}$ = $\dpi{100} \small \frac{16}{18}$$\dpi{100} \small \frac{12}{18}$ + $\dpi{100} \small \frac{3}{18}$ = $\dpi{100} \small \frac{1}{18}$
$$\frac{8}{9}$$ – ($$\frac{2}{3}$$ + $$\frac{1}{6}$$) = $$\frac{1}{18}$$

Question 10.
You have $$\frac{1}{3}$$ yard of wire. You use $$\frac{1}{3}$$ yard to make an electric circuit. How much wire do you have left?

$\dpi{100} \small \frac{1}{3}$$\dpi{100} \small \frac{1}{3}$ = 0

Question 11.
Your friend finds $$\frac{5}{8}$$ – $$\frac{2}{5}$$. Explain why his answer is unreasonable. What did he do wrong?

Answer: For subtracting two fractions, the denominators must be same. Here, denominators are different.
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 8 is not a multiple of 5, so rewrite each fraction with a denominator of 8 $\dpi{100} \small \times$ 5 = 40
Rewrite $\dpi{100} \small \frac{5}{8}$ as $\dpi{100} \small \frac{5 \times 5}{8 \times 5}$ = $\dpi{100} \small \frac{25}{40}$
$\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 8}{5 \times 8}$ = $\dpi{100} \small \frac{16}{40}$
$\dpi{100} \small \frac{5}{8}$$\dpi{100} \small \frac{2}{5}$ = $\dpi{100} \small \frac{25}{40}$$\dpi{100} \small \frac{16}{40}$
= $\dpi{100} \small \frac{9}{40}$
$$\frac{5}{8}$$ – $$\frac{2}{5}$$ = $$\frac{9}{40}$$

Question 12.
Number Sense
Which two fractions have a difference of $$\frac{1}{8}$$ ?

$\dpi{100} \small \frac{1}{2}$ and $\dpi{100} \small \frac{3}{8}$
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 8 is a multiple of 2, so rewrite $\dpi{100} \small \frac{1}{2}$ with a denominator of 8.
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 4}{2 \times 4}$ = $\dpi{100} \small \frac{4}{8}$
$\dpi{100} \small \frac{1}{2}$$\dpi{100} \small \frac{3}{8}$ = $\dpi{100} \small \frac{4}{8}$$\dpi{100} \small \frac{3}{8}$
= $\dpi{100} \small \frac{1}{8}$
So, fractions $\dpi{100} \small \frac{1}{2}$ and $\dpi{100} \small \frac{3}{8}$ have the difference of $\dpi{100} \small \frac{1}{8}$.

Think and Grow: Modeling Real Life

Example
A geologist needs $$\frac{1}{2}$$ cup of volcanic sand to perform an experiment. She has $$\frac{3}{2}$$ cups of quartz sand. She has $$\frac{2}{3}$$ cup more quartz sand than volcanic sand. Can she perform the experiment?

Find how many cups of volcanic sand the geologist has by subtracting $$\frac{2}{3}$$ from $$\frac{3}{2}$$.
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 3 is not a multiple of 2, so rewrite each fraction with a denominator 2 × 3 = 6.

Show and Grow

Question 13.
The world record for the longest dog tail is $$\frac{77}{100}$$ meter. The previous record was $$\frac{1}{20}$$ meter. shorter than the current record. Was the previous record longer than $$\frac{3}{4}$$ meter?

Given that,
World record for the longest dog tail is $\dpi{100} \small \frac{77}{100}$ meter.
Previous record was $\dpi{100} \small \frac{1}{20}$ meter shorter than the current record
So subtract $\dpi{100} \small \frac{1}{20}$ meter from the current record to find previous record.
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Rewrite $\dpi{100} \small \frac{1}{20}$ as $\dpi{100} \small \frac{5}{100}$
Previous record = $\dpi{100} \small \frac{77}{100}$$\dpi{100} \small \frac{5}{100}$ = $\dpi{100} \small \frac{72}{100}$ = $\dpi{100} \small \frac{18}{25}$ meter
So the previous record is not longer than $\dpi{100} \small \frac{3}{4}$ meter.

Question 14.
DIG DEEPER!
A woodworker has 1 gallon of paint for a tree house. He uses $$\frac{3}{8}$$ gallon to paint the walls and $$\frac{1}{5}$$ gallon to paint the ladder. He needs $$\frac{1}{4}$$ gallon to paint the roof. Does he have enough paint? Explain.
Given that,
Woodworker has 1 gallon of paint
$\dpi{100} \small \frac{3}{8}$ gallon is used to paint the walls.
$\dpi{100} \small \frac{1}{5}$ gallon is used to paint the ladder.
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
$\dpi{100} \small \frac{3}{8}$ = $\dpi{100} \small \frac{15}{40}$
$\dpi{100} \small \frac{1}{5}$ = $\dpi{100} \small \frac{8}{40}$
1 – $\dpi{100} \small \frac{15}{40}$$\dpi{100} \small \frac{8}{40}$ = $\dpi{100} \small \frac{17}{40}$
Therefore, he has more than $\dpi{100} \small \frac{1}{4}$ gallon to paint the roof.

### Subtract Fractions with Unlike Denominators Homework & Practice 8.5

Subtract

Question 1.
$$\frac{3}{4}$$ – $$\frac{1}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 8 is a multiple of 4, so rewrite $$\frac{3}{4}$$ with a denominator of 8
Rewrite $\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 2}{4 \times 2}$ = $\dpi{100} \small \frac{6}{8}$
$\dpi{100} \small \frac{3}{4}$$\dpi{100} \small \frac{1}{8}$ = $\dpi{100} \small \frac{6}{8}$$\dpi{100} \small \frac{1}{8}$
= $\dpi{100} \small \frac{5}{8}$
$$\frac{3}{4}$$ – $$\frac{1}{8}$$ = $\dpi{100} \small \frac{5}{8}$

Question 2.
$$\frac{4}{5}$$ – $$\frac{6}{15}$$ = __
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 15 is a multiple of 5, so rewrite $$\frac{4}{5}$$ with a denominator of 15
Rewrite $\dpi{100} \small \frac{4}{5}$ as $\dpi{100} \small \frac{4 \times 3}{5 \times 3}$ = $\dpi{100} \small \frac{12}{15}$
$\dpi{100} \small \frac{4}{5}$$\dpi{100} \small \frac{6}{15}$ = $\dpi{100} \small \frac{12}{15}$$\dpi{100} \small \frac{6}{15}$
= $\dpi{100} \small \frac{6}{15}$
$$\frac{4}{5}$$ – $$\frac{6}{15}$$ = $\dpi{100} \small \frac{6}{15}$

Question 3.
$$\frac{1}{2}$$ – $$\frac{1}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 8 is a multiple of 2, so rewrite $$\frac{1}{2}$$ with a denominator of 8
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 4}{2 \times 4}$ = $\dpi{100} \small \frac{4}{8}$
$\dpi{100} \small \frac{1}{2}$$\dpi{100} \small \frac{1}{8}$ = $\dpi{100} \small \frac{4}{8}$$\dpi{100} \small \frac{1}{8}$
= $\dpi{100} \small \frac{3}{8}$
$$\frac{1}{2}$$ – $$\frac{1}{8}$$ = $\dpi{100} \small \frac{3}{8}$

Question 4.
$$\frac{5}{3}$$ – $$\frac{3}{4}$$ = _____
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 4 is not a multiple of 3, so rewrite each fraction with a denominator of 4 x 3 =12
Rewrite $\dpi{100} \small \frac{5}{3}$ as $\dpi{100} \small \frac{5 \times 4}{3 \times 4}$ = $\dpi{100} \small \frac{20}{12}$
$\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 3}{4 \times 3}$ = $\dpi{100} \small \frac{9}{12}$
$\dpi{100} \small \frac{5}{3}$$\dpi{100} \small \frac{3}{4}$ = $\dpi{100} \small \frac{20}{12}$$\dpi{100} \small \frac{9}{12}$
= $\dpi{100} \small \frac{11}{12}$
$$\frac{5}{3}$$ – $$\frac{3}{4}$$ = $\dpi{100} \small \frac{11}{12}$

Question 5.
$$\frac{6}{8}$$ – $$\frac{7}{10}$$ = _____
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 10 is not a multiple of 8, so rewrite each fraction with a denominator of 10 x 8 =80
Rewrite $\dpi{100} \small \frac{6}{8}$ as $\dpi{100} \small \frac{6 \times 10}{8 \times 10}$ = $\dpi{100} \small \frac{60}{80}$
$\dpi{100} \small \frac{7}{10}$ as $\dpi{100} \small \frac{7 \times 8}{10 \times 8}$ = $\dpi{100} \small \frac{56}{80}$
$\dpi{100} \small \frac{6}{8}$$\dpi{100} \small \frac{7}{10}$ = $\dpi{100} \small \frac{60}{80}$$\dpi{100} \small \frac{56}{80}$
= $\dpi{100} \small \frac{4}{80}$
$$\frac{6}{8}$$ – $$\frac{7}{10}$$ = $\dpi{100} \small \frac{4}{80}$ = $\dpi{100} \small \frac{1}{20}$

Question 6.
$$\frac{5}{6}$$ – $$\frac{1}{4}$$ – $$\frac{3}{12}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 12 is a multiple of 4 and 6, so rewrite each fraction with a denominator of 12
Rewrite $\dpi{100} \small \frac{5}{6}$ as $\dpi{100} \small \frac{5 \times 2}{6 \times 2}$ = $\dpi{100} \small \frac{10}{12}$
$\dpi{100} \small \frac{1}{4}$ as $\dpi{100} \small \frac{1 \times 3}{4 \times 3}$ = $\dpi{100} \small \frac{3}{12}$
$\dpi{100} \small \frac{5}{6}$$\dpi{100} \small \frac{1}{4}$$\dpi{100} \small \frac{3}{12}$ = $\dpi{100} \small \frac{10}{12}$$\dpi{100} \small \frac{3}{12}$$\dpi{100} \small \frac{3}{12}$
= $\dpi{100} \small \frac{4}{12}$
$$\frac{5}{6}$$ – $$\frac{1}{4}$$ – $$\frac{3}{12}$$ = $\dpi{100} \small \frac{4}{12}$

Question 7.
You eat $$\frac{1}{12}$$ of a vegetable casserole. Your friend eats $$\frac{1}{6}$$ of the same casserole. How much more does your friend eat than you?

$\dpi{100} \small \frac{1}{6}$$\dpi{100} \small \frac{1}{12}$
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 12 is a multiple of 6, so rewrite $\dpi{100} \small \frac{1}{6}$ with a denominator of 12
Rewrite $\dpi{100} \small \frac{1}{6}$ as $\dpi{100} \small \frac{1 \times 2}{6 \times 2}$ = $\dpi{100} \small \frac{2}{12}$
$\dpi{100} \small \frac{1}{6}$$\dpi{100} \small \frac{1}{12}$ = $\dpi{100} \small \frac{2}{12}$$\dpi{100} \small \frac{1}{12}$ = $\dpi{100} \small \frac{1}{12}$
So, my friend eats $\dpi{100} \small \frac{1}{12}$ of a vegetable casserole than me.

Question 8.
Writing
Why do fractions need a common denominator before you can add or subtract them?
In order to add fractions, the fractions must have a common denominator. We need the pieces of each fraction to be the same size to combine them together. These two fractions have the same denominator, so the equal parts that the whole has been split into are the same size.

Question 9.
Logic
Find a.

a = $\dpi{100} \small \frac{7}{10}$$\dpi{100} \small \frac{1}{2}$
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 10 is a multiple of 2, so rewrite $\dpi{100} \small \frac{1}{2}$ with a denominator of 10
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 5}{2 \times 5}$ = $\dpi{100} \small \frac{5}{10}$
a = $\dpi{100} \small \frac{7}{10}$$\dpi{100} \small \frac{1}{2}$ = $\dpi{100} \small \frac{7}{10}$$\dpi{100} \small \frac{5}{10}$
a = $\dpi{100} \small \frac{2}{10}$ = $\dpi{100} \small \frac{1}{5}$

Question 10.
DIG DEEPER!
Write and solve an equation to find the difference between Length A and Length B on the ruler.

$$\frac{1}{10}$$ × $$\frac{2}{2}$$ = $$\frac{2}{20}$$
$$\frac{9}{10}$$ × $$\frac{2}{2}$$ = $$\frac{18}{20}$$
$$\frac{18}{20}$$ – $$\frac{2}{20}$$ = $$\frac{16}{20}$$

Question 11.
Modeling Real Life
You want to stack cups in $$\frac{1}{4}$$ minute. Your first attempt takes $$\frac{1}{2}$$ minute. Your second attempt takes $$\frac{3}{10}$$ minute less than your first attempt. Do you meet your goal?
$\dpi{100} \small \frac{1}{2}$$\dpi{100} \small \frac{3}{10}$
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 10 is a multiple of 2, so rewrite $\dpi{100} \small \frac{1}{2}$ with a denominator of 10
Rewrite $\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 5}{2 \times 5}$ = $\dpi{100} \small \frac{5}{10}$
$\dpi{100} \small \frac{1}{2}$$\dpi{100} \small \frac{3}{10}$ = $\dpi{100} \small \frac{5}{10}$$\dpi{100} \small \frac{3}{10}$
= $\dpi{100} \small \frac{2}{10}$
= $\dpi{100} \small \frac{1}{5}$
So my second attempt takes $\dpi{100} \small \frac{1}{5}$ minute and I did not meet my goal.

Question 12.
Modeling Real Life
You and your friend each have a canvas of the same size. You divide your canvas into 5 sections and paint 3 of them. Your friend divides her canvas into 7 sections and paints 4 of them. Who paints more? How much more?
My canvas = $\dpi{100} \small \frac{3}{5}$
My friend canvas = $\dpi{100} \small \frac{4}{7}$
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 7 is a multiple of 5, so rewrite both with a denominator of 7 x 5 = 35
Rewrite $\dpi{100} \small \frac{3}{5}$ as $\dpi{100} \small \frac{3 \times 7}{5 \times 7}$ = $\dpi{100} \small \frac{21}{35}$
$\dpi{100} \small \frac{4}{7}$ as $\dpi{100} \small \frac{4 \times 5}{7 \times 5}$ = $\dpi{100} \small \frac{20}{35}$
$\dpi{100} \small \frac{3}{5}$$\dpi{100} \small \frac{4}{7}$ = $\dpi{100} \small \frac{21}{35}$$\dpi{100} \small \frac{20}{35}$ = $\dpi{100} \small \frac{1}{35}$
I paint $\dpi{100} \small \frac{1}{35}$ more canvas than my friend.

Review & Refresh

Question 13.
1.7 + 5 + 4.3 = ___

Question 14.
15.24 + 6.13 – 7 = ___

### Lesson 8.6 Add Mixed Numbers

Explore and Grow

Use a model to find the sum.

Construct Arguments
How can you add mixed numbers with unlike denominators without using a model? Explain why your method makes sense.

Think and Grow: Add Mixed Numbers

Key Idea
A proper fraction is a fraction less than 1. An improper fraction is a fraction greater than 1. A mixed number represents the sum of a whole number and a proper fraction. You can use equivalent fractions to add mixed numbers.
Example
Find 1$$\frac{1}{2}$$ + 2$$\frac{5}{6}$$

Show and Grow

Question 1.
2$$\frac{2}{3}$$ + 2$$\frac{1}{6}$$ = ___
To add the fractional parts, use a common denominator.
2 $\dpi{100} \small \frac{2}{3}$  = 2 $\dpi{100} \small \frac{4}{6}$
2 $\dpi{100} \small \frac{1}{6}$  = 2 $\dpi{100} \small \frac{1}{6}$
2 $\dpi{100} \small \frac{4}{6}$  + 2 $\dpi{100} \small \frac{1}{6}$ = 4 $\dpi{100} \small \frac{5}{6}$

Question 2.
1$$\frac{5}{12}$$ + 3$$\frac{3}{4}$$ = ___
To add the fractional parts, use a common denominator.
1 $\dpi{100} \small \frac{5}{12}$ = 1 $\dpi{100} \small \frac{5}{12}$
3 $\dpi{100} \small \frac{3}{4}$ = 3 $\dpi{100} \small \frac{9}{12}$
1 $\dpi{100} \small \frac{5}{12}$ + 3 $\dpi{100} \small \frac{9}{12}$ = 4 $\dpi{100} \small \frac{14}{12}$ = 4 $\dpi{100} \small \frac{7}{6}$

Apply and Grow: Practice

Question 3.
5$$\frac{4}{9}$$ + 1$$\frac{2}{3}$$ = ___
To add the fractional parts, use a common denominator.
5 $\dpi{100} \small \frac{4}{9}$ = 5 $\dpi{100} \small \frac{4}{9}$
1 $\dpi{100} \small \frac{2}{3}$ = 1 $\dpi{100} \small \frac{6}{9}$
5 $\dpi{100} \small \frac{4}{9}$ + 1 $\dpi{100} \small \frac{6}{9}$ = 6 $\dpi{100} \small \frac{10}{9}$ = $\dpi{100} \small \frac{64}{9}$ = 7 $\dpi{100} \small \frac{1}{9}$

Question 4.
3$$\frac{1}{2}$$ + $$\frac{5}{12}$$ = ___
To add the fractional parts, use a common denominator.
3 $\dpi{100} \small \frac{1}{2}$ = 3 $\dpi{100} \small \frac{6}{12}$
3 $\dpi{100} \small \frac{1}{2}$ + $\dpi{100} \small \frac{5}{12}$ = 3 $\dpi{100} \small \frac{6}{12}$ + $\dpi{100} \small \frac{5}{12}$
= 3 $\dpi{100} \small \frac{11}{12}$

Question 5.
4$$\frac{5}{6}$$ + 3$$\frac{5}{12}$$ = ___
To add the fractional parts, use a common denominator.
4 $\dpi{100} \small \frac{5}{6}$ = 4 $\dpi{100} \small \frac{10}{12}$
3 $\dpi{100} \small \frac{5}{12}$ = 3 $\dpi{100} \small \frac{5}{12}$
4 $\dpi{100} \small \frac{10}{12}$ + 3 $\dpi{100} \small \frac{5}{12}$ = 7 $\dpi{100} \small \frac{15}{12}$ = $\dpi{100} \small \frac{99}{12}$ = 8 $\dpi{100} \small \frac{3}{12}$

Question 6.
$$\frac{4}{5}$$ + 8$$\frac{7}{20}$$ = ___
To add the fractional parts, use a common denominator.
Rewrite $\dpi{100} \small \frac{4}{5}$ as $\dpi{100} \small \frac{16}{20}$
$\dpi{100} \small \frac{4}{5}$ + 8 $\dpi{100} \small \frac{7}{20}$ = $\dpi{100} \small \frac{16}{20}$ + 8 $\dpi{100} \small \frac{7}{20}$
= 8 $\dpi{100} \small \frac{23}{20}$

Question 7.
2$$\frac{1}{3}$$ + $$\frac{1}{6}$$ + 3$$\frac{2}{3}$$ = ___
To add the fractional parts, use a common denominator.
2 $\dpi{100} \small \frac{1}{3}$ = 2 $\dpi{100} \small \frac{2}{6}$
3 $\dpi{100} \small \frac{2}{3}$ = 3 $\dpi{100} \small \frac{4}{6}$
2 $\dpi{100} \small \frac{1}{3}$ + $\dpi{100} \small \frac{1}{6}$ + 3 $\dpi{100} \small \frac{2}{3}$ = 2 $\dpi{100} \small \frac{2}{6}$ + $\dpi{100} \small \frac{1}{6}$ + 3 $\dpi{100} \small \frac{4}{6}$
= 5 $\dpi{100} \small \frac{7}{6}$

Question 8.
5$$\frac{1}{2}$$ + 4$$\frac{3}{4}$$ + 6$$\frac{5}{8}$$ = ___
To add the fractional parts, use a common denominator.
5 $\dpi{100} \small \frac{1}{2}$ = 5 $\dpi{100} \small \frac{4}{8}$
4 $\dpi{100} \small \frac{3}{4}$ = 4 $\dpi{100} \small \frac{6}{8}$
5 $\dpi{100} \small \frac{1}{2}$ + 4 $\dpi{100} \small \frac{3}{4}$  + 6 $\dpi{100} \small \frac{5}{8}$ = 5 $\dpi{100} \small \frac{4}{8}$ + 4 $\dpi{100} \small \frac{6}{8}$ + 6 $\dpi{100} \small \frac{5}{8}$
= 15 $\dpi{100} \small \frac{15}{8}$

Question 9.
Your science class makes magic milk using 1$$\frac{1}{8}$$ cups of watercolor paint and 1$$\frac{3}{4}$$ cups of milk. How many cups of magic milk does your class make?

Given data,
Watercolor paint = 1 $\dpi{100} \small \frac{1}{8}$ cups
Milk = 1 $\dpi{100} \small \frac{3}{4}$ cups
To add the fractional parts, use a common denominator.
1 $\dpi{100} \small \frac{3}{4}$ = 1 $\dpi{100} \small \frac{6}{8}$
1 $\dpi{100} \small \frac{1}{8}$ + 1 $\dpi{100} \small \frac{6}{8}$ = 2 $\dpi{100} \small \frac{7}{8}$ = $\dpi{100} \small \frac{23}{8}$
So the class makes 2 $\dpi{100} \small \frac{7}{8}$ cups of magic milk

Question 10.
Structure
Find 2$$\frac{3}{10}$$ + 4$$\frac{2}{5}$$ two different ways.
Method 1:
To add the fractional parts, use a common denominator.
4 $\dpi{100} \small \frac{2}{5}$ = 4 $\dpi{100} \small \frac{4}{10}$
2 $\dpi{100} \small \frac{3}{10}$ + 4 $\dpi{100} \small \frac{2}{5}$ = 2 $\dpi{100} \small \frac{3}{10}$ + 4 $\dpi{100} \small \frac{4}{10}$
= 6 $\dpi{100} \small \frac{7}{10}$
Method 2:
Write the mixed numbers as improper fractions with a common denominator and then add.
2 $\dpi{100} \small \frac{3}{10}$ = 2 + $\dpi{100} \small \frac{3}{10}$ = $\dpi{100} \small \frac{20}{10}$ + $\dpi{100} \small \frac{3}{10}$ = $\dpi{100} \small \frac{23}{10}$
4 $\dpi{100} \small \frac{2}{5}$ = 4 + $\dpi{100} \small \frac{2}{5}$ = $\dpi{100} \small \frac{22}{5}$ = $\dpi{100} \small \frac{44}{10}$
2 $\dpi{100} \small \frac{3}{10}$ + 4 $\dpi{100} \small \frac{2}{5}$ = $\dpi{100} \small \frac{23}{10}$ + $\dpi{100} \small \frac{44}{10}$
= $\dpi{100} \small \frac{67}{10}$
= 6 $\dpi{100} \small \frac{7}{10}$

Question 11.
DIG DEEPER!
Find the missing numbers.

To add the fractional parts, use a common denominator.
Rewrite 2 $\dpi{100} \small \frac{3}{4}$  as 2 $\dpi{100} \small \frac{6}{8}$
we can write 4 $\dpi{100} \small \frac{3}{8}$ as $\dpi{100} \small \frac{35}{8}$ = 3 $\dpi{100} \small \frac{11}{8}$
2 $\dpi{100} \small \frac{6}{8}$ + 1 $\dpi{100} \small \frac{5}{8}$ = 3 $\dpi{100} \small \frac{11}{8}$ = 4 $\dpi{100} \small \frac{3}{8}$
So the missing numbers are 1 and 5.
2 $\dpi{100} \small \frac{6}{8}$ + 1 $\dpi{100} \small \frac{5}{8}$ = 4 $\dpi{100} \small \frac{3}{8}$

Think and Grow: Modeling Real Life

Example
You kayak 1$$\frac{8}{10}$$ miles and then take a break. You kayak 1$$\frac{1}{4}$$ more miles. How many miles do you kayak altogether?

Show and Grow

Question 12.
You listen to a song that is 2$$\frac{3}{4}$$ minutes long. Then you listen to a song that is 3$$\frac{1}{3}$$ minutes long. How many minutes do you spend listening to the two songs altogether?
To add the fractional parts, use a common denominator 4 x 3 = 12.
2 $\dpi{100} \small \frac{3}{4}$ = 2 $\dpi{100} \small \frac{9}{12}$
3 $\dpi{100} \small \frac{1}{3}$ = 3 $\dpi{100} \small \frac{4}{12}$
2 $\dpi{100} \small \frac{3}{4}$ + 3 $\dpi{100} \small \frac{1}{3}$ = 2 $\dpi{100} \small \frac{9}{12}$ + 3 $\dpi{100} \small \frac{4}{12}$
= 5 $\dpi{100} \small \frac{13}{12}$ min
So I spend 5 $\dpi{100} \small \frac{13}{12}$ min listening to the two songs altogether.

Question 13.
DIG DEEPER!
A beekeeper collects 3$$\frac{3}{4}$$ more pounds of honey from Hive 3 than Hive 1. Which hive produces the most honey? Explain.

From the given information,
Honey from Hive 3 = Hive 1 honey + 3 $\dpi{100} \small \frac{3}{4}$
= 23 $\dpi{100} \small \frac{5}{8}$ + 3 $\dpi{100} \small \frac{3}{4}$
= 23 $\dpi{100} \small \frac{5}{8}$ + 3 $\dpi{100} \small \frac{6}{8}$
= 26 $\dpi{100} \small \frac{11}{8}$
Use a common denominator for all the hives
Hive 1 honey = 23 $\dpi{100} \small \frac{5}{8}$
Hive 2 honey = 27 $\dpi{100} \small \frac{1}{2}$ = 27 $\dpi{100} \small \frac{4}{8}$
Hive 3 honey = 26 $\dpi{100} \small \frac{11}{8}$
Therefore, Hive 2 produces the most honey.

### Add Mixed Numbers Homework & Practice 8.6

Question 1.
6$$\frac{2}{5}$$ + 1$$\frac{3}{10}$$
To add the fractional parts, use a common denominator.
6 $\dpi{100} \small \frac{2}{5}$= 6 $\dpi{100} \small \frac{4}{10}$
6 $\dpi{100} \small \frac{2}{5}$ + 1 $\dpi{100} \small \frac{3}{10}$ = 6 $\dpi{100} \small \frac{4}{10}$ + 1 $\dpi{100} \small \frac{3}{10}$
= 7 $\dpi{100} \small \frac{7}{10}$

Question 2.
2$$\frac{2}{3}$$ + 5$$\frac{3}{6}$$ = ___
To add the fractional parts, use a common denominator.
2 $\dpi{100} \small \frac{2}{3}$= 2 $\dpi{100} \small \frac{4}{6}$
2 $\dpi{100} \small \frac{2}{3}$ + 5 $\dpi{100} \small \frac{3}{6}$ = 2 $\dpi{100} \small \frac{4}{6}$ + 5 $\dpi{100} \small \frac{3}{6}$
= 7 $\dpi{100} \small \frac{7}{6}$

Question 3.
$$\frac{1}{4}$$ + 3$$\frac{2}{5}$$ = ___
To add the fractional parts, use a common denominator 4 x 5 = 20
$\dpi{100} \small \frac{1}{4}$ = $\dpi{100} \small \frac{5}{20}$
3 $\dpi{100} \small \frac{2}{5}$ = 3 $\dpi{100} \small \frac{8}{20}$
$\dpi{100} \small \frac{1}{4}$ + 3 $\dpi{100} \small \frac{2}{5}$  = $\dpi{100} \small \frac{5}{20}$ + 3 $\dpi{100} \small \frac{8}{20}$ = 3 $\dpi{100} \small \frac{13}{20}$

Question 4.
9$$\frac{5}{7}$$ + $$\frac{2}{3}$$ = ___
To add the fractional parts, use a common denominator 7 x 3 = 21
9 $\dpi{100} \small \frac{5}{7}$ = 9 $\dpi{100} \small \frac{15}{21}$
$\dpi{100} \small \frac{2}{3}$ = $\dpi{100} \small \frac{14}{21}$
9 $\dpi{100} \small \frac{5}{7}$ + $\dpi{100} \small \frac{2}{3}$ = 9 $\dpi{100} \small \frac{15}{21}$ + $\dpi{100} \small \frac{14}{21}$ = 9 $\dpi{100} \small \frac{29}{21}$

Question 5.
2$$\frac{1}{2}$$ + 1$$\frac{3}{4}$$ + $$\frac{1}{2}$$ = ___
To add the fractional parts, use a common denominator
2 $\dpi{100} \small \frac{1}{2}$ = 2 $\dpi{100} \small \frac{2}{4}$
1 $\dpi{100} \small \frac{3}{4}$ = 1 $\dpi{100} \small \frac{3}{4}$
$\dpi{100} \small \frac{1}{2}$ = $\dpi{100} \small \frac{2}{4}$
2 $\dpi{100} \small \frac{1}{2}$ + 1 $\dpi{100} \small \frac{3}{4}$  + $\dpi{100} \small \frac{1}{2}$ = 2 $\dpi{100} \small \frac{2}{4}$ +1 $\dpi{100} \small \frac{3}{4}$ + $\dpi{100} \small \frac{2}{4}$ = 3 $\dpi{100} \small \frac{7}{4}$

Question 6.
2$$\frac{2}{3}$$ + 4$$\frac{1}{2}$$ + 3$$\frac{5}{6}$$ = ___
To add the fractional parts, use a common denominator
2 $\dpi{100} \small \frac{2}{3}$ = 2 $\dpi{100} \small \frac{4}{6}$
4 $\dpi{100} \small \frac{1}{2}$ = 4 $\dpi{100} \small \frac{3}{6}$
2 $\dpi{100} \small \frac{2}{3}$ + 4 $\dpi{100} \small \frac{1}{2}$  + 3 $\dpi{100} \small \frac{5}{6}$ = 2 $\dpi{100} \small \frac{4}{6}$ + 4 $\dpi{100} \small \frac{3}{6}$  + 3 $\dpi{100} \small \frac{5}{6}$ = 9 $\dpi{100} \small \frac{12}{6}$
2 $\dpi{100} \small \frac{2}{3}$ + 4 $\dpi{100} \small \frac{1}{2}$  + 3 $\dpi{100} \small \frac{5}{6}$ = 11

Question 7.
A veterinarian spends 3$$\frac{3}{4}$$ hours helping cats and 5$$\frac{1}{2}$$ hours helping dogs. How many hours does she spend helping cats and dogs altogether?

To add the fractional parts, use a common denominator
3 $\dpi{100} \small \frac{3}{4}$
5 $\dpi{100} \small \frac{1}{2}$ = 5 $\dpi{100} \small \frac{2}{4}$
3 $\dpi{100} \small \frac{3}{4}$ + 5 $\dpi{100} \small \frac{1}{2}$ = 3 $\dpi{100} \small \frac{3}{4}$ + 5 $\dpi{100} \small \frac{2}{4}$
= 8 $\dpi{100} \small \frac{5}{4}$
So veterinarian spends 8 $\dpi{100} \small \frac{5}{4}$ hours helping cats and dogs altogether.

Question 8.
Writing
How is adding mixed numbers with unlike denominators similar to adding fractions with unlike denominators? How is it different?
For both adding mixed numbers and adding fractions we have to use a common denominator.
How it is different?

Question 9.
Logic
Can you add two mixed numbers and get a sum of 2? Explain.
Yes, the sum of two mixed numbers can be equal to 2 but only of one of the mixed numbers is negative.

Question 10.
Structure
Shade the model to represent the sum. Then write an equation to represent your model.

Answer: 4 $$\frac{1}{4}$$

Question 11.
Modeling Real Life
An emperor tamarin has a body length of 9$$\frac{5}{10}$$ inches and a tail length of 14$$\frac{1}{4}$$ inches. How long is the emperor tamarin?

To add the fractional parts, use a common denominator 10 x 4 = 40
9 $\dpi{100} \small \frac{5}{10}$ = 9 $\dpi{100} \small \frac{20}{40}$
14 $\dpi{100} \small \frac{1}{4}$ = 14 $\dpi{100} \small \frac{10}{40}$
9 $\dpi{100} \small \frac{5}{10}$ + 14 $\dpi{100} \small \frac{1}{4}$ = 9 $\dpi{100} \small \frac{20}{40}$ + 14 $\dpi{100} \small \frac{10}{40}$
= 23 $\dpi{100} \small \frac{30}{40}$
So, an emperor tamarin is 23 $\dpi{100} \small \frac{3}{4}$ inches long.

Question 12.
DIG DEEPER!
A long jumper jumps 1$$\frac{2}{3}$$ feet farther on her third attempt than her second attempt. On which attempt does she jump the farthest? Explain.

Third attempt = 1 $\dpi{100} \small \frac{2}{3}$ + 13 $\dpi{100} \small \frac{3}{4}$
To add the fractional parts, use a common denominator 3 x 4 = 12
1 $\dpi{100} \small \frac{2}{3}$ = 1 $\dpi{100} \small \frac{8}{12}$
13 $\dpi{100} \small \frac{3}{4}$ = 13 $\dpi{100} \small \frac{9}{12}$
1 $\dpi{100} \small \frac{2}{3}$ + 13 $\dpi{100} \small \frac{3}{4}$ = 1 $\dpi{100} \small \frac{8}{12}$ + 13 $\dpi{100} \small \frac{9}{12}$ = 14 $\dpi{100} \small \frac{17}{12}$
Therefore, she jumps the farthest on her first attempt.

Review & Refresh

Question 13.
354 × 781
The number close to 354 is 350
The number close to 781 is 800.
350 × 800 = 280000
354 × 781 = 276474

Question 14.
4,029 × 276
The number close to 4029 is 4000
The number close to 276 is 300
4000 × 300 = 1200000
4029 × 276 = 1112004

Question 15.
950 × 326
The number close to 950 is 1000
The number close to 326 is 300
1000 × 300 = 300000
950 × 326 = 309700

### Lesson 8.7 Subtract Mixed Numbers

Explore and Grow

Use a model to find the difference
3$$\frac{5}{6}$$ – 2$$\frac{1}{3}$$

Construct Arguments
How can you subtract mixed numbers with unlike denominators without using a model? Explain why your method makes sense.

Think and Grow: Subtract Mixed Numbers

You can use equivalent fractions to subtract mixed numbers that have fractional parts with unlike denominators.
Example
Find 3$$\frac{1}{4}$$ – 1$$\frac{1}{2}$$.

Show and Grow

Subtract.

Question 1.
1$$\frac{4}{5}$$ – 1$$\frac{3}{10}$$
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
1 $\dpi{100} \small \frac{4}{5}$ = 1 $\dpi{100} \small \frac{8}{10}$
1 $\dpi{100} \small \frac{4}{5}$ – 1 $\dpi{100} \small \frac{3}{10}$ = 1 $\dpi{100} \small \frac{8}{10}$ – 1 $\dpi{100} \small \frac{3}{10}$ = 0 + $\dpi{100} \small \frac{5}{10}$
= $\dpi{100} \small \frac{1}{2}$

Question 2.
5$$\frac{7}{12}$$ – 3$$\frac{5}{6}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
3 $\dpi{100} \small \frac{5}{6}$ = 3 $\dpi{100} \small \frac{10}{12}$
5 $\dpi{100} \small \frac{7}{12}$ – 3 $\dpi{100} \small \frac{5}{6}$ = 5 $\dpi{100} \small \frac{7}{12}$ – 3 $\dpi{100} \small \frac{10}{12}$
5 $\dpi{100} \small \frac{7}{12}$  = 4 + $\dpi{100} \small \frac{12}{12}$ + $\dpi{100} \small \frac{7}{12}$ = 4 $\dpi{100} \small \frac{19}{12}$
5 $\dpi{100} \small \frac{7}{12}$ – 3 $\dpi{100} \small \frac{10}{12}$ = 4 $\dpi{100} \small \frac{19}{12}$ – 3 $\dpi{100} \small \frac{10}{12}$
= 1 $\dpi{100} \small \frac{9}{12}$

Apply and Grow: Practice

Subtract.

