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Go Math Grade 8 Chapter 1 Real Numbers Answer Key

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Lesson 1: Rational and Irrational Numbers

• • • Rational and Irrational Numbers – Page No. 12
    • Rational and Irrational Numbers – Page No. 13
    • Rational and Irrational Numbers – Page No. 14

Lesson 2: Sets of real Numbers

• • • Sets of real Numbers – Page No. 18
    • Sets of real Numbers – Page No. 19
    • Sets of real Numbers – Page No. 20

Lesson 3: Ordering Real Numbers

• • • Ordering Real Numbers – Page No. 24
    • Ordering Real Numbers – Page No. 25
    • Ordering Real Numbers – Page No. 26

Model Quiz

• • • Model Quiz – Page No. 27

Mixed Review

• • • Mixed Review – Page No. 28

Guided Practice – Rational and Irrational Numbers – Page No. 12

Write each fraction or mixed number as a decimal.

Question 1.
\(\frac{2}{5}\) =

Answer:
0.4

Explanation:
\(\frac{2}{5}\) = \(\frac{2 × 2}{5 × 2}\) = \(\frac{4}{10}\) = 0.4

Question 2.
\(\frac{8}{9}\) =

Answer:
0.88

Explanation:
\(\frac{8}{9}\) = \(\frac{8 × 10}{9 × 10}\) = \(\frac{80}{9 × 10}\) = \(\frac{8.88}{10}\) = 0.88

Question 3.
3 \(\frac{3}{4}\) =

Answer:
3.75

Explanation:
3 \(\frac{3}{4}\) =\(\frac{15}{4}\) = 3.75

Question 4.
\(\frac{7}{10}\) =

Answer:
0.7

Explanation:
\(\frac{7}{10}\) = 0.7

Question 5.
2 \(\frac{3}{8}\) =

Answer:
2.375

Explanation:
2 \(\frac{3}{8}\) = \(\frac{19}{8}\) = 2.375

Question 6.
\(\frac{5}{6}\) =

Answer:
0.833

Explanation:
\(\frac{5}{6}\) = \(\frac{5 × 10}{6 × 10}\) = \(\frac{50}{6 × 10}\) = \(\frac{8.33}{10}\) = 0.833

Write each decimal as a fraction or mixed number in simplest form

Question 7.
0.675
\(\frac{□}{□}\)

Answer:
\(\frac{27}{40}\)

Explanation:
\(\frac{0.675 × 1000}{1 × 1000}\) = \(\frac{675}{1000}\) = \(\frac{675/25}{1000/25}\) = \(\frac{27}{40}\)

Question 8.
5.6
______ \(\frac{□}{□}\)

Answer:
5 \(\frac{3}{5}\)

Explanation:
\(\frac{5.6 × 10}{10}\) = \(\frac{56}{10}\) = 5 \(\frac{6}{10}\) = 5 \(\frac{6/2}{10/2}\) = 5 \(\frac{3}{5}\)

Question 9.
0.44
\(\frac{□}{□}\)

Answer:
\(\frac{11}{25}\)

Explanation:
\(\frac{0.44 × 100}{1 × 100}\) = \(\frac{44}{100}\) = \(\frac{44/4}{100/4}\) = \(\frac{11}{25}\)

Question 10.
0.\(\bar{4}\)
\(\frac{□}{□}\)

Answer:
\(\frac{4}{9}\)

Explanation:
Let x = 0.\(\bar{4}\)
Now, 10x = 4.\(\bar{4}\)
10x – x = 4.\(\bar{4}\) – 0.\(\bar{4}\)
9x = 4
x = \(\frac{4}{9}\)

Question 11.
0.\(\overline { 26 } \)
\(\frac{□}{□}\)

Answer:
\(\frac{26}{99}\)

Explanation:
Let x = 0.\(\overline {26}\)
Now, 100x = 26.\(\overline{26}\)
100x – x = 26.\(\overline{26}\) – 0.\(\overline {26}\)
99x = 26
x = \(\frac{26}{99}\)

Question 12.
0.\(\overline { 325 } \)
\(\frac{□}{□}\)

Answer:
\(\frac{325}{999}\)

Explanation:
Let x = 0.\(\overline {325}\)
Now, 1000x = 325.\(\overline{325}\)
1000x – x = 325.\(\overline{325}\) – 0.\(\overline {325}\)
999x = 325
x = \(\frac{325}{999}\)

Solve each equation for x

Question 13.
x² = 144
± ______

Answer:
x=±12

Explanation:
x² = 144
Taking square roots on both the sides
$\sqrt{x^2}=\pm \sqrt{144}$x = $\pm 12$

Question 14.
x² = \(\frac{25}{289}\)
± \(\frac{□}{□}\)

Answer:
x = ±\(\frac{5}{17}\)

Explanation:
x² = \(\frac{25}{289}\)
Taking square roots on both the sides
$\sqrt{x^2}=\pm \sqrt{}$x = $\pm \backslash (\backslash frac\{5\}\{17\}\backslash )$

Question 15.
x³ = 216
______

Answer:
x = 6

Explanation:
x³ = 216
Taking cube roots on both the sides
$\sqrt{3^{}{\mathrm{x3}}^{}}=\; 3\sqrt{}$x = 6

Approximate each irrational number to two decimal places without a calculator.

Question 16.
\(\sqrt { 5 } \) ≈ ______

Answer:
2.236

Explanation:
x = \(\sqrt { 5 } \)
The 5 is in between 4 and 6
Take square root of each year
√4 < √5 < √6
2 < √5 < 3
√5 = 2.2
(2.2)² = 4.84
(2.25)² = 5.06
(2.5)³ = 5.29
A good estimate for √5 is 2.25

Question 17.
\(\sqrt { 3 } \) ≈ ______

Answer:
1.75

Explanation:
\(\sqrt { 3 } \)
1 < 3 < 4
√1 < √3 < √4
1 < √3 < 2
√3 = 1.6
(1.65)² = 2.72
(1.7)² = 2.89
(1.75)² = 3.06
A good estimate for √3 is 1.75

Question 18.
\(\sqrt { 10 } \) ≈ ______

Answer:
3.15

Explanation:
\(\sqrt { 10 } \)
9 < 10 < 16
√9 < √10 < √16
3 < √10 < 4
√10 = 3.1
(3.1)² = 9.61
(3.15)² = 9.92
(3.2)² = 10.24
A good estimate for √10 is 3.15

Question 19.
What is the difference between rational and irrational numbers?
Type below:
_____________

Answer:

Rational number can be expressed as a ration of two integers such as 5/2
Irrational number cannot be expressed as a ratio of two integers such as √13

Explanation:
A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational.

1.1 Independent Practice – Rational and Irrational Numbers – Page No. 13

Question 20.
A \(\frac{7}{16}\)-inch-long bolt is used in a machine. What is the length of the bolt written as a decimal?
______ -inch-long

Answer:
0.4375 inch

Explanation:
The length of the bolt is \(\frac{7}{16}\)-inch
Let, x = \(\frac{7}{16}\)
Multiplying by 125 on both nominator and denominator
x = \(\frac{7×125}{16×125}\) = \(\frac{875}{2000}\) =\(\frac{437.5}{1000}\) = 0.4375

Question 21.
The weight of an object on the moon is \(\frac{1}{6}\) its weight on Earth. Write \(\frac{1}{6}\) as a decimal.
______

Answer:
0.1666

Explanation:
The weight of the object on the moon is \(\frac{1}{6}\)
Let, x = \(\frac{1}{6}\)
Multiplying by 100 on both nominator and denominator
x = \(\frac{1×100}{6×100}\) = \(\frac{16.6}{100}\) =0.166

Question 22.
The distance to the nearest gas station is 2 \(\frac{4}{5}\) kilometers. What is this distance written as a decimal?
______

Answer:
2.8

Explanation:
The distance of the nearest gas station is 2 \(\frac{4}{5}\)
Let, x = 2 \(\frac{4}{5}\)
Multiplying by 100 on both nominator and denominator
x = 2 \(\frac{4×100}{5×100}\) = \(\frac{80}{100}\) =0.8

Question 23.
A baseball pitcher has pitched 98 \(\frac{2}{3}\) innings. What is the number of innings written as a decimal?
______

Answer:
98.6

Explanation:
A baseball pitcher has pitched 98 \(\frac{2}{3}\) innings.
98 \(\frac{2}{3}\) = 98 + 2/3
= (294/3) + (2/3)
296/3
98.6

Question 24.
A heartbeat takes 0.8 second. How many seconds is this written as a fraction?
\(\frac{□}{□}\)

Answer:
\(\frac{4}{5}\)

Explanation:
A heartbeat takes 0.8 seconds.
0.8
There are 8 tenths.
8/10 = 4/5

Question 25.
There are 26.2 miles in a marathon. Write the number of miles using a fraction.
\(\frac{□}{□}\)

Answer:
26\(\frac{1}{5}\)

Explanation:
There are 26.2 miles in a marathon.
26.2 miles
262/10
131/5
26 1/5 miles

Question 26.
The average score on a biology test was 72.\(\bar{1}\). Write the average score using a fraction.
\(\frac{□}{□}\)

Answer:
80 \(\frac{1}{9}\)

Explanation:
The average score on a biology test was 72.\(\bar{1}\).
72.\(\bar{1}\)
Let x = 72.\(\bar{1}\)
10x = 10(72.\(\bar{1}\))
10x = 721.1
-x = -0.1
9x = 721
x = 721/9
x = 80 1/9

Question 27.
The metal in a penny is worth about 0.505 cent. How many cents is this written as a fraction?
\(\frac{□}{□}\)

Answer:
\(\frac{101}{200}\)

Explanation:
The metal in a penny is worth about 0.505 cent.
0.505 cent
505 thousandths
505/1000
101/200 cents

Question 28.
Multistep An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting.

a. If x is the length of one side of the painting, what equation can you set up to find the length of a side?
x² = ______

Answer:
x² = 400

Explanation:
The area of a square is the square of its equal side, x
x² = 400

Question 28.
b. Solve the equation you wrote in part a. How many solutions does the equation have?
x = ± ______

Answer:
x = ± 20

Explanation:
Take the square root on both sides. Solve
x = ± 20

Question 28.
c. Do all of the solutions that you found in part b make sense in the context of the problem? Explain.
Type below:
_____________

Answer:
No. Both values of x do not make sense.

Explanation:
The length cannot be negative, hence negative value does not make sense.
No. Both values of x do not make sense.

Question 28.
d. What is the length of the wood trim needed to go around the painting?
P = ______ inches

Answer:
Length P = 20 + 2y

Rational and Irrational Numbers – Page No. 14

Question 29.
Analyze Relationships To find \(\sqrt { 15 } \), Beau found 3² = 9 and 4² = 16. He said that since 15 is between 9 and 16, \(\sqrt { 15 } \) must be between 3 and 4. He thinks a good estimate for \(\sqrt { 15 } \) is \(\frac { 3+4 }{ 2 } \) = 3.5. Is Beau’s estimate high, low, or correct? Explain.
_____________

Answer:
3.85

Explanation:
15 is closer to 16
√15 is closer to √16
Beau’s estimate is low.
(3.8)² = 14.44
(3.85)² = 14.82
(3.9)² = 15.21
√15 is 3.85

Question 30.
Justify Reasoning What is a good estimate for the solution to the equation x³ = 95? How did you come up with your estimate?
x ≈ ______

Answer:
x ≈  4.55

Explanation:
³√x = 95
x = ³√95
64 < 95 < 125
Take the cube root of each number
³√64 < ³√95  < ³√125
4 < ³√95 < 5
³√95 = 4.6
(4.5)³ = 91.125
(4.55)³ = 94.20
(4.6)³ = 97.336
³√95 = 4.55

Question 31.
The volume of a sphere is 36π ft³. What is the radius of the sphere? Use the formula V = \(\frac { 4 }{ 3 } \)πr³ to find your answer.

r = ______

Answer:
r = 3

Explanation:
V = 4/3 πr³
36π = 4/3 πr³
r³ = 36π/π . 3/4
r³ = 27
r = ³√27
r = 3

FOCUS ON HIGHER ORDER THINKING

Question 32.
Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning.
_____________

Answer:
Yes

Explanation:
Yes. The cube root of a negative number would be negative. Because the product of three negative signs is always negative.

Question 33.
Make a Conjecture Evaluate and compare the following expressions.
\(\sqrt { \frac { 4 }{ 25 } } \) and \(\frac { \sqrt { 4 } }{ \sqrt { 25 } } \) \(\sqrt { \frac { 16 }{ 81 } } \) and \(\frac { \sqrt { 16 } }{ \sqrt { 81 } } \) \(\sqrt { \frac { 36 }{ 49 } } \) and\(\frac { \sqrt { 36 } }{ \sqrt { 49 } } \)
Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots.
Expressions are: _____________

Answer:
Evaluating and comparing
√4/25 = 2/5
√16/81 = 4/9
√36/49 = 6/7
Conjecture about a division rule for square roots
√a/√b = √(a/b)
Conjecture about a multiplication rule for square roots
√a × √b

Question 34.
Persevere in Problem Solving
The difference between the solutions to the equation x² = a is 30. What is a? Show that your answer is correct.
_____

Answer:
30

Explanation:
x² = a
x = ±√a
√a – (-√a) = 30
√a + √a = 30
2√a = 30
√a = 15
a = 225
x² = 225
x = ±225
x = ±15
15 – (-15) = 15 + 15 = 30

Guided Practice – Sets of real Numbers – Page No. 18

Write all names that apply to each number.