Question 3.
8$$\frac{11}{12}$$ – 5$$\frac{2}{3}$$ = _____
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
5 $\dpi{100} \small \frac{2}{3}$ = 5 $\dpi{100} \small \frac{8}{12}$
8 $\dpi{100} \small \frac{11}{12}$ – 5 $\dpi{100} \small \frac{2}{3}$ = 8 $\dpi{100} \small \frac{11}{12}$ – 5 $\dpi{100} \small \frac{8}{12}$
8$$\frac{11}{12}$$ – 5$$\frac{2}{3}$$ = 3 $\dpi{100} \small \frac{3}{12}$ = 3 $\dpi{100} \small \frac{1}{4}$

Question 4.
6 – 4$$\frac{3}{4}$$ = ___
6 – 4 $\dpi{100} \small \frac{3}{4}$ = 6 – $\dpi{100} \small \frac{19}{4}$ = $\dpi{100} \small \frac{5}{4}$

Question 5.
21$$\frac{2}{9}$$ – 10$$\frac{1}{3}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
10 $\dpi{100} \small \frac{1}{3}$ = 10 $\dpi{100} \small \frac{3}{9}$
21 $\dpi{100} \small \frac{2}{9}$ = 20 + $\dpi{100} \small \frac{9}{9}$ + $\dpi{100} \small \frac{2}{9}$ = 20 $\dpi{100} \small \frac{11}{9}$
21 $\dpi{100} \small \frac{2}{9}$ – 10 $\dpi{100} \small \frac{1}{3}$ = 20 $\dpi{100} \small \frac{11}{9}$ – 10 $\dpi{100} \small \frac{3}{9}$
= 10 $\dpi{100} \small \frac{8}{9}$

Question 6.
7$$\frac{1}{2}$$ – $$\frac{5}{8}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
7 $\dpi{100} \small \frac{1}{2}$ = 7 $\dpi{100} \small \frac{4}{8}$ = 6 + $\dpi{100} \small \frac{8}{8}$ + $\dpi{100} \small \frac{4}{8}$ = 6 $\dpi{100} \small \frac{12}{8}$
7 $\dpi{100} \small \frac{1}{2}$$\dpi{100} \small \frac{5}{8}$ = 6 $\dpi{100} \small \frac{12}{8}$$\dpi{100} \small \frac{5}{8}$
= 6 $\dpi{100} \small \frac{7}{8}$

Question 7.
9$$\frac{7}{20}$$ – 1$$\frac{3}{5}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
1 $\dpi{100} \small \frac{3}{5}$ = 1 $\dpi{100} \small \frac{12}{20}$
9 $\dpi{100} \small \frac{7}{20}$ = 8 + $\dpi{100} \small \frac{20}{20}$ + $\dpi{100} \small \frac{7}{20}$ = 8 $\dpi{100} \small \frac{27}{20}$
9 $\dpi{100} \small \frac{7}{20}$ – 1 $\dpi{100} \small \frac{3}{5}$ = 8 $\dpi{100} \small \frac{27}{20}$ – 1 $\dpi{100} \small \frac{12}{20}$
= 7 $\dpi{100} \small \frac{15}{20}$
= 7 $\dpi{100} \small \frac{3}{4}$

Question 8.
7$$\frac{5}{6}$$ – 1$$\frac{1}{6}$$ – 2$$\frac{2}{3}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
2 $\dpi{100} \small \frac{2}{3}$ = 2 $\dpi{100} \small \frac{4}{6}$
7 $\dpi{100} \small \frac{5}{6}$ = 6 + $\dpi{100} \small \frac{6}{6}$ + $\dpi{100} \small \frac{5}{6}$ = 6 $\dpi{100} \small \frac{11}{6}$
7 $\dpi{100} \small \frac{5}{6}$ – 1 $\dpi{100} \small \frac{1}{6}$ – 2 $\dpi{100} \small \frac{2}{3}$ = 6 $\dpi{100} \small \frac{11}{6}$ – 1 $\dpi{100} \small \frac{1}{6}$ – 2 $\dpi{100} \small \frac{4}{6}$
= 3 $\dpi{100} \small \frac{6}{6}$
= 3 + 1
7$$\frac{5}{6}$$ – 1$$\frac{1}{6}$$ – 2$$\frac{2}{3}$$ = 4

Question 9.
A volunteer at a food bank buys 3$$\frac{3}{4}$$ pounds of cheese to make sandwiches. She uses 2$$\frac{7}{8}$$ pounds. How much cheese does she have left?

Given that,
A volunteer buys 3 $\dpi{100} \small \frac{3}{4}$ pounds of cheese
She uses 2 $\dpi{100} \small \frac{7}{8}$ pounds
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
3 $\dpi{100} \small \frac{3}{4}$ = 3 $\dpi{100} \small \frac{6}{8}$ = 2 + $\dpi{100} \small \frac{8}{8}$ + $\dpi{100} \small \frac{6}{8}$ = 2 $\dpi{100} \small \frac{14}{8}$
Cheese left = 2 $\dpi{100} \small \frac{14}{8}$ – 2 $\dpi{100} \small \frac{7}{8}$ = 0 + $\dpi{100} \small \frac{7}{8}$
So she left with $\dpi{100} \small \frac{7}{8}$ pounds of cheese.

Question 10.
Writing
How is adding mixed numbers the same as subtracting mixed numbers? How is it different?
Adding and subtracting mixed numbers is same because we need to use a common denominator for both.
Adding and subtracting mixed numbers is different because
For subtracting mixed numbers, the first fraction should be greater than the second number.

Question 11.
Number Sense
Write the words as an expression. Then evaluate.
Subtract the sum of four and three-fourths and two and five-eighths from eleven and seven-eighths.
11 $\dpi{100} \small \frac{7}{8}$ – (4 $\dpi{100} \small \frac{3}{4}$ + 2 $\dpi{100} \small \frac{5}{8}$)
To subtract or add the fractional parts, use a common denominator.
4 $\dpi{100} \small \frac{3}{4}$ = 4 $\dpi{100} \small \frac{6}{8}$
4 $\dpi{100} \small \frac{3}{4}$ + 2 $\dpi{100} \small \frac{5}{8}$ = 4 $\dpi{100} \small \frac{6}{8}$ + 2 $\dpi{100} \small \frac{5}{8}$ = 6 $\dpi{100} \small \frac{11}{8}$
11 $\dpi{100} \small \frac{7}{8}$ – (4 $\dpi{100} \small \frac{3}{4}$ + 2 $\dpi{100} \small \frac{5}{8}$) = 11 $\dpi{100} \small \frac{7}{8}$ – 6 $\dpi{100} \small \frac{11}{8}$
11 $\dpi{100} \small \frac{7}{8}$ = 10 + $\dpi{100} \small \frac{8}{8}$ + $\dpi{100} \small \frac{7}{8}$ = 10 $\dpi{100} \small \frac{15}{8}$
11 $\dpi{100} \small \frac{7}{8}$ – (4 $\dpi{100} \small \frac{3}{4}$ + 2 $\dpi{100} \small \frac{5}{8}$) = 10 $\dpi{100} \small \frac{15}{8}$ – 6 $\dpi{100} \small \frac{11}{8}$
11 $\dpi{100} \small \frac{7}{8}$ – (4 $\dpi{100} \small \frac{3}{4}$ + 2 $\dpi{100} \small \frac{5}{8}$) = 4 $\dpi{100} \small \frac{4}{8}$ = 4 $\dpi{100} \small \frac{1}{2}$

Question 12.
DIG DEEPER!
Find the missing number.

To subtract the fractional parts, use a common denominator.
3 $\dpi{100} \small \frac{1}{4}$ = 3 $\dpi{100} \small \frac{3}{12}$

3 $\dpi{100} \small \frac{1}{4}$ – 1 $\dpi{100} \small \frac{1}{12}$ = 3 $\dpi{100} \small \frac{3}{12}$ – 1 $\dpi{100} \small \frac{1}{12}$
= 2 $\dpi{100} \small \frac{2}{12}$
3 $\dpi{100} \small \frac{1}{4}$ – 1 $\dpi{100} \small \frac{1}{12}$ = 2 $\dpi{100} \small \frac{1}{6}$
So the missing number is 1.

Think and Grow: Modeling Real Life

Example
A dragonfly is 1$$\frac{1}{2}$$ inches long. How much longer is the walking leaf than the dragonfly?

To find how much longer the walking leaf is than the dragonfly, subtract the length of the dragonfly from the length of the walking leaf.

The walking leaf is __ inches longer than the dragonfly.

The walking leaf is 1 $$\frac{1}{6}$$ inches longer than the dragonfly.

Show and Grow

Question 13.
You volunteer 5$$\frac{3}{4}$$ hours in 1 month. You spend 3$$\frac{1}{3}$$ hours volunteering at an animal shelter. You spend the remaining hours picking up litter on the side of the road. How many hours do you spend picking up litter?
Given that,
Volunteering hours in 1 month = 5 $\dpi{100} \small \frac{3}{4}$
Time spent at an animal shelter = 3 $\dpi{100} \small \frac{1}{3}$
Remaining hours in a month are for picking up litter = 5 $\dpi{100} \small \frac{3}{4}$ – 3 $\dpi{100} \small \frac{1}{3}$
To subtract the fractional parts, use a common denominator 4 x 3 =12
5 $\dpi{100} \small \frac{3}{4}$ = 5 $\dpi{100} \small \frac{9}{12}$
3 $\dpi{100} \small \frac{1}{3}$ = 3 $\dpi{100} \small \frac{4}{12}$
Time spend for picking up litter = 5 $\dpi{100} \small \frac{9}{12}$ – 3 $\dpi{100} \small \frac{4}{12}$ = 2 $\dpi{100} \small \frac{5}{12}$.

Question 14.
A professional basketball player is 6$$\frac{3}{4}$$ feet tall. Your friend is 4$$\frac{5}{6}$$ feet tall. How much taller is the basketball player than your friend?
To find how much taller is the basketball player than the friend, subtract the height of the friend from the height of the basketball player.
Given that,
Basketball player is 6 $\dpi{100} \small \frac{3}{4}$ feet tall.
My friend is 4 $\dpi{100} \small \frac{5}{6}$ feet tall.
To subtract the fractional parts, use a common denominator 4 x 6 = 24
6 $\dpi{100} \small \frac{3}{4}$ = 6 $\dpi{100} \small \frac{18}{24}$ = 5 + $\dpi{100} \small \frac{24}{24}$ + $\dpi{100} \small \frac{18}{24}$ = 5 $\dpi{100} \small \frac{42}{24}$
4 $\dpi{100} \small \frac{5}{6}$ = 4 $\dpi{100} \small \frac{20}{24}$
6 $\dpi{100} \small \frac{3}{4}$ – 4 $\dpi{100} \small \frac{5}{6}$ = 5 $\dpi{100} \small \frac{42}{24}$ – 4 $\dpi{100} \small \frac{20}{24}$
= 1 $\dpi{100} \small \frac{22}{24}$
= 1 $\dpi{100} \small \frac{11}{12}$
The basketball player is 1 $\dpi{100} \small \frac{11}{12}$ feet taller than my friend.

Question 15.
Your rain gauge has 2$$\frac{1}{2}$$ inches of water. After a rainstorm, your rain gauge has 1$$\frac{3}{4}$$ more inches of water. It is sunny for a week. Now your rain gauge has 2$$\frac{2}{3}$$ inches of water. How many inches of water evaporated?
After a rainstorm, water in the rain gauge = 2 $\dpi{100} \small \frac{1}{2}$ + 1 $\dpi{100} \small \frac{3}{4}$
To subtract or add the fractional parts, use a common denominator.
2 $\dpi{100} \small \frac{1}{2}$ = 2 $\dpi{100} \small \frac{2}{4}$
Water in the rain gauge = 2 $\dpi{100} \small \frac{1}{2}$ +1 $\dpi{100} \small \frac{3}{4}$ = 2 $\dpi{100} \small \frac{2}{4}$ + 1 $\dpi{100} \small \frac{3}{4}$ = 3 $\dpi{100} \small \frac{5}{4}$ inches
Water evaporated = 3 $\dpi{100} \small \frac{5}{4}$ – 2 $\dpi{100} \small \frac{2}{3}$
3 $\dpi{100} \small \frac{5}{4}$ = 3 $\dpi{100} \small \frac{15}{12}$
2 $\dpi{100} \small \frac{2}{3}$ = 2 $\dpi{100} \small \frac{8}{12}$

3 $\dpi{100} \small \frac{5}{4}$ – 2 $\dpi{100} \small \frac{2}{3}$ = 3 $\dpi{100} \small \frac{15}{12}$ – 2 $\dpi{100} \small \frac{8}{12}$ = 1 $\dpi{100} \small \frac{7}{12}$
So, 1 $\dpi{100} \small \frac{7}{12}$ inches of water evaporated.

### Subtract Mixed Numbers Homework & Practice 8.7

Subtract

Question 1.
9$$\frac{5}{6}$$ – 4$$\frac{1}{2}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
4 $\dpi{100} \small \frac{1}{2}$ = 4 $\dpi{100} \small \frac{3}{6}$
9 $\dpi{100} \small \frac{5}{6}$ – 4 $\dpi{100} \small \frac{1}{2}$ = 9 $\dpi{100} \small \frac{5}{6}$ – 4 $\dpi{100} \small \frac{3}{6}$
= 5 $\dpi{100} \small \frac{2}{6}$
9$$\frac{5}{6}$$ – 4$$\frac{1}{2}$$ = 5 $\dpi{100} \small \frac{2}{6}$ = 5 $\dpi{100} \small \frac{1}{3}$

Question 2.
3$$\frac{2}{3}$$ – $$\frac{1}{9}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
3 $\dpi{100} \small \frac{2}{3}$ = 3 $\dpi{100} \small \frac{6}{9}$
3 $\dpi{100} \small \frac{2}{3}$$\dpi{100} \small \frac{1}{9}$ = 3 $\dpi{100} \small \frac{6}{9}$$\dpi{100} \small \frac{1}{9}$ = 3 $\dpi{100} \small \frac{5}{9}$

Question 3.
6$$\frac{1}{3}$$ – 1$$\frac{11}{12}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
6 $\dpi{100} \small \frac{1}{3}$ = 6 $\dpi{100} \small \frac{4}{12}$ = 5 $\dpi{100} \small \frac{16}{12}$
6 $\dpi{100} \small \frac{1}{3}$ – 1 $\dpi{100} \small \frac{11}{12}$ = 5 $\dpi{100} \small \frac{16}{12}$ – 1 $\dpi{100} \small \frac{11}{12}$
= 4 $\dpi{100} \small \frac{5}{12}$

Question 4.
12$$\frac{5}{6}$$ – 7$$\frac{3}{10}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 6 x 10 = 60
12 $\dpi{100} \small \frac{5}{6}$ = 12 $\dpi{100} \small \frac{50}{60}$
7 $\dpi{100} \small \frac{3}{10}$ = 7 $\dpi{100} \small \frac{18}{60}$
12 $\dpi{100} \small \frac{5}{6}$ – 7 $\dpi{100} \small \frac{3}{10}$ = 12 $\dpi{100} \small \frac{50}{60}$ – 7 $\dpi{100} \small \frac{18}{60}$
= 5 $\dpi{100} \small \frac{32}{60}$
12$$\frac{5}{6}$$ – 7$$\frac{3}{10}$$ = 5 $\dpi{100} \small \frac{32}{60}$ = 5 $\dpi{100} \small \frac{8}{15}$

Question 5.
5 – 2$$\frac{3}{4}$$ = ___
5 – 2 $\dpi{100} \small \frac{3}{4}$ = 5 – $\dpi{100} \small \frac{11}{4}$ = $\dpi{100} \small \frac{9}{4}$

Question 6.
4$$\frac{1}{5}$$ – 2$$\frac{1}{4}$$ = __
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 5 x 4 = 20
4 $\dpi{100} \small \frac{1}{5}$ = 4 $\dpi{100} \small \frac{4}{20}$ = 3 $\dpi{100} \small \frac{24}{20}$
2 $\dpi{100} \small \frac{1}{4}$ = 2 $\dpi{100} \small \frac{5}{20}$

4 $\dpi{100} \small \frac{1}{5}$ – 2 $\dpi{100} \small \frac{1}{4}$ = 3 $\dpi{100} \small \frac{24}{20}$ – 2 $\dpi{100} \small \frac{5}{20}$ = 1 $\dpi{100} \small \frac{19}{20}$

Subtract.

Question 7.
7$$\frac{5}{8}$$ – 1$$\frac{5}{6}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 8 x 6 = 48
7 $\dpi{100} \small \frac{5}{8}$ = 7 $\dpi{100} \small \frac{30}{48}$ = 6 $\dpi{100} \small \frac{78}{48}$
1 $\dpi{100} \small \frac{5}{6}$ = 1 $\dpi{100} \small \frac{40}{48}$
7 $\dpi{100} \small \frac{5}{8}$ – 1 $\dpi{100} \small \frac{5}{6}$ = 6 $\dpi{100} \small \frac{78}{48}$ – 1 $\dpi{100} \small \frac{40}{48}$ = 5 $\dpi{100} \small \frac{38}{48}$
= 5 $\dpi{100} \small \frac{19}{24}$

Question 8.
8$$\frac{1}{9}$$ – 6$$\frac{7}{8}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 8 x 9 = 72
8 $\dpi{100} \small \frac{1}{9}$ = 8 $\dpi{100} \small \frac{8}{72}$ = 7 $\dpi{100} \small \frac{80}{72}$
6 $\dpi{100} \small \frac{7}{8}$ = 6 $\dpi{100} \small \frac{63}{72}$
8 $\dpi{100} \small \frac{1}{9}$ – 6 $\dpi{100} \small \frac{7}{8}$ = 7 $\dpi{100} \small \frac{80}{72}$ – 6 $\dpi{100} \small \frac{63}{72}$
= 1 $\dpi{100} \small \frac{17}{72}$

Question 9.
1$$\frac{6}{7}$$ + 5$$\frac{13}{14}$$ – 2$$\frac{1}{2}$$ = ___
To subtract or add the fractional parts, use a common denominator
1 $\dpi{100} \small \frac{6}{7}$ = 1 $\dpi{100} \small \frac{12}{14}$
2 $\dpi{100} \small \frac{1}{2}$ = 2 $\dpi{100} \small \frac{7}{14}$

1 $\dpi{100} \small \frac{6}{7}$ + 5 $\dpi{100} \small \frac{13}{14}$ – 2 $\dpi{100} \small \frac{1}{2}$ = 1 $\dpi{100} \small \frac{12}{14}$ + 5 $\dpi{100} \small \frac{13}{14}$ – 2 $\dpi{100} \small \frac{7}{14}$
= 4 $\dpi{100} \small \frac{18}{14}$
= 4 $\dpi{100} \small \frac{9}{7}$

Question 10.
Your friend says the difference of 8 and 3$$\frac{7}{10}$$ is 5$$\frac{7}{10}$$. Is your friend correct? Explain.
8 – 3 $\dpi{100} \small \frac{7}{10}$ = 8 – $\dpi{100} \small \frac{37}{10}$ = $\dpi{100} \small \frac{43}{10}$

Question 11.
DIG DEEPER!
Use a symbol card to complete the equation. Then ﬁnd b.

4 $\dpi{100} \small \frac{1}{4}$ – 1 $\dpi{100} \small \frac{17}{20}$ – b = 1 $\dpi{100} \small \frac{1}{2}$
4 $\dpi{100} \small \frac{1}{4}$ = 4 $\dpi{100} \small \frac{5}{20}$ = 2 $\dpi{100} \small \frac{45}{20}$
1 $\dpi{100} \small \frac{1}{2}$ = 1 $\dpi{100} \small \frac{10}{20}$
b = 4 $\dpi{100} \small \frac{1}{4}$ – 1 $\dpi{100} \small \frac{17}{20}$ – 1 $\dpi{100} \small \frac{1}{2}$
= 2 $\dpi{100} \small \frac{45}{20}$ – 1 $\dpi{100} \small \frac{17}{20}$ – 1 $\dpi{100} \small \frac{10}{20}$
= $\dpi{100} \small \frac{18}{20}$
b = $\dpi{100} \small \frac{18}{20}$ = $\dpi{100} \small \frac{9}{10}$

Question 12.
Modeling Real Life
The world record for the heaviest train pulled with a human beard is 2$$\frac{3}{4}$$ metric tons. The world record for the heaviest train pulled by human teeth is 4$$\frac{1}{5}$$ metric tons. How much heavier is the train pulled by teeth than the train pulled with a beard?
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 4 x 5 = 20
2 $\dpi{100} \small \frac{3}{4}$ = 2 $\dpi{100} \small \frac{15}{20}$
4 $\dpi{100} \small \frac{1}{5}$ = 4 $\dpi{100} \small \frac{4}{20}$ = 3 $\dpi{100} \small \frac{24}{20}$
4 $\dpi{100} \small \frac{1}{5}$ – 2 $\dpi{100} \small \frac{3}{4}$ = 3 $\dpi{100} \small \frac{24}{20}$ – 2 $\dpi{100} \small \frac{15}{20}$
= 1 $\dpi{100} \small \frac{9}{20}$
The train pulled by teeth 1 $\dpi{100} \small \frac{9}{20}$ metric tons heavier than the train pulled with a beard.

Question 13.
Modeling Real Life
Your friend’s hair is 50$$\frac{4}{5}$$ centimeters long. Your hair is 8$$\frac{9}{10}$$ centimeters long. How much longer is your friend’s hair than yours?

My friend’s hair = 50 $\dpi{100} \small \frac{4}{5}$ cm
My hair = 8 $\dpi{100} \small \frac{9}{10}$ cm
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator
50 $\dpi{100} \small \frac{4}{5}$ = 50 $\dpi{100} \small \frac{8}{10}$ = 49 $\dpi{100} \small \frac{18}{10}$
50 $\dpi{100} \small \frac{4}{5}$ – 8 $\dpi{100} \small \frac{9}{10}$ = 49 $\dpi{100} \small \frac{18}{10}$ – 8 $\dpi{100} \small \frac{9}{10}$ = 41 $\dpi{100} \small \frac{9}{10}$
My friend’s hair is 41 $\dpi{100} \small \frac{9}{10}$ cms longer than my hair.

Review & Refresh

Question 14.
Round 6.294.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Nearest whole number: 6
Nearest tenth: 60
Nearest hundredth: 600

Question 15.
Round 10.571.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Nearest whole number: 11
Nearest tenth: 110
Nearest hundredth: 1100

### Lesson 8.8 Problem Solving: Fractions

Explore and Grow

Make a plan to solve the problem.

At a state park, every $$\frac{1}{10}$$ mile of walking trail is marked. Every $$\frac{1}{4}$$ mile of a separate biking trail is marked. The table shows the number of mileage markers you and your friend pass while walking and biking on the trails. Who travels farther? How much farther?

Make Sense of Problems
You decide to walk farther and you pass 4 more mileage markers on the walking trail. Does this change your plan to solve the problem? Explain.
At a state park, every $$\frac{1}{10}$$ mile of the walking trail is marked.
8 × $$\frac{1}{10}$$ = $$\frac{8}{10}$$
Every $$\frac{1}{4}$$ mile of a separate biking trail is marked.
9 × $$\frac{1}{4}$$ = 2 $$\frac{1}{4}$$
$$\frac{8}{10}$$ + $$\frac{4}{10}$$ = $$\frac{12}{10}$$
= 1 $$\frac{2}{10}$$

Think and Gow: Problem Solving: Fractions

Example
To repair a skate ramp, you cut a piece of wood from a 9$$\frac{1}{2}$$-foot-long board. Then you cut the remaining piece in half. Each half is 3$$\frac{5}{12}$$ feet long. How long is the first piece you cut?

Understand the Problem

What do you know?

• The board is 9 feet long.2first piece you cut.
• You cut a piece from the board.
• You cut the rest into two pieces that are each 3$$\frac{5}{12}$$ feet long.

What do you need tofind?

• You need to find the length of the first piece you cut.

Make a Plan

How will you solve?
Write and solve an equation: Subtract the sum of the lengths of the last two pieces you cut from the total length of the board.

Solve

Let g represent the length of the first piece you cut.

So, the length of the first piece you cut is __ feet.

So, the length of the first piece you cut is 2 $$\frac{2}{3}$$ feet.

Show and Grow

Question 1.

Apply and Grow: Practice

Understand the problem. What do you know? What do you need to find? Explain.

Question 2.
A racehorse eats 38$$\frac{1}{2}$$ pounds of food each day. He eats 22$$\frac{3}{4}$$ pounds of hay and 7$$\frac{1}{2}$$ pounds of grains. How many pounds of his daily diet is not hay or grains?

Given,
A racehorse eats 38$$\frac{1}{2}$$ pounds of food each day.
He eats 22$$\frac{3}{4}$$ pounds of hay and 7$$\frac{1}{2}$$ pounds of grains.
38$$\frac{1}{2}$$ – 22$$\frac{3}{4}$$
= 15 $$\frac{3}{4}$$
Thus 15 $$\frac{3}{4}$$ pounds of his daily diet is not grains.

Question 3.
In 2015, American Pharoah won all of the horse races shown in the table. How many kilometers did American Pharoah run in the races altogether?

Understand the problem. Then make a plan. How will you solve? Explain.
Add all the lengths to find how many kilometers did American Pharoah run in the races altogether.
2 + 1 $$\frac{9}{10}$$ + 2 $$\frac{2}{5}$$
First add all the whole numbers.
2 + 1 + 2 = 5
$$\frac{9}{10}$$ + $$\frac{4}{10}$$ = 1 $$\frac{3}{10}$$
5 + 1 $$\frac{3}{10}$$ = 6 $$\frac{3}{10}$$

Question 4.
You have 2$$\frac{1}{2}$$ cups of blueberries. You use 1$$\frac{1}{4}$$ cups for pancakes and $$\frac{1}{2}$$ cup for muffins. What fraction of a cup of blueberries do you have left?
Given,
You have 2$$\frac{1}{2}$$ cups of blueberries.
You use 1$$\frac{1}{4}$$ cups for pancakes and $$\frac{1}{2}$$ cup for muffins.
1$$\frac{1}{4}$$ + $$\frac{1}{2}$$ = 1 $$\frac{3}{4}$$
2$$\frac{1}{2}$$ – 1 $$\frac{3}{4}$$ = $$\frac{3}{4}$$

Question 5.
A customer orders 2 pounds of cheese at a deli. The deli worker places some cheese in a bowl and weighs it. The scale shows 1$$\frac{1}{4}$$ pounds. The bowl weighs $$\frac{1}{8}$$ pound. What fraction of a pound of cheese does the worker need to add to the bowl?
Given,
A customer orders 2 pounds of cheese at a deli. The deli worker places some cheese in a bowl and weighs it.
The scale shows 1$$\frac{1}{4}$$ pounds. The bowl weighs $$\frac{1}{8}$$ pound.
1$$\frac{1}{4}$$ + $$\frac{1}{8}$$ = 1 $$\frac{3}{8}$$
2 – 1 $$\frac{3}{8}$$ = $$\frac{5}{8}$$

Question 6.
Reasoning
Student A is 8$$\frac{1}{2}$$ inches shorter than Student B. Student B is 3$$\frac{1}{4}$$ inches taller than Student C. Student C is 56$$\frac{3}{8}$$ inches tall. How tall is Student A? Student B?
Student A is 8$$\frac{1}{2}$$ inches shorter than Student B.
Student B is 3$$\frac{1}{4}$$ inches taller than Student C.
Student C is 56$$\frac{3}{8}$$ inches tall.
Student B is 56$$\frac{3}{8}$$ + 3$$\frac{1}{4}$$ = 59 $$\frac{5}{8}$$
Thus the height of student B is 59 $$\frac{5}{8}$$ inches.
Student A is 59 $$\frac{5}{8}$$ – 8$$\frac{1}{2}$$ = 51 $$\frac{1}{8}$$
Thus the height of Student A is 51 $$\frac{1}{8}$$ inches.

Question 7.
DIG DEEPER!
A police dog spends $$\frac{1}{8}$$ of his workday in a police car, $$\frac{3}{4}$$ of his workday in public, and the rest of his workday at the police station. What fraction of the dog’s day is spent at the police station?
Given,
A police dog spends $$\frac{1}{8}$$ of his workday in a police car, $$\frac{3}{4}$$ of his workday in public, and the rest of his workday at the police station.
$$\frac{1}{8}$$ + $$\frac{3}{4}$$ = $$\frac{7}{8}$$
1 – $$\frac{7}{8}$$ = $$\frac{1}{8}$$
Thus $$\frac{1}{8}$$ fraction of the dog’s day is spent at the police station

Think and Grow: Modeling Real Life

Example
The Magellan spacecraft, launched by the United States, spent 5$$\frac{5}{12}$$ years in space before it burned in Venus’s atmosphere. Its first 4 cycles around Venus each lasted $$\frac{2}{3}$$ year. The remaining cycles around Venus lasted a total of 1$$\frac{1}{2}$$ years. How long did it take to travel from Earth to Venus?

Think: What do you know? What do you need to find? How will you solve?

Show and Grow

Question 8.
You have one of each euro coin shown. Your friend has four euro coins that have a total weight of 21$$\frac{3}{10}$$ grams. Whose coins weigh more? How much more?

2 $$\frac{3}{10}$$ + 5 $$\frac{3}{4}$$ + 7 $$\frac{4}{5}$$ + 7 $$\frac{1}{2}$$
= 23 $$\frac{7}{20}$$
23 $$\frac{7}{20}$$ – 21$$\frac{3}{10}$$
= 2 $$\frac{1}{20}$$

### Problem Solving: Fractions Homework & Practice 8.8

Understand the problem. What do you know? What do you need to find? Explain.

Question 1.
Your goal is to exercise for 15 hours this month. You exercise for 3$$\frac{1}{2}$$ hours the ﬁrst week and 3$$\frac{3}{4}$$ hours the next week. How many more hours do you need to exercise to reach your goal?
Given,
Your goal is to exercise for 15 hours this month.
You exercise for 3$$\frac{1}{2}$$ hours the ﬁrst week and 3$$\frac{3}{4}$$ hours the next week.
3$$\frac{1}{2}$$ + 3$$\frac{3}{4}$$ = 7 $$\frac{1}{4}$$
15 – 7 $$\frac{1}{4}$$ = 7 $$\frac{3}{4}$$ hours

Question 2.
A taxi driver travels 4$$\frac{5}{8}$$ miles to his ﬁrst stop. He travels 1$$\frac{3}{4}$$ miles less to his second stop. How many miles does the taxi driver travel for the two stops?

Understand the problem. Then make a plan. How will you solve? Explain.
Given,
A taxi driver travels 4$$\frac{5}{8}$$ miles to his ﬁrst stop. He travels 1$$\frac{3}{4}$$ miles less to his second stop.
4$$\frac{5}{8}$$ + 1$$\frac{3}{4}$$ = 7 $$\frac{1}{4}$$

Question 3.
During the U.S. Civil War, $$\frac{5}{9}$$ of the states fought for the Union, and $$\frac{11}{36}$$ of the states fought for the Confederacy. The rest of the states were border states. What fraction of the states were border states?
Given,
During the U.S. Civil War, $$\frac{5}{9}$$ of the states fought for the Union, and $$\frac{11}{36}$$ of the states fought for the Confederacy. The rest of the states were border states.
$$\frac{5}{9}$$ + $$\frac{11}{36}$$ = $$\frac{31}{36}$$

Question 4.
You have 6$$\frac{3}{4}$$ pounds of clay. You use 4$$\frac{1}{6}$$ pounds to make a medium-sized bowl and 1$$\frac{1}{2}$$ pounds to make a small bowl. How many pounds of clay do you have left?
Given,
You have 6$$\frac{3}{4}$$ pounds of clay.
You use 4$$\frac{1}{6}$$ pounds to make a medium-sized bowl and 1$$\frac{1}{2}$$ pounds to make a small bowl.
4$$\frac{1}{6}$$ + 1$$\frac{1}{2}$$ = 5 $$\frac{2}{3}$$
6$$\frac{3}{4}$$ – 5 $$\frac{2}{3}$$ = 1 $$\frac{1}{12}$$

Question 5.
DIG DEEPER!
Newton and Descartes have a 70-day summer vacation. They go to camp for $$\frac{23}{70}$$ of their vacation, and they travel for $$\frac{6}{35}$$ of their vacation. They stay home the rest of their vacation. How many weeks do Newton and Descartes spend at home?
Given,
Newton and Descartes have a 70-day summer vacation. They go to camp for $$\frac{23}{70}$$ of their vacation, and they travel for $$\frac{6}{35}$$ of their vacation.
$$\frac{23}{70}$$ + $$\frac{6}{35}$$ = $$\frac{1}{2}$$
That means Newton and Descartes spend 5 weeks at home.

Question 6.
Modeling Real Life
A farmer plants beets in a square garden with side lengths of 12$$\frac{2}{3}$$ feet. He plants squash in a garden with a perimeter of 50$$\frac{1}{2}$$ feet. Which garden has a greater perimeter? How much greater is it?
Given,
A farmer plants beets in a square garden with side lengths of 12$$\frac{2}{3}$$ feet.
The perimeter of the square = 4s
P = 4 × 12$$\frac{2}{3}$$
P = 50 $$\frac{2}{3}$$
50 $$\frac{2}{3}$$ is greater than 50$$\frac{1}{2}$$

Question 7.
DIG DEEPER!
Which grade uses more leafy greens daily for its classroom rabbits? How much more does it use?

Review & Refresh

Find the product.

Question 1.
0.43 × 1,000 = 430
43 x 10-2 x 103 = 43 x 101
= 430

Question 2.
25.8 × 0.1 = 2.58
258 x 10-1​​​​​​​ x 10-1= 2.58

Many historic landmarks are located in Washington, D.C.

Question 1.
Initial construction of the Washington Monument began in 1848. When the height of the monument reached 152 feet, construction halted due to lack of funds. How many feet were added to the height of the monument when construction resumed 23 years later?

The initial construction of the Washington Monument began in 1848. When the height of the monument reached 152 feet, construction halted due to lack of funds.
554 $$\frac{3}{4}$$ – 152 = 402 $$\frac{3}{4}$$

Question 2.
You visit several historic landmarks. You start at the Capitol Building and walk to the Washington Monument, then the Lincoln Memorial, then the White House, and then back to the Capitol Building.

a. You walk 3 miles each hour. It takes you $$\frac{1}{2}$$ hour to walk from the Lincoln Memorial to the White House. What is the distance from the Lincoln Memorial to the White House? Label the map.

b. What is the total distance you walk visiting the landmarks?
Answer: 1 $$\frac{1}{5}$$ + $$\frac{4}{5}$$ + 1 $$\frac{4}{5}$$ + 1 $$\frac{3}{5}$$
= 5 $$\frac{2}{5}$$

Question 3.
A law in Washington, D.C., restricts a new building’s height to no more than 20 feet taller than the width of the street it faces. You design a building with stories that are each 15 feet tall for a street that is 88$$\frac{2}{3}$$ feet wide. What is the greatest number of stories your building can have? How much shorter is your building than the height restriction?
Given,
A law in Washington, D.C., restricts a new building’s height to no more than 20 feet taller than the width of the street it faces.
20 × 88$$\frac{2}{3}$$ = 1773 $$\frac{1}{3}$$
You design a building with stories that are each 15 feet tall for a street that is 88$$\frac{2}{3}$$ feet wide.
15 × 88$$\frac{2}{3}$$ = 1330
1773 $$\frac{1}{3}$$ – 1330 = 443 $$\frac{1}{3}[/latex ### Add and Subtract Fractions Activity Mixed Number Number Subtract and Add Directions: 1. Each player flips four Mixed Number Cards. 2. Each player arranges the cards to create two differences that will have the greatest possible sum. 3. Each player records the two differences, and then adds the differences. 4. Players repeat Steps 1–3. 5. Each player adds Sum A and Sum B to find the total. The player with the greatest total wins! Answer: ### Add and Subtract Fractions Performance Chapter Practice 8.1 Simplest Form Write the fraction in simplest form. Question 1. [latex]\frac{2}{12}$$
Step 1: Find the common factors of 2 and 12.
Factors of 2:    1, 2
Factors of 12:  1, 2, 3, 4, 6, 12
The common factors of 2 and 12 are 1 and 2.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{2}{12} = \frac{2 \div 2}{12 \div 2} = \frac{1}{6}$
Because 1 and 6 have no common factors other than 1, $$\frac{2}{12}$$ is in simplest form.

Question 2.
$$\frac{15}{30}$$
Step 1: Find the common factors of 15 and 30.
Factors of 15:    1, 3, 5, 15
Factors of 30:  1, 2, 3, 5, 6, 10, 15, 30
The common factors of 15 and 30 are 1, 3, 5 and 15.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

$\dpi{100} \small \frac{15}{30} = \frac{15 \div 15}{30 \div 15} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1, $$\frac{15}{30}$$ is in simplest form.

Question 3.
$$\frac{16}{24}$$
Step 1: Find the common factors of 16 and 24.
Factors of 16:    1, 2, 4, 8, 16
Factors of 24:  1, 2, 3, 4, 6, 8, 12, 24
The common factors of 16 and 24 are 1, 2, 4 and 8.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3}$
Because 2 and 3 have no common factors other than 1, $$\frac{16}{24}$$ is in simplest form.

Question 4.
$$\frac{18}{36}$$
Step 1: Find the common factors of 18 and 36.
Factors of 18:  1, 2, 3, 6, 9, 18
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The common factors of 18 and 36 are 1, 2, 3, 6, 9 and 18.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.

$\dpi{100} \small \frac{18}{36} = \frac{18 \div 18}{36 \div 18} = \frac{1}{2}$
Because 1 and 2 have no common factors other than 1, $$\frac{18}{36}$$ is in simplest form.

Question 5.
$$\frac{8}{32}$$
Step 1: Find the common factors of 8 and 32.
Factors of 8:  1, 2, 4, 8
Factors of 32: 1, 2, 4, 8, 16, 32
The common factors of 8 and 32 are 1, 2, 4 and 8.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{8}{32} = \frac{8 \div 8}{32 \div 8} = \frac{1}{4}$
Because 1 and 4 have no common factors other than 1, $$\frac{8}{32}$$ is in simplest form.

Question 6.
$$\frac{25}{10}$$
Step 1: Find the common factors of 25 and 10.
Factors of 25:  1, 5, 25
Factors of 10: 1, 2, 5, 10
The common factors of 25 and 10 are 1 and 5.
Step 2: Write an equivalent fraction by dividing the numerator and the denominator by the greatest of the common factors.
$\dpi{100} \small \frac{25}{10} = \frac{25 \div 5}{10 \div 5} = \frac{5}{2}$
Because 5 and 2 have no common factors other than 1, $$\frac{25}{10}$$ is in simplest form.

8.2 Estimate Sums and Differences of Fractions

Estimate the sum or difference.

Question 7.
$$\frac{7}{8}$$ – $$\frac{1}{5}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{7}{8}$ is about
Think: The numerator is about the same as the denominator.
$\dpi{100} \small \frac{1}{5}$ is about
Think: The numerator is near to zero.
Step 2: Estimate the difference.
An estimate of $\dpi{100} \small \frac{7}{8}$$\dpi{100} \small \frac{1}{5}$  is 1 – 0 = 1.

Question 8.
$$\frac{5}{6}$$ + $$\frac{9}{10}$$
Step 1: Estimate each fraction.
$\dpi{100} \small \frac{5}{6}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1.
$\dpi{100} \small \frac{9}{10}$ is between $\dpi{100} \small \frac{1}{2}$ and 1, but is closer to 1.
Step 2: Estimate the sum.
An estimate of $$\frac{5}{6}$$ + $$\frac{9}{10}$$ = 1 + 1 = 2.

Question 9.
$$\frac{11}{12}$$ – $$\frac{89}{100}$$
Step 1: Use mental math to estimate each fraction.
$\dpi{100} \small \frac{11}{12}$ is about
Think : The numerator is about the same as the denominator.
$\dpi{100} \small \frac{89}{100}$ is about
Think : The numerator is closer to denominator.
Step 2: Estimate the difference.
An estimate of $\dpi{100} \small \frac{11}{12}$$\dpi{100} \small \frac{89}{100}$  is 1 – 1 = 0.

Question 10.
Precision
Your friend says $$\frac{7}{8}$$ – $$\frac{5}{12}$$ is about 0. Find a closer estimate. Explain why your estimate is closer.
$$\frac{7}{8}$$ – $$\frac{5}{12}$$ = $$\frac{11}{24}$$
$$\frac{11}{24}$$ = 0.45
The number 0.45 is close to 0.

8.3 Find Common Denominators

Use a common denominator to write an equivalent fraction for each fraction.

Question 11.
$$\frac{1}{4}$$ and $$\frac{1}{2}$$
Use the product of the denominators: 4 $\dpi{100} \small \times$ 2 = 8
Write equivalent fractions with denominators of 8
$\dpi{100} \small \frac{1}{4} = \frac{1\times 2}{4\times 2} = \frac{2}{8}$
$\dpi{100} \small \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{2}{8}$  and $\dpi{100} \small \frac{4}{8}$.

Question 12.
$$\frac{2}{3}$$ and $$\frac{2}{9}$$
Use the product of the denominators : 3 $\dpi{100} \small \times$ 9 = 27
Write equivalent fractions with denominators of 27
$\dpi{100} \small \frac{2}{3} = \frac{2 \times 9}{3 \times 9} = \frac{18}{27}$
$\dpi{100} \small \frac{2}{9} = \frac{2 \times 3}{9 \times 3} = \frac{6}{27}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{18}{27}$  and $\dpi{100} \small \frac{6}{27}$.

Question 13.
$$\frac{2}{3}$$ and $$\frac{5}{6}$$
Use the product of the denominators : 3 $\dpi{100} \small \times$ 6 = 18
Write equivalent fractions with denominators of 18
$\dpi{100} \small \frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}$
$\dpi{100} \small \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{12}{18}$  and $\dpi{100} \small \frac{15}{18}$.

Question 14.
$$\frac{4}{5}$$ and $$\frac{1}{3}$$
Use the product of the denominators : 5 $\dpi{100} \small \times$ 3 = 15
Write equivalent fractions with denominators of 15
$\dpi{100} \small \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}$
$\dpi{100} \small \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{12}{15}$  and $\dpi{100} \small \frac{5}{15}$.

Question 15.
$$\frac{5}{6}$$ and $$\frac{8}{9}$$
Use the product of the denominators : 6 $\dpi{100} \small \times$ 9 = 54
Write equivalent fractions with denominators of 54
$\dpi{100} \small \frac{5}{6} = \frac{5 \times 9}{6 \times 9} = \frac{45}{54}$
$\dpi{100} \small \frac{8}{9} = \frac{8 \times 6}{9 \times 6} = \frac{48}{54}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{45}{54}$  and $\dpi{100} \small \frac{48}{54}$.

Question 16.
$$\frac{4}{5}$$ and $$\frac{3}{4}$$
Use the product of the denominators : 5 $\dpi{100} \small \times$ 4 = 20
Write equivalent fractions with denominators of 20
$\dpi{100} \small \frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}$
$\dpi{100} \small \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$
Therefore, equivalent fractions are $\dpi{100} \small \frac{16}{20}$  and $\dpi{100} \small \frac{15}{20}$.

8.4 Add Fractions with Unlike Denominators

Question 17.
$$\frac{2}{15}$$ + $$\frac{2}{3}$$ = __
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 15 is a multiple of 3, so rewrite $$\frac{2}{3}$$ with a denominator of 15.
Rewrite $\dpi{100} \small \frac{2}{3}$  as $\dpi{100} \small \frac{2 \times 5}{3 \times 5}$ = $\dpi{100} \small \frac{10}{15}$
$\dpi{100} \small \frac{2}{15}$ + $\dpi{100} \small \frac{2}{3}$ = $\dpi{100} \small \frac{2}{15}$ + $\dpi{100} \small \frac{10}{15}$
= $\dpi{100} \small \frac{12}{15}$

Question 18.
$$\frac{3}{4}$$ + $$\frac{1}{8}$$ = __
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 8 is a multiple of 4, so rewrite $$\frac{3}{4}$$ with a denominator of 8.
Rewrite $\dpi{100} \small \frac{3}{4}$  as $\dpi{100} \small \frac{3 \times 2}{4 \times 2}$ = $\dpi{100} \small \frac{6}{8}$
$\dpi{100} \small \frac{3}{4}$ + $\dpi{100} \small \frac{1}{8}$ = $\dpi{100} \small \frac{6}{8}$ + $\dpi{100} \small \frac{1}{8}$
= $\dpi{100} \small \frac{7}{8}$

Question 19.
$$\frac{7}{2}$$ + $$\frac{1}{6}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 6 is a multiple of 2, so rewrite $$\frac{7}{2}$$ with a denominator of 6.
Rewrite $\dpi{100} \small \frac{7}{2}$  as $\dpi{100} \small \frac{7 \times 3}{2 \times 3}$ = $\dpi{100} \small \frac{21}{6}$

$\dpi{100} \small \frac{7}{2}$ + $\dpi{100} \small \frac{1}{6}$ = $\dpi{100} \small \frac{21}{6}$ + $\dpi{100} \small \frac{1}{6}$
= $\dpi{100} \small \frac{22}{6}$
= $\dpi{100} \small \frac{11}{3}$

Question 20.
$$\frac{5}{9}$$ + $$\frac{1}{2}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 9 is not a multiple of 2, so rewrite each fraction with a denominator of 9 x 2 = 18.
Rewrite $\dpi{100} \small \frac{5}{9}$  as $\dpi{100} \small \frac{5 \times 2}{9 \times 2}$ = $\dpi{100} \small \frac{10}{18}$
$\dpi{100} \small \frac{1}{2}$ as $\dpi{100} \small \frac{1 \times 9}{2 \times 9}$ = $\dpi{100} \small \frac{9}{18}$
$\dpi{100} \small \frac{5}{9}$ + $\dpi{100} \small \frac{1}{2}$ = $\dpi{100} \small \frac{10}{18}$ + $\dpi{100} \small \frac{9}{18}$
= $\dpi{100} \small \frac{19}{18}$

Question 21.
$$\frac{7}{10}$$ + $$\frac{5}{6}$$ = ___

Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 10 is not a multiple of 6, so rewrite each fraction with a denominator of 10 x 6 = 60.

Rewrite $\dpi{100} \small \frac{7}{10}$  as $\dpi{100} \small \frac{7 \times 6}{10 \times 6}$ = $\dpi{100} \small \frac{42}{60}$

$\dpi{100} \small \frac{5}{6}$ as $\dpi{100} \small \frac{5 \times 10}{6 \times 10}$ = $\dpi{100} \small \frac{50}{60}$

$\dpi{100} \small \frac{7}{10}$ + $\dpi{100} \small \frac{5}{6}$ = $\dpi{100} \small \frac{42}{60}$ + $\dpi{100} \small \frac{50}{60}$

= $\dpi{100} \small \frac{92}{60}$

Question 22.
$$\frac{1}{6}$$ + $$\frac{11}{12}$$ + $$\frac{4}{6}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the sum.
Think: 12 is a multiple of 6, so rewrite each fraction with a denominator of 12.

Rewrite $\dpi{100} \small \frac{1}{6}$  as $\dpi{100} \small \frac{1 \times 2}{6 \times 2}$ = $\dpi{100} \small \frac{2}{12}$

$\dpi{100} \small \frac{4}{6}$ as $\dpi{100} \small \frac{4 \times 2}{6 \times 2}$ = $\dpi{100} \small \frac{8}{12}$

$\dpi{100} \small \frac{1}{6}$ + $\dpi{100} \small \frac{11}{12}$ + $\dpi{100} \small \frac{4}{6}$ = $\dpi{100} \small \frac{2}{12}$ + $\dpi{100} \small \frac{11}{12}$ + $\dpi{100} \small \frac{8}{12}$

= $\dpi{100} \small \frac{21}{12}$

= $\dpi{100} \small \frac{7}{4}$

8.5 Subtract Fractions with Unlike Denominators

Subtract

Question 23.
$$\frac{1}{4}$$ – $$\frac{1}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 8 is a multiple of 4, so rewrite $$\frac{1}{4}$$ with a denominator of 8.
Rewrite $\dpi{100} \small \frac{1}{4}$  as $\dpi{100} \small \frac{1 \times 2}{4 \times 2}$ = $\dpi{100} \small \frac{2}{8}$
$\dpi{100} \small \frac{1}{4}$$\dpi{100} \small \frac{1}{8}$ = $\dpi{100} \small \frac{2}{8}$$\dpi{100} \small \frac{1}{8}$
= $\dpi{100} \small \frac{1}{8}$

Question 24.
$$\frac{3}{2}$$ – $$\frac{7}{10}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 10 is a multiple of 2, so rewrite $$\frac{3}{2}$$ with a denominator of 10.
Rewrite $\dpi{100} \small \frac{3}{2}$  as $\dpi{100} \small \frac{3 \times 5}{2 \times 5}$ = $\dpi{100} \small \frac{15}{10}$
$\dpi{100} \small \frac{3}{2}$$\dpi{100} \small \frac{7}{10}$ = $\dpi{100} \small \frac{15}{10}$$\dpi{100} \small \frac{7}{10}$
= $\dpi{100} \small \frac{8}{10}$
= $\dpi{100} \small \frac{4}{5}$

Question 25.
$$\frac{15}{16}$$ – $$\frac{7}{8}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 16 is a multiple of 8, so rewrite $$\frac{7}{8}$$ with a denominator of 16.
Rewrite $\dpi{100} \small \frac{7}{8}$  as $\dpi{100} \small \frac{7 \times 2}{8 \times 2}$ = $\dpi{100} \small \frac{14}{16}$
$\dpi{100} \small \frac{15}{16}$$\dpi{100} \small \frac{7}{8}$ = $\dpi{100} \small \frac{15}{16}$$\dpi{100} \small \frac{14}{16}$
= $\dpi{100} \small \frac{1}{16}$

Question 26.
$$\frac{4}{3}$$ – $$\frac{2}{5}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 5 is not a multiple of 3, so rewrite each fraction with a denominator of 5 x 3 = 15.
Rewrite $\dpi{100} \small \frac{4}{3}$  as $\dpi{100} \small \frac{4 \times 5}{3 \times 5}$ = $\dpi{100} \small \frac{20}{15}$
$\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 3}{5 \times 3}$ = $\dpi{100} \small \frac{6}{15}$
$\dpi{100} \small \frac{4}{3}$$\dpi{100} \small \frac{2}{5}$ = $\dpi{100} \small \frac{20}{15}$$\dpi{100} \small \frac{6}{15}$
= $\dpi{100} \small \frac{14}{15}$

Question 27.
$$\frac{5}{6}$$ – $$\frac{3}{4}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 6 is not a multiple of 4, so rewrite each fraction with a denominator of 6 x 4 = 24.
Rewrite $\dpi{100} \small \frac{5}{6}$  as $\dpi{100} \small \frac{5 \times 4}{6 \times 4}$ = $\dpi{100} \small \frac{20}{24}$
$\dpi{100} \small \frac{3}{4}$ as $\dpi{100} \small \frac{3 \times 6}{4 \times 6}$ = $\dpi{100} \small \frac{18}{24}$
$\dpi{100} \small \frac{5}{6}$$\dpi{100} \small \frac{3}{4}$ = $\dpi{100} \small \frac{20}{24}$$\dpi{100} \small \frac{18}{24}$
= $\dpi{100} \small \frac{2}{24}$
= $\dpi{100} \small \frac{1}{12}$

Question 28.
$$\frac{7}{10}$$ – $$\frac{2}{5}$$ + $$\frac{11}{20}$$ = ___
Use equivalent fractions to write the fractions with a common denominator. Then find the difference.
Think: 20 is a multiple of 10 and 5, so rewrite each fraction with a denominator of 20.
Rewrite $\dpi{100} \small \frac{7}{10}$  as $\dpi{100} \small \frac{7 \times 2}{10 \times 2}$ = $\dpi{100} \small \frac{14}{20}$
$\dpi{100} \small \frac{2}{5}$ as $\dpi{100} \small \frac{2 \times 4}{5 \times 4}$ = $\dpi{100} \small \frac{8}{20}$

$\dpi{100} \small \frac{7}{10}$$\dpi{100} \small \frac{2}{5}$ + $\dpi{100} \small \frac{11}{20}$ = $\dpi{100} \small \frac{14}{20}$$\dpi{100} \small \frac{8}{20}$ + $\dpi{100} \small \frac{11}{20}$

Question 29.
1$$\frac{3}{4}$$ + 7$$\frac{5}{8}$$ = ___
To add the fractional parts, use a common denominator
1 $\dpi{100} \small \frac{3}{4}$ = 1 $\dpi{100} \small \frac{6}{8}$
1 $\dpi{100} \small \frac{3}{4}$ + 7 $\dpi{100} \small \frac{5}{8}$ = 1 $\dpi{100} \small \frac{6}{8}$ + 7 $\dpi{100} \small \frac{5}{8}$
= 8 $\dpi{100} \small \frac{11}{8}$

Question 30.
3$$\frac{3}{10}$$ + 2$$\frac{7}{20}$$ = ___
To add the fractional parts, use a common denominator
3 $\dpi{100} \small \frac{3}{10}$ = 3 $\dpi{100} \small \frac{6}{20}$
3 $\dpi{100} \small \frac{3}{10}$ + 2 $\dpi{100} \small \frac{7}{20}$ = 3 $\dpi{100} \small \frac{6}{20}$ + 2 $\dpi{100} \small \frac{7}{20}$
= 5 $\dpi{100} \small \frac{13}{20}$

Question 31.
$$\frac{1}{3}$$ + 6$$\frac{4}{5}$$ = ___
To add the fractional parts, use a common denominator 3 x 5 = 15
$\dpi{100} \small \frac{1}{3}$ = $\dpi{100} \small \frac{5}{15}$
6 $\dpi{100} \small \frac{4}{5}$ = 6 $\dpi{100} \small \frac{12}{15}$
$\dpi{100} \small \frac{1}{3}$ + 6 $\dpi{100} \small \frac{4}{5}$  = $\dpi{100} \small \frac{5}{15}$ + 6 $\dpi{100} \small \frac{12}{15}$
= 6 $\dpi{100} \small \frac{17}{15}$

Question 32.
5$$\frac{8}{9}$$ + $$\frac{5}{6}$$ = _____
To add the fractional parts, use a common denominator 9 x 6 = 54
5 $\dpi{100} \small \frac{8}{9}$ = 5 $\dpi{100} \small \frac{48}{54}$
$\dpi{100} \small \frac{5}{6}$ = $\dpi{100} \small \frac{45}{54}$
5 $\dpi{100} \small \frac{8}{9}$ + $\dpi{100} \small \frac{5}{6}$  = 5 $\dpi{100} \small \frac{48}{54}$ + $\dpi{100} \small \frac{45}{54}$
= 5 $\dpi{100} \small \frac{93}{54}$

Question 33.
2$$\frac{2}{3}$$ + $$\frac{4}{9}$$ + 4$$\frac{1}{3}$$ = ___
To add the fractional parts, use a common denominator
2 $\dpi{100} \small \frac{2}{3}$ = 2 $\dpi{100} \small \frac{6}{9}$
4 $\dpi{100} \small \frac{1}{3}$ = 4 $\dpi{100} \small \frac{3}{9}$
2 $\dpi{100} \small \frac{2}{3}$ + $\dpi{100} \small \frac{4}{9}$ + 4 $\dpi{100} \small \frac{1}{3}$   = 2 $\dpi{100} \small \frac{6}{9}$ + $\dpi{100} \small \frac{4}{9}$ + 4 $\dpi{100} \small \frac{3}{9}$
= 6 $\dpi{100} \small \frac{13}{9}$

Question 34.
5$$\frac{1}{2}$$ + 2$$\frac{5}{8}$$ + 3$$\frac{3}{4}$$ = _____
To add the fractional parts, use a common denominator
5 $\dpi{100} \small \frac{1}{2}$ = 5 $\dpi{100} \small \frac{4}{8}$
3 $\dpi{100} \small \frac{3}{4}$ = 3 $\dpi{100} \small \frac{6}{8}$
5 $\dpi{100} \small \frac{1}{2}$ + 2 $\dpi{100} \small \frac{5}{8}$ + 3 $\dpi{100} \small \frac{3}{4}$   =5 $\dpi{100} \small \frac{4}{8}$ + 2 $\dpi{100} \small \frac{5}{8}$ + 3 $\dpi{100} \small \frac{6}{8}$
= 10 $\dpi{100} \small \frac{15}{8}$

8.7 Subtract Mixed Numbers

Subtract

Question 35.
8$$\frac{7}{10}$$ – 1$$\frac{2}{5}$$ = ___

Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
1 $\dpi{100} \small \frac{2}{5}$ = 1 $\dpi{100} \small \frac{4}{10}$
8 $\dpi{100} \small \frac{7}{10}$ – 1 $\dpi{100} \small \frac{2}{5}$  = 8 $\dpi{100} \small \frac{7}{10}$ – 1 $\dpi{100} \small \frac{4}{10}$ = 7 $\dpi{100} \small \frac{3}{10}$

Question 36.
15$$\frac{97}{100}$$ – 10$$\frac{7}{20}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
10 $\dpi{100} \small \frac{7}{20}$ = 10 $\dpi{100} \small \frac{35}{100}$
15 $\dpi{100} \small \frac{97}{100}$ – 10 $\dpi{100} \small \frac{7}{20}$  = 15 $\dpi{100} \small \frac{97}{100}$ – 10 $\dpi{100} \small \frac{35}{100}$ = 5 $\dpi{100} \small \frac{62}{100}$

Question 37.
4 – 3$$\frac{5}{6}$$ = ___
4 – 3 $\dpi{100} \small \frac{5}{6}$ = 4 – $\dpi{100} \small \frac{23}{6}$ = $\dpi{100} \small \frac{1}{6}$

Question 38.
5$$\frac{1}{3}$$ – 2$$\frac{1}{2}$$ = _____
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 3 x 2 = 6.
5 $\dpi{100} \small \frac{1}{3}$ = 5 $\dpi{100} \small \frac{2}{6}$ = 4 $\dpi{100} \small \frac{8}{6}$
2 $\dpi{100} \small \frac{1}{2}$ = 2 $\dpi{100} \small \frac{3}{6}$
5 $\dpi{100} \small \frac{1}{3}$ – 2 $\dpi{100} \small \frac{1}{2}$ = 4 $\dpi{100} \small \frac{8}{6}$ – 2 $\dpi{100} \small \frac{3}{6}$
= 2 $\dpi{100} \small \frac{5}{6}$

Question 39.
9$$\frac{2}{5}$$ – 6$$\frac{3}{4}$$ = ___
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator 5 x 4 = 20.
9 $\dpi{100} \small \frac{2}{5}$ = 9 $\dpi{100} \small \frac{8}{20}$ = 8 $\dpi{100} \small \frac{28}{20}$
6 $\dpi{100} \small \frac{3}{4}$ = 6 $\dpi{100} \small \frac{15}{20}$
9 $\dpi{100} \small \frac{2}{5}$ – 6 $\dpi{100} \small \frac{3}{4}$ = 8 $\dpi{100} \small \frac{28}{20}$ – 6 $\dpi{100} \small \frac{15}{20}$
= 2 $\dpi{100} \small \frac{13}{20}$

Question 40.
2$$\frac{3}{8}$$ + 7$$\frac{1}{2}$$ – 1$$\frac{11}{16}$$ = ____
To add or subtract the fractional parts, use a common denominator.
2 $\dpi{100} \small \frac{3}{8}$ = 2 $\dpi{100} \small \frac{6}{16}$
7 $\dpi{100} \small \frac{1}{2}$ = 7 $\dpi{100} \small \frac{8}{16}$
2 $\dpi{100} \small \frac{3}{8}$ + 7 $\dpi{100} \small \frac{1}{2}$  – 1 $\dpi{100} \small \frac{11}{16}$ = 2 $\dpi{100} \small \frac{6}{16}$ + 7 $\dpi{100} \small \frac{8}{16}$ – 1 $\dpi{100} \small \frac{11}{16}$
= 8 $\dpi{100} \small \frac{3}{16}$

Question 41.
Modeling Real Life
A family adopts a puppy that weighs 7$$\frac{7}{8}$$ pounds. They take him to the vet 2 weeks later, and he weighs 12$$\frac{3}{16}$$ pounds. How much weight did the puppy gain?
Subtract the fractional parts and subtract the whole number parts.
To subtract the fractional parts, use a common denominator.
7 $\dpi{100} \small \frac{7}{8}$ = 7 $\dpi{100} \small \frac{14}{16}$
12 $\dpi{100} \small \frac{3}{16}$ – 7 $\dpi{100} \small \frac{7}{8}$ = 12 $\dpi{100} \small \frac{3}{16}$  – 7 $\dpi{100} \small \frac{14}{16}$
= 11 $\dpi{100} \small \frac{19}{16}$ – 7 $\dpi{100} \small \frac{14}{16}$
= 4 $\dpi{100} \small \frac{5}{16}$
So the puppy gains 4 $\dpi{100} \small \frac{5}{16}$ pounds weight.

8.8 Problem Solving: Fractions

Question 42.
A radio station plays three commercials between two songs. The commercials play for 2 minutes altogether. The first commercial is $$\frac{1}{2}$$ minute, and the second commercial is 1$$\frac{1}{4}$$ minutes. How long is the third commercial?
Given,
A radio station plays three commercials between two songs.
The commercials play for 2 minutes altogether. The first commercial is $$\frac{1}{2}$$ minute, and the second commercial is 1$$\frac{1}{4}$$ minutes.
$$\frac{1}{2}$$ + 1$$\frac{1}{4}$$ = 1 $$\frac{3}{4}$$
2 – 1 $$\frac{3}{4}$$ = $$\frac{1}{4}$$

Question 43.
Your friend plants a tree seedling on Earth Day that is 1$$\frac{1}{3}$$ feet tall. In 1 year, the tree grows 1$$\frac{5}{6}$$ feet. After 2 years, the tree is 4$$\frac{11}{12}$$ feet tall. How much did the tree grow in the second year?

Given,
Your friend plants a tree seedling on Earth Day that is 1$$\frac{1}{3}$$ feet tall.
In 1 year, the tree grows 1$$\frac{5}{6}$$ feet. After 2 years, the tree is 4$$\frac{11}{12}$$ feet tall.
4$$\frac{11}{12}$$ – 1$$\frac{5}{6}$$  = 3 $$\frac{1}{12}$$
3 $$\frac{1}{12}$$ – 1$$\frac{1}{3}$$ = 1 $$\frac{3}{4}$$
Thus the growth of the tree in the second year is 1 $$\frac{3}{4}$$ feet.

Conclusion:

Enhance your math skills and performance skills by referring to our Big Ideas Math Grade 5 Answer Key for Chapter 8 Add and Subtract Fractions. Make use of the links to complete your homework and assignments. Practice the given questions number of times to score the highest marks in the exams. Also, keep in touch with our site to get the Solution Key for all Big Ideas Math Grade 5 Chapters from 1 to 14.

## Big Ideas Math Answers Grade 2 Chapter 1 Numbers and Arrays

Big Ideas Math Answers 2nd Grade 1st Chapter Numbers and Arrays PDF is provided here. We have given the solutions to all the questions in Big Ideas Math Answers Grade 2 Chapter 1 Numbers and Arrays in pdf format. BIM Grade 2 Textbook Chapter 1 Numbers and Arrays Answer Key helps the students to complete their homework in time and also to enhance their knowledge. It will also useful to learn the concepts in depth.

Apply the real-time math examples by learning the tricks using BIM Grade 2 Chapter 1 Numbers and Arrays. Have a look at the topics before you start the preparation. The quick method of solving math questions si helpful to the students to save their time. We have provided the answers for each and every section of chapter 1 numbers and arrays in the following sections.

The topics covered in Bid Ideas Math Book Grade 2 Answer Key Chapter 1 Numbers and Arrays are Even and Odd Numbers, Model Even and Odd Numbers, Equal Groups, Use Arrays, and Make Arrays. The performance task given at the end helps you to test your skills. Make use of the below links and learn the basic topics covered here.

Vocabulary

Lesson 1: Even and Odd Numbers

Lesson 2: Model Even and Odd Numbers

Lesson 3: Equal Groups

Lesson 4: Use Arrays

Lesson 5: Make Arrays

### Numbers and Arrays Vocabulary

Organize It

Use the review words to complete the graphic organizer.

‘+’ is the mathematical symbol used to represent the notion of positive as well as the operation of addition.

Define It

Use your vocabulary cards to identify the word.

Question 1.

Question 2.

Question 3.

Chapter 1 Vocabulary Cards

### Lesson 1.1 Even and Odd Numbers

Explore and Grow

Use linking cubes to model each story.
There are 6 students in the gym. Does each student have a partner?
There are 5 students in the library. Does each student have a partner?

Think and Grow

Show and Grow

Question 1.

Explanation: There are 6 parts, An even number can be shown as 2 equal parts.

Question 2.

Explanation: There are 9 parts, An odd number cannot be shown as 2 equal parts.

Color cubes to show the number. Circle even or odd.

Question 3.

Explanation: 11 is odd number and cannot be shown as 2 equal parts.

Question 4.

Explanation: 16 is even number and can be shown as 2 equal parts.

Apply and Grow: Practice

Color cubes to show the number. Circle even or odd.

Question 5.

Explanation: 13 is odd number and cannot be shown as 2 equal parts.

Question 6.

Explanation: 10 is even number and can be shown as 2 equal parts.

Is the number even or odd?

Question 7.

Explanation: 1 is odd number and cannot be shown as 2 equal parts.

Question 8.

Explanation: 4 is even number and can be shown as 2 equal parts.

Question 9.

Explanation: 18 is even number and can be shown as 2 equal parts.

Question 10.

Explanation: 17 is odd number and cannot be shown as 2 equal parts.

Question 11.

Explanation: 19 is odd number and cannot be shown as 2 equal parts.

Question 12.

Explanation: 20 is even number and can be shown as 2 equal parts.

Question 13.
Number Sense
Circle even or odd to describe each group. Then write each number in the correct group.

Think and Grow: Modeling Real Life

There is an even number of students in your class. There are more than 16 but fewer than 20 students. How many students are in your class?

Explanation: Between 16 and 20, Even number is 18.

Show how you know:

Show and Grow

Question 14.
There is an odd number of cows in a ﬁeld. There are more than 13 but fewer than 17 cows. How many cows are in the ﬁeld?

Explanation: Between 13 and 17, Odd number is 15.

Question 15.
There are 14 geese on a farm. There are 2 more chickens than geese. Is there an even or odd number of chickens?

Explanation: There are total 16 Chickens on a farm, which is Even number.

### Even and Odd Numbers Home & Practice 1.1

Question 1.

Explanation: Total 11 parts, which cannot be shown as 2 equal parts.

Question 2.

Explanation: Total 10 parts, which can be shown as 2 equal parts.

Color cubes to show the number. Circle even or Odd

Question 3.

Explanation: 9 is Odd number which cannot be shown in 2 equal parts.

Question 4.

Explanation: 14 is Even number which can be shown in 2 equal parts.

Question 5.

Explanation: 18 is Even number which can be shown in 2 equal parts.

Question 6.

Explanation: 15 is Odd number which cannot be shown in 2 equal parts.

Is the number even or odd?

Question 7.

Explanation: 2 is Even number which can be shown in 2 equal parts.

Question 8.

Explanation: 5 is Odd number which cannot be shown in 2 equal parts.

Review & Refresh

Question 9.
DIG DEEPER!
You break apart a linking cube train to make two equal parts. There is 1 cube left over. Is the number of cubes even or odd? Explain.

Explanation: There are 5 parts in cube train, when it is broken in 2 equal parts then there is 1 cube left (2+2+1) which is Odd number.

Question 10.
Modeling Real Life
There is an even number of eggs in a nest. There are more than 10 but fewer than 14 eggs. How many eggs are in the nest?

Explanation: 12 is even number and also between 10 and 14, 12 is the only even number.

Question 11.
Modeling Real Life
You have 6 green crayons. You have 1 more blue crayon than green crayons. Do you have an even or an odd number of blue crayons?

Explanation:  Total 7 blue crayon which is Odd number so it cannot be shown in 2 equal parts.

Review & Refresh

Question 12.
6 + 6 = ___
Explanation: 12 is Even number which can be shown in 2 equal parts.

Question 13.
9 + 9 = ___
Explanation: 18 is Even number which can be shown in 2 equal parts.

Question 14.
7 + 7 = ___
Explanation: 14 is Even number which can be shown in 2 equal parts.

Question 15.
8 + 8 = ___
Explanation: 16 is Even number which can be shown in 2 equal parts.

### Lesson 1.2 Model Even and Odd Numbers

Explore and Grow

Use linking cubes to model each sum. Is the sum even or odd?

Explanation: 4+4=8, 8 is Even number which can be shown in 2 equal parts.

Explanation: 5+4=9, 9 is Odd number which cannot be shown in 2 equal parts.

Show and Grow

Question 1.

Explanation: 7=4+3, Odd number

Question 2.

Explanation: 10=5+5, Even number

Question 3.

Explanation: 14=7+7, Even number

Question 4.

Explanation: 17=9+8, Odd number

Question 5.

Explanation: 18=9+9, Even number

Question 6.

Explanation: 13=7+6, Odd number

Question 7.

Explanation: 15=8+7, Odd number

Question 8.

Explanation: 20=10+10, Even number

Question 9.
YOU BE THE TEACHER
Descartes uses doubles plus 1 to model an odd number. Is he correct? Explain.

Explanation: 3+4=7, Odd number which cannot be shown as 2 equal parts.

Question 10.
You do 6 sit-ups on Saturday and 7 on Sunday. Do you do an even or odd number of sit-ups in all?

Explanation: 6+7=13, It is an Odd number

Think and Grow: Modeling Real Life

There is an even number of marbles in one bag and an odd number of marbles in another bag. Is there an even or an odd number of marbles in all?

Which equation could match the story?

There is an ___ number of marbles in all.
Explanation: 8+7=15, 8 is even and 7 is odd number, in total 15 is Odd number of marbles.

Show and Grow

Question 11.
Two buckets each have an odd number of seashells. Is there an even or an odd number of seashells in all?

Which equation could match the story?

There is an __ number of seashells in all.
Explanation: 9+5=14, 9 and 5 are odd numbers which in total 14 is Even number.

Question 12.
DIG DEEPER!
You have an odd number of ﬂowers. You and your friend have an even number of ﬂowers in all. Does your friend have an even or an odd number of ﬂowers?

Your friend has an ___ number of ﬂowers.
Explanation: Friend has only even number of flowers.

### Model Even and Odd Numbers Homework & Practice 1.2

Question 1.

Explanation: 8=4+4, Even number

Question 2.

Explanation: 17=9+8, Odd number

Question 3.

Explanation: 9=5+4, Odd number

Question 4.

Explanation: 16=8+8, Even number

Question 5.
Reasoning
Fill in the blanks using even or odd.
The sum of two even numbers is ___.
The sum of two odd numbers is ___.
The sum of an even number and an odd number is __.

Question 6.
You do 10 jumping jacks on Saturday and 10 on Sunday. Do you do an even or odd number of jumping jacks in all?

Explanation: 10+10=20, which is Even number.

Question 7.
Modeling Real Life
You and your friend each have an even number of googly eyes. Do you and your friend have an even or an odd number of googly eyes in all?

Which equation could match the story?

You have an __ number of googly eyes in all.
Explanation: 4 and 6 are Even numbers. So totally 10 googly eyes which is even number.

Question 8.
DIG DEEPER!
You hop an even number of times. You and your friend hop an odd number of times in all. Does your friend hop an even or an odd number of times?
Your friend hops an ___ number of times.
Explanation: you hop an even and overall odd number, so your friend has odd number of times.

Review & Refresh

Circle the shape that shows equal shares.

Question 9.

Question 10.

### Lesson 1.3 Equal Groups

Explore and Grow

Circle groups of two oranges. Complete the sentence.

Answer: 4 groups of 2 is 8.

Show and Grow

Question 1.

5+5=10

Question 2.

2+2+2=6

Question 3.
Circle groups of 3. Write a repeated addition equation.

5+5+5+5+5=25

Apply and Grow: Practice

Question 4.
Circle groups of 5. Write a repeated addition equation.

5+5=15

Question 5.
Circle groups of 4. Write a repeated addition equation.

Question 6.
YOU BE THE TEACHER
Newton says he can circle 5 equal groups. Is he correct? Explain.

Yes he is correct. Newton can circle 3 groups of 5 parts. So, 3+3+3+3+3=15

Think and Grow: Modeling Real Life

You have 3 boxes. There are 5 pencils in each box. How many pencils are there in all?

Model:

Answer: 15 pencils in total. 5+5+5=15.

Show and Grow

Question 7.
You have 5 bags. There are 4 notebooks in each bag. How many notebooks are there in all?

Answer: 20 notebooks in total. 5+5+5+5=20.