Question 1.
\(\frac{7}{8}\)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 2.
\(\sqrt { 36 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers

Explanation:
\(\sqrt { 36 } \) = 6

Question 3.
\(\sqrt { 24 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
e. Irrational Numbers

Question 4.
0.75
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 5.
0
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers

Question 6.
−\(\sqrt { 100 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers

Explanation:
−\(\sqrt { 100 } \) = – 10

Question 7.
5.\(\overline { 45 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 8.
−\(\frac{18}{6}\)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers

Explanation:
−\(\frac{18}{6}\) = -3

Tell whether the given statement is true or false. Explain your choice.

Question 9.
All whole numbers are rational numbers.
i. True
ii. False

Answer:
i. True

Explanation:
All whole numbers are rational numbers.
Whole numbers are a subset of the set of rational numbers and can be written as ratio of the whole number to 1.

Question 10.
No irrational numbers are whole numbers.
i. True
ii. False

Answer:
i. True

Explanation:
True. Whole numbers are ration numbers.

Identify the set of numbers that best describes each situation. Explain your choice.

Question 11.
the change in the value of an account when given to the nearest dollar
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
c. Integer Numbers

Explanation:
The change can be a whole dollar amount and can be positive, negative or zero.

Question 12.
the markings on a standard ruler

Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers

Explanation:
The ruler is marked every 1/16t inch.

ESSENTIAL QUESTION CHECK-IN

Question 13.
What are some ways to describe the relationships between sets of numbers?

Answer:
There are two ways that we have been using until now to describe the relationships between sets of numbers

• Using a scheme or a diagram as the one on page 15.
• Verbal description, for example, “All irrational numbers are real numbers.”

1.2 Independent Practice – Sets of real Numbers – Page No. 19

Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram.

Question 14.
\(\sqrt { 9 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Explanation:
\(\sqrt { 9 } \) = 3

Question 15.
257
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Question 16.
\(\sqrt { 50 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
e. Irrational Numbers

Question 17.
8 \(\frac{1}{2}\)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 18.
16.6
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 19.
\(\sqrt { 16 } \)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Explanation:
\(\sqrt { 16 } \) = 4

Identify the set of numbers that best describes each situation. Explain your choice.

Question 20.
the height of an airplane as it descends to an airport runway
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
d. Whole Numbers

Explanation:
Whole. The height of an airplane as it descents to an airport runway is a whole number greater than 0

Question 21.
the score with respect to par of several golfers: 2, – 3, 5, 0, – 1
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
c. Integer Numbers

Explanation:
Integers. The scores are counting numbers, their opposites, and zero.

Question 22.
Critique Reasoning Ronald states that the number \(\frac{1}{11}\) is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain.
i. Ronald
ii. Nathaniel

Answer:
ii. Nathaniel

Explanation:
Nathaniel is correct.
A fraction is a rational real number, even if it is not a terminating decimal.

Sets of real Numbers – Page No. 20

Question 23.
Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain your answer.

Options:
a. Real Numbers
b. Rational Numbers
c. Irrational Numbers
d. Integers
e. Whole Numbers

Answer:
e. Whole Numbers

Explanation:
Circumference of the circle
A = 2πr
π = 2πr
Diameter is twice the radius
2r = 1
Whole

Question 24.
Critical Thinking A number is not an integer. What type of number can it be?
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers
e. Irrational Numbers

Question 25.
A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons?
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers

FOCUS ON HIGHER ORDER THINKING

Question 26.
Explain the Error Katie said, “Negative numbers are integers.” What was her error?
Type below:
_______________

Answer:
Her error is that she stated that all negative numbers are integers. Some negative numbers are integers such as -4 but some are not such an -0.8

Question 27.
Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain.
Type below:
_______________

Answer:
Not always.

Explanation:
Not always.
If the calculator shows a terminating decimal, the number is rational but otherwise, it is not possible as you can only see a few digits.

Question 28.
Draw Conclusions The decimal 0.\(\bar{3}\) represents \(\frac{1}{3}\). What type of number best describes 0.\(\bar{9}\) , which is 3 × 0.\(\bar{3}\)? Explain.
Type below:
_______________

Answer:
1

Explanation:
let x = 0.9999999
10x = 9.99999999
10x = 9 + 0.999999999
10x = 9 + x
9x = 9
x=1.

Question 29.
Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this?

Answer:
Because irrational numbers are nonrepeating, otherwise they could be represented as a fraction. Although a potential counter-example to this claim is that some irrational numbers can only be represented in decimal form, for example, 0.1234567891011121314151617…, 0.24681012141618202224…, 0.101101110111101111101111110… are all irrational numbers.

Guided Practice – Ordering Real Numbers – Page No. 24

Compare. Write <, >, or =.

Question 1.
\(\sqrt { 3 } \) + 2 ________ \(\sqrt { 3 } \) + 3

Answer:
\(\sqrt { 3 } \) + 2 < \(\sqrt { 3 } \) + 3

Explanation:
\(\sqrt { 3 } \) is between 1 and 2
\(\sqrt { 3 } \) + 2 is between 3 and 4
\(\sqrt { 3 } \) + 3 is between 4 and 5
\(\sqrt { 3 } \) + 2 < \(\sqrt { 3 } \) + 3

Question 2.
\(\sqrt { 11 } \) + 15 _______ \(\sqrt { 8 } \) + 15

Answer:
\(\sqrt { 11 } \) + 15 > \(\sqrt { 8 } \) + 15

Explanation:
\(\sqrt { 11 } \) is between 3 and 4
\(\sqrt { 8 } \) is between 2 and 3
\(\sqrt { 11 } \) + 15 is between 18 and 19
\(\sqrt { 8 } \) + 15 is between 17 and 18
\(\sqrt { 11 } \) + 15 > \(\sqrt { 8 } \) + 15

Question 3.
\(\sqrt { 6 } \) + 5 _______ 6 + \(\sqrt { 5 } \)

Answer:
\(\sqrt { 6 } \) + 5 < 6 + \(\sqrt { 5 } \)

Explanation:
\(\sqrt { 6 } \) is between 2 and 3
\(\sqrt { 5 } \) is between 2 and 3
\(\sqrt { 6 } \) is between 7 and 8
\(\sqrt { 5 } \) is between 8 and 9
\(\sqrt { 6 } \) + 5 < 6 + \(\sqrt { 5 } \)

Question 4.
\(\sqrt { 9 } \) + 3 _______ 9 + \(\sqrt { 3 } \)

Answer:
\(\sqrt { 9 } \) + 3 < 9 + \(\sqrt { 3 } \)

Explanation:
\(\sqrt { 9 } \) + 3
9 + \(\sqrt { 3 } \)
\(\sqrt { 3 } \) is between 1 and 2
\(\sqrt { 9 } \) + 3 = 3 + 3 = 6
9 + \(\sqrt { 3 } \) is between 10 and 11
\(\sqrt { 9 } \) + 3 < 9 + \(\sqrt { 3 } \)

Question 5.
\(\sqrt { 17 } \) – 3 _______ -2 + \(\sqrt { 5 } \)

Answer:
\(\sqrt { 17 } \) – 3 > -2 + \(\sqrt { 5 } \)

Explanation:
\(\sqrt { 17 } \) is between 4 and 5
\(\sqrt { 5 } \) is between 2 and 3
\(\sqrt { 17 } \) – 3 is between 1 and 2
-2 + \(\sqrt { 5 } \) is between 0 and 1
\(\sqrt { 17 } \) – 3 > -2 + \(\sqrt { 5 } \)

Question 6.
10 – \(\sqrt { 8 } \) _______ 12 – \(\sqrt { 2 } \)

Answer:
10 – \(\sqrt { 8 } \) < 12 – \(\sqrt { 2 } \)

Explanation:
\(\sqrt { 8 } \) is between 2 and 3
\(\sqrt { 2 } \) is between 1 and 2
10 – \(\sqrt { 8 } \) is between 8 and 7
12 – \(\sqrt { 2 } \) is between 11 and 10
10 – \(\sqrt { 8 } \) < 12 – \(\sqrt { 2 } \)

Question 7.
\(\sqrt { 7 } \) + 2 _______ \(\sqrt { 10 } \) – 1

Answer:
\(\sqrt { 7 } \) + 2 > \(\sqrt { 10 } \) – 1

Explanation:
\(\sqrt { 7 } \) is between 2 and 3
\(\sqrt { 10 } \) is between 3 and 4
\(\sqrt { 7 } \) + 2 is between 4 and 5
\(\sqrt { 10 } \) – 1 is between 2 and 3
\(\sqrt { 7 } \) + 2 > \(\sqrt { 10 } \) – 1

Question 8.
\(\sqrt { 17 } \) + 3 _______ 3 + \(\sqrt { 11 } \)

Answer:
\(\sqrt { 17 } \) + 3 > 3 + \(\sqrt { 11 } \)

Explanation:
\(\sqrt { 17 } \) is between 4 and 5
\(\sqrt { 11 } \) is between 3 and 4
\(\sqrt { 17 } \) + 3 is between 7 and 8
3 + \(\sqrt { 11 } \) is between 6 and 7
\(\sqrt { 17 } \) + 3 > 3 + \(\sqrt { 11 } \)

Question 9.
Order \(\sqrt { 3 } \), 2 π, and 1.5 from least to greatest. Then graph them on the number line.
\(\sqrt { 3 } \) is between _________ and _____________ , so \(\sqrt { 3 } \) ≈ ____________.
π ≈ 3.14, so 2 π ≈ _______________.

From least to greatest, the numbers are ______________, _____________________ ,_________________.
Type below:
___________

Answer:
1.5, \(\sqrt { 3 } \), 2 π

Explanation:
\(\sqrt { 3 } \) is between 1.7 and 1.75
π = 3.14; 2 π = 6.28

1.5, \(\sqrt { 3 } \), 2 π

Question 10.
Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least.

Type below:
___________

Answer:
\(\sqrt { 17 } \) – 2, 1+ π/2, 2.5, 12/5

Explanation:
\(\sqrt { 17 } \) – 2
\(\sqrt { 17 } \) is between 4 and 5
Since, 17 is closer to 16, the estimated value is 4.1
1+ π/2
1 + (3.14/2) = 2.57
12/5 = 2.4
2.5
\(\sqrt { 17 } \) – 2, 1+ π/2, 2.5, 12/5

ESSENTIAL QUESTION CHECK-IN

Question 11.
Explain how to order a set of real numbers.
Type below:
___________

Answer:
Evaluate the given numbers and write in decimal form. Plot on number line and arrange the numbers accordingly.

Independent Practice – Ordering Real Numbers – Page No. 25

Order the numbers from least to greatest.

Question 12.
\(\sqrt { 7 } \), 2, \(\frac { \sqrt { 8 } }{ 2 } \)
Type below:
____________

Answer:
\(\frac { \sqrt { 8 } }{ 2 } \), 2, \(\sqrt { 7 } \)

Explanation:
\(\sqrt { 7 } \), 2, \(\frac { \sqrt { 8 } }{ 2 } \)
\(\sqrt { 7 } \) is between 2 and 3
Since 7 is closer to 9, (2.65)² = 7.02, hence the estimated value is 2.65
\(\frac { \sqrt { 8 } }{ 2 } \)
\(\sqrt { 8 } \) is between 2 and 3
Since 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85
2.85/2 = 1.43

\(\frac { \sqrt { 8 } }{ 2 } \), 2, \(\sqrt { 7 } \)

Question 13.
\(\sqrt { 10 } \), π, 3.5
Type below:
____________

Answer:
π, \(\sqrt { 10 } \), 3.5

Explanation:
\(\sqrt { 10 } \), π, 3.5
\(\sqrt { 10 } \) is between 3 and 4
Since, 10 is closer to 9, (3.15)² = 9.92, hence the estimated value is 3.15
π = 3.14
3.5

π, \(\sqrt { 10 } \), 3.5

Question 14.
\(\sqrt { 220 } \), −10, \(\sqrt { 100 } \), 11.5
Type below:
____________

Answer:
-10, √100, 11.5, √220

Explanation:
\(\sqrt { 220 } \), −10, \(\sqrt { 100 } \), 11.5
196 < 220 < 225
√196 < √220 < √225
14 < √220 < 15
√220 = 14.5
√100 = 10

-10, √100, 11.5, √220

Question 15.
\(\sqrt { 8 } \), −3.75, 3, \(\frac{9}{4}\)
Type below:
____________

Answer:
−3.75, \(\frac{9}{4}\), \(\sqrt { 8 } \)

Explanation:
\(\sqrt { 8 } \), −3.75, 3, \(\frac{9}{4}\)
\(\sqrt { 8 } \) is between 2 and 3
Since, 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85
-3.75 = 3
9/4 = 2.25

−3.75, \(\frac{9}{4}\), \(\sqrt { 8 } \)

Question 16.
Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters.
a. Find the area of the square.
_______ m²

Answer:
(3.5)² = 12.25

Explanation:
Area of the square = x²
Area = (3.5)² = 12.25

Question 16.
b. Find the area of the circle.
_______ m²

Answer:
π(2)² = 12.56

Explanation:
Area of the circle = πr² where r = d/2 = 4/2 = 2
Area = π(2)² = 12.56

Question 16.
c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant?
___________

Answer:
12.25 < 12.56
The circle will give more space

Question 17.
Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table.
a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate.

______

Answer:
7.4815

Explanation:
\(\sqrt { 60 } \) = 7.75
58/8 = 7.25
7.3333
7 3/5 = 7.60
Average = (7.75 + 7.25 + 7.33 + 7.60)/4 = 7.4815

Question 17.
b. Winnie’s father estimated the distance across his ranch to be \(\sqrt { 56 } \) km. How does this distance compare to Winnie’s estimate?
____________

Answer:
They are nearly identical

Explanation:
\(\sqrt { 56 } \) = 7.4833
They are nearly identical

Give an example of each type of number.