Question 8.
DIG DEEPER!
There are 4 boxes. Each box has the same number of glue sticks. There are 16 in all. How many glue sticks are in each box?

Answer: There are 4 glue sticks in each box.

Question 9.
Explain how you solved Exercises 7 and 8. What did you do differently?
_______________________________________
_______________________________________

### Equal Groups Homework & Practice 1.3

Question 1.

3+3=6

Question 2.

4+4+4=12

Question 3.
Circle groups of 2. Write a repeated addition equation.

Question 4.
Structure
Show two different ways to put the buttons in equal groups.
One Way:

Another Way:

Question 5.
Modeling Real Life
You have 3 jars of paint brushes.There are 6 paint brushes in each jar. How many paint brushes are there in all?

Explanation: 6+6+6=18
Review & Refresh

Question 6.

How many students chose baseball? ___

Which sport is the most favorite?

### Lesson 1.4 Use Arrays

Explore and Grow

How many equal groups are there? Write an addition equation to tell how many cars there are in all.

Number of equal groups: ___
3+3=6

Show and Grow

Question 1.

4+4=8

Question 2.

4+4+4=12

Question 3.

4+4+4+4=16

Apply and Grow: Practice

Question 4.

5+5=10

Question 5.

3+3+3=9

Question 6.

5+5+5=15

Question 7.

5+5+5+5=20

Question 8.
Logic
Which arrays show the same number of circles?

Answer: Red and Blue circles show the same number.

Think and Grow: Modeling Real Life

The arrays show the desks in two classrooms. Which classroom has more desks?

Classroom A: 5+5+5+5+5=25
Classroom B: 7+7+7+7=28
Answer: Classroom B has more number of desks.

Show and Grow

Question 9.
The arrays show gardens of green and yellow pepper plants. Are there more green pepper plants or yellow pepper plants?

Explanation: There are 15 Green pepper plants and 12 Yellow pepper plants.

### Use Arrays Homework & Practice 1.4

Question 1.

2+2+2=6

Question 2.

2+2=4

Question 3.

1+1+1+1=4

Question 4.

6+6+6= 18

Question 5.
Number Sense
Use the array to complete the equation.

Question 6.
Modeling Real Life
The arrays show toy cars. Are there more orange cars or blue cars?

Explanation: There are 16 orange cars and 15 blue cars.

Question 7.
DIG DEEPER!
The arrays show a sheet of stickers separated into two pieces. How many rows and columns of stickers did the sheet have before it was separated?

Answer: 5 rows and 5 columns

Review & Refresh

Question 8.

8 vertices
12 edges

Question 9.

0  vertices
0 edges

### Lesson 1.5 Make Arrays

Explore and Grow

Use counters to model the story. Write an addition equation to match.

There are 4 rows of students. There are 3 students in each row. How many students are there in all?

Explanation: 3+3+3+3=12 students.

Show and Grow

Question 1.
A photo album has 3 rows of photos. There are 2 photos in each row. How many photos are there in all?

Explanation: 2+2+2=6

Question 2.
You have 4 rows of stickers. There are 5 stickers in each row. How many stickers do you have in all?

Explanation: 5+5+5+5=20

Apply and Grow: Practice

Question 3.
An ice cube tray has 5 rows. There are 2 ice cubes in each row. How many ice cubes are there in all?

Explanation: 2+2+2+2+2=10

Question 4.
A bookcase has 3 shelves. There are 5 stuffed animals on each shelf. How many stuffed animals are there in all?

Explanation: 5+5+5=15 stuffed animals

Question 5.
A closet has 4 shelves. There are 2 games on each shelf. How many games are there in all?

Explanation: 2+2+2+2=8

Question 6.
Structure
Make an array to match the equation.
5 + 5 + 5 + 5 = 20

Think and Grow: Modeling Real Life

A marching band has 3 equal rows of drummers. There are 15 drummers in all. How many drummers are in each row?

Model:

Explanation: 5+5+5=15

Show and Grow

Question 7.
A quilt has 4 equal rows of patches. There are 24 patches in all. How many patches are in each row?

Explanation: 6+6+6+6=24

Question 8.
A building has 3 equal rows of windows. There are 18 windows in all. How many columns are there?

Explanation: 3+3+3+3+3+3=18

### Make Arrays Homework & Practice 1.5

Question 1.
A parking lot has 3 rows. There are 5 parking spots in each row. How many parking spots are there in all?

Explanation: 5+5+5=15

Question 2.
A bookcase has 4 shelves. There are 3 books on each shelf. How many books are there in all?

Explanation: 3+3+3+3=12

Question 3.
Reasoning
Newton has 10 tokens. Which equations can Newton use to make an array with his tokens?

Question 4.
Modeling Real Life
A theater has 4 equal rows of seats. There are 16 seats in all. How many seats are in each row?

Explanation: 4+4+4+4=16

Question 5.
Modeling Real Life
A chorus has 5 equal rows of singers. There are 30 singers in all. How many singers are in each row?

Explanation: 6+6+6+6+6=30

Review & Refresh

Question 6.
There are 9 lions in all. How many lions are in the den?

Answer: 0 lions are in den.
Explanation: All 9 lions are outside the den. Therefore, 0 lions are in the den.

### Numbers and Arrays Performance Task

Question 1.
Your art supplies are packaged in boxes as described below.

a. What do you have the most of ?

Explanation: Colored Pencils= 5+5+5=15
Markers= 7+7=14
Crayons= 4+4+4+4=16
Crayons are more than Colored Pencils and Markers.
b. Do you have an even or an odd number of markers?

Question 2.
a. You have 5 equal rows of paint bottles. You have 20 paint bottles in all. How many paint bottles are in each row?

Answer: 4 paint bottles are in each row.
b. You add another row of paint bottles. How many paint bottles do you have now?
c. Describe another way to arrange the paint bottles you have now.

### Numbers and Arrays Activity

Array Flip and Find

To Play: Place the Array Flip and Find Cards face down in the boxes. Take turns ﬂipping two cards. If your cards show the same total, keep the cards. If your cards show different totals, ﬂip the cards back over. Play until all matches are made.

### Numbers and Arrays Chapter Practice

1.1 Even and Odd Numbers

Question 1.

Explanation: 5 cube part

Question 2.

Explanation: 14 cube part

Color cubes to show the number. Circle even or odd.

Question 3.

Question 4.

Question 5.
Modeling Real Life
You see an odd number of boats. There are more than 15 but fewer than 19 boats. How many boats do you see?

Show how you know:

Explanation: Between 15 and 19 there is Odd number 17.

1.2 Model Even and Odd Numbers

Question 6.

Question 7.

Question 8.

Question 9.

1.3 Equal Groups

Question 10.

Question 11.
Circle groups of 3. Write a repeated addition equation.

Question 12.
Structure
Show two different ways to put the balls in equal groups.
One Way:

Another Way:

1.4 Use Arrays

Question 13.

Question 14.

Question 15.

Question 16.

1.5 Make Arrays

Question 17.
A cupboard has 4 shelves. There are 3 glasses on each shelf. How many glasses are there in all?

Question 18.
A bingo card has 5 rows. There are 5 squares in each row. How many squares are there in all?

Question 19.
Modeling Real Life
A pet store has 5 equal rows of ﬁsh tanks. There are 20 ﬁsh tanks in all. How many ﬁsh tanks are in each row?

Explanation: 4+4+4+4+4=20

Conclusion:

We wish that the data provided in Big Ideas Math Book Grade 2 Chapter 1 Numbers and Arrays Answer Key is satisfactory for all students. This Big Ideas Math Answers Grade 2 Chapter 1 Numbers and Arrays is not only useful for the students but also for the parents. To clear your doubts, please write a comment in the below comment section. Stay tuned to our site bigideasmathanswers.com to get the answer key of other grades and grade 2 chapters.

## Big Ideas Math Answers Grade 2 Chapter 7 Understand Place Value to 1,000

Big Ideas Math Grade 2 Chapter 7 Understand Place Value to 1,000 Answer Key is provided in a comprehensive manner for better understanding. Students must solve as many questions as possible in BIM 2nd Grade Chapter 7 Understand Place Value to 1,000 Book and cross-check method and solutions here. If you wanted to become a pro in the place value concept, then refer to Big Ideas Math Answers Grade 2 Chapter 7 Understand Place Value to 1,000.

## Big Ideas Math Book 2nd Grade Answer Key Chapter 7 Understand Place Value to 1,000

The list of concepts covered in Big Ideas Math Book 2nd Grade 7th Chapter Understand Place Value to 1,000 is given in the below section. It is one of the important concepts that must be learned at the elementary. So, no student is supposed to skip this chapter. Download BIM Grade 2 Chapter 7 Understand Place Value to 1,000 PDF for free of cost and start preparation.

Parents and teachers can have a look at this free Big Ideas Math 2nd Grade Chapter 7 Understand Place Value to 1,000 Answers for a better understanding. After solving the questions available here, you will be able to solve any kind of question framed on the concept and clear your tests with a good score. If you have a doubt just tap on the respective link you wanted to prepare and learn the fundamental topics included within it easily.

Vocabulary

Lesson: 1 Hundreds

Lesson: 2 Model Numbers to 1,000

Lesson: 3 Understand Place Value

Lesson: 4 Write Three-Digit Numbers

Lesson: 5 Represent Numbers in Different Ways

Understand Place Value to 1,000

### Understand Place Value to 1,000 Vocabulary

Organize It
Use the review words to complete the graphic organizer.

Explanation:
Tens Place: The first digit on the right of the decimal point means tenths.
Ones Place The ones place is just to the left of the decimal point.

Define It
Use your vocabulary cards to identify the word.

### Lesson 7.1 Hundreds

Explore and Grow
How many unit cubes and rods are in a flat?

Answer: 100 cubes and 10 rods
Explanation
Given that, For counting cubes, should count all the rows and columns. By counting all the boxes there are
10 rows and 10 columns= 100 cubes To find out the rods count only the columns, by counting columns there are 10 rods.
Show and Grow
Write how many tens. Circle groups of 10 tens. Write how many hundreds. Then write the number.
Question 1.

______ tens
______ hundred
______
Answer: 20 tens & 2 hundred (200)
Explanation
Given that 20 tens are there by circling 10 tens
For counting tens, count columns and for counting hundreds count all the boxes.
Question 2.

______ tens
______ hundred
______
Answer: 40 tens & 4 hundred (400)
Explanation
Given that 40 tens are there by circling 10 tens
For counting tens, count columns and for counting hundreds count all the boxes.

Apply and Grow: Practice

Write how many tens. Circle groups of 10 tens. Write how many hundreds. Then write the number.
Question 3.

______ tens
______ hundred
______
Answer: 30 tens &  3 hundreds (300)
Explanation
Given that 30 tens are there by circling 10 tens
for counting tens, count columns and for counting hundreds count all the boxes.

Question 4.

______ tens
______ hundred
______
Answer: 60 tens & 6 hundreds
Explanation
Given that 60 tens are there by circling 10 tens.
for counting tens, count columns and for counting hundreds count all the boxes.

Question 5.
DIG DEEPER!
How many hundreds are in 700? How many tens?
_____ hundreds
______ tens
Answer: 7 hundreds are in 700 and 70 tens are in 700.
Explanation
there are 7 hundreds in 700 and 70 tens in 700. 10 tens gives  1 hundreds totally there are 70 tens is 700.

Question 6.
You have 80 bags of crayons. There are 10 crayons in each bag. How many crayons do you have in all?
_____ crayons
Explanation
Given that, There are 80 bags of crayons each bags contains 10 crayons.10 x 80=800 ( 10 crayons in each bags x 80 totally bags = 800 )
Thus, There are 800 crayons do we have in all.

Think and Grow: Modeling Real Life

A store sells oranges in bags of 10. The store sells 500 oranges. How many bags do they sell?

Make a quick sketch:
_____ bags
Explanation
Given that, The store sells 500 oranges each bags contains 10 oranges.10 X 50 = 500  (10 oranges in each bags x 50 bags = 500 oranges)
Thus, The store sell the 50 bags of oranges.

Question 7.
A store sells bottles of glitter glue in boxes of 10. The store sells 600 bottles. How many boxes do they sell?

_____ boxes
Explanation
Given that, The store sells 600 bottles of glitter glue each boxes contains 10 glitter glue bottles.
10 X 60 = 600  (10 bottle of glitter glue in each boxes x 60 boxes = 600 bottles of glitter glue)
Thus, The store sell the 60 boxes of glitter glue.

Question 8.
DIG DEEPER!
You have 10 packages of invitations. Each package has 10 invitations. You need 300 invitations. How many more packages do you need?

_____ more packages
Explanation
Given that, There are 10 packages of invitations each packages has 10 invitations you needed 300 invitations
10 x 30 = 300 (10 invitations of  packages x 30 packages = 300 invitations)
Thus, 30 packages of invitations are needed.

### Hundreds Homework & Practice 7.1

Write how many tens. Circle groups of 10 tens. Write how many hundreds. Then write the number.
Question 1.

_____ tens
______ hundreds
______
Answer: 50 tens & 5 hundreds.
Explanation
Given that 50 tens are there by circling 10 tens
For counting tens, count columns and for counting hundreds count all the boxes.

Question 2.

_____ tens
______ hundreds
______
Answer: 80 tens & 8 hundreds
Explanation
Given that 80 tens are there by circling 10 tens
For counting tens, count columns and for counting hundreds count all the boxes.

Question 3.
Which One Doesn’t Belong?
Which does not belong with the other three?

Question 4.
Modeling Real Life
A class has 30 boxes of pencils. Each box has 10 pencils. The class needs 600 pencils. How many more boxes does the class need?

______ boxes
Explanation
Given that, the class has 30 boxes of pencils each boxes contains 10 pencils. The class needs 600 pencils.
10 x 60 (30 +30)=600 (10 Pencils in each boxes X 60 boxes (30 boxes which you have + 30 boxes of pencils you need)= 600 pencils)
Thus, the class needs another 30 boxes.

Question 5.
Modeling Real Life
You have 10 packages of cards. Each package has 10 cards. You need 200 cards. How many more packages do you need?

______ more packages
Explanation
Given that, 10 packages of cards each packages contains 10 cards, you need 200 cards
10 x 20= 200 (10 cards in each packages X 20 packages of cards (10 packages which you have+10 packages you needed) =200 cards)
Thus, you need another 10 packages of cards.

Review & Refresh

Question 6.

Question 7.

### Lesson 7.2 Model Numbers to 1,000

Explore and Grow

Model the number. Make a quick sketch to match.

Show and Grow

Question 1.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 7 hundreds, 4 tens and 5 ones is 745
Explanation
There are 7 unit of cubes each unit of cubes contains 10 tens which gives 700(10 tens in one unit of cubes x 70 tens in the whole unit of cubes = 700)
and there are 4 rod each contains 10 cubes that gives 4 tens is 40 and plus the ones 5 is 745.

Apply and Grow: Practice

Question 2.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 4 hundreds, 6 tens and 3 ones is 463.
Explanation
There are 4 unit of cubes each unit of cubes contains 10 tens which gives 400(10 tens in one unit of cubes x 40 tens in the whole unit of cubes = 400)
and there are 6 rod each contains 10 cubes that gives 6 tens is 60 and plus the ones 3 is 463.

Question 3.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 6 hundreds, 0 tens and 9 ones is 609.
Explanation
There are 6 unit of cubes each unit of cubes contains 10 tens which gives 600(10 tens in one unit of cubes x 70 tens in the whole unit of cubes = 700)
The place value of 0 (0 x 10 tens = 0) and plus the ones 9 is 609.

Question 4.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 3 hundreds, 4 tens and 6 ones is 346.
Explanation
There are 3 unit of cubes each unit of cubes contains 10 tens which gives 300(10 tens in one unit of cubes x 30 tens in the whole unit of cubes = 300)
and there are 4 rod each contains 10 cubes that gives 4 tens is 40 and plus the ones 6 is 346.

Question 5.
DIG DEEPER!
What number is Newton thinking about?

Think and Grow: Modeling Real Life

Write the missing numbers:
_______ hundreds, _______ tens, and ______ ones
_______ markers
Answer: 3 hundreds, 2 tens and 4 ones is 324 markers.
Explanation
The above images shows that 3 packs of 100 markers(100+100+100=300) 2 packs of 10 markers(10+10=20) with additional of 4 markers, that gives

Show and Grow

Question 6.

______ balloons
Explanation
The above images shows that 5 packs of 100 balloons(100+100+100+100+100= 500) 7 packs of 10 balloons(10+10+10+10+10+10+10=70),
that gives ( 500+70=570)Totally 570 balloons you buy.

### Model Numbers to 1,000 Homework & Practice 7.2

Question 1.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 4 hundreds,7 tens 2 ones is 472
Explanation
There are 4 unit of cubes each unit of cubes contains 10 tens which gives 400(10 tens x 40 tens of whole cubes =400)
There are 7 rod gives you a 7 tens is 70 and plus 2 ones is 472.

Question 2.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 7 hundreds 4 tens 0 ones is 740.
Explanation
There are 7 unit of cubes each unit of cubes contains 10 tens which gives 700(10 tens in one unit of cubes x 70 tens in the whole unit of cubes = 700)
and there are 4 rod each contains 10 cubes that gives 4 tens is 40 and plus the ones place above picture there were no ones 0 is 740

Question 3.

______ hundreds, _______ tens, and ______ ones is ______.
Answer: 2 hundreds 8 tens and 6 ones is 286.
Explanation
There are 2 unit of cubes each unit of cubes contains 10 tens which gives you 200 (10 tens in one unit of cubes x 20 tens in the whole unit of cubes = 200) and there are 8 rod each contains 10 cubes that gives 8 tens is 80 and plus the ones 6 is 286.

Question 4.
YOU BE THE TEACHER
Your friend writes 8 hundreds and 3 ones as 83. Is your friend correct? Explain.
_________________________
_________________________
Answer:  No, 8 is tens and 3 is ones as 83.
Explanation
The first digit is the tens place it tells you that there are 8 tens in the number 83. The next digit is the ones place which is 3.
Therefore there are 8 sets of 10 tens and plus 3 ones in the number 83.

Question 5.
Modeling Real Life

______ stickers
Explanation
Given that, There are 6 packs of stickers each contains 100 stickers that is (100+100+100+100+100+100=600
and 2 packs of stickers each contains 10 stickers that is (10+10= 20) Therefore 600 +20= 620

Review & Refresh

Question 6.

Question 7.

Question 8.

### Lesson 7.3 Understand Place Value

Explore and Grow

Make quick sketches to model each number.

What do you notice about the numbers?
_________________________
_________________________

Show and Grow

Circle the value of the underlined digit.
Question 1.
483
300
30
3
Explanation
Given that there are 3 digit number which is 483. A number can have many digits and each
digit has a special place and value. the 8 is in ones place and its place value is 8.

Question 2.
791
9 hundreds
9 tens
9 ones
Explanation
Given that there are 3 digit number which is 791. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 9 is in tens place value is 90.
Question 3.
612
6
60
600
Explanation
Given that there are 3 digit number which is 612. A number can have many digits and each
digit has a special place and value. the first digit at hundreds place. The 6 is in hundreds place value is 600.

Question 4.
578
7 hundreds
7 tens
7 ones
Explanation
Given that there are 3 digit number which is 578. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 7 is in tens place value is 70.

Apply and Grow: Practice

Circle the value of the underlined digit.
Question 5.
354
300
30
3
Explanation
Given that there are 3 digit number which is 354. A number can have many digits and each
digit has a special place and value. the first digit at hundreds place. The 3 is in hundreds place value is 300.

Question 6.
726
2 hundreds
2 tens
2 ones
Explanation
Given that there are 3 digit number which is 726. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 2 is in tens place value is 20.

Question 7.
594
4
40
400
Explanation
Given that there are 3 digit number which is 594. A number can have many digits and each
digit has a special place and value. the 4 is in ones place and its place value is 4.

Question 8.
475
5 hundreds
50
5 ones
Explanation
Given that there are 3 digit number which is 475. A number can have many digits and each
digit has a special place and value. the 5 is in ones place and its place value is 5.

Circle the values of the underlined digit.
Question 9.
639
3 tens
300
30
Explanation
Given that there are 3 digit number which is 639. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 3 is in tens place value is 30.

Question 10.
872
8 hundreds
80
800
Explanation
Given that there are 3 digit number which is 872. A number can have many digits and each
digit has a special place and value. the first digit at hundreds place. The 8 is in hundreds place value is 800.

Question 11.
Number Sense
Write the number that has the following values.
The tens digit has a value of 40.
The ones digit has a value of 2.
The hundreds digit has a value of 600. ______
Explanation
The combination of hundreds, tens, ones of 600, 40, 2 respectively.

Think and Grow: Modeling Real Life

How many points is one ball worth in each bucket?

Write the score:
______ hundreds, ______ tens, and _______ ones
Blue bucket: ______ points
Yellow bucket: _____ point
Red bucket: ______ points
Answer: Blue bucket 3 points,yellow bucket 4 points and red bucket 5 points

Show and Grow

Question 12.
How many points is one ring worth on each peg?

Green peg: ______ points
Blue peg: _______ point
Purple peg: ______ points
Green peg: 1 points
Blue peg: 3 point
Purple peg: 2 points

### Understand Place Value Homework & Practice 7.3

Circle the value of the underlined digit.
Question 1.
523
500
50
5
Explanation
Given that there are 3 digit number which is 523. A number can have many digits and each
digit has a special place and value. the first digit at hundreds place. The 5 is in hundreds place value is 500.

Question 2.
738
8 hundreds
8 ones
8 tens
Explanation
Given that there are 3 digit number which is 738. A number can have many digits and each
digit has a special place and value. the 8 is in ones place and its place value is 8.

Question 3.
364
60
6 ones
6 hundreds
Explanation
Given that there are 3 digit number which is 364. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 6 is in tens place value is 60.

Circle the values of the underlined digit.
Question 4.
434
4
4 ones
4 hundreds
Explanation
Given that there are 3 digit number which is 434. A number can have many digits and each
digit has a special place and value. the 4 is in ones place and its place value is 4.

Question 5.
920
2
2 tens
20
Explanation
Given that there are 3 digit number which is 920. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 2 is in tens place value is 20.

Question 6.
DIG DEEPER!
Write the number that matches the clues.
The value of the hundreds digit is 800.
The value of the tens digit is 10 less than 70.
The value of the ones digit is an even number greater than 7.
_________
i. 8
80
8

Question 7.
Structure
Write each number in the correct circle.

Question 8.
Modeling Real Life
How many points is one ball worth in each hoop?

Blue hoop: ______ points
Orange hoop: _______ points
Red hoop: _______ point
Blue hoop: 4 points
Orange hoop: 2 points
Red hoop: 5 points

Review & Refresh

Question 9.

Question 10.

Question 11.

### Lesson 7.4 Write Three-Digit Numbers

Explore and Grow

Identify the value of the base ten blocks.

What is the total value of the base ten blocks? ________

How can you write the value of the base ten blocks as an equation?

Standard Form is 246
Expanded Form is 200 + 40 + 6 = 246
Word Form is Two Hundred and Forty Six

Show and Grow

Write the number in standard form, expanded form, and word form.
Question 1.

Expanded form 500+20+8,
Word form Five Hundred Twenty Eight.

Question 2.

Expanded form 700+9,
Word form Seven Hundred Nine.

Apply and Grow: Practice

Write the number in expanded form and word form.
Question 3.
837
_____ + _____ + _____
______________
Answer: 800+30+7 and Eight hundred thirty seven
Explanation
In expanded form, write the number by showing the value of each digit (800+30+7)
In Word form write the number by using words but no numbers (Eight hundred thirty seven)
Question 4.
954
_____ + _____ + ______
_____________
Answer:900+50+4 and Nine hundred fifty four
Explanation
In expanded form, write the number by showing the value of each digit (900+50+4)
In Word form write the number by using words but no numbers ( Nine hundred fifty four)
Write the number in standard form and word form.
Question 5.
500 + 60
______
__________
Answer: 560 and Five hundred sixty
Explanation
In standard form, numbers are written using only numbers. There are no words (560)
In Word form write the number by using words but no numbers ( Nine hundred fifty four)
Question 6.
700 + 20 + 1
_____
____________
Answer:721 and Seven hundred twenty one
Explanation
In standard form, numbers are written using only numbers. There are no words (721)
In Word form write the number by using words but no numbers ( Seven hundred twenty one)

Write the number in expanded form and standard form.
Question 7.
six hundred seventy-four
_____ + ______ + ______
__________
Explanation
In expanded form, write the number by showing the value of each digit (600+70+4)
In standard form, numbers are written using only numbers. There are no words (674)
Question 8.
four hundred seven
_____ + ______ + ______
__________
Explanation
In expanded form, write the number by showing the value of each digit (400+7)
In standard form, numbers are written using only numbers. There are no words (407)
Question 9.
Structure
Which number did Newton model?

Think and Grow: Modeling Real Life

There are 819 pets in a pet store. 800 are fish. 9 are cats. The rest are birds. How many birds are there?

Expanded form:
_____ + _____ + ______
_____ birds
Answer: 10 birds are there and 800+10+9
Explanation
Given that, There are 819 pets in pets store
800 are fish, 9 are cats and rest are birds.
Thus,819 the right the first digit will be at ones place 9 is in ones place and its place value is 9 ,the second digit at tens place 1 is in tens place its place value is 10 and the third digit at hundreds place.8 is in hundreds place its place value is 800.

Show and Grow

Question 10.
There are 21
7 flowers at a flower stand. 200 are roses. 7 are sunflowers. The rest are tulips. How many tulips are there?

______ tulips
Answer: 10 are tulips in flower stand.
Explanation
Given that, There are 217 flowers in flowers stand
200 are roses, 7 are sunflowers and rest are tulips (200 – 7 = 10)
Thus,10 are the tulips.
Question 11.
There are 185 books at a book fair. 80 are chapter books. 5 are comic books. The rest are picture books. How many picture books are there?

______ picture books
Answer: 100 books are picture books.
Explanation
Given that, 185 books are at book fair.
80 books are chapter books and 5 are comic books and rest books are picture books
(80 – 5 = 100)
Thus, 100 books are picture books in book fair.
Question 12.
You sell 326 candles. You sell 6 small candles. You sell 20 medium candles. The rest are large candles. How many large candles did you sell?

______ large candles
Explanation:
Given that, 326 candles are sell by you,
6 are small candles and 20 are medium candles and rest are large candles
326-6(small candles)=320
320-20(medium candles)=300
Thus,300 large candles are you sell.
Three-Digit Numbers Homework & Practice 7.4

Write the number in expanded form and word form.
Question 1.
137
______ + _____ + _____
____________
Answer: 100+30+7 and One hundred thirty seven.
Explanation:
In expanded form, write the number by showing the value of each digit (100+30+7)
In Word form write the number by using words but no numbers ( One hundred twenty four)
Question 2.
280
_____ + _____ + ______
___________
Answer: 200+80 and Two hundred eighty
Explanation
In expanded form, write the number by showing the value of each digit (200+80)
In Word form write the number by using words but no numbers ( Two hundred eighty )
Write the number in standard form and word form.
Question 3.
600 + 10 + 5 ______
___________
Answer:  615 and Six hundred fifteen
Explanation
In standard form, numbers are written using only numbers. There are no words (615)
In Word form write the number by using words but no numbers ( Six hundred fifteen )
Question 4.
900 + 70 + 6 _____
___________
Answer: 976 and Nine hundred seventy six
In standard form, numbers are written using only numbers. There are no words (976)
In Word form write the number by using words but no numbers ( Nine hundred seventy six )
Write the number in expanded form and standard form.
Question 5.
three hundred nine
_____ + ______ + _____
______
Explanation
In expanded form, write the number by showing the value of each digit (300+9)
In standard form, numbers are written using only numbers. There are no words (309)
Question 6.
eight hundred sixty-two
_____ + ______ + ______
_______
Explanation
In expanded form, write the number by showing the value of each digit (800+60+2)
In standard form, numbers are written using only numbers. There are no words (862))
Question 7.
Which One Doesn’t Belong?
Which does not belong with the other three?

Answer: 1 + 9 + 6 does not belong to the other three.

Question 8.
DIG DEEPER!
A number has 7 hundreds. The tens digit is 5 less than the hundreds digit. The ones digit is 2 more than the hundreds digit. What is the number?
______
Explanation
given that, a number has 7 hundred
The tens digit is 5 less than hundreds digit

Question 9.
Modeling Real Life
There are 438 vegetables planted. 400 are carrots. 8 are beets. The rest are onions. How many onions are there?

______ onions
Explanation
As per the statement, There are 438 vegetables are planted
400 are carrot (438-400=38)
8 are beets (38 – 8=30)
Thus, the Remaining 30 are onions.

Question 10.
Modeling Real Life
There are 593 students in after-school programs. 3 students take dance class. 90 students take art class. The rest take karate class. How many students take karate class?

______ students
Answer: 500 students are in karate class.
Explanation:
As per statements, Totally there are 593 students
3 students are in dance class (593-3=590)
90 students are in art class (590-90=500)
Thus, the Remaining 500 students are in karate class.

Review & Refresh

Question 11.
Find the sum. Then change the order of the addends. Write the new equation.
4 + 7 = ______
_____ + _____ = ______

### Lesson 7.5 Represent Numbers in Different Ways

Explore and Grow

Circle the models that show each number.

Show and Grow

Question 1.
Show 261 two ways.

Question 2.
Show 345 two ways.

Apply and Grow: Practice

Question 3.
Show 432 two ways.

Question 4.
Show 527 two ways.

Question 5.
YOU BE THE TEACHER
Your friend says that 800 + 40 + 11 is the same as 800 + 50 +1. Is your friend correct? Explain.
_________________________
__________________________

Think and Grow: Modeling Real Life

The models show how many dinosaur toys you and your friend have. Does your friend have the same number of dinosaur toys as you? Explain.

_________________________
__________________________

Show and Grow

Question 6.
The models show how many trading cards you and your friend have. Does your friend have the same number of trading cards as you? Explain.

______________________
______________________

### Represent Numbers in Different Ways Homework & Practice 7.5

Question 1.
Show 134 two ways.

Question 2.
Show 319 two ways.

Question 3.
Number Sense
Which ways show 948?
8 hundreds, 4 tens, and 8 ones
9 hundreds, 4 tens, and 8 ones
8 hundreds, 4 tens, and 9 ones
8 hundreds, 14 tens, and 9 ones
94 tens and 8 ones
948 ones

Question 4.
Modeling Real Life
The models show how many bouncy balls you and your friend have. Does your friend have the same number of bouncy balls as you? Explain.

________________________
__________________________

Review & Refresh

Question 5.
70 + 30 = ______

Question 6.
53 + 19 = _____

Question 7.
90 − 50 = _____

Question 8.
64 − 40 = _______

### Understand Place Value to 1,000 Performance Task

You make trail mix using the sunflower seeds, almonds, and raisins shown.

Question 1.
How many sunflower seeds, almonds, and raisins do you use?
______ + _____ + _____
_____
Answer: 400+ 30+ 5 is equal to 435.
Explanation
Given that, 400 sunflower seeds, 30 almonds and 5 raisins  for trail mix
thus, 400+30+5= 435

Question 2.
You add another bag of almonds and another bag of raisins to your trail mix. How many sunflower seeds, almonds, and raisins are there now?
_____ + _____ + _____
______
Explanation
Given that, 400 sunflower seeds, 30 of almonds add another bags of almonds is contains 10 totally 40 and 5 of raisins add another bags of raisins contains 5 totally 10 raisins.
400+40+10=450

Question 3.
Your friend makes trail mix with 3 bags of sunflower seeds,13 bags of almonds, and 4 bags of raisins. Does your friend use the same number of ingredients as you? Explain.
Yes No
__________________
___________________
Answer: No, friend makes a trail mix with only 3 bags of sunflower 13 bags of almonds and 4 bag of raisins. The ingredients you used is 400 sunflower seeds,40 almonds bags and 10 raisins bags. Friend and your’s are not the same ingredients.

### Understand Place Value to 1,000 Activity

Naming Numbers Flip and Find
To Play: Place the Naming Numbers Flip and Find Cards face down in the boxes. Take turns flipping 2 cards. If your cards show the same number, keep the cards. If your cards show different numbers, ﬂip the cards back over. Play until all matches are made.

7.1 Hundreds

Write how many tens. Circle groups of 10 tens. Write how many hundreds. Then write the number.
Question 1.

_____ tens
_______ hundreds
________
Answer: 40 tens and 4 hundreds
Explanation
Given that 40 tens are there by circling 10 tens
For counting tens, count columns and for counting hundreds count all the boxes.

Question 2.

_____ tens
_______ hundreds
________
Answer:  50 tens and 5 hundred
Explanation
Given that 50 tens are there by circling 10 tens
For counting tens, count columns and for counting hundreds count all the boxes.

Question 3.

Modeling Real Life
A cafeteria has 80 bags of pretzels. Each bag has 10 pretzels. The cafeteria needs 900 pretzels. How many more bags does the cafeteria need?
______ more bags

Explanation:
Given that, the cafeteria has 80 bags of pretzels  each bags contains 10 pretzels, cafeteria needs 900 pretzels,
10 bags x 10 pretzels =100
Thus 10 bags of pretzels needs for the cafeteria.

7.2 Model Numbers to 1,000

Question 4.

______ hundreds, _____ tens, and _____ ones is ______.
Answer:5 hundreds 2 tens, 5 ones is 525.
Explanation
There are 5 unit of cubes each unit of cubes contains 10 tens which gives 500(10 tens in one unit of cubes x 50 tens in the whole unit of cubes = 500)
and there are 2 rod each contains 10 cubes that gives 2 tens is 20 and plus the ones 5 is 525.

Question 5.

______ hundreds, _____ tens, and _____ ones is ______.
Answer: 8 hundred,0 tens, and 3 ones is 803.
Explanation
There are 8 unit of cubes each unit of cubes contains 10 tens which gives 800(10 tens in one unit of cubes x 80 tens in the whole unit of cubes = 800)
the second digit at tens place 0 in the tens place and the right first digit is ones plus the ones 3 is 803.

Question 6.
Number Sense
What number is Descartes thinking about?

______
Explanation
The above answer represents that 4 hundreds is 400 and the second digit at tens place. The 2 is in tens place value is
20 and the right first digit is in ones place the 5 is in ones place value 5 is 425.

7.3 Understand Place Value

Circle the value of the underlined digit
Question 7.
429
200
2
2 tens
Explanation
Given that there is 3 digit number which is 429. A number can have many digits and each
digit has a special place and value. the second digit at tens place. The 2 is in tens place value is 20.
Question 8.
751
70
700
7 tens
Explanation
Given that there is 3 digit number which is 751. A number can have many digits and each
digit has a special place and value. the first digit at hundreds place. The 7 is in hundreds place value is 700

Question 9.
Number Sense
Write the number that has the following values.
The tens digit has a value of 60.
The ones digit has a value of 3.
The hundreds digit has a value of 900. _______
Explanation:
A number can have many digits and each digit has a special place and value. the first digit at hundreds place.
The 9 is in hundreds place value is 900. The second digit at tens place the 6 is in tens place value is 60 and
the right first digit is in ones place the 3 is in ones place value 3 is 963.

7.4 Write Three-Digit Numbers

Write the number in expanded form and word form.
Question 10.
605
_____ + _____ + ______
________
Answer: 600+5 and Six hundred five
Explanation
In expanded form, write the number by showing the value of each digit (600+05)
In Word form write the number by using words but no numbers ( Six hundred five)

Question 11.
541
______ + ______ + ______

___________
Answer: 500+40+1 and Five hundred forty-one
Explanation
In expanded form, write the number by showing the value of each digit (500+40+1)
In Word form write the number by using words but no numbers (Five hundred forty-one)

Write the number in standard form and word form.
Question 12.
100 + 20 + 4
______
___________
Answer: 124 and One hundred twenty-four
Explanation
In standard form, numbers are written using only numbers. There are no words (124)
In Word form write the number by using words but no numbers (One hundred twenty-four)

Question 13.
700 + 8
_______
_____________
Answer: 708 and Seven hundred eight
Explanation
In standard form, numbers are written using only numbers. There are no words (708)
In Word form write the number by using words but no numbers (Seven hundred eight)

Write the number in expanded form and standard form.
Question 14.
three hundred thirty
_____ + _____ + _____
_____
Explanation
In expanded form, write the number by showing the value of each digit (300+30)
In standard form, numbers are written using only numbers. There are no words. (330)
Question 15.
two hundred fifty-six
_____ + ______ + _____
______
Explanation
In expanded form, write the number by showing the value of each digit (200+50+6)
In standard form, numbers are written using only numbers. There are no words. (256)
7.5 Represent Numbers in Different Ways

Question 16.
Show 345 two ways.

Answer: Three hundred forty-five and 300+40+5
Explanation
There are 3 digit number(345)  1 digit is for hundreds which is 3, 2nd digit is for tens which is 2 and the last digit for once 5 is 345
In word form for 345 is Three hundreds forty-five,
The expanded form for 345 is 300+40+5 which gives 345.

Question 17.
Show 562 two ways.

Answer: Five hundred sixty-two and 500+60+2
Explanation
There are 3 digit numbers 1 digit number is hundreds which are 5, 2nd digit number is tens which is 6 tens and the last number 2 ones is 562
In word form for 562 is Five hundred sixty-two,
The expanded form for 562 is 500+60+2 is 562.

Question 18.
Your friend says that 600 + 30 + 1 is the same as 500 + 130 + 1. Is your friend correct? Explain.
___________________
___________________
Answer: Yes correct, given that 600+30+1= 631 ( which gives) and 500+130+1 = 631( which also gives the same answer)
both equation shows the same answers.

Conclusion:

In this chapter, a brief explanation of the lessons are discussed in Answer Key of Big Ideas Math book Grade 2 Chapter 7 Understand Place Value to 1,000. Here we have provided the exercise problems and the solutions to help in practising the lessons. you can get the different and simple methods of solving problems in Big Ideas Math Grade 2 Answer Key Chapter 7 Understand Place Value to 1,000. Stay in touch with our site to get the latest edition solutions of all other Big Ideas Math Grade 2 Answers.

## Big Ideas Chapter 7 Divide Decimals 5th Grade Math Book Answer Key

Find the best tips that make your students love to practice maths and encourage them to use them for easy practicing. We have also provided different tricks to solve all the math problems. Big Ideas math 5th grade Chapter 7 Divide Decimals textbook Answer Key is the best source for the students. All the relevant links of Divide Decimals are given below. Check the links and begin your practice now.

Lesson: 1 Division Pattern with Decimals

Lesson: 2 Estimate Decimals Quotients

Lesson: 3 Use Models to Divide Decimals by Whole Numbers

Lesson: 4 Divide Decimals by One-Digit Numbers

Lesson: 5 Divide Decimals by Two-Digit Numbers

Lesson: 6 Use Models to Divide Decimals

Lesson: 7 Divide Decimals

Lesson: 8 Insert Zeros in the Dividend

Lesson: 9 Problem Solving: Decimal Operations

Chapter: 7 – Divide Decimals

### Lesson 7.1 Division Pattern with Decimals

Explore and Grow

Use the relationship between positions in a place value chart to find each quotient.

What patterns do you notice?

Structure

Describe the placement of the decimal point when dividing a decimal by 10, 100, 0.1, and 0.01.

Think and Grow: Division Pattern with Decimals

Example
Find 74 ÷ 103.
Use place value concepts. Every time you multiply a number by $$\frac{1}{10}$$ or divide a number by 10, each digit in the number shifts one position to the right in a place value chart.

Notice the pattern: In each quotient, the number of places the decimal point moves to the left is the same as the exponent.

Example
Find 5.8 ÷ 0.01.
Use place value concepts. Every time you multiply a number by 10 or divide a number by 0.1, each digit in the number shifts one position to the left in a place value chart.

Notice the pattern: When you divide by 0.1, the decimal point moves one place to the right. When you divide by 0.01, the decimal point moves two places to the right.

Show and Grow

Find the quotient.
Question 1.
62.5 ÷ 102 = ______
Explanation:  First Simplify the 102 which means  10X10 =100 then we need to calculate the fraction to a decimal just divide the numerator(62.5) by the denominator (100): 62.5 ÷ 100 =0.625 so,  62.5/100 =0.625
Question 2.
1.84 ÷ 0.1 = ______
Explanation: To convert this simple fraction to a decimal just divide the numerator (1.84) by the denominator (0.1): 1.84 ÷ 0.1 = 18.4 so, 1.84/0.1 = 18.4

Apply and Grow: Practice

Find the quotient.
Question 3.
76 ÷ 10 = ______
Explanation: To convert this simple fraction to a decimal just divide the numerator (76) by the denominator (10): 76 ÷ 10 = 7.6 so, 76/10 = 7.6
Question 4.
3.65 ÷ 0.1 = _______
Explanation: To convert this simple fraction to a decimal just divide the numerator (3.65) by the denominator (0.1): 3.65 ÷ 0.1 = 36.5. so, 3.65/0.1 = 36.5
Question 5.
2.9 ÷ 0.01 = ______
Explanation: To convert this simple fraction to a decimal just divide the numerator (2.9) by the denominator (0.01): 2.9 ÷ 0.01 = 290. so, 2.9/0.01 = 290
Question 6.
18.7 ÷ 102 = ______
Explanation: First Simplify the 102 which means  10X10 =100 then we need to calculate the fraction to a decimal just divide the numerator(18.7) by the denominator (100): 18.7 ÷ 100 =0.187 so,  18.7/100 =0.187

Find the value of k.
Question 7.
95.8 ÷ k = 958
Explanation: Lets solve your equation step by step 95.8/k = 958
Multiply both side by side K.
95.8 = 958K
958k = 95.8 (Flip the equation)
958k/958 = 95.8/958(Divide both sides by 958)
K=0.1
Question 8.
k ÷ 103 = 0.35
Explanation: K÷103 =0.35
Step 1: calculate the value of the power which means 103 = 10x10x10=1000
k/1000=0.35
step 2: multiply both side by 1000
1000X K/1000 = 1000X0.35
Step 3: simplify
1000 X K/1000 = 1000X0.35
K = 350
Question 9.
245 ÷ k = 24,500
Explanation: variable K cannot be equal to 0 since division by zero is not defined. Multiply both side of equation by K
245 = 24500K
swap sides so that all variables terms are on the left hand side
24500K = 245
Divide both sides by 24500.
K =245/24500
Reduces the fraction 245/24500 to lowest terms by extracting and cancelling out 245
K = 1/100 ,Therefore K = 0.01
Question 10.
Newton goes on a 10-day road trip. He takes $435 with him. He spends all of his money and spends the same amount each day. How much money does he spend each day? Answer:$43.5/per day
Explanation: Newton takes $435 for 10 days road trip. 435/10 = 43.5 Newton Spend the money per day is =$43.5/day
Question 11.
Number Sense
For which equations does b = 100?
49 ÷ b = 0.49
247 ÷ b = 0.247
1.3 ÷ b = 0.013
0.5 ÷ b = 0.05

49 ÷ b = 0.49

1.3 ÷ b = 0.013

For these two equations b value should be 100.

Question 12.
YOU BE THE TEACHER
Your friend says 8,705 ÷ 103 is equivalent 8,705 × 0.001. Is your friend correct? Explain.

First simplify the 103 which means  10 $\dpi{100} \bg_white \small \times$ 10 $\dpi{100} \bg_white \small \times$ 10 = 1000

8,705 ÷ 103

= 8,705 ÷ 1000
= 8,705 $\dpi{100} \bg_white \small \times$ $\dpi{100} \bg_white \small \frac{1}{1000}$
= 8,705 $\dpi{100} \bg_white \small \times$ 0.001
8,705 ÷ 103 is equivalent 8,705 × 0.001
So, my friend answer is correct.

Think and Grow: Modeling Real Life

Example
A contractor buys 2 adjacent lots of land. One lot is 0.55 acre and the other is 1.65 acres. The contractor divides the land equally for 10 new homes. How much land does each home have?

To find how much land each home has, divide the sum of the lot sizes by 10.
Add the sizes of the lots.

Divide the total number of acres by 10. Dividing 2.20 by 10, or 101, shifts the digits ______ position to the right in a place value chart. So, the decimal point moves ______ place to the left.
2.20 ÷ 10 = 2.20 ÷ 101 = ______
Each home has ________ acre.

Show and Grow

Question 13.
An art teacher has 68.5 pounds of clay and orders 56.5 more pounds. The teacher equally divides the clay among 100 students. How much clay does each student get?
To find how much clay each student get, divide the sum of the clay by 100.
Add the quantities of the clay.
68.5 + 56.5 = 125
Divide the total clay by 100. Dividing 125 by 100, or 102
125 ÷ 100 = 125 ÷ 102 = 1.25
Each student gets 1.25 pounds clay.

Question 14.
A museum has a replica of the Space Needle that is 6.05 feet tall. It is one-hundredth of the height of the actual Space Needle. How tall is the actual Space Needle?

Replica of the Space Needle height = 6.05 feet
Let actual Space Needle height = h
$\dpi{100} \small \frac{1}{100}$ (h) = 6.05
h = 6.05 $\dpi{100} \small \times$ 100 = 605
So actual Space Needle height is 605 feet.

Question 15.
DIG DEEPER!
A pile of 102 loonies weighs 627 grams and a pile of 102 toonies weighs 730 grams. How much more does a toonie weigh than a loonie? Is there more than one way to solve the problem? Explain.

A pile of 102 loonies weight = 627 grams
A pile of 102 toonies weight = 730 grams
730 – 627 = 103
Toonie weighs 103 grams more than a loonie.
Method – 2
1 loonie weight = $\dpi{100} \small \frac{627}{10^{2}}$ = 6.27
1 toonie weight = $\dpi{100} \small \frac{730}{10^{2}}$ = 7.30
7.30 – 6.27 = 1.03
For 102 toonies and loonies = 1.03 x 102 = 103
Toonie weighs 103 grams more than a loonie.

### Division Pattern with Decimals Homework & Practice 7.1

Find the quotient.
Question 1.
810 ÷ 10 = ______

Explanation:
To convert this simple fraction to a decimal just divide the numerator (810) by the denominator (10):
When we divide by 10, the decimal point moves one place to the left.
810 ÷ 10 = 81.

Question 2.
7.4 ÷ 0.01 = ______

Explanation: To convert this simple fraction to a decimal just divide the numerator (7.4) by the denominator (0.01). When we divide by 0.01, the decimal point moves two places to the right. : 7.4 ÷ 0.01 = 740.

Question 3.
903 ÷ 103 = ______

First Simplify the 103 which means  10 x 10 x 10 =1000, then we need to calculate the fraction to a decimal just divide the numerator (903) by the denominator (1000).
When we divide by 1000, the decimal point moves three places to the left.

Question 4.
267.1 ÷ 0.01 = ______

Explanation: To convert this simple fraction to a decimal just divide the numerator (267.1) by the denominator (0.01).
When we divide by 0.01, the decimal point moves two places to the right :
267.1 ÷ 0.01 = 26710

Question 5.
5.6 ÷ 0.1 = ______

Explanation: To convert this simple fraction to a decimal just divide the numerator (5.6) by the denominator (0.1).
When we divide by 0.1, the decimal point moves one place to the right :
5.6 ÷ 0.1 = 56

Question 6.
0.4 ÷ 102 = ______

First Simplify the 102 which means  10 x 10 = 100, then we need to calculate the fraction to a decimal just divide the numerator (0.4) by the denominator (100).
When we divide by 100, the decimal point moves two places to the left :
0.4 ÷ 100 = 0.004

Find the value of k.
Question 7.
89 ÷ k = 8.9

Explanation: Lets solve your equation step by step 89 ÷ k = 8.9
Multiply both sides by K.
89 = 8.9 K
8.9 K = 89 (Flip the equation)

$\dpi{100} \small \frac{8.9 k}{8.9}$  = $\dpi{100} \small \frac{89}{8.9}$  (Divide both sides by 8.9)
k = 10

Question 8.
k ÷ 0.01 = 36

$\dpi{100} \small \frac{k}{0.01}$ = 36
Multiply both sides by 0.01
$\dpi{100} \small \frac{k}{0.01}$ x 0.01 = 36 x 0.01
k = 0.36

Question 9.
72.4 ÷ 0.724

To convert this simple fraction to a decimal just divide the numerator (72.4) by the denominator (0.724).

Question 10.
A box of 100 sanitizing wipes costs $12. How much does one wipe cost? Answer: 100 sanitizing wipes =$12
one wipe cost = $\dpi{100} \bg_white \small \frac{12}{100}$ = $0.12 When we divide by 100, the decimal point moves two places to the left. Question 11. Patterns How does the value of a number change when you divide by 10? 100? 1,000? Answer: When we divide by 10, the decimal point moves one place to the left. When we divide by 100, the decimal point moves two places to the left. When we divide by 1000, the decimal point moves three places to the left. Question 12. Writing How can you determine where to place the decimal point when dividing 61 by 1,000? Answer: $\dpi{100} \small \frac{61}{1000}$ When we divide by 1000, the decimal point moves three places to the left. so, $\dpi{100} \small \frac{61}{1000}$ = 0.061 Question 13. DIG DEEPER! What is Newton’s number? Answer: 3.4 is the number. 57 – 23 = 34 34 x 0.1 = 3.4 Question 14. Modeling Real Life A family buys 2 personal watercrafts for$3,495 each. The family makes 10 equal payments for the watercrafts. What is the amount of each payment?
To find amount of each payment, divide the sum of the personal watercrafts by 10.
3,495 + 3,495 = 6990
Divide the total sum by 10. Dividing 6990 by 10, or 101
6990 ÷ 10 = 6990 ÷ 101 = 699
So, the amount of each payment = $699. Question 15. Modeling Real Life A group of people attempts to bake the largest vegan cake. They use 17 kilograms of cocoa powder, which is one-tenth the amount of kilograms of dates they use. How many kilograms of cocoa power and dates do they use altogether? Answer: Cocoa powder = 17 kilograms Let dates amount = d (1/10)d = 17 dates(d) = 17 x 10 = 170 kilograms Sum of cocoa power and dates = 17 + 170 = 187 kilograms Review & Refresh Find the sum or difference. Question 16. 0.75 – 0.23 = ______ Answer: 0.52 Question 17. 1.46 + 1.97 = ______ Answer: 3.43 ### Lesson 7.2 Estimate Decimals Quotients Explore and Grow Choose an expression to estimate each quotient. Write the expression. You may use an expression more than once. Compare your answers with a partner. Did you choose the same expressions? Answer: Construct Arguments Which estimated quotient do you think will be closer to the quotient 8.3 ÷ 2.1? Explain your reasoning. Answer: Think and Grow: Estimate Decimals Quotients Key Idea You can use compatible numbers to estimate quotients involving decimals. When the divisor is greater than the dividend, rename the dividend as tenths or hundredths, then divide. Example Estimate 146.26 ÷ 41.2. Round the divisor 41.2 to 40. Think: What numbers close to 146.26 are easily divided by 40? Choose 160 because 146.26 is closer to 160. So, 146.26 ÷ 41.2 is about _____. Example Estimate 4.2 ÷ 8. Rename 4.2 as tenths. 4.2 is 42 tenths. 42 tenths is close to40 tenths. 40 and 8 are compatible numbers. 40 tenths ÷ 8 = _______ tenths, or ______ So, 4.2 ÷ 8 is about ______. Show and Grow Estimate the quotient. Question 1. 17.4 ÷ 3.1 Answer: Round the divisor 3.1 to 3. Think: What numbers close to 17.4 are easily divided by 3? Use 18. 18 ÷ 3 = 6 So, 17.4 ÷ 3.1 is about 6. Question 2. 57.5 ÷ 6.89 Answer: Round the divisor 6.89 to 7. Think: What numbers close to 57.5 are easily divided by 7? Use 56. 56 ÷ 7 = 8 So, 57.5 ÷ 6.89 is about 8. Question 3. 3.7 ÷ 5 Answer: Rename 3.7 as tenths 3.7 is 37 tenths. 37 is close to 35. 35 tenths ÷ 5 = 7 tenths or 0.7 So, 3.7 ÷ 5 is about 0.7 Question 4. 25.8 ÷ 30 Answer: Rename 25.8 as tenths 25.8 is 258 tenths. 258 is close to 270. 270 tenths ÷ 30 = 9 tenths or 0.9 So, 25.8 ÷ 30 is about 0.9 Apply and Grow: Practice Estimate the quotient. Question 5. 3.5 ÷ 6 Answer: Rename 3.5 as tenths 3.5 is 35 tenths. 35 is close to 36. 36 tenths ÷ 6 = 6 tenths or 0.6 So, 3.5 ÷ 6 is about 0.6 Question 6. 1.87 ÷ 9 Answer: Rename 1.87 as tenths 1.87 is 18.7 tenths. 18.7 is close to 18. 18 tenths ÷ 9 = 2 tenths or 0.2 So, 1.87 ÷ 9 is about 0.2 Question 7. 46 ÷ 2.3 Answer: Round the divisor 2.3 to 2. 46 ÷ 2 = 23 Question 8. 31.1 ÷ 6.5 Answer: Round the divisor 6.5 to 6. 31.1 is closer to 30. 30 ÷ 6 = 5 So, 31.1 ÷ 6.5 is about 5. Question 9. 91.08 ÷ 5.2 Answer: Round the divisor 5.2 to 5. 91.08 is closer to 90. 90 ÷ 5 = 18 So, 91.08 ÷ 5.2 is about 18. Question 10. 137.14 ÷ 12.2 Answer: Round the divisor 12.2 to 12. 137.14 is closer to 144. 12 and 144 are compatible numbers. 144 ÷ 12 = 12 So, 137.14 ÷ 12.2 is about 12. Question 11. A group of 6 friends goes ice skating. They pay$43.50 altogether for admission and skate rental. The friends share the cost equally. How much does each friend pay?

Total amount paid = $43.50 6 friends goes ice skating. 43.5 is closer to 42. 42 ÷ 6 = 7 So, each friend pay about$7.

Question 12.
Reasoning
Descartes estimates 43.2 ÷ 7.3 using mental math. Do you think he uses 43 ÷ 7 or 42 ÷ 7? Explain.
Round the divisor 7.3 to 7
Think: What numbers close to 43.2 are easily divided by 7?
Use 42.
42 and 7 are compatible numbers.
42 ÷ 7 = 6
So, 42 ÷ 7 is correct.

Question 13.
DIG DEEPER!
Describe a division situation in which an estimate of two decimals is appropriate.

Think and Grow: Modeling Real Life

Example
Your friend types 25 words each minute. About how many more words can your friend type each minute than you?

To find how many words you can type each minute, divide the number of words you type in 15 minutes by 15.
Think: What numbers close to 307.5 are easily divided by 15?

Choose 300 because 307.5 is closer to 300. So, 307.5 ÷ 15 is about _______.
So, you type about _______ words each minute.
Subtract the words you type each minute from the words your friend types each minute.

Your friend can type about ______ more words each minute than you.

Show and Grow

Question 14.
Newton subscribes to a television streaming service and buys a gym membership. He spends $143.99 on the streaming service for 12 months. About how much more does it cost each month for the gym membership than the streaming service? Answer: To find much more does it cost each month, divide how much he spends for 12 months by 12. Think: What numbers close to$143.99 are easily divided by 12?
Use 144
144 ÷ 12 = 12
Gym Membership each month = $19.99 =$20
20 – 12 = $8 The gym membership costs$8 more than the streaming service.

Question 15.
A fish tank pump filters 158.5 gallons of water each hour. About how many gallons of water does the pump filter each minute?
Fish tank pump filters 158.5 gallons of water
1 hour = 60 minutes
Think: What numbers close to 158.5 are easily divided by 60?
Use 180
180 ÷ 60 = 3
Pump filters about 3 gallons of water each minute.

Question 16.
DIG DEEPER!
A group of 32 students goes to a museum and a play. The total cost for the museum is $358.98 and the total cost for the play is$256.48. About how much does it cost for each student to go to the museum and the play?
Cost for museum = $358.98 Cost for the play =$256.48
358.98 + 256.48 = $615.46 Think: What numbers close to 615.46 are easily divided by 32? Use 608. It is closer to 615.46 608 ÷ 32 =$19
Each student go to the museum and the play costs about $19. ### Estimate Decimals Quotients Homework & Practice 7.2 Estimate the quotient. Question 1. 2.3 ÷ 6 Answer: Rename 2.3 as tenths 2.3 is 23 tenths. 23 is close to 24. 24 tenths ÷ 6 = 4 tenths or 0.4 So, 2.3 ÷ 6 is about 0.4 Question 2. 1.67 ÷ 8 Answer: Rename 1.67 as hundredths 1.67 is 167 hundredths. 167 is close to 168. 168 hundredths ÷ 8 = 21 hundredths or 0.21 So, 1.67 ÷ 8 is about 0.21 Question 3. 28 ÷ 4.7 Answer: Round the divisor 4.7 to 5 28 is closer to 30 30 ÷ 5 = 6 So, 28 ÷ 4.7 is about 6. Question 4. 13.8 ÷ 4.9 Answer: Round the divisor 4.9 to 5 Think: What numbers close to 13.8 are easily divided by 5? Use 15. 15 ÷ 5 = 3 So, 13.8 ÷ 4.9 is about 3. Question 5. 42.1 ÷ 7.3 Answer: Round the divisor 7.3 to 7 Think: What numbers close to 42.1 are easily divided by 7? Use 42. 42 ÷ 7 = 6 So, 42.1 ÷ 7.3 is about 6. Question 6. 201.94 ÷ 18.1 Answer: Round the divisor 18.1 to 18 Think: What numbers close to 201.94 are easily divided by 18? Use 198. 198 ÷ 18 = 11 So, 201.94 ÷ 18.1 is about 11. Question 7. A carpenter has a plank of wood that is 121.92 centimeters long. He cuts the plank into 4 equal pieces. About how long is each piece? Answer: Given that, Plank of wood = 121.92 cm long 121.92 is closer to 120. 120 ÷ 4 = 30 So, each piece is 30 cm long. Question 8. Reasoning A family used 9.8 gallons of gasoline to drive 275.5 miles. To determine how far they drove using one gallon of gasoline, can they use an estimate, or is an exact answer required? Explain. Answer: Given that, 9.8 gallons of gasoline drives = 275.5 miles 1 gallon = 275.5 ÷ 9.8 Divisor 9.8 is rounded to 10. 275.5 is closer to 276. 276 ÷ 10 is about 27.6 Question 9. YOU BE THE TEACHER Your friend says 9 ÷ 2.5 is about 3. Is your friend’s estimate reasonable? Explain. Answer: Round the divisor 2.5 to 3. 9 ÷ 3 =3 So, my friend’s estimate is reasonable. Number Sense Without calculating, tell whether the quotient is greater than or less than 1. Explain. Question 10. 4.58 ÷ 0.3 Answer: When the dividend is greater than the divisor, the quotient is greater than 1. Question 11. 0.6 ÷ 12 Answer: When the divisor is greater than the dividend, the quotient is less than 1. Question 12. Modeling Real Life The maximum allowed flow rate for a shower head in California is 42.5 gallons of water in 17 minutes. About how much greater is this than the maximum allowed flow rate for a kitchen faucet in California? Answer: To find much much greater it is, divide how much gallons of water in 17 minutes by 17. Think: What numbers close to 42.5 are easily divided by 17? Use 34. 34 is closer to 42.5. 34 ÷ 17 =2 Kitchen faucet = 2.2 gallons 2.2 – 2 = 0.2 Shower head in California is about 0.2 gallons greater than the maximum allowed flow rate for a kitchen faucet in California. Question 13. Modeling Real Life To compare the amounts in the table, you assume the same amount of snow fell each hour for 24 hours. About how many more inches of snow fell in Colorado each hour than in Utah? Answer: Time t = 24 hours Colorado snowfall = 75.8 is closer to 72 Illinois snowfall = 37.8 Utah snowfall = 55.5 is closer to 48 (72 – 48)/24 = 1 Snow fall in Colorado each hour is about 1 inch more than in Utah. Review & Refresh Find the product. Check whether your answer is reasonable. Question 14. 56 × 78 = _____ Answer: 4368 Question 15. 902 × 27 = ______ Answer: 24,354 Question 16. 4,602 × 35 = _______ Answer: 1,61,070 ### Lesson 7.3 Use Models to Divide Decimals by Whole Numbers Explore and Grow Complete the table. Answer: Reasoning When you divide a decimal by a whole number, what does the quotient represent? Answer: Think and Grow: Use Models to Divide Decimals Example Use a model to find 2.16 ÷ 3. Think: 2.16 is 2 ones, 1 tenth, and 6 hundredths. • 21 tenths can be divided equally as 3 groups of _______ tenths. • 6 hundredths can be divided equally as 3 groups of _______ hundredths. So, 216 hundredths can be divided equally as 3 groups of _______ hundredths. So, 2.16 ÷ 3 = _______ Show and Grow Question 1. Use the model to find 3.25 ÷ 5. 3.25 ÷ 5 = ______ Answer: Think: 3.25 is 3 ones, 2 tenths and 5 hundredths. 32 tenths can be divided equally as 5 groups So, 325 hundredths can be divided equally as 5 groups So, 3.25 ÷ 5 = 0.65 Apply and Grow: Practice Use the model to find the quotient. Question 2. 2.4 ÷ 4 Answer: Think: 2.4 is 2 ones and 4 tenths 24 tenths can be divided equally as 4 groups of 6 tenths. So, 2.4 ÷ 4 = 6 tenths = 0.6 Question 3. 1.36 ÷ 2 Answer: Think: 1.36 is 1 ones, 3 tenth and 6 hundredths. 13 tenths can be divided equally as 2 groups So, 136 hundredths can be divided equally as 2 groups So, 1.36 ÷ 2 = 0.68 Use a model to find the quotient. Question 4. 1.5 ÷ 3 Answer: Think: 1.5 is 1 ones and 5 tenths 15 tenths can be divided equally as 3 groups of 5 tenths. So, 1.5 ÷ 3 = 5 tenths = 0.5 Question 5. 2.7 ÷ 9 Answer: Think: 2.7 is 2 ones and 7 tenths 27 tenths can be divided equally as 9 groups of 3 tenths. So, 2.7 ÷ 9 = 3 tenths = 0.3 Question 6. 1.44 ÷ 8 Answer: Think: 1.44 is 1 ones, 4 tenth and 4 hundredths. 14 tenths can be divided equally as 8 groups So, 144 hundredths can be divided equally as 8 groups So, 1.44 ÷ 8 = 0.18 Question 7. 3.12 ÷ 6 Answer: Think: 3.12 is 3 ones, 1 tenth and 2 hundredths. 31 tenths can be divided equally as 6 groups So, 312 hundredths can be divided equally as 6 groups So, 3.12 ÷ 6 = 0.52 Question 8. Reasoning Do you start dividing the ones first when finding 5.95 ÷ 7? Explain. Answer: Think: 5.95 is 5 ones, 9 tenth and 5 hundredths. We have to start dividing the tenths first because 5 ones is less than 7. 59 tenths can be divided equally as 7 groups So, 595 hundredths can be divided equally as 7 groups So, 5.95 ÷ 7 = 0.85 Question 9. Number Sense Without dividing, determine whether the quotient of 9.85 and 5 is greater than or less than 2. Explain. Answer: Quotient of 9.85 and 5 is less than 2, because 5 x 2 =10 and 9.85 is less than 10. Think and Grow: Modeling Real Life Example A bag of 3 racquetballs weighs 4.2 ounces. What is the weight of each racquetball? Divide the weight of the bag by 3 to find the weight of each racquetball. Think: 4.2 is 4 ones and 2 tenths. Shade 42 tenths to represent 4.2. Divide the model to show 3 equal groups. 42 tenths can be divided equally as 3 groups of ______ tenths. 4.2 ÷ 3 = ______ So, each racquetball weighs ______ ounces. Show and Grow Question 10. You cut a 3.75-foot-long string into 5 pieces of equal length to make a beaded wind chime. What is the length of each piece of string? Answer: Divide the length of the string by 5 to find the length of each piece of string. Think: 3.75 is 3 ones, 7 tenths and 5 hundredths. 37 tenths can be divided equally as 5 groups of 7 tenths with remainder 2. Remainder has to place before 5 hundredths. 25 hundredths can be divided equally as 5 groups of 5 hundredths. So, 375 hundredths can be divided equally as 5 groups of 75 hundredths. 3.75 ÷ 5 = 0.75 Question 11. DIG DEEPER! You pay$5.49 for 3 pounds of plums and $6.36 for 4 pounds of peaches. Which fruit costs more per pound? How much more? Answer: Think: 5.49 is 5 ones, 4 tenths and 9 hundredths. 5 ones can be divided equally as 3 groups of 1 ones with remainder 2. Remainder has to place before 4 tenths. 24 tenths can be divided equally as 3 groups of 8 tenths 9 hundredths can be divided equally as 3 groups of 3 hundredths So, 549 hundredths can be divided equally as 3 groups of 183 hundredths. Plums = 5.49 ÷ 3 = 1.83 Think: 6.36 is 6 ones, 3 tenths and 6 hundredths. 6 ones can be divided equally as 4 groups of 1 ones with remainder 2. Remainder has to place before 3 tenths. 23 tenths can be divided equally as 4 groups of 5 tenths with remainder 3. Remainder has to place before 6 hundredths. 36 hundredths can be divided equally as 4 groups of 9 hundredths So, 636 hundredths can be divided equally as 4 groups of 159 hundredths. Peaches = 6.36 ÷ 4 = 1.59 1.83 – 1.59 = 0.24 So, plums costs 0.24 more per pound than peaches. ### Use Models to Divide Decimals by Whole Numbers Homework & Practice 7.3 Use the model to find the quotient. Question 1. 1.5 ÷ 5 Answer: Think: 1.5 is 1 ones and 5 tenths 15 tenths can be divided equally as 5 groups of 3 tenths. So, 1.5 ÷ 5 = 3 tenths = 0.3 Question 2. 2.55 ÷ 3 Answer: Think: 2.55 is 2 ones, 5 tenths and 5 hundredths. 25 tenths can be divided equally as 3 groups of 8 tenths with remainder 1. Remainder has to place before 5 hundredths. 15 hundredths can be divided equally as 3 groups of 5 hundredths. So, 255 hundredths can be divided equally as 3 groups of 85 hundredths. 2.55 ÷ 3 = 0.85 Use a model to find the quotient. Question 3. 1.6 ÷ 8 Answer: Think: 1.6 is 1 ones and 6 tenths 16 tenths can be divided equally as 8 groups of 2 tenths. So, 1.6 ÷ 8 = 2 tenths = 0.2 Question 4. 2.1 ÷ 7 Answer: Think: 2.1 is 2 ones and 1 tenths 21 tenths can be divided equally as 7 groups of 3 tenths. So, 2.1 ÷ 7 = 3tenths = 0.3 Question 5. 1.56 ÷ 2 Answer: Think: 1.56 is 1 ones, 5 tenths and 6 hundredths. 15 tenths can be divided equally as 2 groups of 7 tenths with remainder 1. Remainder has to place before 6 hundredths. 16 hundredths can be divided equally as 2 groups of 8 hundredths. So, 156 hundredths can be divided equally as 2 groups of 78 hundredths. 1.56 ÷ 2 = 0.78 Question 6. 2.84 ÷ 4 Answer: Think: 2.84 is 2 ones, 8 tenths and 4 hundredths. 28 tenths can be divided equally as 4 groups of 7 tenths. 4 hundredths can be divided equally as 4 groups of 1 hundredths. So, 284 hundredths can be divided equally as 4 groups of 71 hundredths. 2.84 ÷ 4 = 0.71 Question 7. Structure Write a decimal division equation represented by the model. Answer: 1.8 ÷ 3 Question 8. Writing Explain how dividing a decimal by a whole number is similar to dividing a whole number by a whole number. Answer: When dividing a decimal by a whole number, first we will divide the decimal by the whole number ignoring decimal point. Now put the decimal point in the quotient same as the decimal places in the dividend. So , dividing a decimal by a whole number is similar to dividing a whole number by a whole number. Question 9. Modeling Real Life A designer learns there are 5.08 centimeters in 2 inches. How many centimeters are in 1 inch? Answer: 5.08 ÷ 2 Think: 5.08 is 5 ones, 0 tenths and 8 hundredths. 50 tenths can be divided equally as 2 groups of 25 tenths. 8 hundredths can be divided equally as 2 groups of 4 hundredths. So, 508 hundredths can be divided equally as 2 groups of 254 hundredths. So, 2.54 cm are in 1 inch. Question 10. Modeling Real Life Newton buys 4 gallons of gasoline. He pays$8.64. How much does 1 gallon of gasoline cost?
8.64 ÷ 4
Think: 8.64 is 8 ones, 6 tenths and 4 hundredths.
8 ones can be divided equally as 4 groups of 2 ones.
6 tenths can be divided equally as 4 groups of 1 tenths with remainder 2. Remainder has to place before 4 hundredths.
24 hundredths can be divided equally as 4 groups of 6 hundredths.
So, 864 hundredths can be divided equally as 4 groups of 216 hundredths.

### Divide Decimals by One-Digit Numbers Homework & Practice 7.4

Question 1.
$$\sqrt [ 3 ]{ 9.6 }$$
Divide the ones
9 ÷ 3
3 ones x 3 = 9
9 ones – 9 ones
There are 0 ones left over.
Divide the tenths
6 ÷ 3
2 tenths x 3
6 – 6 = 0
There are 0 tenths left over.
So, 9.6 ÷ 3 = 3.2.

Question 2.
$$\sqrt [ 6 ]{ 7.56 }$$
Divide the ones
7 ÷ 6
1 ones x 6 = 6
7 ones – 6 ones
There are 1 ones left over.
Divide the tenths
15 ÷ 6
2 tenths x 6
15 – 12 = 3
There are 3 tenths left over.
Divide the hundredths
36 ÷ 6 = 6 hundredths.
So, 7.56 ÷ 6 = 1.26.

Question 3.
$$\sqrt [ 8 ]{ 42.4 }$$
Divide the ones
42 ÷ 8
5 ones x 8 = 40
42 ones – 40 ones
There are 2 ones left over.
Divide the tenths
24 ÷ 8
3 tenths x 8
24 – 24 = 0
There are 0 tenths left over.
So, 42.4 ÷ 8 = 5.3.

Question 4.
63.6 ÷ 4 = ______
Divide the ones
63 ÷ 4
15 ones x 4 = 60
63 ones – 60 ones
There are 3 ones left over.
Divide the tenths
36 ÷ 4
9 tenths x 4
36 – 36 = 0
There are 0 tenths left over.
63.6 ÷ 4 = 15.9

Question 5.
15.68 ÷ 7 = ______
Divide the ones
15 ÷ 7
2 ones x 7 = 14
15 ones – 14 ones
There are 1 ones left over.
Divide the tenths
16 ÷ 7
2 tenths x 7
16 – 14 = 2
There are 2 tenths left over.
Divide the hundredths
28 ÷ 7 = 4 hundredths
15.68 ÷ 7 = 2.24

Question 6.
143.82 ÷ 9 = _______
Divide the ones
143 ÷ 9
15 ones x 9 = 135
143 ones – 135 ones
There are 8 ones left over.
Divide the tenths
88 ÷ 9
9 tenths x 9
88 – 81 = 7
There are 7 tenths left over.
Divide the hundredths
72 ÷ 9 = 8 hundredths
143.82 ÷ 9 = 15.98

Find the value of y.
Question 7.
y ÷ 6 = 7.8
y = 7.8 x 6
y= 46.8

Question 8.
14.9 ÷ 5 = y
Divide the ones
14 ÷ 5
2 ones x 5 = 10
14 ones – 10 ones
There are 4 ones left over.
Divide the tenths
49 ÷ 5
9 tenths x 5
49 – 45 = 4
There are 4 tenths left over.
Divide the hundredths
40 ÷ 5 = 8 hundredths
14.9 ÷ 5 = 2.98
y = 2.98

Question 9.
y ÷ 2 = 4.7
y = 4.7 x 2
y = 9.4

Question 10.
Number Sense
Evaluate the expression.
(213.3 – 95.7) ÷ 8
(213.3 – 95.7) ÷ 8 = 117.6 ÷ 8
Divide the ones
117 ÷ 8
14 ones x 8 = 112
117 ones – 112 ones
There are 5 ones left over.
Divide the tenths
56 ÷ 8
7 tenths x 8
56 – 56 = 0
There are 0 tenths left over.
(213.3 – 95.7) ÷ 8 = 117.6 ÷ 8 = 14.7

Question 11.
Writing
Write and solve a real-life problem that involves dividing a decimal by a whole number.
In 5 minutes John eats 7.5 chocolates.  how many chocolates can he eat in one minute?
7.5 ÷ 5
Divide the ones
7 ÷ 5
1 ones x 5 = 5
7 ones – 5 ones
There are 2 ones left over.
Divide the tenths
25 ÷ 5 = 5 tenths
7.5 ÷ 5 = 1.5
In 1 minute, he can eat 1.5 chocolates.