Question 18.
a real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)
Type below:
____________

Answer:
A real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)
Example: 3.7

Explanation:
\(\sqrt { 13 } \) = 3.61
\(\sqrt { 13 } \) = 3.74
A real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)
Example: 3.7

Question 19.
an irrational number between 5 and 7
Type below:
____________

Answer:
An irrational number between 5 and 7
Example: \(\sqrt { 29 } \)

Explanation:
5² = 25 and 7² = 49
An irrational number between 5 and 7
Example: \(\sqrt { 29 } \)

Ordering Real Numbers – Page No. 26

Question 20.
A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right?

_____________

Answer:
Neither are correct

Explanation:
\(\sqrt { 115 } \), 115/11, 10.5624
\(\sqrt { 115 } \) is between 10 and 11
Since, 115 is closer to 121, (10.7)² = 114.5, hence the estimated value is 10.7
115/11 = 10.4545
10.5624
Neither are correct

Question 21.
Math History
There is a famous irrational number called Euler’s number, symbolized with an e. Like π, its decimal form never ends or repeats. The first few digits of e are 2.7182818284.
a. Between which two square roots of integers could you find this number?
Type below:
_____________

Answer:
The square of e lies between 7 and 8
2.718281828
(2.72)² = 7.3984
Hence, it lies between \(\sqrt { 7 } \) = 2.65 and \(\sqrt { 8 } \) = 2.82

Question 21.
b. Between which two square roots of integers can you find π?
Type below:
_____________

Answer:
3.142
(3.14)² = 9.8596
Hence. it lies between \(\sqrt { 9 } \) = 3 and \(\sqrt { 10 } \) = 3.16

H.O.T.

FOCUS ON HIGHER ORDER THINKING

Question 22.
Analyze Relationships
There are several approximations used for π, including 3.14 and \(\frac{22}{7}\). π is approximately 3.14159265358979 . . .
a. Label π and the two approximations on the number line.

Type below:
_____________

Answer:

Question 22.
b. Which of the two approximations is a better estimate for π? Explain.
Type below:
_____________

Answer:
As we can see from the number line, 22/7 is closer to π, so we can conclude that 22/7 is a better estimation for π.

Question 22.
c. Find a whole number x so that the ratio \(\frac{x}{113}\) is a better estimate for π than the two given approximations.
Type below:
_____________

Answer:
355/113 is a better estimation for π, because 355/113 = 3.14159292035 = 3.14159265358979 = π

Question 23.
Communicate Mathematical Ideas
What is the fewest number of distinct points that must be graphed on a number line, in order to represent natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers? Explain.
_______ points

Answer:
2 points

Explanation:
There need to be plotting of at least 2 points because a rational number can never be equal to an irrational number. So let’s say 5 points are the same among six but the 6th will be different as there both rational numbers and irrational numbers included.

Question 24.
Critique Reasoning
Jill says that 12.\(\bar{6}\) is less than 12.63. Explain her error.
Type below:
_____________

Answer:
12.\(\bar{6}\) = 12.666
12.\(\bar{6}\) > 12.63

1.1 Rational and Irrational Numbers – Model Quiz – Page No. 27

Write each fraction as a decimal or each decimal as a fraction.

Question 1.
\(\frac{7}{20}\)
_______

Answer:
0.35

Explanation:
\(\frac{7}{20}\) = 0.35

Question 2.
1.\(\overline { 27} \)
______ \(\frac{□}{□}\)

Answer:
1\(\frac{28}{99}\)

Explanation:
1.\(\overline { 27} \)
x = 1.\(\overline { 27} \)
100x = 100(1.\(\overline { 27} \))
100x = 127(\(\overline { 27} \))
x = .\(\overline { 27} \)
99x = 127
x = 127/99
x = 1 28/99

Question 3.
1 \(\frac{7}{8}\)
______

Answer:
1.875

Explanation:
1 \(\frac{7}{8}\)
1 + 7/8
8/8 + 7/8
15/8 = 1.875

Solve each equation for x.

Question 4.
x² = 81
± ______

Answer:
± 9

Explanation:
x² = 81
x = ± 81
x = ± 9

Question 5.
x³ = 343
______

Answer:
x = 7

Explanation:
x³ = 343
x = 7

Question 6.
x² = \(\frac{1}{100}\)
± \(\frac{□}{□}\)

Answer:
± \(\frac{1}{10}\)

Explanation:
x² = \(\frac{1}{100}\)
x = ± \(\frac{1}{10}\)

Question 7.
A square patio has an area of 200 square feet. How long is each side of the patio to the nearest 0.05?
______ feet

Answer:
14.15 feet

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the square, we have to take the square root of the area.
Let’s denote with A the area of the patio and with s each side of the square.
We have:
A = 200
A = s.s
s = \(\sqrt { A } \) = \(\sqrt { 200 } \)
Following the steps as in “Explore Activity” on page 9, we can make an estimation for the irrational number:
196 < 200 < 225
\(\sqrt { 196 } \) < \(\sqrt { 200 } \) < \(\sqrt { 225 } \)
14 < \(\sqrt { 200 } \) < 15
We see that 200 is much closer to 196 than to 225, therefore the square root of it should be between 14 and 14.5. To make a better estimation, we pick some numbers between 14 and 14.5 and calculate their squares:
(14.1)² = 198.81
(14.2)² = 201.64
14.1 < \(\sqrt { 200 } \) < 14.2
\(\sqrt { 200 } \) = 14.15
We see that 200 is much closer to 14.1 than to 14.2, therefore the square root of it should be between 14.1 and 14.15. If we round to the nearest 0.05, we have:
s = 14.15

1.2 Sets of Real Numbers

Write all names that apply to each number.

Question 8.
\(\frac { 121 }{ \sqrt { 121 } }\)
Type below:
___________

Answer:
Rational, whole, integer, real numbers

Explanation:
\(\frac { 121 }{ \sqrt { 121 } }\)
121/11 = 11

Question 9.
\(\frac{π}{2}\)
Type below:
___________

Answer:
Irrational, real numbers

Question 10.
Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice.
___________

Answer:
True

Explanation:
“All integers are rational numbers” is true, because every integer can be expressed as a fraction with a denominator equal to 1. The set of integer A a subset of rational numbers.

1.3 Ordering Real Numbers

Compare. Write <, >, or =.

Question 11.
\(\sqrt { 8 }\) + 3 _______ 8 + \(\sqrt { 3 }\)

Answer:
\(\sqrt { 8 }\) + 3 < 8 + \(\sqrt { 3 }\)

Explanation:
4 < 8 < 9
\(\sqrt { 4 }\) < \(\sqrt { 8 }\) < \(\sqrt { 9 }\)
2 < \(\sqrt { 8 }\) < 3
1 < 3 < 4
\(\sqrt { 1 }\) < \(\sqrt { 3 }\) < \(\sqrt { 4 }\)
1 < \(\sqrt { 3 }\) < 2
\(\sqrt { 8 }\) + 3 is between 5 and 6
8 + \(\sqrt { 3 }\) is between 9 and 10
\(\sqrt { 8 }\) + 3 < 8 + \(\sqrt { 3 }\)

Question 12.
\(\sqrt { 5 }\) + 11 _______ 5 + \(\sqrt { 11 }\)

Answer:
\(\sqrt { 5 }\) + 11 > 5 + \(\sqrt { 11 }\)

Explanation:
\(\sqrt { 5 }\) lies in between 2 and 3
\(\sqrt { 11 }\) lies in between 3 and 4
\(\sqrt { 5 }\) + 11 lies in between 13 and 14
5 + \(\sqrt { 11 }\) lies in between 8 and 9
\(\sqrt { 5 }\) + 11 > 5 + \(\sqrt { 11 }\)

Order the numbers from least to greatest.

Question 13.
\(\sqrt { 99 }\), π², 9.\(\bar { 8 }\)
Type below:
_______________

Answer:
π², 9.\(\bar { 8 }\), \(\sqrt { 99 }\)

Explanation:
\(\sqrt { 99 }\), π², 9.\(\bar { 8 }\)
99 lies between 9² and 10²
99 is closer to 100, hence \(\sqrt { 99 }\) is closer to 10
(9.9)² = 98.01
(9.95)² = 99.0025
(10)² = 100
\(\sqrt { 99 }\) = 9.95
π² = 9.86
9.88888 = 9.89

π², 9.\(\bar { 8 }\), \(\sqrt { 99 }\)

Question 14.
\(\sqrt { \frac { 1 }{ 25 } } \), \(\frac{1}{4}\), 0.\(\bar { 2 }\)
Type below:
____________

Answer:
\(\sqrt { \frac { 1 }{ 25 } } \), 0.\(\bar { 2 }\), \(\frac{1}{4}\)

Explanation:
\(\sqrt { \frac { 1 }{ 25 } } \), \(\frac{1}{4}\), 0.\(\bar { 2 }\)
\(\sqrt { \frac { 1 }{ 25 } } \) = 1/5 = 0.2
1/4 = 0.25
0.\(\bar { 2 }\) = 0.222 = 0.22

\(\sqrt { \frac { 1 }{ 25 } } \), 0.\(\bar { 2 }\), \(\frac{1}{4}\)

Essential Question

Question 15.
How are real numbers used to describe real-world situations?
Type below:
_______________

Answer:
In real-world situations, we use real numbers to count or make measurements. They can be seen as a convention for us to quantify things around, for example, the distance, the temperature, the height, etc.

Selected Response – Mixed Review – Page No. 28

Question 1.
The square root of a number is 9. What is the other square root?
Options:
a. -9
b. -3
c. 3
d. 81

Answer:
a. -9

Explanation:
We know that every positive number has two square roots, one positive and one negative. We are given the principal square root (9), so the other square root would be its negative (-9). To prove that, we square both numbers and we compare the results:
9 • 9 = 81
(-9). (-9)= 81

Question 2.
A square acre of land is 4,840 square yards. Between which two integers is the length of one side?
Options:
a. between 24 and 25 yards
b. between 69 and 70 yards
c. between 242 and 243 yards
d. between 695 and 696 yards

Answer:
b. between 69 and 70 yards

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to Bud the side of the square, we have to take the square root of the area.
Let’s denote with A the area of the land and with each side of the square. We have:
A = 4840
A = s . s
A = s²
s = √A = √4840
Following the steps as in °Explore Activity on page 9, we can make an estimation for the irrational number:
4761 < 4840 < 4900
\(\sqrt { 4761 }\) < \(\sqrt { 4840 }\) < \(\sqrt { 4900 }\)
69 < \(\sqrt { 4840 }\) < 70
Each side of the land is between 69 and 70 yards.

Question 3.
Which of the following is an integer but not a whole number?
Options:
a. -9.6
b. -4
c. 0
d. 3.7

Answer:
b. -4

Explanation:
Whole numbers are not negative
-4 is an integer but not a whole number

Question 4.
Which statement is false?
Options:
a. No integers are irrational numbers.
b. All whole numbers are integers.
c. No real numbers are irrational numbers.
d. All integers greater than 0 are whole numbers.

Answer:
c. No real numbers are irrational numbers.

Explanation:
Rational and irrational numbers are real numbers.

Question 5.
Which set of numbers best describes the displayed weights on a digital scale that shows each weight to the nearest half pound?
Options:
a. whole numbers
b. rational numbers
c. real numbers
d. integers

Answer:
b. rational numbers

Explanation:
The scale weighs nearest to 1/2 pound.

Question 6.
Which of the following is not true?
Options:
a. π² < 2π + 4
b. 3π > 9
c. \(\sqrt { 27 }\) + 3 > 172
d. 5 – \(\sqrt { 24 }\) < 1

Answer:
c. \(\sqrt { 27 }\) + 3 > 172

Explanation:
a. π² < 2π + 4
(3.14)² < 2(3.14) + 4
9.86 < 10.28
True
b. 3π > 9
9.42 > 9
True
c. \(\sqrt { 27 }\) + 3 > 172
5.2 + 3 > 8.5
8.2 > 8.5
False
d. 5 – \(\sqrt { 24 }\) < 1
5 – 4.90 < 1
0.1 < 1
True

Question 7.
Which number is between \(\sqrt { 21 }\) and \(\frac{3π}{2}\) ?
Options:
a. \(\frac{14}{3}\)
b. 2 \(\sqrt { 6 }\)
c. 5
d. π + 1

Answer:

Explanation:
a. \(\sqrt { 21 }\) and \(\frac{3π}{2}\)
\(\sqrt { 21 }\) = 4.58
\(\frac{3π}{2}\) = 4.71
14/3 = 4.67
b. 2\(\sqrt { 6 }\) = 4.90
c. 5
d. π + 1 = 3.14 + 1 = 4.14

Question 8.
What number is shown on the graph?