Question 12.
YOU BE THE TEACHER

Divide the ones
197 ÷ 4
49 ones x 4 = 196
197 ones – 196 ones
There are 1 ones left over.
Divide the tenths
12 ÷ 4
3 tenths x 4 = 12
12 – 12 = 0
There are 0 tenths left over.
197.2 ÷ 4 = 49.3
So, my friend answer is not correct.

Question 13.
Modeling Real Life
You buy 2 packages of ground beef. One package contains 4.5 pounds and the other contains 2.25 pounds. You put equal amounts of meat into 9 freezer bags. How many pounds of meat are in each bag?
To find many pounds of meat are in each bag, divide the total meat by 9.
Add the two packages of meat.
4.5 + 2.25 = 6.75
6.75 ÷ 9
Divide the tenths
67 ÷ 9
7 tenths x 9 = 63
67 tenths – 63 tenths
There are 4 tenths left over.
Divide the hundredths
45 ÷ 9 = 5 hundredths
6.75 ÷ 9 = 0.75

Question 14.
DIG DEEPER!
A homeowner hangs wallpaper on the walls of her bathroom. What is the width of the bathroom?

We know that perimeter of a rectangle = 2(l + w)
8.52 = 2(2.74 + w)
2.74 + w = 8.52 ÷ 2
8.52 ÷ 2
Divide the ones
8 ÷ 2 = 4 ones
Divide the tenths
52 ÷ 2 = 26 tenths
8.52 ÷ 2 = 4.26
2.74 + w = 4.26
Width w = 4.26 – 2.74 = 1.52
So, width of the bathroom = 1.52 m

Review & Refresh

Use partial quotients to divide.
Question 15.
607 ÷ 15 = ______
15 x 40 = 600 with remainder 7.

Question 16.
4,591 ÷ 33 = ______

Question 17.
6,699 ÷ 87 = ______
87 x 50 = 4350
6,699 – 4350 = 2349
87 x 20 = 1740
2349 – 1740 = 609
87 x 5 = 435
609 – 435 = 174
87 x 2 = 174
6,699 ÷ 87 = 50 + 20 + 5 + 2 = 77.

### Lesson 7.5 Divide Decimals by Two-Digit Numbers

Explore and Grow

Write a division problem you can use to find the width of each rectangle. Then find the width of each rectangle.

Precision

Think and Grow: Divide Decimals by Two-Digit Numbers

Example
Find 79.8 ÷ 14. Estimate _________
Regroup 7 tens as 70 ones and combine with 9 ones.

Example
Find 20.54 ÷ 26.
Step 1: Estimate the quotient.
2,000 hundredths ÷ 25 = _______ hundredths

Step 2: Divide as you do with whole numbers.
Step 3: Use the estimate to place the decimal point.
So, 20.54 ÷ 26 = _______.

Show and Grow

Question 1.
$$\sqrt [ 12 ]{ 51.6 }$$
Divide the ones
51 ÷ 12
4 ones x 12 = 48
51 ones – 48 ones
There are 3 ones left over.
Divide the tenths
36 ÷ 12
3 tenths x 12 = 36
36 – 36 = 0
There are 0 tenths left over.
So, 51.6 ÷ 12 = 4.3

Question 2.
$$\sqrt [ 17 ]{ 140.25 }$$
Divide the ones
140 ÷ 17
8 ones x 17 = 136
140 ones – 136 ones
There are 4 ones left over.
Divide the tenths
42 ÷ 17
2 tenths x 17 = 34
42 – 34 = 8
There are 8 tenths left over.
Divide the hundredths
85 ÷ 17 = 5 hundredths.
So, 140.25 ÷ 17 = 8.25

Question 3.
$$\sqrt [ 61 ]{ 32.33 }$$
Divide the tenths
323 ÷ 61
5 ones x 61 = 305
323 tenths – 305 tenths
There are 18 tenths left over.
Divide the hundredths
183 ÷ 61 = 3 hundredths
So, 32.33 ÷ 61 = 0.53

Apply and Grow: Practice

Place a decimal point where it belongs in the quotient.
Question 4.
251.75 ÷ 19 = 1 3 . 2 5
When dividing a decimal by a whole number, first we will divide the decimal by the whole number ignoring decimal point. Now put the decimal point in the quotient same as the decimal places in the dividend.

Question 5.
88.04 ÷ 62 = 1 . 4 2

Question 6.
3.22 ÷ 23 = 0 .1 4

Question 7.
$$\sqrt [ 54 ]{ 97.2 }$$
Divide the ones
97 ÷ 54
1 ones x 54 = 54
97 ones – 54 ones
There are 43 ones left over.
Divide the tenths
432 ÷ 54 = 8 tenths
So, 97.2 ÷ 54 = 1.8

Question 8.
$$\sqrt [ 91 ]{ 200.2 }$$
Divide the ones
200 ÷ 91
2 ones x 91 = 182
200 ones – 182 ones
There are 18 ones left over.
Divide the tenths
182 ÷ 91 = 2 tenths
So, 200.2 ÷ 91 = 2.2

Question 9.
$$\sqrt [ 2 ]{ 56.2 }$$
Divide the ones
56 ÷ 2
28 ones x 2 = 56
56 ones – 56 ones
There are 0 ones left over.
Divide the tenths
2 ÷ 2 = 1 tenths
So, 56.2 ÷ 2 = 28.1

Question 10.
6.08 ÷ 16 = _____
Divide the tenths
60 ÷ 16
3 tenths x 16 = 48
60 tenths – 48 tenths
There are 12 tenths left over.
Divide the hundredths
128 ÷ 16
8 hundredths x 16
128 – 128 = 0
There are 0 hundredths left over.
So, 6.08 ÷ 16 = 0.38

Question 11.
7.45 ÷ 5 = _______
Divide the tenths
74 ÷ 5
14 tenths x 5 = 70
74 tenths – 70 tenths
There are 4 tenths left over.
Divide the hundredths
45 ÷ 5
9 hundredths x 5 = 45
45 – 45 = 0
There are 0 hundredths left over.
So, 7.45 ÷ 5 = 1.49

Question 12.
147.63 ÷ 37 = _______
Divide the ones
147 ÷ 37
3 ones x 37 = 111
147 ones – 111 ones
There are 36 ones left over.
Divide the tenths
366 ÷ 37
9 tenths x 37 = 333
366 – 333 = 33
Divide the hundredths
333 ÷ 37 = 9 hundredths
So, 147.63 ÷ 37 = 3.99

Find the value of y.

Question 13.
y ÷ 44 = 1.82
y = 44 x 1.82
y = 80.08

Question 14.
106.6 ÷ 82 = y
Divide the ones
106 ÷ 82
1 ones x 82 = 82
106 ones – 82 ones
There are 24 ones left over.
Divide the tenths
246 ÷ 82
3 tenths x 82 = 246
246 – 246 = 0
106.6 ÷ 82 = 1.3, y = 1.3

Question 15.
y ÷ 13 = 2.6
y = 13 x 2.6
y = 33.8

Question 16.
Logic
Newton and Descartes find 44.82 ÷ 18. Only one of them is correct. Without solving, who is correct? Explain.

Descartes answer is correct, 44.82 ÷ 18 = 2.49
When dividing a decimal by a whole number, first we will divide the decimal by the whole number ignoring decimal point. Now put the decimal point in the quotient same as the decimal places in the dividend.

Question 17.
DIG DEEPER!
Find a decimal that you can divide by a two-digit whole number to get the quotient shown. Fill in the boxes with your dividend and divisor.

Dividend is 20 and divisor is 12.

Think and Grow: Modeling Real Life

Example
You practice paddle boarding for 3 weeks. You paddle the same amount each day for 5 days each week. You paddle 22.5 miles altogether. How many miles do you paddle each day?

To find the total number of days you paddle in 3 weeks, multiply the days you paddle each week by 3.
5 × 3 = 15 So, you paddle board _______ days in 3 weeks.
To find the number of miles you paddle each day, divide the total number of miles by the number of days you paddle in 3 weeks.

You paddle _______ miles each day.

Show and Grow

Question 18.
Descartes borrows $6,314.76 for an all-terrain vehicle. He pays back the money in equal amounts each month for 3 years. What is his monthly payment? Answer: Time t = 3 years = 3 x 12 = 36 months Descartes borrowed amount =$6,314.76
6,314.76 ÷ 36
63 ÷ 36 = 1 and 27 is left over
271 ÷ 36 = 7 and 19 is left over
194 ÷ 36 = 5 and 14 is left over
147 ÷ 36 = 4 and 3 is left over
36 ÷ 36 = 1 and 0 left over.
6,314.76 ÷ 36 = 175.41
Descartes monthly payment is $175.41 Question 19. A blue car travels 297.6 miles using 12 gallons of gasoline and a red car travels 358.8 miles using 13 gallons of gasoline. Which car travels farther using 1 gallon of gasoline? How much farther? Answer: 297 ones ÷ 12 = 24 ones x 12 = 288 297 ones – 288 ones There are 9 ones left over. 96 ÷ 12 = 8 tenths x 12 = 96 96 – 96 = 0 There are 0 hundredths left over. So, 297.6 ÷ 12 = 24.8 358 ones ÷ 13 = 27 ones x 13 = 351 358 ones – 351 ones There are 7 ones left over. 78 ÷ 13 = 6 tenths x 13 = 78 78 – 78 = 0 There are 0 hundredths left over. So, 358.8 ÷ 13 = 27.6 Red car – blue car = 27.6 – 24.8 = 2.8 Red car travels 2.8 miles farther than blue car using 1 gallon of gasoline. Question 20. DIG DEEPER! The rectangular dog park has an area of 2,616.25 square feet. How much fencing does an employee need to enclose the dog park? Answer: ### Divide Decimals by Two-Digit Numbers Homework & Practice 7.5 Place a decimal point where it belongs in the quotient. Question 1. 127.2 ÷ 24 = 5 . 3 Answer: Question 2. 48.64 ÷ 32 = 1 . 5 2 Answer: Question 3. 514.18 ÷ 47 = 1 0 . 9 4 Answer: Find the quotient. Then check your answer. Question 4. $$\sqrt [ 72 ]{ 93.6 }$$ Answer: Divide the ones 93 ÷ 72 1 ones x 72 = 72 93 ones – 72 ones There are 21 ones left over. Divide the tenths 216 ÷ 72 = 3 tenths. So, 93.6 ÷ 72 = 1.3 Question 5. $$\sqrt [ 7 ]{ 3.92 }$$ Answer: Divide the tenths 39 ÷ 7 5 ones x 7 = 35 39 ones – 35 ones There are 4 ones left over. Divide the hundredths 42 ÷ 7 = 6 tenths. So, 3.92 ÷ 7 = 0.56 Question 6. $$\sqrt [ 29 ]{ 1.74 }$$ Answer: Divide the hundredths 174 ÷ 29 6 ones x 29 = 174 174 hundredths – 174 hundredths There are 0 hundredths left over. So, 1.74 ÷ 29 = 0.06 Question 7. 24.3 ÷ 9 = _______ Answer: Divide the ones 24 ÷ 9 2 ones x 9 = 18 24 ones – 18 ones There are 6 ones left over. Divide the tenths 63 ÷ 9 7 tenths x 9 = 63 63 – 63 = 0 There are 0 tenths left over. So, 24.3 ÷ 9 = 2.7 Question 8. 244.9 ÷ 31 = ______ Answer: Divide the ones 244 ÷ 31 7 ones x 31 = 217 244 ones – 217 ones There are 27 ones left over. Divide the tenths 279 ÷ 31 9 tenths x 31 279 – 279 = 0 There are 0 tenths left over. So, 244.9 ÷ 31 = 7.9 Question 9. 55.62 ÷ 27 = ______ Answer: Divide the ones 55 ÷ 27 2 ones x 27 = 54 55 ones – 54 ones There is 1 ones left over. Divide the tenths 162 ÷ 27 6 tenths x 27 162 – 162 = 0 There are 0 tenths left over. So, 55.62 ÷ 27 = 2.06 Find the value of y. Question 10. y ÷ 16 = 0.23 Answer: y = 16 x 0.23 y = 3.68 Question 11. 44.1 ÷ 21 = y Answer: Divide the ones 44 ÷ 21 2 ones x 21 = 42 44 ones – 42 ones There are 2 ones left over. Divide the tenths 21 ÷ 21 1 tenths x 21 21 – 21 = 0 There are 0 tenths left over. So, 44.1 ÷ 21 = 2.1 Question 12. y ÷ 28 = 11.04 Answer: y = 28 x 11.04 y = 309.12 Question 13. YOU BE THE TEACHER Your friend finds 21.44 ÷ 16. Is your friend correct? Explain. Answer: My friend answer is not correct. When dividing a decimal by a whole number, first we will divide the decimal by the whole number ignoring decimal point. Now put the decimal point in the quotient same as the decimal places in the dividend. Divide the ones 21 ÷ 16 1 ones x 16 = 16 21 ones – 16 ones There are 5 ones left over. Divide the tenths 54 ÷ 16 3 tenths x 16 54 tenths – 48 tenths There are 6 tenths left over. Divide the hundredths 64 ÷ 16 4 hundredths x 16 64 hundredths- 64 hundredths There are 0 hundredths left over. So, 21.44 ÷ 16 = 1.34 Question 14. DIG DEEPER! A banker divides the amount shown among 12 people. How can she regroup the money? How much money does each person get? Answer: Question 15. Modeling Real Life You have hip-hop dance practice for 5 weeks. You attend practice 5 days each week. Each practice is the same length of time. You practice for 37.5 hours altogether. How many hours do you practice each day? Answer: To find the total number of days you practice in 5 weeks, multiply the days you practice each week by 5. 5 × 5 = 25 So, you practice 25 days in 5 weeks. To find the number of hours you practice each day, divide the total number of hours by the number of days you practice in 5 weeks. 37.5 ÷ 25 Divide the ones 37 ÷ 25 1 ones x 25 = 25 37 ones – 25 ones There are 12 ones left over. Divide the tenths 125 ÷ 25 5 tenths x 25 125 tenths – 125 tenths There are 0 tenths left over. So, 37.5 ÷ 25 = 1.5 So, I practice dance 1.5 hours each day. Question 16. DIG DEEPER! Your rectangular classroom rug has an area of 110.5 square feet. What is the perimeter of the rug? Answer: Review & Refresh Find the product. Question 17. 0.52 × 0.4 = _______ Answer: 0.208 Question 18. 0.7 × 21.3 = _______ Answer: 14.91 Question 19. 1.52 × 8.6 = ______ Answer: 13.072 ### Lesson 7.6 Use Models to Divide Decimals Explore and Grow Use the model to find each quotient. Answer: Structure When using a model to divide decimals, how do you determine the number of rows and columns to shade? How do you divide the shaded region? Answer: Think and Grow: Use Models to Divide Decimals Example Use a model to find 1.2 ÷ 0.3. Shade 12 columns to represent 1.2. Divide the model to show groups of 0.3. There are ______ groups of ______ tenths. So, 1.2 ÷ 0.3 = ________. Example Use a model to find 0.7 ÷ 0.14. Shade 7 columns to represent 0.7. Divide the model to show groups of 0.14. There are ______ groups of _______ hundredths. So, 0.7 ÷ 0.14 = ______. Show and Grow Use the model to find the quotient. Question 1. 1.5 ÷ 0.5 = _____ Answer: Shade 15 columns to represent 1.5. Divide the model to show groups of 0.5. There are 3 groups of 5 tenths. So, 1.5 ÷ 0.5 = 3 Question 2. 1.72 ÷ 0.86 = ______ Answer: Shade 17.2 columns to represent 1.72. Divide the model to show groups of 0.86. There are 2 groups of 86 hundredths. So, 1.72 ÷ 0.86 = 2 Apply and Grow: Practice Use the model to find the quotient. Question 3. 0.32 ÷ 0.04 = ______ Answer: Shade 3.2 columns to represent 0.32. Divide the model to show groups of 0.04. There are 8 groups of 4 hundredths. So, 0.32 ÷ 0.04 = 8 Question 4. 0.9 ÷ 0.15 = ______ Answer: Shade 9 columns to represent 0.9. Divide the model to show groups of 0.15. There are 6 groups of 15 hundredths. So, 0.9 ÷ 0.15 = 6 Question 5. 1.4 ÷ 0.07 = _____ Answer: Shade 14 columns to represent 1.4. Divide the model to show groups of 0.07. There are 20 groups of 7 hundredths. So, 1.4 ÷ 0.07 = 20 Question 6. 1.08 ÷ 0.09 = _____ Answer: Shade 10.8 columns to represent 1.08. Divide the model to show groups of 0.09. There are 12 groups of 9 hundredths. So, 1.08 ÷ 0.09 = 12 Question 7. You have$1.50 in dimes. You exchange all of your dimes for quarters. How many quarters do you get?
Quarter = 0.25
1.50 ÷ 0.25
Shade 15 columns to represent 1.50.
Divide the model to show groups of 0.25.
There are 6 groups of 25 hundredths.
So, 1.50 ÷ 0.25 = 6 quarters.

Question 8.
YOU BE THE TEACHER
Your friend uses the model below and says 1.6 ÷ 0.08 = 2. Is your friend correct? Explain.

1.6 ÷ 0.08
Shade 16 columns to represent 1.6.
Divide the model to show groups of 0.08.
There are 20 groups of 8 hundredths.
So, 1.6 ÷ 0.08 = 20
So, my friend answer is wrong.

Question 9.
Structure
Use the model to find the missing number.
0.72 ÷ ____ = 8

Shade 7.2 columns to represent 0.72.
Divide the model to show groups of 8.
There are 0.09 groups of 800 hundredths.
So, 0.72 ÷ 0.09 = 8
Missing number is 0.09.

Think and Grow: Modeling Real Life

Example
Is aluminum more than 5 times as dense as neon?
Divide the density of aluminum by the density of neon to find how many times as dense it is.
Use a model. Shade 27 columns to represent 2.7.
Divide the model to show groups of 0.9.

There are ______ groups of ______ tenths.
So, 2.7 ÷ 0.9 = _______.
Compare the quotient to 5.
So, aluminum ________ more than 5 times as dense as neon.

Show and Grow

Question 10.
Use the table above. Is neon more than 9 times as dense as hydrogen?
Divide the density of neon by the density of hydrogen to find how many times as dense it is.
Use a model. Shade 9 columns to represent 0.9.
Divide the model to show groups of 0.09.
There are 10 groups of 9 hundredths.
So, 0.9 ÷ 0.09 = 10
Compare the quotient to 9.
So, neon is more than 9 times as dense as hydrogen.

Question 11.
You fill a bag with peanuts, give the cashier $5, and receive$3.16 in change. How many pounds of peanuts do you buy?

Amount to buy peanuts = 5 – 3.16 = 1.84
peanuts per pound = $0.23 1.84 ÷ 0.23 Shade 18.4 columns to represent 1.84. Divide the model to show groups of 0.23. There are 8 groups of 23 hundredths. So, 1.84 ÷ 0.23 = 8 I can buy 8 pounds of peanuts. Question 12. DIG DEEPER! You have 2.88 meters of copper wire and 5.85 meters of aluminum wire. You need 0.24 meter of copper wire to make one bracelet and 0.65 meter of aluminum wire to make one necklace. Can you make more bracelets or more necklaces? Explain. Answer: Copper wire = 2.88 ÷ 0.24 Shade 28.8 columns to represent 2.88. Divide the model to show groups of 0.24. There are 12 groups of 24 hundredths. So, 2.88 ÷ 0.24 = 12 Aluminum wire = 5.85 ÷ 0.65 Shade 58.5 columns to represent 5.85. Divide the model to show groups of 0.65. There are 9 groups of 65 hundredths. So, 5.85 ÷ 0.65 = 9 So, we can make more bracelets. ### Use Models to Divide Decimals Homework & Practice 7.6 Use the model to find the quotient. Question 1. 0.08 ÷ 0.02 = _____ Answer: Shade 8 columns to represent 0.08. Divide the model to show groups of 0.02. There are 4 groups of 2 hundredths. So, 0.08 ÷ 0.02 = 4 Question 2. 0.4 ÷ 0.05 = ______ Answer: Shade 5 columns to represent 0.4. Divide the model to show groups of 0.05. There are 8 groups of 5 hundredths. So, 0.4 ÷ 0.05 = 8 Question 3. 1.7 ÷ 0.85 = ______ Answer: Shade 17 columns to represent 1.7. Divide the model to show groups of 0.85. There are 2 groups of 85 hundredths. So, 1.7 ÷ 0.85 = 2 Question 4. 1.5 ÷ 0.3 = _______ Answer: Shade 15 columns to represent 1.5. Divide the model to show groups of 0.3. There are 5 groups of 3 tenths. So, 1.5 ÷ 0.3 = 5 Question 5. You have a piece of scrapbook paper that is 1.5 feet long. You cut it into pieces that are each 0.5 foot long. How many pieces of scrap book paper do you have now? Answer: 1.5 ÷ 0.5 Shade 15 columns to represent 1.5. Divide the model to show groups of 0.5. There are 3 groups of 5 tenths. So, 1.5 ÷ 0.5 = 3 So, I have 3 pieces of scrap book paper. Question 6. YOU BE THE TEACHER Your friend uses the model below and says 0.12 ÷ 0.04 = 0.03. Is your friend correct? Explain. Answer: 0.12 ÷ 0.04 Shade 1.2 columns to represent 0.12. Divide the model to show groups of 0.04. There are 3 groups of 4 hundredths. So, 0.12 ÷ 0.04 = 3 My friend is not correct. Question 7. Writing Write a real-life problem that involves dividing a decimal by another decimal. Answer: Question 8. Modeling Real Life Does the watercolor paint cost more than 3 times as much as the paintbrush? Explain. Answer: Divide the price of watercolor paint by the price of paintbrush to find how many times as cost it is. Use a model. Shade 29.6 columns to represent 2.96. Divide the model to show groups of 0.74. There are 4 groups of 74 hundredths. So, 2.96 ÷ 0.74 Compare the quotient to 3. So, watercolor paint costs more than 3 times as much as the paintbrush. Question 9. DIG DEEPER! You have 3.75 cups of popcorn kernels. You fill a machine with 0.25 cup of kernels 3 times each hour. How many hours pass before you run out of kernels? Answer: Filling kernels each hour = 0.25 x 3 = 0.75 Total cups of popcorn kernels = 3.75 3.75 ÷ 0.75 Shade 37.5 columns to represent 3.75. Divide the model to show groups of 0.75. There are 5 groups of 75 hundredths. So, 3.75 ÷ 0.75 = 5 hours Review & Refresh Complete the equation. Identify the property shown. Question 10. 3 × 14 = 14 × 3 Answer: Commutative Property of Multiplication Question 11. 8 × (3 + 10) = (8 × 3) + (8 × 10) Answer: Distributive Property ### Lesson 7.7 Divide Decimals Explore and Grow Use the model to find 0.96 ÷ 0.32. Find 96 ÷ 32. Answer: Structure How can multiplying by a power of 10 help you divide decimals? Answer: Think and Grow: Divide Decimals by Decimals Key Idea To divide by a decimal, multiply the divisor by a power of 10 to make it a whole number. Multiply the dividend by the same power of 10. Then divide as you would with whole numbers. Example Find 6.12 ÷ 1.8. Estimate _______ Example Find 2.43 ÷ 0.09. So, 2.43 ÷ 0.09 = ______. Show and Grow Multiply the divisor by a power of 10 to make it a whole number. Then write the equivalent expression. Question 1. 3.5 ÷ 0.5 Answer: Step 1: Multiply 0.5 by a power of 10 to make it a whole number. Then multiply 3.5 by the same power of 10. 0.5 x 10 = 5 3.5 x 10 = 35 35 ÷ 5 = 7 So, 3.5 ÷ 0.5 = 7 Question 2. 9.84 ÷ 2.4 Answer: Step 1: Multiply 2.4 by a power of 10 to make it a whole number. Then multiply 9.84 by the same power of 10. 2.4 x 10 = 24 9.84 x 10 = 98.4 Step 2: Divide 98.4 ÷ 24 98 ÷ 24 = 4 with remainder 2. 24 ÷ 24 = 1 with remainder 0. So, 9.84 ÷ 2.4 = 4.1 Question 3. 4.68 ÷ 0.78 Answer: Step 1: Multiply 0.78 by a power of 10 to make it a whole number. Then multiply 4.68 by the same power of 10. 0.78 x 100 = 78 4.68 x 100 = 468 Step 2: Divide 468 ÷ 78 = 6 So, 4.68 ÷ 0.78 = 6 Apply and Grow: Practice Place a decimal point where it belongs in the quotient. Question 4. 28.47 ÷ 0.39 = 7 3 . 0 Answer: Question 5. 75.85 ÷ 3.7 = 2 0 . 5 Answer: Question 6. 4.51 ÷ 4.1 = 1 . 1 Answer: Find the quotient. Then check your answer. Question 7. $$\sqrt [ 1.5 ]{ 7.5 }$$ Answer: Step 1: Multiply 7.5 by a power of 10 to make it a whole number. Then multiply 1.5 by the same power of 10. 7.5 x 10 = 75 1.5 x 10 = 15 75 ÷ 15 = 5 So, 7.5 ÷ 1.5 = 5 Question 8. $$\sqrt [ 0.13 ]{ 0.91 }$$ Answer: Step 1: Multiply 0.91 by a power of 100 to make it a whole number. Then multiply 0.13 by the same power of 100. 0.91 x 100 = 91 0.13 x 100 = 13 91 ÷ 13 = 7 So, 0.91 ÷ 0.13 = 7 Question 9. $$\sqrt [ 2.4 ]{ 2.88 }$$ Answer: Step 1: Multiply 2.88 by a power of 10 to make it a whole number. Then multiply 2.4 by the same power of 10. 2.88 x 10 = 28.8 2.4 x 10 = 24 Step 2: Divide 28.8 ÷ 24 28 ÷ 24 = 1 with remainder 4. 48 ÷ 24 = 2 with remainder 0. So, 2.88 ÷ 2.4 = 1.2 Question 10. $$\sqrt [ 0.6 ]{ 7.8 }$$ Answer: Step 1: Multiply 7.8 by a power of 10 to make it a whole number. Then multiply 0.6 by the same power of 10. 7.8 x 10 = 78 0.6 x 10 = 6 78 ÷ 6 = 13 So, 7.8 ÷ 0.6 = 13 Question 11. $$\sqrt [ 3.6 ]{ 4.32 }$$ Answer: Step 1: Multiply 4.32 by a power of 10 to make it a whole number. Then multiply 3.6 by the same power of 10. 4.32 x 10 = 43.2 3.6 x 10 = 36 Step 2: Divide 43.2 ÷ 36 43 ÷ 36 = 1 with remainder 7. 72 ÷ 36 = 2 with remainder 0. So, 4.32 ÷ 3.6 = 1.2 Question 12. $$\sqrt [ 0.1 ]{ 11.2 }$$ Answer: Step 1: Multiply 11.2 by a power of 10 to make it a whole number. Then multiply 0.1 by the same power of 10. 11.2 x 10 = 112 0.1 x 10 = 1 112 ÷ 1 = 112 So, 11.2 ÷ 0.1 = 112 Question 13. 40.42 ÷ 8.6 = ______ Answer: Step 1: Multiply 8.6 by a power of 10 to make it a whole number. Then multiply 40.42 by the same power of 10. 8.6 x 10 = 86 40.42 x 10 = 404.2 Step 2: Divide 404.2 ÷ 86 404 ÷ 86 = 4 with remainder 60. 602 ÷ 86 = 7 with remainder 0. So, 40.42 ÷ 8.6 = 4.7 Question 14. 7.2 ÷ 2.4 = _______ Answer: Step 1: Multiply 2.4 by a power of 10 to make it a whole number. Then multiply 7.2 by the same power of 10. 2.4 x 10 = 24 7.2 x 10 = 72 Step 2: Divide 72 ÷ 24 = 3 So, 7.2 ÷ 2.4 = 3 Question 15. 5.76 ÷ 1.8 = _______ Answer: Step 1: Multiply 1.8 by a power of 10 to make it a whole number. Then multiply 5.76 by the same power of 10. 1.8 x 10 = 18 5.76 x 10 = 57.6 Step 2: Divide 57.6 ÷ 18 57 ÷ 18 = 3 with remainder 3. 36 ÷ 18 = 2 with remainder 0. So, 5.76 ÷ 1.8 = 3.2 Question 16. YOU BE THE TEACHER Descartes says 4.14 ÷ 2.3 = 1.8. Is he correct? Explain. Answer: Step 1: Multiply 2.3 by a power of 10 to make it a whole number. Then multiply 4.14 by the same power of 10. 2.3 x 10 = 23 4.14 x 10 = 41.4 Step 2: Divide 41.4 ÷ 23 41 ÷ 23 = 1 with remainder 18. 184 ÷ 23 = 8 with remainder 0. So, 4.14 ÷ 2.3 = 1.8. Descartes answer is correct. Question 17. Logic What can you conclude about Newton’s quotient? Answer: The quotient will be above 5.72. Because if the divisor is less than 1 then the quotient must be greater than the dividend. Think and Grow: Modeling Real Life Example A farmer sells a bag of papayas for$5.46. How much does the bag of papayas weigh?

Divide the price of the papayas by the price per pound to find how much the bag of papayas weighs.
5.46 ÷ 1.3 = ? Estimate _______

So, the bag of papayas weighs _______ pounds.

Show and Grow

Use the table above.
Question 18.
You buy a honeydew for $6.08. What is the weight of the honeydew? Answer: Honeydew price =$0.8
6.08 ÷ 0.8
Step 1: Multiply 0.8 by a power of 10 to make it a whole number. Then multiply 6.08 by the same power of 10.
0.8 x 10 = 8
6.08 x 10 = 60.8
Step 2: Divide 60.8 ÷ 8
60 ÷ 8 = 7 with remainder 4.
48 ÷ 8 = 6 with remainder 0.
So, 6.08 ÷ 0.8 = 7.6
Weight of the honeydew = 7.6 pounds

Question 19.
You buy a pumpkin for $7.20 and a watermelon for$5.94. Does the watermelon or the pumpkin weigh more? How much more?
Pumpkin = 7.20 ÷ 0.45
Watermelon = 5.94 ÷ 0.33
7.20 ÷ 0.45
Step 1: Multiply 0.45 by a power of 10 to make it a whole number. Then multiply 7.20 by the same power of 10.
0.45 x 100 = 45
7.20 x 100 = 720
Step 2: Divide 720 ÷ 45 = 16
Pumpkin weight = 16 pounds
5.94 ÷ 0.33
Step 1: Multiply 0.33 by a power of 10 to make it a whole number. Then multiply 5.94 by the same power of 10.
0.33 x 100 = 33
5.94 x 100 = 594
Step 2: Divide 594 ÷ 33 = 18
Watermelon weight = 18 pounds
Watermelon weighs 2 pounds more than the pumpkin.

Question 20.
DIG DEEPER!
You pay $5 for a pineapple and receive$2.48 in change. The inedible parts of the pineapple weigh 1.75 pounds. How many pounds of edible pineapple do you have? Explain.

Amount paid = 5 – 2.48 = $2.52 pineapple price per pound =$0.63
2.52 ÷ 0.63
Step 1: Multiply 0.63 by a power of 10 to make it a whole number. Then multiply 2.52 by the same power of 10.
0.63 x 100 = 63
2.52 x 100 = 252
Step 2: Divide 252 ÷ 63 = 4
Total weight = 4 pounds
Edible pineapple = total weight – inedible pineapple weight
= 4 – 1.75
= 2.25
Edible pineapple weight = 2.25 pounds.

### Divide Decimals Homework & Practice 7.7

Multiply the divisor by a power of 10 to make it a whole number. Then write the equivalent expression.
Question 1.
16.15 ÷ 1.9
Step 1: Multiply 1.9 by a power of 10 to make it a whole number. Then multiply 16.15 by the same power of 10.
1.9 x 10 = 19
16.15 x 10 = 161.5
Step 2: Divide 161.5 ÷ 19
161 ÷ 19 = 8 with remainder 9.
95 ÷ 19 = 5 with remainder 0.
So, 16.15 ÷ 1.9 = 8.5

Question 2.
0.36 ÷ 0.09
Step 1: Multiply 0.09 by a power of 10 to make it a whole number. Then multiply 0.36 by the same power of 10.
0.09 x 100 = 9
0.36 x 100 = 36
Step 2: Divide 36 ÷ 9 = 4
So, 0.36 ÷ 0.09 = 4

Question 3.
2.04 ÷ 1.7
Step 1: Multiply 1.7 by a power of 10 to make it a whole number. Then multiply 2.04 by the same power of 10.
1.7 x 10 = 17
2.04 x 10 = 20.4
Step 2: Divide 20.4 ÷ 17
20 ÷ 17 = 1 with remainder 3.
34 ÷ 17 = 2 with remainder 0.
So, 2.04 ÷ 1.7 = 1.2

Place a decimal point where it belongs in the quotient.
Question 4.
81.27 ÷ 13.5 = 6 . 0 2

Question 5.
5.76 ÷ 3.2 = 1 . 8

Question 6.
47.15 ÷ 2.3 = 2 0 . 5

Question 7.
$$\sqrt [ 5.3 ]{ 21.2 }$$
Step 1: Multiply 5.3 by a power of 10 to make it a whole number. Then multiply 21.2 by the same power of 10.
5.3 x 10 = 53
21.2 x 10 = 212
212 ÷ 53 = 4
So, 21.2 ÷ 5.3 = 4

Question 8.
$$\sqrt [ 0.03 ]{ 76.38 }$$
Step 1: Multiply 0.03 by a power of 10 to make it a whole number. Then multiply 76.38 by the same power of 10.
0.03 x 100 = 3
76.38 x 100 = 7,638
Step 2: Divide 7638 ÷ 3
76 ÷ 3 = 25 with remainder 1.
138 ÷ 3 = 46 with remainder 0.
So, 76.38 ÷ 0.03 = 25.46

Question 9.

$$\sqrt [ 6.2 ]{ 33.48 }$$
Step 1: Multiply 6.2 by a power of 10 to make it a whole number. Then multiply 33.48 by the same power of 10.
6.2 x 10 = 62
33.48 x 10 = 334.8
Step 2: Divide 334.8 ÷ 62
334 ÷ 62 = 5 with remainder 24.
248 ÷ 62 = 4 with remainder 0.
So, 33.48 ÷ 6.2 = 5.4

Question 10.
0.63 ÷ 0.09 = ______
Step 1: Multiply 0.09 by a power of 10 to make it a whole number. Then multiply 0.63 by the same power of 10.
0.09 x 100 = 9
0.63 x 100 = 63
Step 2: Divide 63 ÷ 9 = 7
So, 0.63 ÷ 0.09 = 7

Question 11.
10.53 ÷ 3.9 = ______
Step 1: Multiply 3.9 by a power of 10 to make it a whole number. Then multiply 10.53 by the same power of 10.
3.9 x 10 = 39
10.53 x 10 = 105.3
Step 2: Divide 105.3 ÷ 39
105 ÷ 39 = 2 with remainder 27.
273 ÷ 39 = 7 with remainder 0.
So, 10.53 ÷ 3.9 = 2.7

Question 12.
33.8 ÷ 2.6 = ______
Step 1: Multiply 2.6 by a power of 10 to make it a whole number. Then multiply 33.8 by the same power of 10.
2.6 x 10 = 26
33.8 x 10 = 338
Step 2: Divide 338 ÷ 26 = 13
So, 33.8 ÷ 2.6 = 13

Question 13.
Logic
Without calculating, determine whether 5.4 ÷ 0.9 is greater than or less than 5.4. Explain.
5.4 ÷ 0.9 is greater than 5.4
If the divisor is less than 1 then the quotient must be greater than the dividend.

Question 14.
Structure
Explain how 35.64 ÷ 2.97 compares to 3,564 ÷ 297.
Both 35.64 ÷ 2.97 and 3,564 ÷ 297 are same.
Both dividend and divisor are multiplied by same power of 10.
35.64 x 100 = 3564
2.97 x 100 = 297.