Options:
a. π+3
b. \(\sqrt { 4 }\) + 2.5
c. \(\sqrt { 20 }\) + 2
d. 6.\(\overline { 14 } \)

Answer:
c. \(\sqrt { 20 }\) + 2

Explanation:
6.48
a. π+3 = 3.14 + 3 = 6.14
b. \(\sqrt { 4 }\) + 2.5 = 2 + 2.5 = 4.5
c. \(\sqrt { 20 }\) + 2 = 4.47 + 2 = 6.47
d. 6.\(\overline { 14 } \) = 6.1414

Question 9.
Which is in order from least to greatest?
Options:
a. 3.3, \(\frac{10}{3}\), π, \(\frac{11}{4}\)
b. \(\frac{10}{3}\), 3.3, \(\frac{11}{4}\), π
c. π, \(\frac{10}{3}\), \(\frac{11}{4}\), 3.3
d. \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Answer:
d. \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Explanation:
10/3 = 3.3333333
11/4 = 2.75

Mini-Task

Question 10.
The volume of a cube is given by V = x³, where x is the length of an edge of the cube. The area of a square is given by A = x², where x is the length of a side of the square. A given cube has a volume of 1728 cubic inches.
a. Find the length of an edge.
______ inches

Answer:
12 inches

Explanation:
V = x³
A = x²
1728 = x³
x = 12
The length of an edge = 12 in

Question 10.
b. Find the area of one side of the cube.
______ in²

Answer:
144 in²

Explanation:
A = (12)² = 144
Area of the side of the cube = 144 in²

Question 10.
c. Find the surface area of the cube.
______ in²

Answer:
864 in²

Explanation:
SA = 6 (12)² = 864
The surface area of the cube = 864 in²

Question 10.
d. What is the surface area in square feet?
______ ft²

Answer:
6 ft²

Explanation:
SA = 864/144 = 6
The surface area of the cube = 6 ft²

Conclusion:

We hope the details prevailed in this Grade 8 Go Math Answer Key Chapter 1 Real Numbers is helpful for you guys. Make use of the above links and try to solve all the problems. This HMH Go Math Grade 8 Answer Key also helps to complete the homework within the time without any mistakes. Stick to our ccssmathanswers.com site to get the pdf links of all the chapters. Feel free to post your comments in the below box. We will try to clarify your doubts as early as possible. All the Best Guys!!!

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Go Math Grade 6 Chapter 5 Model Percents Answer Key

Make yourself comfortable by using HMH Go math Grade 6 Answer Key Chapter 5 Model Percents. So, make use of the resources of Go Math Answer Key to score good marks in the exams. Test your skills by solving the problems given at the end of the chapter. Just click on the links and start solving the problems.

Lesson 1: Investigate • Model Percents

• Model Percents – Page No. 271
• Model Percents – Page No. 272
• Model Percents – Page No. 273
• Model Percents Lesson Check – Page No. 274

Lesson 2: Write Percents as Fractions and Decimals

• Write Percents as Fractions and Decimals – Page No. 277
• Write Percents as Fractions and Decimals – Page No. 278
• Write Percents as Fractions and Decimals – Page No. 279
• Write Percents as Fractions and Decimals Lesson Check – Page No. 280

Lesson 3: Write Fractions and Decimals as Percents

• Write Fractions and Decimals as Percents – Page No. 283
• Write Fractions and Decimals as Percents – Page No. 284
• Write Fractions and Decimals as Percents – Page No. 285
• Write Fractions and Decimals as Percents Lesson Check – Page No. 286

Mid-Chapter Checkpoint

• Mid-Chapter Checkpoint – Page No. 287
• Mid-Chapter Checkpoint – Page No. 288

Lesson 4: Percent of a Quantity

• Percent of a Quantity – Page No. 290
• Percent of a Quantity – Page No. 291
• Percent of a Quantity – Page No. 292
• Percent of a Quantity – Page No. 293
• Percent of a Quantity Lesson Check – Page No. 294

Lesson 5: Problem Solving • Percents

• Percents – Page No. 297
• Percents – Page No. 298
• Percents – Page No. 299
• Percents Lesson Check – Page No. 300

Lesson 6: Find the Whole from a Percent

• Find the Whole from a Percent – Page No. 303
• Find the Whole from a Percent – Page No. 304
• Find the Whole from a Percent – Page No. 305
• Find the Whole from a Percent Lesson Check – Page No. 306

Chapter 5 Review/Test

• Review/Test – Page No. 307
• Review/Test – Page No. 308
• Review/Test – Page No. 309
• Review/Test – Page No. 310
• Review/Test – Page No. 311
• Review/Test – Page No. 312

Share and Show – Page No. 271

Write a ratio and a percent to represent the shaded part.

Question 1.

Type below:
_____________

Answer:
53% and \(\frac{53}{100}\)

Explanation:
53 squares are shaded out of 100.
So, 53% and 35/100 are the answers.

Question 2.

Type below:
_____________

Answer:
1% and \(\frac{100}{100}\)

Explanation:
100 out of 100 squares are shaded
So, So, 1% and 100/100 are the answers.

Question 3.

Type below:
_____________

Answer:
40% and \(\frac{40}{100}\)

Explanation:
40 squares are shaded out of 100.
So, 40% and 40/100 are the answers.

Model the percent and write it as a ratio.

Question 4.
30%
\(\frac{□}{□}\)

Answer:

Explanation:
30% is 30 out of 100
30 out of 100 squares is 30/100
30% = \(\frac{30}{100}\)

Question 5.
5%
\(\frac{□}{□}\)

Answer:

Explanation:
5% is 5 out of 100
5 out of 100 squares is 5/100
5% = \(\frac{5}{100}\)

Question 6.
75%
\(\frac{□}{□}\)

Answer:

Explanation:
75% is 75 out of 100
75 out of 100 squares is 75/100
75% = \(\frac{75}{100}\)

Problem Solving + Applications

Question 7.
Use a Concrete Model Explain how to model 32% on a 10-by-10 grid. How does the model represent the ratio of 32 to 100?
Type below:
_____________

Answer:

Question 8.
A floor has 100 tiles. There are 24 black tiles and 35 brown tiles. The rest of the tiles are white. What percent of the tiles are white?
_______ %

Answer:
41%

Explanation:
A floor has 100 tiles. There are 24 black tiles and 35 brown tiles.
24 + 35 = 59
100 – 59 = 41 tiles are white
41 tiles out of 100 are white tiles

Pose a Problem – Page No. 272

Question 9.
Javier designed a mosaic wall mural using 100 tiles in 3 different colors: yellow, blue, and red. If 64 of the tiles are yellow, what percent of the tiles are either red or blue?

To find the number of tiles that are either red or blue, count the red and blue squares. Or subtract the number of yellow squares, 64, from the total number of squares, 100.
36 out of 100 tiles are red or blue.
The ratio of red or blue tiles to all tiles is \(\frac{36}{100}\).
So, the percent of the tiles that are either red or blue is 36%.
Write another problem involving a percent that can be solved by using the mosaic wall mural.
Type below:
_____________

Answer:
Sam designed a mosaic wall mural using 100 squares using two colors. She represented the squares with red and blue colors. She has 54 red tiles. What percent of other tiles she can use with blue color?
100 – 54 = 46 blue tiles.

Question 10.
Select the 10-by-10 grids that model 45%. Mark all that apply.
Options:
a.
b.
c.
d.
e.

Answer:
a.
c.
e.

Model Percents – Page No. 273

Write a ratio and a percent to represent the shaded part.

Question 1.

Type below:
_____________

Answer:
31% and \(\frac{31}{100}\)

Explanation:
31 squares are shaded out of 100.
So, 31% and 31/100 are the answers.

Question 2.

Type below:
_____________

Answer:
70% and \(\frac{70}{100}\)

Explanation:
70 squares are shaded out of 100.
So, 70% and 70/100 are the answers.

Question 3.

Type below:
_____________

Answer:
48% and \(\frac{48}{100}\)

Explanation:
48 squares are shaded out of 100.
So, 48% and 48/100 are the answers.

Model the percent and write it as a ratio.

Question 4.
97%
\(\frac{□}{□}\)

Answer:

Explanation:
97% is 97 out of 100
97 out of 100 squares is 97/100
97% = \(\frac{97}{100}\)

Question 5.
24%
\(\frac{□}{□}\)

Answer:

Explanation:
24% is 24 out of 100
24 out of 100 squares is 24/100
24% = \(\frac{24}{100}\)

Question 6.
50%
\(\frac{□}{□}\)

Answer:

Explanation:
50% is 50 out of 100
50 out of 100 squares is 50/100
50% = \(\frac{50}{100}\)

Problem Solving

The table shows the pen colors sold at the school supply store one week. Write the ratio comparing the number of the given color sold to the total number of pens sold. Then shade the grid.

Question 7.
Black
\(\frac{□}{□}\)

Answer:
\(\frac{49}{100}\)

Explanation:
The total number of pens sold = 36 + 49 + 15 = 100
Black : total number of pens sold = 49:100
49 out of 100 squares need to shade the grid

Question 8.
Not Blue
\(\frac{□}{□}\)

Answer:
\(\frac{64}{100}\)

Explanation:
Not Blue = Black + Red = 49 + 15 = 64

Question 9.
Is every percent a ratio? Is every ratio a percent? Explain.
Type below:
_____________

Answer:
Every percent is a ratio but not all ratios are percent. All ratios can be expressed as percents, decimals, or fractions or in ratio form.

Lesson Check – Page No. 274

Question 1.
What percent of the large square is shaded?

_______ %

Answer:
63%

Explanation:
63 squares are shaded out of 100.
So, 63% and 63/100 are the answers.

Question 2.
Write a ratio to represent the shaded part.

\(\frac{□}{□}\)

Answer:
\(\frac{10}{100}\)

Explanation:
63 squares are shaded out of 100.
63/100 is the answer.

Spiral Review

Question 3.
Write a number that is less than −2 \(\frac{4}{5}\) and greater than −3 \(\frac{1}{5}\).
Type below:
_____________

Answer:
-2.9, -3.0, -3.1

Explanation:
−2 \(\frac{4}{5}\) = -14/5 = -2.8
−3 \(\frac{1}{5}\) = -16/5 = -3.2
-2.9, -3.0, -3.1 are the numbers less than −2 \(\frac{4}{5}\) and greater than −3 \(\frac{1}{5}\)

Question 4.
On a coordinate grid, what is the distance between (2, 4) and (2, –3)?
_______ units

Answer:
7 units

Explanation:
|-3| = 3
4+ 0 = 4; 0 + 3 = 3
4 + 3 = 7

Question 5.
Each week, Diana spends 4 hours playing soccer and 6 hours babysitting. Write a ratio to compare the time Diana spends playing soccer to the time she spends babysitting.
\(\frac{□}{□}\)

Answer:
\(\frac{2}{3}\)

Explanation:
Each week, Diana spends 4 hours playing soccer and 6 hours babysitting.
The ratio to compare the time Diana spends playing soccer to the time she spends babysitting is 4:6 or 4/6 = 2/3

Question 6.
Antwone earns money at a steady rate mowing lawns. The points (1, 25) and (5, 125) appear on a graph of the amount earned versus number of lawns mowed. What are the coordinates of the point on the graph with an x-value of 3?
Type below:
_____________

Answer:
(3, 75)

Explanation:
y2-y1/x2-x1.
Y2 is 125, Y1 is 25, X2 is 5, and X1 is 1.
You then plug the numbers in, 125-25=100. 5-1=4.
Then you divide 100/4, in which you get 25. So you time 25 by 3, getting 75.

Share and Show – Page No. 277

Write the percent as a fraction.

Question 1.
80%
\(\frac{□}{□}\)

Answer:
\(\frac{80}{100}\)

Explanation:
80% is 80 out of 100
80 out of 100 squares is 80/100

Question 2.
150%
______ \(\frac{□}{□}\)

Answer:
1\(\frac{1}{2}\)

Explanation:
150% is 150 out of 100
150 out of 100 squares is 150/100 = 3/2 = 1 1/2

Question 3.
0.2%
\(\frac{□}{□}\)

Answer:
\(\frac{2}{1,000}\)

Explanation:
0.2% is 0.2 out of 100
0.2 out of 100 squares is 0.2/100 = 2/1,000

Write the percent as a decimal.

Question 4.
58%
______

Answer:
0.58

Explanation:
58% is 58 out of 100
58 out of 100 squares is 58/100
58/100 = 0.58

Question 5.
9%
______

Answer:
0.09

Explanation:
9% is 9 out of 100
9 out of 100 squares is 9/100
9/100 = 0.09

On Your Own

Write the percent as a fraction or mixed number.

Question 6.
17%
\(\frac{□}{□}\)

Answer:
\(\frac{17}{100}\)

Explanation:
17% is 17 out of 100
17 out of 100 squares is 17/100

Question 7.
20%
\(\frac{□}{□}\)

Answer:
\(\frac{1}{5}\)

Explanation:
20% is 20 out of 100
20 out of 100 squares is 20/100 = 2/10 = 1/5

Question 8.
125%
______ \(\frac{□}{□}\)

Answer:
1\(\frac{1{4}\)

Explanation:
125% is 125 out of 100
125 out of 100 squares is 125/100 = 1 1/4

Question 9.
355%
______ \(\frac{□}{□}\)

Answer:
3\(\frac{11}{20}\)

Explanation:
355% is 355 out of 100
355 out of 100 squares is 355/100 = 3 11/20

Question 10.
0.1%
\(\frac{□}{□}\)

Answer:
\(\frac{1}{1,000}\)

Explanation:
0.1% is 0.1 out of 100
0.1 out of 100 squares is 0.1/100 = 1/1,000

Question 11.
2.5%
\(\frac{□}{□}\)

Answer:
\(\frac{1}{40}\)

Explanation:
2.5% is 2.5 out of 100
2.5 out of 100 squares is 2.5/100 = 25/1,000 = 1/40

Write the percent as a decimal.