Question 15.
Modeling Real Life
A farmer sells a bag of grapes for $5.88. How much do the grapes weigh? Answer: Bag of grapes price =$5.88
Grapes price per pound = $2.80 5.88 ÷ 2.8 Step 1: Multiply 2.8 by a power of 10 to make it a whole number. Then multiply 5.88 by the same power of 10. 2.8 x 10 = 28 5.88 x 10 = 58.8 Step 2: Divide 58.8 ÷ 28 58 ÷ 28 = 2 with remainder 2. 28 ÷ 28 = 1 with remainder 0. So, 5.88 ÷ 2.8 = 2.1 Grapes weight = 2.1 pounds Question 16. DIG DEEPER! Descartes makes 2.5 times as many ounces of applesauce as Newton. Newton eats 8 ounces of his applesauce, and then divides the rest equally into 3 containers. How much applesauce is in each of Newton’s containers? Answer: From the given information, Newton makes applesauce = 72.5 ÷ 2.5 Step 1: Multiply 2.5 by a power of 10 to make it a whole number. Then multiply 72.5 by the same power of 10. 2.5 x 10 = 25 72.5 x 10 = 725 Step 2: Divide 725 ÷ 25 = 29 So, newton makes 29 ounces of applesauce He eats 8 ounces = 29 – 8 = 21 21 ÷ 3 = 7 ounces. 7 ounces of applesauce is in each of Newton’s containers. Review & Refresh Question 17. Write the number in two other forms. Standard form: Word form: two hundred thirty thousand, eighty-two Expanded form: Answer: Standard form is 230,082 Expanded form is 200000 + 30000 + 80 + 2. ### Lesson 7.8 Insert Zeros in the Dividend Explore and Grow Use the model to find each quotient. Answer: Reasoning Why is the number of digits in the quotients you found above different than the number of digits in the dividends? Answer: Think and Grow: Inserting Zeros in the Dividend Example Find 52.6 ÷ 4. Estimate ________ Example Find 1 ÷ 0.08. Show and Grow Find the quotient. Then check your answer. Question 1. $$\sqrt [ 0.5 ]{ 85 }$$ Answer: Multiply 0.5 by a power of 10 to make it a whole number. Then multiply 85 by the same power of 10. 0.5 x 10 = 5 85 x 10 = 850 850 ÷ 5 = 170 So, 85 ÷ 0.5 = 170. Question 2. $$\sqrt [ 15 ]{ 9.6 }$$ Answer: Insert a zero in the dividend and continue to divide. 96 ÷ 15 = 6 with remainder 6. 60 ÷ 15 = 4 with remainder 0. So, 9.6 ÷ 15 = 0.64 Question 3. $$\sqrt [ 0.24 ]{ 2.52 }$$ Answer: Multiply 0.24 by a power of 10 to make it a whole number. Then multiply 2.52 by the same power of 10. 0.24 x 100 = 24 2.52 x 100 = 252 252 ÷ 24 252 ÷ 24 = 10 with remainder 12. Insert a zero in the dividend and continue to divide. 120 ÷ 24 = 5 with remainder 0. So, 2.52 ÷ 0.24 = 10.5. Apply and Grow: Practice Place a decimal point where it belongs in the quotient. Question 4. 3.24 ÷ 0.48 = 6 . 7 5 Answer: Question 5. 35 ÷ 0.5 = 7 0. Answer: Question 6. 12.8 ÷ 2.5 = 5 .1 2 Answer: Find the quotient. Then check your answer. Question 7. $$\sqrt [ 2.4 ]{ 0.84 }$$ Answer: Multiply 2.4 by a power of 10 to make it a whole number. Then multiply 0.84 by the same power of 10. 2.4 x 10 = 24 0.84 x 10 = 8.4 Insert a zero in the dividend and continue to divide. 84 ÷ 24 = 3 with remainder 12. 120 ÷ 24 = 5 with remainder 0. So, 0.84 ÷ 2.4 = 0.35 Question 8. $$\sqrt [ 0.32 ]{ 2.08 }$$ Answer: Multiply 0.32 by a power of 10 to make it a whole number. Then multiply 2.08 by the same power of 10. 0.32 x 100 = 32 2.08 x 100 = 208 208 ÷ 32 = 6 with remainder 16. Insert a zero in the dividend and continue to divide. 160 ÷ 32 = 5 with remainder 0. So, 2.08 ÷ 0.32 = 6.5 Question 9. $$\sqrt [ 4 ]{ 45.8 }$$ Answer: 45.8 ÷ 4 45 ÷ 4 = 11 with remainder 1. 18 ÷ 4 = 4 with remainder 2. Insert a zero in the dividend and continue to divide. 20 ÷ 4 = 5 with remainder 0. So, 45.8 ÷ 4 = 11.45. Question 10. 9 ÷ 1.2 = ______ Answer: Multiply 1.2 by a power of 10 to make it a whole number. Then multiply 9 by the same power of 10. 1.2 x 10 = 12 9 x 10 = 90 90 ÷ 12 Insert a zero in the dividend and continue to divide. 12 ) 90 ( 7.5 84 ——- 60 – 60 ——- 0 So, 9 ÷ 1.2 = 7.5 Question 11. 3.5 ÷ 2.5 = ______ Answer: Multiply 2.5 by a power of 10 to make it a whole number. Then multiply 3.5 by the same power of 10. 2.5 x 10 = 25 3.5 x 10 = 35 35 ÷ 25 Insert a zero in the dividend and continue to divide. 25 ) 35 ( 1.4 25 ——- 100 -100 ——- 0 So, 3.5 ÷ 2.5 = 1.4 Question 12. 1.8 ÷ 12 = ______ Answer: Insert a zero in the dividend and continue to divide. 12 ) 18 ( 1.5 12 ——- 60 – 60 ——- 0 So, 1.8 ÷ 12 = 0.15 Question 13. You read 2.5 chapters of the book each night. How many nights does it take you to finish the book? Answer: Total chapters in the book = 15 15 ÷ 2.5 Multiply 2.5 by a power of 10 to make it a whole number. Then multiply 15 by the same power of 10. 2.5 x 10 = 25 15 x 10 = 150 150 ÷ 25 = 6 nights to finish the book. Question 14. Precision Why does Newton place zeros to the right of the dividend but Descartes does not? Answer: Newton’s dividend does not have enough digits to divide completely, so he placed zeros to the right of the dividend. Descartes dividend is a multiple of divisor and it is divided completely, so no need of placing zeros. Think and Grow: Modeling Real Life Example The John Muir Trail in Yosemite National Park is 210 miles long. A hiker completes the trail in 20 days by hiking the same distance each day. How many miles does the hiker travel each day? Divide 210 miles by 20 to find how many miles the hiker travels each day. So, the hiker travels _______ miles each day. Show and Grow Question 15. A box of 15 tablets weighs 288 ounces. Each tablet weighs the same number of ounces. What is the weight of each tablet? Answer: Divide 288 ounces by 15 to find the weight of each tablet. 288 ÷ 15 15 ) 288 ( 19.2 15 ——- 138 -135 ——- 30 – 30 ——- 0 288 ÷ 15 = 19.2 The weight of each tablet = 19.2 ounces. Question 16. Which bag of dog food costs less per pound? Explain why it makes sense to write each quotient as a decimal in this situation. Answer: Question 17. DIG DEEPER! A farmer sells a pound of rice for$0.12 and a pound of oats for $0.08. Can you buy more pounds of rice or oats with$3? How much more? Explain.
Rice = 3 ÷ 0.12
Multiply 0.12 by a power of 10 to make it a whole number. Then multiply 3 by the same power of 10.
0.12 x 100 = 12
3 x 100 = 300
300 ÷ 12 = 25
Oats = 3 ÷ 0.08
Multiply 0.08 by a power of 10 to make it a whole number. Then multiply 3 by the same power of 10.
0.08 x 100 = 8
3 x 100 = 300
300 ÷ 8 = 37.5
I can buy 12.5 pounds oats more than the rice.

### Insert Zeros in the Dividend Homework & Practice 7.8

Place a decimal point where it belongs in the quotient.
Question 1.
9.3 ÷ 0.31 = 3 0.

Question 2.
10 ÷ 0.8 = 1 2 . 5

Question 3.
0.76 ÷ 0.25 = 3 . 0 4

Question 4.
$$\sqrt [ 0.8 ]{ 30 }$$
Multiply 0.8 by a power of 10 to make it a whole number. Then multiply 30 by the same power of 10.
0.8 x 10 = 8
30 x 10 = 300
300 ÷ 8
30 ÷ 8 = 3 with remainder 6.
60 ÷ 8 = 7 with remainder 4.
Insert a zero in the dividend and continue to divide.
40 ÷ 8 = 5 with remainder 0.
So, 30 ÷ 0.8 = 37.5.

Question 5.
$$\sqrt [ 15 ]{ 91.2 }$$
91.2 ÷ 15
91 ÷ 15 = 6 with remainder 1.
Insert a zero in the dividend and continue to divide.
120 ÷ 15 = 8 with remainder 0.
So, 91.2 ÷ 15 = 6.08

Question 6.
$$\sqrt [ 35 ]{ 97.3 }$$
97.3 ÷ 35
97 ÷ 35 = 2 with remainder 27.
273 ÷ 35 = 7 with remainder 28.
Insert a zero in the dividend and continue to divide.
280 ÷ 35 = 8 with remainder 0.
So, 97.3 ÷ 35 = 2.78.

Question 7.
3.57 ÷ 0.84 = ______
Multiply 0.84 by a power of 10 to make it a whole number. Then multiply 3.57 by the same power of 10.
0.84 x 100 = 84
3.57 x 100 = 357
357 ÷ 84
Insert a zero in the dividend and continue to divide.

84 ) 357 ( 4.25

336

——-

210

-168

——-

420

– 420

——-

0
3.57 ÷ 0.84 = 4.25

Question 8.
20.2 ÷ 4 = _____
Insert a zero in the dividend and continue to divide.

4 ) 20.2 ( 5.05

20

——

20

20

——

0
20.2 ÷ 4 = 5.05

Question 9.
1.74 ÷ 0.25 = _______
Multiply 0.25 by a power of 10 to make it a whole number. Then multiply 1.74 by the same power of 10.
0.25 x 100 = 25
1.74 x 100 = 174
174 ÷ 25
Insert a zero in the dividend and continue to divide.

25 ) 174 ( 6.96

150

——-

240

-225

——-

150

-150

——-

0
1.74 ÷ 0.25 = 6.96

Question 10.
A painter has 5 gallons of paint to use in a room. He uses 2.5 gallons of paint for 1 coat. How many coats can he paint?

Multiply 2.5 by a power of 10 to make it a whole number. Then multiply 5 by the same power of 10.
2.5 x 10 = 25
5 x 10 = 50
50 ÷ 25 = 2
He can paint 2 coats.

Question 11.
YOU BE THE TEACHER
Your friend say she can find 5.44 ÷ 0.64 by dividing both the divisor and dividend by 0.01 to make an equivalent problem with a whole-number divisor. Is he correct? Explain.
To divide this 5.44 ÷ 0.64, multiply 0.64 by a power of 10 to make it a whole number. Then multiply 5.44 by the same power of 10.
Multiplying by 100 and dividing by 0.01 both are same.
So, my friend is correct.

Question 12.
Writing
Explain when you need to insert a zero in the dividend when dividing.
When dividend does not have enough digits to divide completely, then we need to insert a zero in the dividend.
For example, 35 ÷ 25
Here 35 is not a multiple of 25, so we have to add a zero to 35.

Question 13.
Modeling Real Life
You cut a 12-foot-long streamer into 8 pieces of equal length. How long is each piece?
Length of each piece = 12 ÷ 8

8 ) 12 ( 1.5

8

—–

40

40

—–

0
So, length of each piece = 1.5

Question 14.
DIG DEEPER!
How many days longer does the bag of dog food last for the 20-pound dog than the 40-pound dog? Explain.

Total cups of dog food = 200
20-pound dog eats per day = 1.25 cups
40-pound dog eats per day = 1.25 x 2 = 2.5 cups
200 ÷ 1.25
Multiply 1.25 by a power of 10 to make it a whole number. Then multiply 200 by the same power of 10.
1.25 x 100 = 125
200 x 100 = 20,000
20,000 ÷ 125
200 ÷ 125 = 1 with remainder 75
7500 ÷ 125 = 60 with remainder 0.
So, 200 ÷ 1.25 = 160
Food lasts for the 20-pound dog = 160 days
40-pound dog = 200 ÷ 2.5
Multiply 2.5 by a power of 10 to make it a whole number. Then multiply 200 by the same power of 10.
2.5 x 10 = 25
200 x 10 = 2000
2000 ÷ 25 = 80
Food lasts for the 40-pound dog = 80 days

Review & Refresh

Question 15.
1.7 + 6.8 = ________

Question 16.
150.23 + 401.79 = _______

### Lesson 7.9 Problem Solving: Decimal Operations

Explore and Grow

Make a plan to solve the problem.
Three friends take a taxi ride that costs $4.75 per mile. They travel 10.2 miles and tip the driver$8. They share the total cost equally. How much does each friend pay?

Reasoning

Think and Grow: Problem Solving: Decimal Operations

Example
You spend $67.45 on the video game controller, the gaming headset, and 3 video games. The video games each cos the same amount. How much does each video game cost? Understand the Problem What do you know? • You spend a total of$67.45.
• The controller costs $15.49 and the headset costs$21.99.
• You buy 3 video games that each cost the same amount.

What do you need to find?
• You need to find the cost of each video game.

Make a Plan
How will you solve?
Write and solve an equation to find the cost of each video game.

Solve

So, each video game costs ______.

Show and Grow

Question 1.
v = (67.45 – 15.49 – 21.99) ÷ 3
= 29.97 ÷ 3
v = 9.99
So, each video game costs $9.99. Apply and Grow: Practice Understand the problem. What do you know? What do you need to find? Explain. Question 2. Your friend pays$84.29 for a sewing machine and 6 yards of fabric. The sewing machine costs $59.99. How much does each yard of fabric cost? Answer: What do you know? • You spend a total of$84.29 for a sewing machine and 6 yards of fabric.
• The sewing machine costs $59.99 and the 6 yards of fabric costs$24.3.

What do you need to find?

We have to find each yard of fabric cost.
1 yard of fabric cost = 24.3 ÷ 6
So, each yard of fabric costs = $4.05 Question 3. There are 25.8 grams of fiber in 3 cups of cooked peas. There are 52.5 grams of fiber in 5 cups of avocados. Which contains more fiber in 1 cup, cooked peas or avocados? Answer: Cooked peas = 25.8 ÷ 3 = 8.6 grams Avocados = 52.5 ÷ 3 = 10.5 grams So, avocados contains more fiber in 1 cup. Understand the problem. Then make a plan. How will you solve? Explain. Question 4. Your friend makes a hexagonal frame with a perimeter of 7.5 feet. You make a triangular frame with a perimeter of 5.25 feet. Whose frame has longer side lengths? How much longer? Answer: Hexagonal perimeter = 6a = 7.5 feet Triangular perimeter = 3sides(3a) = 5.25 feet So, 6a ÷ 2 = 3a 7.5 ÷ 2 = 3.75 feet 5.25 – 3.75 = 1.5 feet So, triangular frame has 1.5 feet longer side lengths. Question 5. You spend$119.92 on the wet suit, the snorkeling equipment, and 2 research books. The books each cost the same amount. How much does each book cost?

Write and solve an equation to find the cost of each book.
cost of each book = (Amount spend – wet suit cost – snorkeling equipment cost) ÷ 2
= (119.92 – 64.95 – 14.99) ÷ 2
= 39.98 ÷ 2
= 19.99
So, each book costs $19.99. Question 6. DIG DEEPER! You pour goop into molds and bake them to make plastic lizards. You run out of goop and go shopping for more. Which package costs less per ounce of goop? Explain. Answer: Fluorescent package = 40.5 ÷ 2.25 =$18
Color-Changing package = 16.2 ÷ 1.8 = $9 So, Color-Changing package costs less per ounce of goop. Think and Grow: Modeling Real Life Example Descartes spends$16.40 on the game app, an e-book, and 5 songs. The e-book costs 4 times as much as the game app. The songs each cost the same amount. How much does each song cost?

Think: What do you know? What do you need to find? How will you solve?
Step 1: Multiply the cost of the app by4 to find the cost of the e-book.
1.99 × 4 = 7.96 The e-book costs _______.
Step 2: Write and solve an equation to find the cost of each song.

Let c represent the cost of each song.
c = (16.40 – 1.99 – 7.96) ÷ 5c
= _____ ÷ 5
= _____
So, each song costs $______. Show and Grow Question 7. You spend$2.24 on a key chain, a bookmark, and 2 pencils. The key chain costs 3 times as much as the bookmark. The pencils each cost the same amount. How much does each pencil cost?

Given that,
Bookmark cost = $0.45 Key chain cost = 3 x 0.45 =$1.35
Write and solve an equation to find the cost of each pencil.
cost of each pencil = (Amount spend – keychain cost – bookmark cost) ÷ 2
= (2.24 – 1.35 – 0.45) ÷ 2
= 0.44 ÷ 2
= $0.22 So, cost of each pencil =$0.22.

Question 8.
Newton buys an instant-print camera, a camera bag, and 2 packs of film. He pays $113.96 after using a$5 coupon. The camera costs $69.40, which is 5 times as much as the camera case. How much does each pack of film cost? Answer: Total cost = 113.96 + 5 =$118.96
Camera cost = $69.40 Camera case cost = 69.40 ÷ 5 =$13.88
Write and solve an equation to find the cost of each pack of film cost.
cost of each pack of film = (amount spend – camera cost – camera case cost) ÷ 2
= (118.96 – 69.40 – 13.88) ÷ 2
= 35.68 ÷ 2
= 17.84
So, cost of each pack of film = $17.84 ### Problem Solving: Decimal Operations Homework & Practice 7.9 Understand the problem. What do you know? What do you need to find? Explain. Question 1. A 20-ounce bottle of ketchup costs$2.80. A 14-ounce bottle of mustard costs $2.38. Which item costs less per ounce? How much less? Answer: 20-ounce bottle of ketchup costs =$2.80
14-ounce bottle of mustard costs = $2.38 1 ounce ketchup = 2.8 ÷ 20 =$0.14
1 ounce mustard = 2.38 ÷ 14 = $0.17 Ketchup costs$0.03 less per ounce than mustard.

Question 2.
Gymnast A scores the same amount in each of his 4 events. Gymnast B scores the same amount in each of his 3 events. Which gymnast scores more in each of his events? How much more?

Gymnast A in each of his events = 33.56 ÷ 4 = 8.39 points
Gymnast B in each of his events = 25.05 ÷ 3 = 8.35 points
Gymnast A scores 0.04 points more in each of his events than gymnast B.

Understand the problem. Then make a plan. How will you solve? Explain.
Question 3.
Three children’s tickets to the circus cost $53.85. Two adult tickets to the circus cost$63.90. How much more does 1 adult ticket cost than 1 children’s ticket? Which item costs more per ounce? How much more?
3 children’s tickets cost = $53.85 2 adult tickets cost =$63.90
1 adult ticket cost = 63.90 ÷ 2 = $31.95 1 children’s ticket cost = 53.85 ÷ 3 =$17.95
One adult ticket cost is $14 more than 1 children’s ticket. Question 4. A chef at a restaurant buys 50 pounds of red potatoes for$27.50 and 30 pounds of sweet potatoes for $22.50. Which kind of potato costs more per pound? How much more? Answer: Red potatoes per pound = 27.5 ÷ 50 =$0.55
Sweet potatoes per pound = 22.5 ÷ 30 = $0.75 Sweet potatoes costs$0.2 more per pound than red potatoes.

Question 5.
Modeling Real Life
You download 2 music videos, a TV series, and a movie for $42.95 total. The TV series costs 2 times as much as the movie. How much does each music video cost? Answer: Total cost =$42.95
Movie cost = $12.99 TV series cost =$25.98
Write and solve an equation to find the cost of each music video cost.
cost of each music video = (total cost – movie cost – TV series cost) ÷ 2
= (42.95 – 12.99 – 25.98) ÷ 2
= 3.98 ÷ 2
= 1.99
So, cost of each music video = $1.99. Question 6. DIG DEEPER! Which item costs more per ounce? How much more? Answer: Glue cost = 23.04 ÷ 1 = 23.04 Paste cost = 4.00 ÷ 2 = 2 Glue costs$21.04 more than paste.

Review & Refresh

Find the quotient.
Question 7.
4,000 ÷ 20 = ______

Question 8.
900 ÷ 300 = _______

Question 9.
5,600 ÷ 800 = _______

Question 1.
Multiple teams adopt different sections of a state highway to clean. The teams must clean both sides of their adopted section of the highway.

a. The teams clean their section of the highway over 4 days. They clean the same distance each day.How many miles of the highway does each team clean each day?
b. Each team divides their daily distance equally among each team member. Which team’s members clean the greatest distance each day?
c. The team that collects the greatest amount of litter per team member wins a prize.Which team wins the prize?

Question 2.
In a community, 25 people volunteer to clean the rectangular park shown. The park is divided into sections of equal area. One section is assigned to each volunteer. What is the area of the section that each volunteer cleans? What is one possible set of dimensions for 24.5 m each section?

### Divide Decimals Activity

Race Around the World: Division
Directions:
1. Players take turns.
2. On your turn, flip a Race Around the World: Division Card and find the quotient.
3. Move your piece to the next number on the board that is highlighted in the quotient.
4. The first player to make it back to North America wins!

### Divide Decimals Chapter Practice

7.1 Division Patterns with Decimals

Find the quotient.
Question 1.
25 ÷ 102 = ______
First Simplify the 102 which means  10 x 10  =100, then we need to calculate the fraction to a decimal just divide the numerator (25) by the denominator (100).
When we divide by 100, the decimal point moves two places to the left.
25 ÷ 102 = 0.25.

Question 2.
1.69 ÷ 0.01 = ______
Answer: To convert this simple fraction to a decimal just divide the numerator (1.69) by the denominator (0.01). When we divide by 0.01, the decimal point moves two places to the right.

1.69 ÷ 0.01 = 169.

Question 3.
681 ÷ 103 = ______
First Simplify the 103 which means  10 x 10 x 10 =1000, then we need to calculate the fraction to a decimal just divide the numerator (681) by the denominator (1000).
When we divide by 1000, the decimal point moves three places to the left.
681 ÷ 103 = 0.681.

Question 4.
5.7 ÷ 0.1 = _____
To convert this simple fraction to a decimal just divide the numerator (5.7) by the denominator (0.1). When we divide by 0.1, the decimal point moves one places to the right.
5.7 ÷ 0.1 = 57

Question 5.
200 ÷ 0.01 = _____
To convert this simple fraction to a decimal just divide the numerator 200 by the denominator (0.01). When we divide by 0.01, the decimal point moves one places to the right.

Question 6.
41.3 ÷ 10 = _____
Answer: To convert this simple fraction to a decimal just divide the numerator (41.3) by the denominator (10). When we divide by 10, the decimal point moves one places to the left.
41.3 ÷ 10 = 4.13

Find the value of k.
Question 7.
74 ÷ k = 7,400
74 ÷ 7400 = k

Explanation: To convert this simple fraction to a decimal just divide the numerator (74) by the denominator (7400). When we divide by 100, the decimal point moves two places to the left.
74 ÷ 7400 = 0.01
k = 0.01.

Question 8.
k ÷ 0.1 = 8.1
k = 8.1 x 0.1
k = 0.81.

Question 9.
0.35 ÷ k = 0.035
0.35 ÷ 0.035 = k
Explanation: To convert this simple fraction to a decimal just divide the numerator (0.35) by the denominator (0.035). When we divide by 0.01, the decimal point moves two places to the right.
0.35 ÷ 0.035 = 10
k = 10.

7.2 Estimate Decimal Quotients

Estimate the quotient.
Question 10.
9.6 ÷ 2
9.6 is closer to 10.
10 ÷ 2 = 5
9.6 ÷ 2 is about 5.

Question 11.
37.2 ÷ 6.4
Round the divisor 6.4 to 6.
Think: What numbers close to 37.2 are easily divided by 6?
Use 36.
36 ÷ 6 = 6
So, 37.2 ÷ 6.4 is about 6.

Question 12.
44.8 ÷ 4.7
Round the divisor 4.7 to 5.
Think: What numbers close to 44.8 are easily divided by 5?
Use 45.
45 ÷ 5 = 9
So, 44.8 ÷ 4.7 is about 9.

Question 13.
78.2 ÷ 10.8
Round the divisor 10.8 to 11.
Think: What numbers close to 78.2 are easily divided by 11?
Use 77.
77 ÷ 11 = 7
So, 78.2 ÷ 10.8 is about 7.

7.3 Use Models to Divide Decimals by Whole Numbers

Use the model to find the quotient.
Question 14.
1.4 ÷ 2

Think: 1.4 is 1 ones and 4 tenths.
14 tenths can be divided equally as 2 groups of 7 tenths.
1.4 ÷ 2 = 0.7

Question 15.
2.85 ÷ 3

Think: 2.85 is 2 ones, 8 tenths and 5 hundredths.
28 tenths can be divided equally as 3 groups of 9 tenths with remainder 1. Remainder has to place before 5 hundredths.
15 hundredths can be divided equally as 3 groups of 5 hundredths.
So, 285 hundredths can be divided equally as 3 groups of 95 hundredths.
2.85 ÷ 3 = 0.95

Use a model to find the quotient.
Question 16.
1.28 ÷ 4
Think: 1.28 is 1 ones, 2 tenths and 8 hundredths.
12 tenths can be divided equally as 4 groups of 3 tenths
8 hundredths can be divided equally as 4 groups of 2 hundredths.
So, 128 hundredths can be divided equally as 4 groups of 32 hundredths.
1.28 ÷ 4 = 0.32

Question 17.
3.5 ÷ 5
Think: 3.5 is 3 ones and 5 tenths.
35 tenths can be divided equally as 5 groups of 7 tenths.
3.5 ÷ 5 = 0.7

7.4 Divide Decimals by One-Digit Numbers

Question 18.
$$\sqrt [ 3 ]{ 14.1 }$$
Divide the ones
14 ÷ 3
4 ones x 3 = 12
14 ones – 12 ones
There are 2 ones left over.
Divide the tenths
21 ÷ 3 = 7 tenths.
So, 14.1 ÷ 3 = 4.7

Question 19.
$$\sqrt [ 6 ]{ 67.68 }$$
Divide the ones
67 ÷ 6
11 ones x 6 = 66
67 ones – 66 ones
There are 1 ones left over.
Divide the tenths
16 ÷ 6
2 tenths x 6 = 12
16 – 12 = 4
There are 4 tenths left over.
Divide the hundredths
48 ÷ 6 = 8 hundredths.
So, 67.68 ÷ 6 = 11.28

Question 20.
$$\sqrt [ 8 ]{ 105.6 }$$
Divide the ones
105 ÷ 8
13 ones x 8 = 104
105 ones – 104 ones
There are 1 ones left over.
Divide the tenths
16 ÷ 8 = 2 tenths.
So, 105.6 ÷ 8 = 13.2

Question 21.
Number Sense
Evaluate (84.7 + 79.8) ÷ 7.
(84.7 + 79.8) ÷ 7 = 164.5 ÷ 7
Divide the ones
164 ÷ 7
23 ones x 7 = 161
164 ones – 161 ones
There are 3 ones left over.
Divide the tenths
35 ÷ 7
5 tenths x 7
35 – 35 = 0
There are 0 tenths left over.
So, 164.5 ÷ 7 = 23.5

7.5 Divide Decimals by Two-Digit Numbers

Question 22.
$$\sqrt [ 32 ]{ 45.12 }$$
Divide the ones
45 ÷ 32
1 ones x 32 = 32
45 ones – 32 ones
There are 13 ones left over.
Divide the tenths
131 ÷ 32
4 tenths x 32 = 128
131 – 128 = 3
There are 3 tenths left over.
Divide the hundredths
32 ÷ 32 = 1 hundredths.
So, 45.12 ÷ 32 = 1.41

Question 23.
$$\sqrt [ 15 ]{ 9.15 }$$
Divide the tenths
91 ÷ 15
6 tenths x 15 = 90
91 – 90 = 1
There are 1 tenths left over.
Divide the hundredths
15 ÷ 15 = 1 hundredths.
So, 9.15 ÷ 15 = 0.61

Question 24.
$$\sqrt [ 73 ]{ 102.2 }$$
Divide the ones
102 ÷ 73
1 ones x 73 = 73
102 ones – 73 ones
There are 29 ones left over.
Divide the tenths
292 ÷ 73 = 4 tenths.
So, 102.2 ÷ 73 = 1.4

Question 25.
17.4 ÷ 87 = ______
Divide the tenths
174 ÷ 87
2 tenths x 87
174 – 174 = 0
There are 0 tenths left over.
17.4 ÷ 87 = 0.2

Question 26.
245.82 ÷ 51 = _______
Divide the ones
245 ÷ 51
4 ones x 51 = 204
245 ones – 204 ones
There are 41 ones left over.
Divide the tenths
418 ÷ 51
8 tenths x 51
418 – 408 = 10
There are 10 tenths left over.
Divide the hundredths
102 ÷ 51 = 2 hundredths
So, 245.82 ÷ 51 = 4.82

Question 27.
5.88 ÷ 42 = ______
Divide the tenths
58 ÷ 42
1 tenths  x 42 = 42
58 tenths – 42 tenths
There are 16 tenths left over.
Divide the hundredths
168 ÷ 42
4 hundredths x 42
168 – 168 = 0
There are 0 hundredths left over.
So, 5.88 ÷ 42 = 0.14

7.6 Use Models to Divide Decimals

Use the model to find the quotient.
Question 28.
0.9 ÷ 0.45 = ______

Shade 9 columns to represent 0.9.
Divide the model to show groups of 0.45.
There are 2 groups of 45 hundredths.
So, 0.9 ÷ 0.45 = 2

Question 29.
0.1 ÷ 0.05 = ______

Shade 1 column to represent 0.1.
Divide the model to show groups of 0.05.
There are 2 groups of 5 hundredths.
So, 0.1 ÷ 0.05 = 2

Question 30.
1.6 ÷ 0.4 = ______

Shade 16 columns to represent 1.6.
Divide the model to show groups of 0.4.
There are 4 groups of 4 tenths.
So, 1.6 ÷ 0.4 = 4

Question 31.
1.9 ÷ 0.38 = ______

Shade 19 columns to represent 1.9.
Divide the model to show groups of 0.38.
There are 5 groups of 38 hundredths.
So, 1.9 ÷ 0.38 = 5

7.7 Divide Decimals

Question 32.
$$\sqrt [ 2.57 ]{ 20.56 }$$
Multiply 2.57 by a power of 10 to make it a whole number. Then multiply 20.56 by the same power of 10.
2.57 x 100 = 257
20.56 x 100 = 2056
2056 ÷ 257 = 8
So, 20.56 ÷ 2.57 = 8.

Question 33.
$$\sqrt [ 4.7 ]{ 16.92 }$$
Multiply 4.7 by a power of 10 to make it a whole number. Then multiply 16.92 by the same power of 10.
4.7 x 10 = 47
16.92 x 10 = 169.2
Step 2 : Divide 169.2 ÷ 47
169 ÷ 47 = 3 with remainder 28.
282 ÷ 47 = 6 with remainder 0.
So, 16.92 ÷ 4.7 = 3.6.

Question 34.
$$\sqrt [ 5.3 ]{ 63.6 }$$
Multiply 5.3 by a power of 10 to make it a whole number. Then multiply 63.6 by the same power of 10.
5.3 x 10 = 53
63.6 x 10 = 636
636 ÷ 53 = 12
So, 63.6 ÷ 5.3 = 12.

7.8 Insert Zeros in the Dividend

Question 35.
$$\sqrt [ 4 ]{ 36.2 }$$
36.2 ÷ 4
36 ÷ 4 = 9
Insert a zero in the dividend and continue to divide.
20 ÷ 4 = 5
So, 36.2 ÷ 4 = 9.05.

Question 36.
$$\sqrt [ 4.8 ]{ 85.2 }$$
Multiply 4.8 by a power of 10 to make it a whole number. Then multiply 85.2 by the same power of 10.
4.8 x 10 = 48
85.2 x 10 = 852
852 ÷ 48
85 ÷ 48 = 1 with remainder 37.
372 ÷ 48 = 7 with remainder 36.
Insert a zero in the dividend and continue to divide.
360 ÷ 48 = 7 with remainder 24.
240 ÷ 48 = 5 with remainder 0.
So, 85.2 ÷ 4.8 = 17.75.

Question 37.
$$\sqrt [ 12 ]{ 52.2 }$$
52.2 ÷ 12
52 ÷ 12 = 4 with remainder 4.
42 ÷ 12 = 3 with remainder 6.
Insert a zero in the dividend and continue to divide.
60 ÷ 12 = 5 with remainder 0.
So, 52.2 ÷ 12 = 4.35.

Question 38.
5 ÷ 0.8 = ______
Multiply 0.8 by a power of 10 to make it a whole number. Then multiply 5 by the same power of 10.
0.8 x 10 = 8
5 x 10 = 50
50 ÷ 8
Insert a zero in the dividend and continue to divide.

8 ) 50 ( 6.25

48

——-

20

-16

——-

40

40

——–

0
So, 5 ÷ 0.8 = 6.25

Question 39.
23.7 ÷ 6 = ______
Insert a zero in the dividend and continue to divide.

6 ) 23.7 ( 3.95

18

——-

57

– 54

——-

30

30

——–

0
23.7 ÷ 6 = 3.95

Question 40.
138.4 ÷ 16 = ______
Insert a zero in the dividend and continue to divide.
16 ) 138.4 ( 8.65

128

——-

104

– 96

——–

80

80

7.9 Problem Solving: Decimal Operations

Question 41.
You spend $28.08 on the fabric scissors, buttons, and two craft kits. The kits each cost the same amount. How much does ASSORTED$6.13 each kit cost?

Given that,
Total amount spent = $28.08 Fabric scissors cost =$6.13
Buttons cost = $3.97 Write and solve an equation to find the cost of each kit. cost of each kit = (amount spend – fabric scissors cost – buttons cost) ÷ 2 = (28.08 – 6.13 – 3.97) ÷ 2 = 17.98 ÷ 2 = 8.99 So, cost of each craft kit =$8.99

### Divide Decimals Cumulative Practice

Question 1.
Which statement is true?

According to BODMAS rule,
Statement c is correct.

Question 2.
You round 23 × 84 and get an underestimate. How did you estimate?
A. 20 × 80
B. 30 × 90
C. 25 × 90
D. 25 × 90
23 × 84 round to 20 × 80 because it is closer to given equation.
84 is close to 80 and all the others options having number 90.
Difference between the numbers in remaining options is greater than the option A numbers.

Question 3.
Which expressions have a product that is shown?

Except option 1, remaining all the other options have the product(0.4) shown in the image.

Question 4.
What number is $$\frac{1}{10}$$ of 800?
A. 0.8
B. 8
C. 80
D. 8,000
$\dpi{100} \small \frac{1}{10}$ (800) = 80

Question 5.
Which number divided by 0.01 is 14 more than 37?
A. 0.51
B. 5.1
C. 51
D. 5,100
14 more than 37 = 37 + 14 = 51
51 x 0.01 = 0.51

Question 6.
Which expressions have a quotient of 40?

2800 ÷ 70, 160 ÷ 4, 3600 ÷ 900 and 8000 ÷ 200 have a quotient of 40.

Question 7.
Which equation is shown by the quick sketch?

Question 8.
What is the value of k?
0.036 × k = 36
A. 10
B. 103
C. 100
D. 36
k = 36 ÷ 0.036
k = 1000 = 103(option B).

Question 9.
What is the quotient of 11.76 and 8?
A. 1.47
B. 1.97
C. 14.7
D. 94.08
11.76 ÷ 8
11 ÷ 8 = 1 with remainder 3.
37 ÷ 8 = 4 with remainder 5.
56 ÷ 8 = 7 with remainder 0.
So, quotient of 11.76 and 8 = 1.47(option A).

Question 10.
Newton wins a race by seven thousandths of a second. What is this number in standard form?
A. 0.007
B. 0.07
C. 0.7
D. 7,00
0.007 is in standard form.

Question 11.
Evaluate 30 – (9 + 6) ÷ 3.
A. 5
B. 19
C. 9
D. 25
According to BODMAS rule.
30 – (9 + 6) ÷ 3
= 30 – (15 ÷ 3)
= 30 – 5
= 25

Question 12.
A food truck owner sells 237 gyros in 1 day. Each gyro costs $7.How much money does the owner collect in 1 day? A.$659
B. $1,419 C.$1,659
D. $11,249 Answer: Each gyro costs$7
237 gyros in 1 day cost = 237 x 7 = $1659 So, the owner collects$1659 in 1 day.

Question 13.
What is the quotient of 4,521 and 3?

4521 ÷ 3 = 1507
So, quotient of 4,521 and 3 is 1507.

Question 14.
What is the value of b?
104 = 10b × 10
A. 3
B. 4
C. 5
D. 10
104 = 10b × 10
If b= 3,
10b × 10 = 103 × 10
= 103+1
= 104
So, b= 3.

Question 15.
Part A What is the area of the sandbox?

Part B The playground committee wants to make the area of the sandbox 2 times the original area. What is the new area? Explain.

Question 16.
Which expressions have a product of 1,200?

30 x 40, 12 x 102 and 120 x 10 have a product of 1,200.

Question 17.
A 5-day pass to a theme park costs $72.50. A 2-day pass to the same park costs$99.50. How much more does the 2-day pass cost each day than the 5-day pass each day?
A. $14.50 B.$35.25
C. $49.75 D.$64.25
5-day pass costs each day = 72.5 ÷ 5 = $14.5 2-day pass costs each day = 99.5 ÷ 2 =$49.75
2-day pass cost each day \$35.25 more than the 5-day pass each day.

Question 18.
Which expressions have a quotient with the first digit in the tens place?

4,536 ÷ 56 = 81
6,750 ÷ 45 = 150
2,403 ÷ 89 = 27
1,496 ÷ 17 = 88
Except option 2, all the other options have a quotient with the first digit in the tens place.

### Divide Decimals STEAM Performance Task

You experiment with levers for your school’s science fair.

Question 1.
You balance the seesaw lever by placing different weights on either side at different distances from the middle. You find the formula for balancing the seesaw lever is (left weight) × (left distance) = (right weight) × (right distance). You test the formula using various combinations of weights.

a. Use the formula to complete the table for the 2nd and 3rd attempts.
b. For your 4th attempt, you have up to 25 pounds in weights to place on each side of the lever. Choose a whole pound weight for the left side and balance the lever to complete the table.
c. The total length of your seesaw lever is 40 inches. Can you balance a 50-pound weight with a 1-pound weight? Explain.
d. For your science fair display, you balance the lever by placing another gram weight on the right side. Which gram weight should you use?

e. How can you apply what you learn from the science fair project to a playground?

You help set up tables for the science fair. There are 93 science fair displays. You use the display boards to determine how many tables to use.

Question 2.
Each display board opens up to form three sides of a trapezoid as shown.
a. How much room do you think each display board needs to open up? Explain.
b. You place the display boards next to each other on 12-foot long tables. How many display boards can you fit on one table?
c. You use one table for snacks and one table for award ribbons. What is the least number of tables you can use? Explain.
d. The diagram shows the room where the science fair is held. Each table for the science fair is 3 feet wide. Your teacher says the ends of the tables can touch to save space. Complete the diagram to arrange the tables so that visitors and judges can see each display board.

Question 3.
Use the Internet or some other resource to learn about other types of science fair projects. Describe one interesting science fair project you want to complete.

Answer:  One of the interesting science fair project is :-
How To Make A Bottle Rocket
-> Did you know you can make and launch a water bottle rocket using a plastic bottle, water, cork,           needle adaptor and pump ?
How do water bottle rockets work?
As you pump air into the bottle the pressure inside the bottle builds up until the force of the air pushing on the water is enough to  force the cork out of the end of the bottle. The water rushes out of the bottle in one direction whilst the bottle pushes back in the other. This results in the bottle shooting upwards.
What you need to make a bottle rocket
->  An empty plastic bottle
->  Cardboard made into a cone and 4 fins
->  A cork
->  A pump with a needle adaptor
->  Water
You can buy a kit with the parts apart from the pump and the bottle-please check the contents before buying.
Instructions – How to make a bottle rocket
Push the needle adaptor of the pump through the cork, it needs to go all the way through so you might have to trim the cork a little bit.
-> Decorate the bottle with the cone and fins.
-> Fill the bottle one quarter full of water and push the cork in tightly.
-> Take the bottle outside and connect the pump to the needle adaptor. Ours wouldn’t stand up on the fins so we rested it on a box, but if you make some strong fins it should stand up by itself.
-> Pump air into the bottle, making sure all spectators stand back, the bottle will lift off with force after a few seconds.
Why does the water bottle rocket launch?
As you pump air into the bottle pressure builds up inside. If you keep pumping, the force of the air pushing on the water eventually becomes strong enough to force the cork out of the bottle allowing water to rush out in one direction while the bottle pushes back in the other direction. This forces the rocket upwards.
Space

Conclusion:

I wish the information provided in the above article regarding Big Ideas Math Book 5th Grade Answer Key Chapter 7 Divide Decimals is helpful for you. For any queries, you can post the comments in the below section.

## Big Ideas Math Book Grade 2 Answer Key Chapter 11 Measure And Estimate Lengths

Students who want to improve their math skills can refer to this Big Ideas Math 2nd Grade 11th Chapter Measure And Estimate Lengths Solutions and prepare well. By solving the questions, you can know how to use a ruler to measure lengths, compare the measurements of different objects. One can finish their homework easily by checking Big Ideas Math Answers Grade 2 Chapter 11 Measure And Estimate Lengths. Click on the topic-wise links mentioned below to get the answers for all types of questions related to Measure And Estimate Lengths.

Vocabulary

Lesson: 1 Measure Lengths in Centimeters

Lesson: 2 Measure Objects Using Metric Length Units

Lesson: 3 Estimate Lengths in Metric Units

Lesson: 4 Measure Lengths in Inches

Lesson: 5 Measure Objects Using Customary Length Units

Lesson: 6 Estimate Lengths in Customary Units

Lesson: 7 Measure Objects Using Different Length Units

Lesson: 8 Measure and Compare Lengths

Chapter 11: Measure And Estimate Lengths

### Measure And Estimate Lengths Vocabulary

Organize It
Use the review words to complete the graphic organizer.

Answer: Tree is measured bu Height and Bird is measured by Length

Define It
Use your vocabulary cards to match.

### Lesson 11.1 Measure Lengths in Centimeters

Explore and Grow

Use a centimeter cube to find the length of each string.

Explain how you measured.
______________________
______________________
______________________
Answer: The centimeters were measured by Scale.

Show and Grow

Measure.
Question 1.

Question 2.

Question 3.

Apply and Grow: Practice

Measure.
Question 4.

Question 5.

Question 6.
Draw a pencil that is about 9 centimeters long.

Question 7.
YOU BE THE TEACHER
Newton says the ribbon is about 14 centimeters long. Is he correct? Explain.

_______________________
_______________________
Answer: No. It is not correctly measured on a scale. It should be measured from 0 cm to 14 cms.

Think and Grow: Modeling Real Life

Will the hammer fit inside a toolbox that is 40 centimeters long? Explain.

______________________
______________________
Answer: The toolbox will be around 50 centimeters long, So the Hammer with 40 centimeters will fit in it.

Show and Grow

Question 8.
Will the sunglasses fit inside a case that is 10 centimeters long? Explain.

________________________
________________________
Answer: Yes, Sunglasses will fit inside the case that is 10 centimeters as Sunglasses will be around 9 centimeters.

### Measure Lengths in Centimeters Homework & Practice 11.1

Measure.
Question 1.

Question 2.

Question 3.

Question 4.

Question 5.
Precision
Which crayon is shorter than 8 centimeters?

Answer: Yellow and Green crayons are shorter than 8 centimeters.

Question 6.
Modeling Real Life
Will the pen fit inside a pouch that is 18 centimeters long? Explain.

______________________
_______________________
Answer: Yes the pen will fit inside a pouch that is 18 centimeters long as the pen is around 15 centimeters.

Review & Refresh

Question 7.

Question 8.

Question 9.

### Lesson 11.2 Measure Objects Using Metric Length Units

Explore and Grow

Which real-life objects are shorter than a centimeter ruler?

Show and Grow

Find and measure the object shown in your classroom.
Question 1.

Question 2.

meter ______ centimeters

Question 3.

Apply and Grow: Practice

Find and measure the object shown in your classroom.
Question 4.

Question 5.

Question 6.

Question 7.

Question 8.
Choose Tools
Would you measure the length of a bus with a centimeter ruler or a meter stick? Why?

________________________
________________________
Answer: The length of a bus can be measured with a meter stick because the bus is big inside.
The centimeter-scale is small and the meter scale is big.

Think and Grow: Modeling Real Life

Your friend says a car has a length of about 4. Is the car about 4 meters long or about 4 centimeters long? Explain.

The car is about 4 ________ long.
________________________
________________________

Show and Grow

Question 9.
Your friend says a shoe has a length of about 12. Is the shoe about 12 centimeters long or about 12meters long? Explain.

The shoe is about 12 ____________ long.
______________________
______________________

Question 10.
DIG DEEPER!
Your friend places 2 of the same real objects end to end. Together, they have a length of about 18 centimeters. Which object did your friend use? Explain.

_________________________
_________________________
Answer: 1st and 2nd object together will be 18 centimeters. As 3rd object will be larger than 18 centimeters.

### Measure Objects Using Metric Length Units Homework & Practice 11.2

Find and measure the object.
Question 1.

Question 2.

Question 3.

Question 4.

Question 5.
Number Sense
Complete the sentences using centimeters or meters.
A window is about 2 ________ long.
A finger is about 8 ______ long.
A zucchini is about 12 _______ long.
An airplane is about 20 _______ long.

Question 6.
Modeling Real Life
Your friend says that the length of a soccer field is about 91. Is the soccer field about 91 centimeters long or about 91 meters long? Explain.

The soccer field is about 91 _______ long.
______________________
_______________________

Question 7.
DIG DEEPER!
Order the lengths from shortest to longest.

_________, _________, _________
Answer: 3 centimeters, 1 meter, 200 centimeter.

Review & Refresh

Question 8.
3 + 7 = _____ + 3
The commutative property of addition says that changing the order of addends does not change the sum.
So, 3 + 7 = 7 + 3

Question 9.
5 + 4 = 4 + ______
The commutative property of addition says that changing the order of addends does not change the sum.
So, 5 + 4 = 4 + 5

Question 10.
6 + 0 = ____ + 6
The commutative property of addition says that changing the order of addends does not change the sum.
So, 6 + 0 = 0 + 6

Question 11.
1 + 2 = 2 + ____
The commutative property of addition says that changing the order of addends does not change the sum.
So, 1 + 2 = 2 + 1

### Lesson 11.3 Estimate Lengths in Metric Units

Explore and Grow

Find an object that is shorter than the string.

Without using a ruler, tell how long you think the object is.

______ centimeters

Explain.
________________________
________________________
Answer: The Pencil is around 12 centimeters which is shorter than the 13 centimeters string.

Show and Grow

Question 1.
The chalk is about 8 centimeters long. What is the best estimate of the length of the toothpick?

Question 2.
The fishing pole is about 1 meter long. What is the best estimate of the length of the alligator?

Apply and Grow: Practice

Question 3.
The hover board is about 1 meter long. What is the best estimate of the length of the surfboard?

Question 4.
The pineapple is about 25 centimeters long. What is the best estimate of the length of the asparagus?

Question 5.
What is the best estimate of the length of a piece of notebook paper?
21 centimeters
1 meter
5 centimeters

Question 6.
What is the best estimate of the height of a traffic light?
1 meter
5 meter
30 centimeters

Question 7.
Precision
Match.

Think and Grow: Modeling Real Life

The leaf is about 8 centimeters long. Draw a tree branch that is about 16 centimeters long.

Show and Grow

Question 8.
The piece of celery is about 10 centimeters long. Draw a carrot that is about 5 centimeters long.

Question 9.
DIG DEEPER!

How did you use the length of the given beads to draw the rectangular bead?
________________________
__________________________
The length of given rectangular beads were measured with scale.

### Estimate Lengths in Metric Units Homework & Practice 11.3

Question 1.
The swimming pool is about 12 meters long. What is the best estimate of the length of the raft?

Explanation: The length of the Swimming pool is 12 meters long, the raft length will be 2 meters as it is very small.

Question 2.
What is the best estimate of the height of a tulip?

Explanation: The height of the tulip is big so, It can be estimated by 4 meters.

Question 3.
What is the best estimate of the height of a giraffe?

Explanation: The height of the giraffe is big so it can be estimated by 50 meters.

Question 4.
Logic
Newton says the best estimate for the height of a skyscraper is 4 meters. Do you agree? Explain.

_______________________
_______________________
Answer: No, The skyscraper’s estimated height is 150 meters.

Question 5.
Modeling Real Life
A granola bar is about 9 centimeters long. Draw its wrapper that is about 12 cm long.

Review & Refresh

Is the equation true or false?
Question 6.

Explanation: 13-5=8 and 15-7=8, they are equal.

Question 7.

3+6=9 and 11-3=8, They are not equal.

Question 8.

Explanation: 2+10=12 and 6+6=12

Question 9.

14-9=5 and 4+9=13

### Lesson 11.4 Measure Lengths in Inches

Explore and Grow

Use an inch tile to find the length of each string.

Explain how you measured.
______________________
______________________
Answer: It is measured with inch scale.

Show and Grow

Measure.
Question 1.

Question 2.

Question 3.

Apply and Grow: Practice

Measure.
Question 4.

Question 5.

Question 6.
Draw a crayon that is about 4 inches long.

Question 7.
YOU BE THE TEACHER

Answer: It is 6 inches but it should be measured from 0 inch to 6 inches.

Think and Grow: Modeling Real Life

Will the toothbrush fit inside a case that is 4 inches long? Explain.

_________________________
_________________________
Explantion: It cannot be fit in 4 inches case because the toothbrush is about 8 inches long.

Show and Grow

Question 8.
Will the colored pencil fit inside a pencil box that is 8 inches long? Explain.

________________________
__________________________
Explanation: The colored pencil can be fit in pencil box as colored pencil will be about 5 inches long.

### Measure Lengths in Inches Homework & Practice 11.4

Measure.
Question 1.

Question 2.

Question 3.

Question 4.
YOU BE THE TEACHER
Newton says the highlighter is about 5 centimeters long. Is he correct? Explain.

______________________
_______________________
Answer: No, The highlighter will be about 11 centimeters long,  So Newton is wrong as 5 centimeters is very small.

Question 5.
Modeling Real Life
Will the screwdriver fit inside a case that is 5 inches long? Explain.

____________________________
____________________________
Explanation: The screwdriver will be about 4.5 inches long so it will fit in 5 inches case.

Review & Refresh

Question 6.
Write how many tens. Circle groups of 10 tens. Write how many hundreds. Then write the number.
_____ tens ______ hundreds _____

Answer: 50 tens 5 hundreds 500

### Lesson 11.5 Measure Objects Using Customary Length Units

Explore and Grow

Which real-life objects are longer than an inch ruler?

Show and Grow
Find the object shown in your classroom. Choose an inch ruler, a yardstick, or a measuring tape to measure the object. Then measure.
Question 1.

Tool: _______

Question 2.

Tool: _______
Length: 2 meters

Question 3.

Tool: _______

Apply and Grow: Practice

Find the object shown in your classroom. Choose an inch ruler, a yardstick, or a measuring tape to measure the object. Then measure.
Question 4.

Tool: __________

Question 5.

Tool: _________

Question 6.
Find and measure an object using a measuring tape.
Object: _________

Question 7.
Find and measure an object using an inch ruler.
Object: ________

Question 8.
Choose Tools
Would you measure the length of the playground with an inch ruler or a yardstick? Explain.

______________________
______________________
Answer: The length of the playground can be measured with a yardstick because the yardstick is long and it will be easy to measure long objects like length of playground.

Think and Grow: Modeling Real Life

She is about 4 ______ tall.
______________________
______________________
Explanation: The height of a person is measured in feet.

Show and Grow

Question 9.
Your friend says the length of a baseball bat is about 1. Is the bat about 1 inch long, about 1 foot long, or about 1 yard long? Explain.
________ long.
_________________________
_________________________
Explanation: The length a baseball bat is 1 foot long.

Question 10.
Your friend says the length around an orange is about 9. Is the length about 9 inches long, about 9 feet long, or about 9 yards long? Explain.

The length is about 9 ________ long.
________________________
_________________________
Explanation: The orange is small object so it can be measured in inches.

### Measure Objects Using Customary Length Units Homework & Practice 11.5

Find the object shown. Choose an inch ruler, a yardstick, or a measuring tape to measure the object. Then measure.
Question 1.

Tool: ______

Question 2.

Tool: _______
Tool: Measuring tape

Question 3.
Find and measure an object using an inch ruler.
Object: _________

Question 4.
Find and measure an object using a yard stick.
Object: ____________

Question 5.
YOU BE THE TEACHER
Descartes says the best tool to measure the length around a basketball is an inch ruler. Is he correct? Explain.
________________________
________________________
Explanation: The length of the basketball cannot be measured with inch ruler, it can be measured with measuring tape.

Question 6.
Modeling Real Life

The toothbrush is about 8 _______ long.
________________________
________________________
Explanation: The toothbrush is small object so it can be measured with inch ruler.

Question 7.
DIG DEEPER!
Order the lengths from shortest to longest.

Answer: 2 feet       1 yard       39 inches
Explanation: 1 feet=12 inches, 12+12=24 inches
1 yard=36 inches

Review & Refresh

Question 8.
_____ minutes

Explanation:
Given,
28 – 55 = 27

Question 9.
You score 23 points. Your two friends score 56 and 18 points. How many points do you and your friends score in all?
______ points

Explanation:
Given,
You score 23 points. Your two friends score 56 and 18 points.
23 + 56 + 18 = 97 points
Thus you and your friend score 97 points in all/

### Lesson 11.6 Estimate Lengths in Customary Units

Explore and Grow

Find an object that is shorter than the string. Draw the object.

Without using a ruler, tell how long you think the object is.
______ inches
Explain
____________________________
____________________________
Explanation: The crayon is about 4 inches and it is shorter than 5 inches string.

Show and Grow

Question 1.
The pipe cleaner is about 3 inches long. What is the best estimate of the length of the craft stick?

Answer: The length of craft stick is 4 inches.

Question 2.
The dog leash is about 5 feet long. What is the best estimate of the length of the dog collar?

Apply and Grow: Practice

Question 3.
The poster is about 18 inches long. What is the best estimate for the length of the bed?

Question 4.
The jump rope is about 6 feet long. What is the best estimate of the length of the dog?

Question 5.
What is the best estimate of the length of a garage?
8 inches
8 feet
8 yards

Question 6.
What is the best estimate of the height of a flag pole?
20 inches
20 feet
20 yards

Question 7.
Precision
Match

Convert from feet and yards to inches.
1 feet = 12 inches
5 feet = 60 inches
1 yard = 36 inches
5 yard = 180 inches

Think and Grow: Modeling Real Life

The sticker is about 1 inch long. Draw another sticker that is about 2 inches long.

Show and Grow

Question 8.
The worm is about 4 inches long. Draw a caterpillar that is about 2 inches long.

Question 9.
DIG DEEPER!
Each toy truck is about 2 inches long. Draw a building block that is about 3 inches long.

How did you use the length of the toy trucks to draw the building block?
__________________________
____________________________
Answer: You can expand the length of the toy trucks to draw the building block.

### Estimate Lengths in Customary Units Homework & Practice 11.6

Question 1.
The driveway is about 20 yards long. What is the best estimate of the length of the truck?

Question 2.
What is the best estimate of the height of a basketball hoop?

Question 3.
What is the best estimate of the length of a hair brush?

Question 4.
Logic
Descartes says the best estimate for the height of the Statue of Liberty is 5 yards. Do you agree? Explain.

_______________________
_______________________
Answer: No, The height of the Statue of Liberty is 5 yards.
Explanation: The height of the statue of liberty is very tall, it is about 100 yards.

Question 5.
Modeling Real Life
The bug is about 4 inches long. Draw a bug that is about 2 inches long.

Question 6.
DIG DEEPER!
Each paperclip is about 2 inches long. Draw a pen that is about 5 inches long.

Review & Refresh

Question 7.
12 − 4 = ______

Question 8.
15 − 6 = ________

### Lesson 11.7 Measure Objects Using Different Length Units

Explore and Grow

Measure the length of the string in inches then in centimeters.

______ inches _______ centimeters
Are there more inches or centimeters? Why?
_____________________________
______________________________

Show and Grow

Find and measure the object shown in your classroom two ways.
Question 1.

Did you use fewer centimeters or fewer inches to measure?
centimeters inches

Question 2.

Did you use fewer meters or fewer feet to measure?
meters feet
meters.

Apply and Grow: Practice

Find and measure the object shown in your classroom two ways.
Question 3.

Did you use fewer centimeters or fewer inches to measure?
centimeters inches
inch

Question 4.

Did you use more meters or more feet to measure?
meters feet
feet

Question 5.
Would you use more centimeters, inches, or feet to measure the length of a calculator?

Question 6.
Writing
What do you notice about the relationship between inches and centimeters? feet and meters?
________________________
_________________________
Answer: 1 inch is equal to 0.39 centimeters
1 foot is equal to 0.30 meters

Think and Grow: Modeling Real Life

Do you use fewer centimeters or fewer meters to measure the length of your house? Explain.

centimeters meters
__________________________
__________________________
Explanation: 1 meter is equal to 100 centimeter, The length of the house is big So, it is easy to measure with meter.

Show and Grow

Question 7.
Do you use fewer feet or fewer yards to measure the length of a football field? Explain.
feet yards
_________________________
_________________________
1 yard=36 inches
1 feet= 12 inches
The length of the football field is big. So, it is measured with yards.

Question 8.
Do you use more meters or more feet to measure the length of your school? Explain.

meters feet
______________________
______________________
Explanation: 1 feet=12 inches
1 meter=39 inches
We use more feet to measure the length of the school.

### Measure Objects Using Different Length Units Homework & Practice 11.7

Find and measure the object shown in two ways.
Question 1.

Did you use more centimeters or more inches to measure the length of the umbrella?
centimeters inches
We use more centimeters to measure the length of the umbrella.

Question 2.

Did you use fewer meters or fewer feet to measure the height of the cabinet?
meters feet
meter

Question 3.
Would you use more centimeters, inches, or feet to measure the height of a lamp?
centimeters
inches
feet

Question 4.
Would you use fewer inches, meters, or feet to measure the length of a sink?
inches
meters
feet

Question 5.
Reasoning
Order the lengths from shortest to longest.

_______, ________, _________
Answer: 12 centimeter, 12 inches, 12 feet

Question 6.

________, ________, ________
Answer: 2 centimeter, 2 feet, 2 meters

Question 7.
Precision
What is the best estimate of the height of a maraca?

9 inches
10 centimeters
3 centimeters

Question 8.
Modeling Real Life
Do you use fewer feet or fewer yards to measure the length of a hallway? Explain.
______________________
______________________

Review & Refresh

Circle the values of the underlined digit.
Question 9.
634
4
4 ones
4 hundreds

Question 10.
918
900
9 hundreds
100

Question 11.
257
0
5 tens
50

### Lesson 11.8 Measure and Compare Lengths

Explore and Grow

Measure the fish. Circle the longer fish.

Show and Grow

Question 1.
How many centimeters longer is the marker than the paper clip?

Apply and Grow: Practice

Question 2.
How many centimeters shorter is the binder clip than the stick of gum?

Question 3.
A finger is 4 centimeters longer than the finger nail. How long is the finger?

_______ centimeters

Question 4.
Writing
Explain how you found the length of the finger in Exercise 3.
______________________
______________________

Think and Grow: Modeling Real Life

Whose path to school is longer? How much longer is it?

Whose path is longer: Your path Friend’s path
Subtraction equation:
______ yards
Whose path is longer: Your path Friend’s path
Subtraction equation: 60-28=32 yards

Show and Grow

Question 5.
Whose path to the pond is shorter? How much shorter is it?

______ meters
Your friend’s path to the pond is shorter.
It is shorter than 12 meters.

### Measure and Compare Lengths Homework & Practice 11.8

Question 1.
How many inches longer is the branch than the worm?

5 inches worm
15-5=10 inches

Question 2.
DIG DEEPER!
The length of a piece of string is 8 inches long. You cut off 5 inches. Draw the length of the string that is left.
___________________________________________________ – 5 inches
8-5 inches=3 inches.

Question 3.
Modeling Real Life
Whose path to the playground is longer? How much longer is it?

_____ yards
68-65=3 yards, It is longer by 3 yards.

Review & Refresh

Question 4.
Count by ones.
_____, _____, 71, _____, 73, _____, _____, 76
Answer: 69, 70, 71, 72, 73, 74, 75, 76
Explanation: There is difference of one number arranged in ascending order.

Question 5.
Count by fives.
85, 90, ____, _____, ______, _____, ______, ______
Answer: 85, 90, 95, 100, 105, 110, 115, 120.
Explanation: There is difference of five numbers arranged in ascending order.

Question 6.
Count by tens.
_____, 43, 53, _____, _____, ______, 93, ______
Answer: 33, 43, 53, 63, 73, 83, 93, 103.
Explanation: There is difference of ten numbers arranged in ascending order.

Measure And Estimate Lengths Performance Task

You are planting a rooftop garden. You want to build a fence around the garden. You have a piece of wood that is 16 feet long.

Question 1.
a. Which designs can you make?

b. You choose the rectangular design. Use repeated addition to find the length of each side in inches.
5 ft = _____ + ____ + ____ + _____ + _____ = in.
2 ft = ______ + ______ = in.
Answer: 5 ft =12 + 12 + 12 + 12 + 12= in
2 ft= 12 + 12 = in

Question 2.
Each seed you plant must be 6 inches apart and 6 inches away from the sides. Draw to find the number of seeds you can plant in your garden.

_____ seeds

### Measure And Estimate Lengths Activity

Spin and Cover
To Play: Players take turns. On your turn, spin one spinner. Then cover the item you would measure using that unit. Continue playing until all objects are covered.

### Measure And Estimate Lengths Chapter Practice

11.1 Measure Lengths in Centimeters

Measure.
Question 1.

Question 2.

11.2 Measure Objects Using Metric Lengths

Find and measure the object
Question 3.

Question 4.

11.3 Estimate Lengths in Metric Units

Question 5.
The book is about 20 centimeters long. What is the best estimate of the length of the bookmark?

Question 6.
What is the best estimate of the length of a paintbrush?

11.4 Measure Lengths in Inches

Measure.
Question 7.

Question 8.

11.5 Measure Objects Using Customary Length Units

Find the object shown. Choose an inch ruler, a yardstick, or a measuring tape to measure the object. Then measure.
Question 9.

Tool: ______
Tool: Measuring tape

Question 10.

Tool: _____
Tool: inch ruler

11.6 Estimate Lengths in Customary Units

Question 11.
The couch is about 8 feet long. What is best estimate of the length of the end table?

Answer: The length of the end table is 2 feet.

Question 12.
What is the best estimate of the length of a pond?

Answer: The length of a pond is 30 feet.

11.7 Measure Objects Using Different Length Units

Question 13.
Would you use more centimeters, meters, or inches to measure the length of a pencil?
centimeters
meters
inches

Question 14.
Would you use fewer centimeters, meters, or feet to measure the length of the teacher’s desk?
centimeters
meters
feet

Question 15.
YOU BE THE TEACHER
Newton says he uses more feet than meters to measure the length of a bicycle. Is he correct? Explain.

______________________
________________________
Answer: Yes Newton uses more feet than meters to measure the length of a bicycle.

11.8 Measure and Compare Lengths

Question 16.
A guinea pig cage is 51 centimeters longer than the guinea pig. How long is the cage?

_______ centimeters

Conclusion:

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## Big Ideas Math Book 2nd Grade Answer Key Chapter 15 Identify and Partition Shapes

Every student must have Big Ideas Math 2nd Grade 15th Chapter Identify and Partition Shapes Answer Key to improve their math skills. By referring solution key, no one feels solving problems is difficult. Students have to practice all the topics included in Big Ideas Math Answers Grade 2 Chapter 15 Identify and Partition Shapes to get better marks in the exam.

The different lessons of Big Ideas Math Book Grade 2 Chapter 15 Identify and Partition Shapes Answer Key are Describe Two-Dimensional Shapes, Identify Angles of Polygons, Draw Polygons, Identify and Draw Cubes, Compose Rectangles, Identify Two, Three, or Four Equal Shares, Partition Shapes into Equal Shares, and Analyze Equal Shares of the Same Shape. Click on the quick link attached here to get the questions and answers for all those topics.

Vocabulary

Lesson: 1 Describe Two-Dimensional Shapes

Lesson: 2 Identify Angles of Polygons

Lesson: 3 Draw Polygons

Lesson: 4 Identify and Draw Cubes

Lesson: 5 Compose Rectangles

Lesson: 6 Identify Two, Three, or Four Equal Shares

Lesson: 7 Partition Shapes into Equal Shares

Lesson: 8 Analyze Equal Shares of the Same Shape

Chapter – 15: Identify and Partition Shapes

### Identify and Partition Shapes Vocabulary

Organize It
Use the review words to complete the graphic organizer.

Define It
Use your vocabulary cards to identify the word. Find the word in the word search

### Lesson 15.1 Describe Two-Dimensional Shapes

Explore and Grow

Create a shape with 3 sides on your geoboard. Draw your shape. Did everyone in your class make the same shape?

Circle the word that makes the sentence true.
_______ are shapes with 3 sides.
Circles Squares Triangles

Show and Grow

Question 1.

______ sides
_______ vertices
Shape : _________
4 vertices

Question 2.

______ sides
_______ vertices
Shape : _________
5 vertices
Shape: Irregular Pentagon

Apply and Grow: Practice

Question 3.

______ sides
_______ vertices
Shape : _________
6 vertices
Shape: Irregular Hexagon

Question 4.

______ sides
_______ vertices
Shape : _________
4 vertices

Question 5.

______ sides
_______ vertices
Shape : _________
8 vertices
Shape: Octagon

Question 6.

______ sides
_______ vertices
Shape : _________
3 vertices
Shape: Scalene Triangle

Question 7.

______ sides
_______ vertices
Shape : _________
5 vertices
Shape: Concave Pentagon

Question 8.

______ sides
_______ vertices
Shape : _________
4 vertices

Question 9.
Writing
How are a pentagon and an octagon different?
____________________
_____________________
Answer: Pentagon has 5 sides and Octagon has 8 sides.

Think and Grow: Modeling Real Life

Draw a pentagon to make a house. Draw 2 quadrilaterals to make windows and 1 quadrilateral to make a door. Draw an octagon to make a chimney.

Show and Grow

Question 10.
Draw a pentagon to make a fish. Draw 4 triangles to make the fins. Draw a hexagon to make an eye.

Question 11.
You draw 5 quadrilaterals. How many sides and vertices do you draw in all?
______ sides _______ vertices

Question 12.
DIG DEEPER!
You draw an octagon and two pentagons. How many sides and vertices do you draw in all?
______ sides ______ vertices

### Describe Two-Dimensional Shapes Homework & Practice 15.1

Question 1.

______ sides
_______ vertices
Shape : _________
6 vertices
Shape: Irregular hexagon

Question 2.

______ sides
_______ vertices
Shape : _________
5 vertices
Shape: Irregular Pentagon

Question 3.

______ sides
_______ vertices
Shape : _________
8 vertices
Shape: Irregular Octagon

Question 4.

______ sides
_______ vertices
Shape : _________
3 vertices
Shape: Right angle Triangle

Question 5.

______ sides
_______ vertices
Shape : _________
4 vertices

Question 6.

______ sides
_______ vertices
Shape : _________
5 vertices
Shape: Irregular Pentagon

Question 7.
Precision
Describe the shape in 3 ways.

Answer: A 5 sided shape called Pentagon
It has 2 parallel sides
All sides are not equal.

Question 8.
Modeling Real Life
Draw a hexagon to make a dog’s body. Draw quadrilaterals for the head and tail. Draw two triangles for the ears.

Question 9.
DIG DEEPER!
You draw a triangle and two hexagons. How many sides and vertices do you draw in all?
______ sides _______ vertices

Review & Refresh

Question 10.
You are building a 34-foot fence. You build 15 feet on Saturday and 13 feet on Sunday. How many feet are left to build?
______ feet
Answer: 6 feet are left to build
Explanation: Total 34 foot fence
15 feet on Saturday
13 feet on Sunday
15+13= 28 built
34-28= 6 feet left to build

### Lesson 15.2 Identify Angles of Polygons

Explore and Grow

Color the triangle blue. Color the quadrilateral red. Color the pentagon green. Color the hexagon orange.

Which shape is not colored? How is it different from the other shapes?
________________________
________________________
________________________
Answer: Circle is not colored. Circle has no straight lines.

Show and Grow

Question 1.

_____ angles
How many right angles? ______
Shape: ______
2 right angles
Shape: Irregular hexagon

Question 2.

______ angles
How many right angles? ______
Shape: _______
1 right angles
Shape: Right angle triangle

Question 3.

_____ angles
How many right angles? ______
Shape: ______
4 right angles
Shape: Square

Question 4.

______ angles
How many right angles? ______
Shape : _______
No right angles
Shape: Trapezium

Apply and Grow: Practice

Question 5.

_____ angles
How many right angles? ______
Shape: ______
1 right angle
Shape: Right angle triangle

Question 6.

_____ angles
How many right angles? ______
Shape: ______
2 right angles
Shape: irregular pentagon

Question 7.

_____ angles
How many right angles? ______
Shape: ______
No right angle
Shape: Irregular Octagon

Question 8.

_____ angles
How many right angles? ______
Shape: ______
2 right angles

Question 9.
Draw and name a polygon with 6 angles.
________

Question 10.
Draw and name a polygon with 2 right angles
_________

Question 11.
Writing
Can you draw a polygon with 4 sides and 5 angles? Explain.
__________________
____________________
Answer: No, A polygon with 4 sides and 5 angles cannot be drawn as the number of sides and angles are always equal.

Think and Grow: Modeling Real Life

You are designing a road sign. The new sign must be a pentagon with only 2 right angles. Which signs might be yours?

Show and Grow

Question 12.
You are making a sign for your lemonade stand. Your sign must be a quadrilateral with 4 right angles. Which signs might be yours?

Question 13.
You draw 3 pentagons. How many angles do you draw in all?
______ angles

Question 14.
DIG DEEPER!
You draw a quadrilateral and three triangles. Your friend draws an octagon and a hexagon. Who draws more angles in all? How many more?
You Friend ______ more angles
Answer: I draw 13 angles in all. My friend draws 14 angles in all. Friend draws more angles in all with 1 more angle.

### Identify Angles of Polygons Homework & Practice 15.2

Question 1.

______ angles
How many right angles? ______
Shape: ______
2 right angles
Shape: Irregular hexagon

Question 2.

______ angles
How many right angles? _____
Shape: ______
4 right angles
Shape: Rhombus

Question 3.

______ angles
How many right angles? ______
Shape: ______
No right angles
Shape: Irregular octagon

Question 4.

_____ angles
How many right angles? ______
Shape: _______
1 right angle
Shape: Irregular pentagon

Question 5.
Draw and name a polygon with 4 sides and 1 right angle.
_______

Question 6.
Draw and name a polygon with 6 angles.
________

Question 7.

DIG DEEPER!
Draw two polygons that have 9 angles in all.

Question 8.
Modeling Real Life
You are designing a company logo. Your logo must be a hexagon with 2 right angles. Which logos might be yours?

Question 9.
DIG DEEPER!
You draw an octagon and two triangles. Your friend draws two quadrilaterals and a pentagon. Who draws more angles in all? How many more?
You Friend _______ more angles
Answer: You draw more angles. 1 angle more
You draw 14 angle in total, your friend draw 13 angle in total.

Review & Refresh

Draw to show the time.
Question 10.

Question 11.

Question 12.

### Lesson 15.3 Draw Polygons

Explore and Grow

Compare the shapes.

How are the shapes the same? How are they different?
_____________________
_____________________
_____________________
Answer: Both the shapes have 4 sides and 4 angles
They are different as length of the shapes are not equal.

Show and Grow

Question 1.
Draw a polygon with 6 sides. Two of the sides are the same length.
_______ angles
Polygon: ________