Question 12.
89%
______

Answer:
0.89

Explanation:
89% is 89 out of 100
89 out of 100 squares is 89/100
89/100 = 0.89

Question 13.
30%
______

Answer:
0.3

Explanation:
30% is 30 out of 100
30 out of 100 squares is 30/100
30/100 = 0.3

Question 14.
2%
______

Answer:
0.02

Explanation:
2% is 2 out of 100
2 out of 100 squares is 2/100
2/100 = 0.02

Question 15.
122%
______

Answer:
1.22

Explanation:
122% is 122 out of 100
122 out of 100 squares is 122/100
122/100 = 1.22

Question 16.
3.5%
______

Answer:
0.035

Explanation:
3.5% is 3.5 out of 100
3.5 out of 100 squares is 3.5/100
3.5/100 = 0.035

Question 17.
6.33%
______

Answer:
0.0633

Explanation:
6.33% is 6.33 out of 100
6.33 out of 100 squares is 6.33/100
6.33/100 = 0.0633

Question 18.
Use Reasoning Write <, >, or =.
21.6% ______ \(\frac{1}{5}\)

Answer:
21.6% > \(\frac{1}{5}\)

Explanation:
1/5 × 100/100 = 100/500 = 0.2/100 = 0.2%
21.6% > 0.2%

Question 19.
Georgianne completed 60% of her homework assignment. Write the portion of her homework that she still needs to complete as a fraction.
\(\frac{□}{□}\)

Answer:
\(\frac{2}{5}\)

Explanation:
Georgianne completed 60% of her homework assignment.
60/100
She needs to complete 40% of her homework = 40/100 = 2/5

Problem Solving + Applications – Page No. 278

Use the table for 20 and 21.

Question 20.
What fraction of computer and video game players are 50 years old or more?
\(\frac{□}{□}\)

Answer:
\(\frac{13}{50}\)

Explanation:
computer and video game players,
50 or more are of 26% = 26/100 = 13/50

Question 21.
What fraction of computer and video game players are 18 years old or more?
\(\frac{□}{□}\)

Answer:
\(\frac{49}{100}\)

Explanation:
18 years old or more are of 49% = 49/100

Question 22.
Box A and Box B each contain black tiles and white tiles. They have the same total number of tiles. In Box A, 45% of the tiles are black. In Box B, \(\frac{11}{20}\) of the tiles are white. Compare the number of black tiles in the boxes. Explain your reasoning.
Type below:
_____________

Answer:
In Box A, 45% of the tiles are black.
In Box B, \(\frac{11}{20}\) of the tiles are white.
11/20 = 0.55 = 55/100 = 55%
100 – 55 = 45%
Both Box A and Box B have an equal number of black tiles

Question 23.
Mr. Truong is organizing a summer program for 6th grade students. He surveyed students to find the percent of students interested in each activity. Complete the table by writing each percent as a fraction or decimal.

Type below:
_____________

Answer:
Sports = 48% = 48/100 = 0.48
Cooking = 23% = 23/100
Music = 20% = 20/100
Art = 9% = 9/100 = 0.09

Write Percents as Fractions and Decimals – Page No. 279

Write the percent as a fraction or mixed number.

Question 1.
44%
\(\frac{□}{□}\)

Answer:
\(\frac{11}{25}\)

Explanation:
44% is 44 out of 100
44 out of 100 squares is 44/100 = 11/25

Question 2.
32%
\(\frac{□}{□}\)

Answer:
\(\frac{8}{25}\)

Explanation:
32% is 32 out of 100
32 out of 100 squares is 32/100 = 8/25

Question 3.
116%
______ \(\frac{□}{□}\)

Answer:
1 \(\frac{4}{25}\)

Explanation:
116% is 116 out of 100
116 out of 100 squares is 116/100 = 1 4/25

Question 4.
250%
______ \(\frac{□}{□}\)

Answer:
2\(\frac{1}{2}\)

Explanation:
250% is 250 out of 100
250 out of 100 squares is 250/100 = 2 1/2

Question 5.
0.3%
\(\frac{□}{□}\)

Answer:
\(\frac{3}{1,000}\)

Explanation:
0.3% is 0.3 out of 100
0.3 out of 100 squares is 0.3/100
3/1,000

Question 6.
0.4%
\(\frac{□}{□}\)

Answer:
\(\frac{1}{250}\)

Explanation:
0.4% is 0.4 out of 100
0.4 out of 100 squares is 0.4/100 = 4/1,000 = 1/250

Question 7.
1.5%
\(\frac{□}{□}\)

Answer:
\(\frac{3}{200}\)

Explanation:
1.5% is 1.5 out of 100
1.5 out of 100 squares is 1.5/100 = 15/1,000 = 3/200

Question 8.
12.5%
\(\frac{□}{□}\)

Answer:
\(\frac{1}{8}\)

Explanation:
12.5% is 12.5 out of 100
12.5 out of 100 squares is 12.5/100 = 125/1,000 = 25/200 = 5/40 = 1/8

Write the percent as a decimal.

Question 9.
63%
______

Answer:
0.63

Explanation:
63% is 63 out of 100
63 out of 100 squares is 63/100
63/100 = 0.63

Question 10.
110%
______

Answer:
1.1

Explanation:
110% is 110 out of 100
110 out of 100 squares is 110/100 = 1.1

Question 11.
42.15%
______

Answer:
0.4215

Explanation:
42.15% is 42.15 out of 100
42.15 out of 100 squares is 42.15/100 = 0.4215

Question 12.
0.1%
______

Answer:
0.001

Explanation:
0.1% is 0.1 out of 100
0.1 out of 100 squares is 0.1/100  = 0.001

Problem Solving

Question 13.
An online bookstore sells 0.8% of its books to foreign customers. What fraction of the books are sold to foreign customers?
\(\frac{□}{□}\)

Answer:
\(\frac{1}{125}\)

Explanation:
An online bookstore sells 0.8% of its books to foreign customers.
0.8% = 0.8/100 = 8/1,000 = 1/125

Question 14.
In Mr. Klein’s class, 40% of the students are boys. What decimal represents the portion of the students that are girls?
______

Answer:
0.4

Explanation:
In Mr. Klein’s class, 40% of the students are boys.
40/100 = 0.4

Question 15.
Explain how percents, fractions, and decimals are related. Use a 10-by-10 grid to make a model that supports your explanation.
Type below:
_____________

Answer:

53 squares are shaded out of 100.
53% or \(\frac{53}{100}\) or 0.53

Lesson Check – Page No. 280

Question 1.
The enrollment at Sonya’s school this year is 109% of last year’s enrollment. What decimal represents this year’s enrollment compared to last year’s?
______

Answer:
1.09 represents this year’s enrollment compared to last year’s

Explanation:
The enrollment at Sonya’s school this year is 109% of last year’s enrollment.
109% = 109/100 = 1.09

Question 2.
An artist’s paint set contains 30% watercolors and 25% acrylics. What fraction represents the portion of the paints that are watercolors or acrylics? Write the fraction in simplest form.
\(\frac{□}{□}\)

Answer:
\(\frac{11}{20}\)

Explanation:
An artist’s paint set contains 30% watercolors and 25% acrylics.
30 + 25 = 55% = 55/100 = 11/20

Spiral Review

Question 3.
Write the numbers in order from least to greatest.
-5.25 1.002 -5.09
Type below:
_____________

Answer:
-5.25, -5.09, 1.002

Question 4.
On a coordinate plane, the vertices of a rectangle are (2, 4), (2, −1), (−5, −1), and ( −5, 4). What is the perimeter of the rectangle?
______ units

Answer:
24 units

Explanation:
(2, 4) to (2, −1) is 4 + 1 = 5
(2, −1) to (−5, −1) is 2 + 5 = 7
5 + 7 + 5 + 7 = 24

Question 5.
The table below shows the widths and lengths, in feet, for different playgrounds. Which playgrounds have equivalent ratios of width to length?

Type below:
_____________

Answer:
12/20 and 16.5/27.5 are equal

Explanation:
12/20 = 0.6
15/22.5 = 0.666
20/25 = 0.8
16.5/27.5 = 0.6

Question 6.
What percent represents the shaded part?

_______ %

Answer:
85%

Explanation:
85 squares are shaded out of 100.
85%

Share and Show – Page No. 283

Write the fraction or decimal as a percent.

Question 1.
\(\frac{3}{25}\)
_______ %

Answer:
12%

Explanation:
3/25 ÷ 25/25 = 0.12/1 = 12/100 = 12%

Question 2.
\(\frac{3}{10}\)
_______ %

Answer:
30%

Explanation:
3/10 ÷ 10/10 = 0.3 = 0.3 × 100/100 = 30/100 = 30%

Question 3.
0.717
_______ %

Answer:
71.7%

Explanation:
0.717 = 717/100 = 71.7%

Question 4.
0.02
_______ %

Answer:
2%

Explanation:
0.02 = 2/100 = 2%

On Your Own

Write the number in two other forms ( fraction, decimal, or percent). Write the fraction in simplest form.

Question 5.
0.01
Type below:
_____________

Answer:
1% and \(\frac{1}{100}\)

Explanation:
0.01 as a fraction 1/100
0.01 as percent 1%

Question 6.
\(\frac{13}{40}\)
Type below:
_____________

Answer:
0.325 and 32.5%

Explanation:
\(\frac{13}{40}\) as decimal 0.325
\(\frac{13}{40}\) as percent 32.5/100 = 32.5%

Question 7.
\(\frac{6}{5}\)
Type below:
_____________

Answer:
1.2 and 120%

Explanation:
\(\frac{6}{5}\) as decimal 1.2
\(\frac{6}{5}\) as percent 120/100 = 120%

Question 8.
0.08
Type below:
_____________

Answer:
8% and \(\frac{8}{100}\)

Explanation:
0.08 as a fraction 8/100
0.08 as percent 8%

The table shows the portion of Kim’s class that participates in each sport. Use the table for 9–10.

Question 9.
Do more students take part in soccer or in swimming? Explain your reasoning.
Type below:
_____________

Answer:
Soccer = 1/5 = 0.2
Swimming = 0.09
0.2 > 0.09
more students take part in Soccer

Question 10.
Explain What percent of Kim’s class participates in one of the sports listed? Explain how you found your answer
_______ %

Answer:
23%

Explanation:
Kim’s class participates in Baseball that is mentioned with 23%

Question 11.
For their reading project, students chose to either complete a character study, or write a book review. \(\frac{1}{5}\) of the students completed a character study, and 0.8 of the students wrote a book review. Joia said that more students wrote a book review than completed a character study. Do you agree with Joia? Use numbers and words to support your answer
Type below:
_____________

Answer:
1/5 = 0.2
0.2 < 0.8
More students completed writing a book review.
I agree with Joia

Sand Sculptures – Page No. 284

Every year, dozens of teams compete in the U.S. Open Sandcastle Competition. Recent winners have included complex sculptures in the shape of flowers, elephants, and racing cars.

Teams that participate in the contest build their sculptures using a mixture of sand and water. Finding the correct ratios of these ingredients is essential for creating a stable sculpture.

The table shows the recipes that three teams used. Which team used the greatest percent of sand in their recipe?

Convert to percents. Then order from least to greatest.

From least to greatest, the percents are 75%, 84%, 95%.
So, Team B used the greatest percent of sand.
Solve.

Question 12.
Which team used the greatest percent of water in their recipe?
Type below:
_____________

Answer:
Team A used the greatest percent of water in their recipe

Explanation:
Team A, 10/10+30 = 10/40 = 0.25 = 25%
Team B, 1/20 × 5/5 = 5/100 = 5%
Team C, 0.16 = 16%

Question 13.
Some people say that the ideal recipe for sand sculptures contains 88.9% sand. Which team’s recipe is closest to the ideal recipe?
Type below:
_____________

Answer:
Team C

Question 14.
Team D used a recipe that consists of 20 cups of sand, 2 cups of flour, and 3 cups of water. How does the percent of sand in Team D’s recipe compare to that of the other teams?
Type below:
_____________

Answer:
Total number of cups together = 20 + 2+ 3 =25 cups
20/25 × 100 = 80/100 = 80%

Write Fractions and Decimals as Percents – Page No. 285

Write the fraction or decimal as a percent.

Question 1.
\(\frac{7}{20}\)
_______ %

Answer:
35%

Explanation:
7/20 = 0.35 = 35%

Question 2.
\(\frac{3}{50}\)
_______ %

Answer:
6%

Explanation:
3/50 = 0.06 = 6%

Question 3.
\(\frac{1}{25}\)
_______ %

Answer:
4%

Explanation:
1/25 = 0.04 = 4%

Question 4.
\(\frac{5}{5}\)
_______ %

Answer:
0.01%

Explanation:
5/5 = 1 = 0.01%

Question 5.
0.622
_______ %

Answer:
6.22%

Explanation:
0.622 = 6.22/100 = 6.22%

Question 6.
0.303
_______ %

Answer:
3.03%

Explanation:
0.303 = 3.03/100 = 3.03%

Question 7.
0.06
_______ %

Answer:
6%

Explanation:
0.06 = 6/100 = 6%

Question 8.
2.45
_______ %

Answer:
245%

Explanation:
2.45 × 100/100 = 245/100 = 245%

Write the number in two other forms (fraction, decimal, or percent). Write the fraction in simplest form

Question 9.
\(\frac{19}{20}\)
Type below:
_____________

Answer:
0.95 and 95%

Explanation:
\(\frac{19}{20}\) as a decimal 0.95
\(\frac{19}{20}\) as a percentage 95%

Question 10.
\(\frac{9}{16}\)
Type below:
_____________

Answer:
0.5625 and 56.25%

Explanation:
\(\frac{9}{16}\) as a decimal 0.5625
\(\frac{9}{16}\) as a percentage 56.25%

Question 11.
0.4
Type below:
_____________

Answer:
\(\frac{2}{5}\) and 40%

Explanation:
0.4 as a fraction 2/5
0.4 as a percentage 40/100 = 40%

Question 12.
0.22
Type below:
_____________

Answer:
\(\frac{11}{50}\) and 22%

Explanation:
0.22 as a fraction 11/50
0.22 as a percentage 22/100 = 22%

Problem Solving

Question 13.
According to the U.S. Census Bureau, \(\frac{3}{25}\) of all adults in the United States visited a zoo in 2007. What percent of all adults in the United States visited a zoo in 2007?
_______ %

Answer:
12%

Explanation:
According to the U.S. Census Bureau, \(\frac{3}{25}\) of all adults in the United States visited a zoo in 2007.
\(\frac{3}{25}\) = 0.12 = 12%

Question 14.
A bag contains red and blue marbles. Given that \(\frac{17}{20}\) of the marbles are red, what percent of the marbles are blue?
_______ %

Answer:
15%

Explanation:
The total number of marbles = 20
If 17 marbles are red, the remaining 3 marbles out of 20 are blue marbles
3/20 = 0.15 = 15%

Question 15.
Explain two ways to write \(\frac{4}{5}\) as a percent.
Type below:
_____________

Answer:
Decimal =0.8.
Percentage =80%

Explanation:
4/5 = 0.8 = 80/100 = 80%

Lesson Check – Page No. 286

Question 1.
The portion of shoppers at a supermarket who pay by credit card is 0.36. What percent of shoppers at the supermarket do NOT pay by credit card?
_______ %

Answer:
36%

Explanation:
The portion of shoppers at a supermarket who pay by credit card is 0.36.
0.36 = 0.36 × 100/100 = 36/100 = 36%

Question 2.
About \(\frac{23}{40}\) of a lawn is planted with Kentucky bluegrass. What percent of the lawn is planted with Kentucky bluegrass?
_______ %

Answer:
57.5%

Explanation:
About \(\frac{23}{40}\) of a lawn is planted with Kentucky bluegrass.
23/40 = 0.575 = 0.575 × 100/100 = 57.5/100 = 57.5%

Spiral Review

Question 3.
A basket contains 6 peaches and 8 plums. What is the ratio of peaches to total pieces of fruit?
Type below:
_____________

Answer:
6:14

Explanation:
total pieces of fruit 6 + 8 = 14
the ratio of peaches to total pieces of fruit is 6:14

Question 4.
It takes 8 minutes for 3 cars to move through a car wash. At the same rate, how many cars can move through the car wash in 24 minutes?
_______ cars

Answer:
9 cars

Explanation:
It takes 8 minutes for 3 cars to move through a car wash.
3/8 × 24 = 9 cars

Question 5.
A 14-ounce box of cereal sells for $2.10. What is the unit rate?
$ _______ per ounce

Answer:
$0.15 per ounce

Explanation:
$2.10/14 × 14/14 = $0.15 per ounce

Question 6.
A model railroad kit contains curved tracks and straight tracks. Given that 35% of the tracks are curved, what fraction of the tracks are straight? Write the fraction in simplest form.
\(\frac{□}{□}\)

Answer:
\(\frac{7}{20}\)

Explanation:
A model railroad kit contains curved tracks and straight tracks. Given that 35% of the tracks are curved,
35% = 35/100 = 7/20

Vocabulary – Page No. 287

Choose the best term from the box to complete the sentence.

Question 1.
A _____ is a ratio that compares a quantity to 100.
Type below:
_____________

Answer:
percent

Concepts and Skills

Write a ratio and a percent to represent the shaded part.

Question 2.

Type below:
_____________

Answer:
17% and \(\frac{17}{100}\)

Explanation:
17 squares are shaded out of 100.
So, 17% and 17/100 are the answers.

Question 3.

Type below:
_____________

Answer:
60% and \(\frac{60}{100}\)

Explanation:
60 squares are shaded out of 100.
So, 60% and 60/100 are the answers.

Question 4.

Type below:
_____________

Answer:
7% and \(\frac{7}{100}\)

Explanation:
7 squares are shaded out of 100.
So, 7% and 7/100 are the answers.

Question 5.

Type below:
_____________

Answer:
11% and \(\frac{11}{100}\)

Explanation:
11 squares are shaded out of 100.
So, 11% and 11/100 are the answers.

Question 6.

Type below:
_____________

Answer:
82% and \(\frac{82}{100}\)

Explanation:
82 squares are shaded out of 100.
So, 82% and 82/100 are the answers.

Question 7.

Type below:
_____________

Answer:
36% and \(\frac{36}{100}\)

Explanation:
36 squares are shaded out of 100.
So, 36% and 36/100 are the answers.

Write the number in two other forms (fraction, decimal, or percent).

Write the fraction in simplest form.

Question 8.
0.04
Type below:
_____________

Answer:
\(\frac{1}{25}\) and 4%

Explanation:
0.04 as a fraction 4/100 = 1/25
0.04 as a decimal 0.04 × 100/100 = 4/100 = 4%

Question 9.
\(\frac{3}{10}\)
Type below:
_____________

Answer:
0.3 and 30%

Explanation:
\(\frac{3}{10}\) as a decimal 0.3
\(\frac{3}{10}\) as a percentage 0.3 × 100/100 = 30/100 = 30%

Question 10.
1%
Type below:
_____________

Answer:
\(\frac{1}{100}\) and 0.01

Explanation:
1% as a fraction 1/100
1% as a decimal 1/100 = 0.01

Question 11.
1 \(\frac{1}{5}\)
Type below:
_____________

Answer:
1.2 and 120%

Explanation:
1 \(\frac{1}{5}\) as a decimal = 6/5 = 1.2
1 \(\frac{1}{5}\) as a percentage 1.2 × 100/100 = 120/100 = 120%

Question 12.
0.9
Type below:
_____________

Answer:
\(\frac{90}{100}\) and 90%

Explanation:
0.9 as a fraction 0.9 × 100/100 = 90/100 = 90%

Question 13.
0.5%
Type below:
_____________

Answer:
\(\frac{5}{1,000}\) and 0.005

Explanation:
0.5% as a fraction = 0.5/100 = 5/1,000
0.5% as a decimal = 0.5/100 = 0.005

Question 14.
\(\frac{7}{8}\)
Type below:
_____________

Answer:
0.875 and 87.5%

Explanation:
\(\frac{7}{8}\) as a decimal 0.875
\(\frac{7}{8}\) as a percentage 87.5/100 = 87.5%

Question 15.
355%
Type below:
_____________

Answer:
\(\frac{71}{20}\) and 35.5

Explanation:
355% as a decimal 355/100 = 71/20 = 35.5

Page No. 288

Question 16.
About \(\frac{9}{10}\) of the avocados grown in the United States are grown in California. About what percent of the avocados grown in the United States are grown in California?
_______ %

Answer:
90%

Explanation:
About \(\frac{9}{10}\) of the avocados grown in the United States are grown in California.
9/10 × 10/10 = 90/100 = 90%

Question 17.
Morton made 36 out of 48 free throws last season. What percent of his free throws did Morton make?
_______ %

Answer:
75%

Explanation:
Morton made 36 out of 48 free throws last season.
36/48 = 0.75 = 75/100 = 75%

Question 18.
Sarah answered 85% of the trivia questions correctly. What fraction describes this percent?
\(\frac{□}{□}\)

Answer:
\(\frac{17}{20}\)

Explanation:
Sarah answered 85% of the trivia questions correctly.
85% = 85/100 = 17/20

Question 19.
About \(\frac{4}{5}\) of all the orange juice in the world is produced in Brazil. About what percent of all the orange juice in the world is produced in Brazil?
_______ %

Answer:
80%

Explanation:
About \(\frac{4}{5}\) of all the orange juice in the world is produced in Brazil.
4/5 = 0.8 × 100/100 = 80/100 = 80%

Question 20.
If you eat 4 medium strawberries, you get 48% of your daily recommended amount of vitamin C. What fraction of your daily amount of vitamin C do you still need?
\(\frac{□}{□}\)

Answer:
\(\frac{13}{25}\)

Explanation:
If you eat 4 medium strawberries, you get 48% of your daily recommended amount of vitamin C.
48% = 48/100
100 – 48 = 52
52% = 52/100 = 13/25 of your daily amount of vitamin C do you still need

Share and Show – Page No. 290

Find the percent of the quantity.

Question 1.
25% of 320
_______

Answer:
80

Explanation:
Write the percent as a rate per 100
25% = 25/100
25/100 × 320 = 80

Question 2.
80% of 50
_______

Answer:
40

Explanation:
Write the percent as a rate per 100
80% = 80/100
80/100 × 50 = 40

Question 3.
175% of 24
_______

Answer:
42

Explanation:
Write the percent as a rate per 100
175% = 175/100
175/100 × 24 = 42

Question 4.
60% of 210
_______

Answer:
126

Explanation:
Write the percent as a rate per 100
60% = 60/100
60/100 × 210 = 126

Question 5.
A jar contains 125 marbles. Given that 4% of the marbles are green, 60% of the marbles are blue, and the rest are red, how many red marbles are in the jar?
_______ marbles

Answer:
45 marbles

Explanation:
A jar contains 125 marbles.
4% of the marbles are green = 125 × 4/100 = 5
60% of the marbles are blue = 125 × 60/100 = 75
Red Marbles = Total Number of Marbles -[Number of Green Marbles + Number of Blue Marbles]
Red Marbles = 125 – (5 + 75) = 125 – 80 = 45

Question 6.
There are 32 students in Mr. Moreno’s class and 62.5% of the students are girls. How many boys are in the class?
_______ students

Answer:
12 students

Explanation:
There are 32 students in Mr. Moreno’s class
62.5% of the students are girls = 32 × 62.5/100 = 20
boys = 32 – 20 = 12

On Your Own – Page No. 291

Find the percent of the quantity.

Question 7.
60% of 90
_______

Answer:
54

Explanation:
Write the percent as a rate per 100
60% = 60/100
60/100 × 90 = 54

Question 8.
25% of 32.4
_______

Answer:
8.1

Explanation:
Write the percent as a rate per 100
25% = 25/100
25/100 × 32.4 = 8.1

Question 9.
110% of 300
_______

Answer:
330

Explanation:
Write the percent as a rate per 100
110% = 110/100
110/100 × 300 = 330

Question 10.
0.2% of 6500
_______

Answer:
13

Explanation:
Write the percent as a rate per 100
0.2% = 0.2/100
0.2/100 × 6500 = 13

Question 11.
A baker made 60 muffins for a cafe. By noon, 45% of the muffins were sold. How many muffins were sold by noon?
_______ muffins

Answer:
27 muffins

Explanation:
A baker made 60 muffins for a cafe. By noon, 45% of the muffins were sold.
60 × 45%
60 × 45/100 = 27

Question 12.
There are 30 treasures hidden in a castle in a video game. LaToya found 80% of them. How many of the treasures did LaToya find?
_______ treasures

Answer:
24 treasures

Explanation:
There are 30 treasures hidden in a castle in a video game.
LaToya found 80% of them.
30 × 80/100 = 24

Question 13.
A school library has 260 DVDs in its collection. Given that 45% of the DVDs are about science and 40% are about history, how many of the DVDs are about other subjects?
_______ DVDs

Answer:
39 DVDs

Explanation:
A school library has 260 DVDs in its collection.
45% of the DVDs are about science = 260 × 45/100 = 117
40% are about history = 260 × 40/100 = 104
other subjects = 260 – (117 + 104) = 260 – 221 = 39

Question 14.
Mitch planted cabbage, squash, and carrots on his 150-acre farm. He planted half the farm with squash and 22% with carrots. How many acres did he plant with cabbage?
_______ acres

Answer:

Explanation:
Mitch planted cabbage, squash, and carrots on his 150-acre farm.
He planted half the farm with squash 150/2 = 75
22% with carrots = 150 × 22/100 = 33
cabbage = 150 – (75 + 33) = 150 – 108 = 42

Question 15.
45% of 60 _______ 60% of 45

Answer:
45% of 60 = 60% of 45

Explanation:
45% of 60
45/100 × 60 = 27
60% of 45
60/100 × 45 = 27
45% of 60 = 60% of 45

Question 16.
10% of 90 _______ 90% of 100

Answer:
10% of 90 _______ 90% of 100

Explanation:
10% of 90
10/100 × 90 = 9
90% of 100
90/100 × 100 = 90
10% of 90 < 90% of 100

Question 17.
75% of 8 _______ 8% of 7.5

Answer:
75% of 8 > 8% of 7.5

Explanation:
75% of 8
75/100 × 8 = 6
8% of 7.5
8/100 × 7.5 = 0.6
75% of 8 > 8% of 7.5

Question 18.
Sarah had 12 free throw attempts during a game and made at least 75% of the free throws. What is the greatest number of free throws Sarah could have missed during the game?
_______ free throws

Answer:
3 free throws

Explanation:
Sarah had 12 free throw attempts during a game and made at least 75% of the free throws.
So, she missed 25% of the free throws.
12 × 25/100 = 3

Question 19.
Chrissie likes to tip a server in a restaurant a minimum of 20%. She and her friend have a lunch bill that is $18.34. Chrissie says the tip will be $3.30. Her friend says that is not a minimum of 20%. Who is correct? Explain.
Type below:
_____________

Answer:
100% = $18.34
10% = $18.34 / 10 = 1.834
20% = 1.834 × 2 = 3.66800 = $3.70
Her friend is correct because $3.70 is more than $3.30.

Unlock The Problem – Page No. 292

Question 20.
One-third of the juniors in the Linwood High School Marching Band play the trumpet. The band has 50 members and the table shows what percent of the band members are freshmen, sophomores, juniors, and seniors. How many juniors play the trumpet?

a. What do you need to find?
Type below:
_____________

Answer:
The percent of the band members are freshmen, sophomores, juniors, and seniors. How many juniors play the trumpet

Question 20.
b. How can you use the table to help you solve the problem?
Type below:
_____________

Answer:
percent of the band members that are Juniors: 24%
In 50 members of the band, 50×24/100 = 12 are Juniors. One-third of them play the trumpet, which makes 12×(1/3) = 4 members.

Question 20.
c. What operation can you use to find the number of juniors in the band?
Type below:
_____________

Answer:
percent of the band members that are Juniors: 24%
In 50 members of the band, 50×24/100 = 12 are Juniors.

Explanation:

Question 20.
d. Show the steps you use to solve the problem.
Type below:
_____________

Answer:
percent of the band members that are Juniors: 24%
In 50 members of the band, 50×24/100 = 12 are Juniors. One-third of them play the trumpet, which makes 12×(1/3) = 4 members.

Question 20.
e. Complete the sentences.
The band has _____ members. There are _____ juniors in the band. The number of juniors who play the trumpet is _____.
Type below:
_____________

Answer:
The band has 50 members. There are 12 juniors in the band. The number of juniors who play the trumpet is 4.

Question 21.
Compare. Circle <, >, or =.
a. 25% of 44 Ο 20% of 50
b. 10% of 30 Ο 30% of 100
c. 35% of 60 Ο 60% of 35
25% of 44 _____ 20% of 50
10% of 30 _____ 30% of 100
35% of 60 _____ 60% of 35

Answer:
25% of 44 >  20% of 50
10% of 30 < 30% of 100
35% of 60 = 60% of 35

Explanation:
25% of 44 = 25/100 × 44 = 11
20% of 50 = 20/100 × 50 = 1000/100 = 10
25% of 44  > 20% of 50
10% of 30 = 10/100 × 30 = 3
30% of 100 = 30/100 × 100 = 30
10% of 30 < 30% of 100
35% of 60 = 35/100 × 60 = 21
60% of 35 = 60/100 × 35 = 21
35% of 60 = 60% of 35

Percent of a Quantity – Page No. 293

Find the percent of the quantity.

Question 1.
60% of 140
_____

Answer:
84

Explanation:
60% of 140
60/100 × 140 = 84

Question 2.
55% of 600
_____

Answer:
330

Explanation:
55% of 600
55/100 × 600 = 330

Question 3.
4% of 50
_____

Answer:
2

Explanation:
4% of 50
4/100 × 50 = 2

Question 4.
10% of 2,350
_____

Answer:
235

Explanation:
10% of 2,350
10/100 × 2,350 = 235

Question 5.
160% of 30
_____

Answer:
48

Explanation:
160% of 30
160/100 × 30 = 48

Question 6.
105% of 260
_____

Answer:
273

Explanation:
105% of 260
105/100 × 260 = 273

Question 7.
0.5% of 12
_____

Answer:
0.06

Explanation:
0.5% of 12
0.5/100 × 12 = 0.06

Question 8.
40% of 16.5
_____

Answer:
6.6

Explanation:
40% of 16.5
40/100 × 16.5 =  6.6

Problem Solving

Question 9.
The recommended daily amount of vitamin C for children 9 to 13 years old is 45 mg. A serving of a juice drink contains 60% of the recommended amount. How much vitamin C does the juice drink contain?
_____ mg

Answer:
27 mg

Explanation:
The recommended daily amount of vitamin C for children 9 to 13 years old is 45 mg. A serving of a juice drink contains 60% of the recommended amount.
45% of 60 = 45/100 × 60 = 27

Question 10.
During a 60-minute television program, 25% of the time is used for commercials and 5% of the time is used for the opening and closing credits. How many minutes remain for the program itself?
_____ minutes

Answer:
42 minutes

Explanation:
60 minutes of tv
25% + 5% = 30%
30%= 0.30
60 times 0.30= 18
60-18=42
inly 42 minutes are used for the program itself

Question 11.
Explain two ways you can find 35% of 700.
Type below:
_____________

Answer:
First way
700 : 100 = x : 35
x = 700 × 35 : 100
x = 245
Second way
700 : 100 × 35 =
245

Lesson Check – Page No. 294

Question 1.
A store has a display case with cherry, peach, and grape fruit chews. There are 160 fruit chews in the display case. Given that 25% of the fruit chews are cherry and 40% are peach, how many grape fruit chews are in the display case?
_____ grape fruit chews

Answer:
56 grape fruit chews

Explanation:
A store has a display case with cherry, peach, and grape fruit chews. There are 160 fruit chews in the display case. Given that 25% of the fruit chews are cherry and 40% are peach,
25% + 40% +?% = 100%
65% + ?% = 100%
?% = 35%
.35×160 = 56

Question 2.
Kelly has a ribbon that is 60 inches long. She cuts 40% off the ribbon for an art project. While working on the project, she decides she only needs 75% of the piece she cut off. How many inches of ribbon does Kelly end up using for her project?
_____ inches

Answer:
18 inches

Explanation:
Length of ribbon = 60 inches
Part of ribbon cut off for an art project = 40%
So, the Length of the ribbon remains is given by
40% of 60 = 40/100 × 60 = 24
Part of a piece she only needs from cut off = 75%
so, the Length of ribbon she need end up using in her project is given by
75/100 × 24 = 18

Spiral Review

Question 3.
Three of the following statements are true. Which one is NOT true?
|−12| > 1      |0| > −4      |20| > |−10|        6 < |−3|
Type below:
_____________

Answer:
|−12| > 1
12 > 1; True
|0| > −4
0 > -4; True
|20| > |−10|
20 > 10; True
6 < |−3|
6 < 3; False

Question 4.
Miyuki can type 135 words in 3 minutes. How many words can she expect to type in 8 minutes?
_____ words

Answer:
360 words

Explanation:
Miyuki can type 135 words in 3 minutes.
135/3 = 45
45 × 8 = 360

Question 5.
Which percent represents the model?

_____ %

Answer:
63%

Explanation:
63 squares are shaded out of 100
63%

Question 6.
About \(\frac{3}{5}\) of the students at Roosevelt Elementary School live within one mile of the school. What percent of students live within one mile of the school?
_____ %

Answer:
60%

Explanation:
About \(\frac{3}{5}\) of the students at Roosevelt Elementary School live within one mile of the school.
3/5 × 100/100 = 60/100 = 60%

Share and Show – Page No. 297

Question 1.
A geologist visits 40 volcanoes in Alaska and California. 15% of the volcanoes are in California. How many volcanoes does the geologist visit in California and how many in Alaska?
Type below:
_____________

Answer:
40 volcanoes = 100% of them
100 – 15% = 85%
Number of volcanoes in California = 15% of 40 volcanoes = 0.15 x 40 = 6
Number of volcanoes in Alaska = 85% of 40 volcanoes 0.85 x 40 = 34

Question 2.
What if 30% of the volcanoes were in California? How many volcanoes would the geologist have visited in California and how many in Alaska?
Type below:
_____________

Answer:
Number of volcanoes in California = 30% of 40 = 30/100 x 40 = 12
Number of volcanoes in Alaska = 70% of 40 = 70/100 x 40 = 28

Question 3.
Ricardo has $25 to spend on school supplies. He spends 72% of the money on a backpack and the rest on a large binder. How much does he spend on the backpack? How much does he spend on the binder?
Type below:
_____________

Answer:
$18 on Backpack $7 on binder.
If you turn the percent into a decimal .72 and multiply .72 by 25 you get 18 which is the cost of the backpack.
subtract 18 from 25 and you get $7 left meaning the binder was $7

Question 4.
Kevin is hiking on a trail that is 4.2 miles long. So far, he has hiked 80% of the total distance. How many more miles does Kevin have to hike in order to complete the trail?
Type below:
_____________

Answer:
0.84 miles

Explanation:
Kevin is hiking on a trail that is 4.2 miles long. So far, he has hiked 80% of the total distance.
80% of 4.2 = 80/100 x 4.2 = 3.36
4.2 – 3.36 = 0.84 miles

On Your Own – Page No. 298

Question 5.
Jordan takes 50% of the cherries from a bowl. Then Mei takes 50% of the remaining cherries. Finally, Greg takes 50% of the remaining cherries. There are 3 cherries left. How many cherries were in the bowl before Jordan arrived?
_____ cherries

Answer:
24 cherries

Explanation:
Let total cherries in a bowl=x
Jordan takes cherries=50% of x = 50x/100
Remaining cherries = x – 50x/100 = x/2
Mei takes cherries=50% of 50x/100 = x/4
remaining cherries= x/2 – x/4 = x/4
Greg takes cherries=50% of x/4 = x/8
remaining cherries = x/4 – x/8 = x/8
Now,remaining cherries in a bowl=3
x/8 =3
x = 8 × 3 = 24

Question 6.
Each week, Tasha saves 65% of the money she earns babysitting and spends the rest. This week she earned $40. How much more money did she save than spend this week?
$ _____

Answer:
Tasha saved $26 and spent $14

Explanation:
Since 65% of 40 is 26, that’s how much Tasha saves. Then do 40 – 26 to get 14, which is how much she spends.
So Tasha saved $26 and spent $14.

Question 7.
An employee at a state park has 53 photos of animals found at the park. She wants to arrange the photos in rows so that every row except the bottom row has the same number of photos. She also wants there to be at least 5 rows. Describe two different ways she can arrange the photos
Type below:
_____________

Answer:
5 rows of 10 photos and last row with 3 photos,
6 rows of 8 photos and last row with 5 photos,
7 rows of 7 photos and last row with 4 photos,
Also, reverse the rows and photos in each row (ex 5 rows 10 photos=10 rows 5 photos) to get another 3 sets.

Question 8.
Explain a Method Maya wants to mark a length of 7 inches on a sheet of paper, but she does not have a ruler. She has pieces of wood that are 4 inches, 5 inches, and 6 inches long. Explain how she can use these pieces to mark a length of 7 inches.
Type below:
_____________

Answer:
Maya can put the 5 and 6-inch pieces together to get 11 inches. She can then subtract the length of the 4-inch piece to get 7 inches.

Question 9.
Pierre’s family is driving 380 miles from San Francisco to Los Angeles. On the first day, they drive 30% of the distance. On the second day, they drive 50% of the distance. On the third day, they drive the remaining distance and arrive in Los Angeles. How many miles did Pierre’s family drive each day? Write the number of miles in the correct box.

Type below:
_____________

Answer:
76 miles

Explanation:
Pierre’s family is driving 380 miles from San Francisco to Los Angeles.
On the first day, they drive 30% of the distance. 380 × 30/100 = 114
On the second day, they drive 50% of the distance. 380 × 50/100 = 190
They traveled 80%.
On the third day, they drive the remaining distance and arrive in Los Angeles.
380 × 20/100 = 76 miles

Problem Solving Percents – Page No. 299

Read each problem and solve.

Question 1.
On Saturday, a souvenir shop had 125 customers. Sixty-four percent of the customers paid with a credit card. The other customers paid with cash. How many customers paid with cash?T
_____ costumers

Answer:
45 costumers

Explanation:
On Saturday, a souvenir shop had 125 customers. Sixty-four percent of the customers paid with a credit card.
125 × 64/100 = 80
100 – 64 = 36
125 × 36/100 = 45

Question 2.
A carpenter has a wooden stick that is 84 centimeters long. She cuts off 25% from the end of the stick. Then she cuts the remaining stick into 6 equal pieces. What is the length of each piece?
_____ cm

Answer:
10 1/2 cm

Explanation:
A carpenter has a wooden stick that is 84 centimeters long. She cuts off 25% from the end of the stick. Then she cuts the remaining stick into 6 equal pieces.
84 × 75/100 = 63
63/6 = 10 1/2

Question 3.
A car dealership has 240 cars in the parking lot and 17.5% of them are red. Of the other 6 colors in the lot, each color has the same number of cars. If one of the colors is black, how many black cars are in the lot?
_____ black cars

Answer:
33 black cars

Explanation:
number of red cars 17.5% × 240 = 42
number of cars of other colors = 240 – 42 = 198
number of black cars 1/6 × 198 = 33

Question 4.
The utilities bill for the Millers’ home in April was $132. Forty-two percent of the bill was for gas, and the rest was for electricity. How much did the Millers pay for gas, and how much did they pay for electricity?
Type below:
_____________

Answer:
Amount of money paid for gas = 132 * (42/100) dollars
= 5544/100 dollars
= 55.44 dollars
Then
The amount of money paid for electricity = (132 – 55.44) dollars
= 76.56 dollars
So the Millers paid 55.44 dollars for gas and 76.56 dollars for electricity in the month of April.

Question 5.
Andy’s total bill for lunch is $20. The cost of the drink is 15% of the total bill and the rest is the cost of the food. What percent of the total bill did Andy’s food cost? What was the cost of his food?
Type below:
_____________

Answer:
$17

Explanation:
Andy paid $20 total for his lunch (100%).
15% is for drink.
Therefore, 100 – 15 = 85% is the percent that was constituted by the food.
85% of 20 is equal to 0.85 × 20 is equal to:
17 × 20/20 = 17
Andy’s food cost $17.

Question 6.
Write a word problem that involves finding the additional amount of money needed to purchase an item, given the cost and the percent of the cost already saved.
Type below:
_____________

Answer:
Each week, Tasha saves 65% of the money she earns babysitting and spends the rest. This week she earned $40. How much more money did she save than spend this week?
Tasha saved $26 and spent $14

Lesson Check – Page No. 300

Question 1.
Milo has a collection of DVDs. Out of 45 DVDs, 40% are comedies and the remaining are action-adventures. How many actionadventure DVDs does Milo own?
_____ DVDs

Answer:
27 DVDs

Explanation:
100%-40%=60%
60/100*45=27
27 DVD’s are action-adventure

Question 2.
Andrea and her partner are writing a 12-page science report. They completed 25% of the report in class and 50% of the remaining pages after school. How many pages do Andrea and her partner still have to write?
_____ pages

Answer:
9 pages

Explanation:
first 50% + 25% = 75%
then you can do 75% of 12
75% = 0.75
of = multiplication
0.75 • 12 which should equal 9
so they have 9 pages left

Spiral Review

Question 3.
What is the absolute value of \(\frac{-4}{25}\)?
\(\frac{□}{□}\)

Answer:
\(\frac{4}{25}\)

Explanation:
|\(\frac{-4}{25}\)| = 4/25

Question 4.
Ricardo graphed a point by starting at the origin and moving 5 units to the left. Then he moved up 2 units. What is the ordered pair for the point he graphed?
Type below:
_____________

Answer:
(-5, 2)

Explanation:
In a coordinate system, the coordinates of the origin are (0, 0).
If he moves 5 units to the left, he is moving in the negative direction along the x-axis, and x takes the value -5.
If he moves up 2 units, he is moving in the positive direction along the y-axis, and y takes the value 2.
The ordered pair (x, y) is (-5, 2).

Question 5.
The population of birds in a sanctuary increases at a steady rate. The graph of the population over time has the points (1, 105) and (3, 315). Name another point on the graph.
Type below:
_____________

Answer:
You could do (2, 210) or (4, 420) or (5, 525)

Question 6.
Alicia’s MP3 player contains 1,260 songs. Given that 35% of the songs are rock songs and 20% of the songs are rap songs, how many of the songs are other types of songs?
_____ songs

Answer:
567 songs

Explanation:
Since 55% of the songs are rock and rap, 45% of the songs are other.
To find 45% of 1260 we multiply by the decimal:
1260 x 0.45 = 567
Therefore 567 of the songs are other.

Share and Show – Page No. 303

Find the unknown value.

Question 1.
9 is 25% of _____.
_____

Answer:
36

Explanation:
25/100 ÷ 25/25 = 1/4
1/4 = 9/s
1/4 × 9/9 = 9/36
the unknown value is 36

Question 2.
14 is 10% of _____.
_____

Answer:
140

Explanation:
10/100 ÷ 10/10 = 1/10
1/10 = 14/s
1/10 × 14/14 = 14/140
the unknown value is 140

Question 3.
3 is 5% of _____.
_____

Answer:
6

Explanation:
5/10 ÷ 5/5 = 1/2
1/2 × 3/3 = 3/6
the unknown value is 6

Question 4.
12 is 60% of _____.
_____

Answer:
20

Explanation:
60/100 ÷ 60/60 = 60/100
60/100 ÷ 5/5 = 12/20
the unknown value is 20

On Your Own

Find the unknown value.

Question 5.
16 is 20% of _____.
_____

Answer:
80

Explanation:
20/100 ÷ 20/20 = 1/5
1/5 × 16/16 = 16/80
the unknown value is 80

Question 6.
42 is 50% of _____.
_____

Answer:
84

Explanation:
50/100 ÷ 50/50 = 1/2
1/2 × 42/42 = 42/84
the unknown value is 84

Question 7.
28 is 40% of _____.
_____

Answer:
70

Explanation:
40/100 ÷ 40/40 = 1/2.5
1/2.5 × 28/28 = 28/70
the unknown value is 70

Question 8.
60 is 75% of _____.
_____

Answer:
80

Explanation:
75/100 ÷ 75/75 = 60/s
60 × 100 = 6000/75 = 80
the unknown value is 80

Question 9.
27 is 30% of _____.
_____

Answer:
90

Explanation:
30/100 ÷ 30/30 = 3/10
3/10 × 9/9 = 27/90
the unknown value is 90

Question 10.
21 is 60% of _____.
_____

Answer:
35

Explanation:
60/100 ÷ 60/60 = 3/5
3/5 × 7/7 = 21/35
the unknown value is 35

Question 11.
12 is 15% of _____.
_____

Answer:
80

Explanation:
15/100 ÷ 15/15 = 3/20
3/20 × 4/4 = 12/80
the unknown value is 80

Solve.

Question 12.
40% of the students in the sixth grade at Andrew’s school participate in sports. If 52 students participate in sports, how many sixth graders are there at Andrew’s school?
_____ students

Answer:
130 students

Explanation:
52/s = 40%
52/s = 40/100
s = 40/100 × 52 = 130

Question 13.
There were 136 students and 34 adults at the concert. If 85% of the seats were filled, how many seats are in the auditorium?
_____ seats

Answer:
80 seats

Explanation:
There are 170 seats filled total. 170 is 85% of 200. There are 200 seats in the auditorium.
If you were to solve for x in the equation 40% = 32/x, you would get x = 80.

Use Reasoning Algebra Find the unknown value.

Question 14.
40% = \(\frac{32}{?}\)
_____

Answer:
80

Explanation:
40/100 = 32/?
40/100 ÷ 40/40 = 2/5
2/5 × 16/16 = 32/80
the unknown value is 80

Question 15.
65% = \(\frac{91}{?}\)
_____

Answer:
140

Explanation:
65/100 = 91/?
65/100 ÷ 65/65 = 13/20
13/20 × 7/7 = 91/140
the unknown value is 140

Question 16.
45% = \(\frac{54}{?}\)
_____

Answer:
120

Explanation:
45/100 ÷ 45/45 = 9/20
9/20 × 6/6 = 54/120

Problem Solving + Applications – Page No. 304

Use the advertisement for 17 and 18.

Question 17.
Corey spent 20% of his savings on a printer at Louie’s Electronics. How much did Corey have in his savings account before he bought the printer?
$ _____

Answer:
$800

Explanation:
(printer cost) = 0.20 * (savings)
(printer cost)/0.20 = (savings)
savings = 5*(printer cost)
Corey’s savings was 5 times that amount.
savings = 5 × 160 = 800

Question 18.
Kai spent 90% of his money on a laptop that cost $423. Does he have enough money left to buy a scanner? Explain.
Type below:
_____________

Answer:
$42.3

Explanation:
He spent 90% of his money. So, he left 10% of money with him.
423 × 10/100 = 42.3 left to buy a scanner

Question 19.
Maurice has completed 17 pages of the research paper he is writing. That is 85% of the required length of the paper. What is the required length of the paper?
_____ pages

Answer:
20 pages

Explanation:
Maurice has completed 17 pages of the research paper he is writing. That is 85% of the required length of the paper.
85%=17 ? what about 100%
100multiplied by 17 divided by 85% =20

Question 20.
Of 250 seventh-grade students, 175 walk to school. What percent of seventh-graders do not walk to school?
_____ %

Answer:
30%

Explanation:
it’s either 30 percent or 70. 70 percent walks to school and 30 percent DO NOT walk to school

Question 21.
What’s the Error? Kate has made 20 free throws in basketball games this year. That is 80% of the free throws she has attempted. To find the total number of free throws she attempted, Kate wrote the equation \(\frac{80}{100}=\frac{?}{20}\). What error did Kate make?
Type below:
_____________

Answer:
20 free throws is 80% of the total attempted
80% to decimal is:
80/100 = 0.8
If total attempted is x, we can say:
20 is 80% (0.8) of x
We can now write an algebraic equation:
20 = 0.8x
We simply solve this for x, that is the number of free throws she attempted:
20 = 0.8x
x = 20/0.8 = 25

Question 22.
Maria spent 36% of her savings to buy a smart phone. The phone cost $90. How much money was in Maria’s savings account before she purchased the phone? Find the unknown value.
$ _____

Answer:
$ 250

Explanation:
let her savings be A
A/Q-
36% of A = $90
36/100 of A = $90
A = 90×100/36
A= $ 250

Find the Whole from a Percent – Page No. 305

Find the Whole from a Percent

Question 1.
9 is 15% of _____.
_____

Answer:
60

Explanation:
15/100 ÷ 15/15 = 3/20
3/20 × 3/3 = 9/60
the unknown value is 60

Question 2.
54 is 75% of _____.
_____

Answer:
72

Explanation:
75/100 ÷ 75/75 = 3/4
3/4 × 18/18 = 54/72
the unknown value is 72

Question 3.
12 is 2% of _____.
_____

Answer:
600

Explanation:
2/100 = 1/50
1/50 × 12/12 = 12/600
the unknown value is 600

Question 4.
18 is 50% of _____.

Answer:
36

Explanation:
50/100 = 1/2
1/2 × 18/18 = 18/36
the unknown value is 36

Question 5.
16 is 40% of _____.
_____

Answer:
40

Explanation:
40/100 = 2/5
2/5 × 8/8 = 16/40
the unknown value is 40

Question 6.
56 is 28% of _____.
_____

Answer:
200

Explanation:
28/100 = 14/50 = 7/25
7/25 × 8/8 = 56/200
the unknown value is 200

Question 7.
5 is 10% of _____.
_____

Answer:
50

Explanation:
10/100 = 1/10
1/10 × 5/5 = 5/50
the unknown value is 50

Question 8.
24 is 16% of _____.
_____

Answer:
150

Explanation:
16/100 = 4/25
4/25 × 6/6 = 24/150
the unknown value is 150

Question 9.
15 is 25% of _____.
_____

Answer:
60

Explanation:
25/100 = 1/4
1/4 × 15/15 = 15/60
the unknown value is 60

Problem Solving

Question 10.
Michaela is hiking on a weekend camping trip. She has walked 6 miles so far. This is 30% of the total distance. What is the total number of miles she will walk?
_____ miles

Answer:
20 miles

Explanation:
Since 6mi=30%,
You should find ten percent.
This is how, divide both sides by 3, and this gives you
2m=10% (2m being 2 miles)
So, to find 100%, you need to multiply both sides by 10
20m=100%
So now, Michaela will walk 20 miles this weekend

Question 11.
A customer placed an order with a bakery for muffins. The baker has completed 37.5% of the order after baking 81 muffins. How many muffins did the customer order?
_____ muffins

Answer:
216 muffins

Explanation:
A customer placed an order with a bakery for muffins. The baker has completed 37.5% of the order after baking 81 muffins.
37.5/100=0.375 and 81/0.375=216
so the answer is 216

Question 12.
Write a question that involves finding what number is 25% of another number. Solve using a double number line and check using equivalent ratios. Compare the methods.
Type below:
_____________

Answer:
25% of 15 = 25/100 × 15 = 375/100 = 3.75

Lesson Check – Page No. 306

Question 1.
Kareem saves his coins in a jar. 30% of the coins are pennies. If there are 24 pennies in the jar, how many coins does Kareem have?
_____ coins

Answer:
80 coins

Explanation:
24=30%
find 100%
24=30%
diivde by 3
8=10%
multiply 10
80=100%
80 coins

Question 2.
A guitar shop has 19 acoustic guitars on display. This is 19% of the total number of guitars. What is the total number of guitars the shop has?
_____ guitars

Answer:
100 guitars

Explanation:
Let’s find out how much 1% is worth first.
19 guitars = 19%
therefore 19 ÷ 19 = [ 1 guitar = 1% ]
The total number of guitars is going to be 100%,
so if 1% × 100 = 100%, then 1 guitar × 100 = 100 guitars total.

Spiral Review

Question 3.
On a coordinate grid, in which quadrant is the point (−5, 4) located?
Type below:
_____________

Answer:
Quadrant II

Explanation:
(-5, 4)
-5 is the negative point of the x coordinate
4 is the positive point of the y coordinate
Quadrant II

Question 4.
A box contains 16 cherry fruit chews, 15 peach fruit chews, and 12 plum fruit chews. Which two flavors are in the ratio 5 to 4?
Type below:
_____________

Answer:
peach fruit chews and plum fruit chews are in the ratio 5 to 4

Explanation:
15 peach fruit chews, and 12 plum fruit chews
15/12 = 5/4

Question 5.
During basketball season, Marisol made \(\frac{19}{25}\) of her free throws. What percent of her free throws did Marisol make?
_____ %

Answer:
76%

Explanation:
During the basketball season, Marisol made \(\frac{19}{25}\) of her free throws.
(19 ÷ 25) × 100 = 76%. Marisol made 76% of her free throws.

Question 6.
Landon is entering the science fair. He has a budget of $115. He has spent 20% of the money on new materials. How much does Landon have left to spend?
$ _____

Answer:
$92

Explanation:
Landon has $92 left because if you divide 115/.20 you get 23 and then you subtract 115-23=92 or $92.

Chapter 5 Review/Test – Page No. 307

Question 1.
What percent is represented by the shaded part?

Options:
a. 46%
b. 60%
c. 64%
d. 640%

Answer:
c. 64%

Explanation:
64 squares are shaded out of 100.
So, 64% and 64/100 are the answers.

Question 2.
Write a percent to represent the shaded part.

_____ %

Answer:
42%

Explanation:
42 squares are shaded out of 100.
So, 42% and 42/100 are the answers.

Question 3.
Rosa made a mosaic wall mural using 42 black tiles, 35 blue tiles and 23 red tiles. Write a percent to represent the number of red tiles in the mural.
_____ %

Answer:
23%

Explanation:
42+35+23= 100
So plug it in.
23/100
23%
Your answer is 23%.

Question 4.
Model 39%.
Type below:
_____________

Answer:

Explanation:
39 squares out of 100 need to shaded

Page No. 308

Question 5.
For 5a–5d, choose Yes or No to indicate whether the percent and the fraction represent the same amount.
5a. 50% and \(\frac{1}{2}\)
5b. 45% and \(\frac{4}{5}\)
5c. \(\frac{3}{8}\) and 37.5%
5d. \(\frac{2}{10}\) and 210%
5a. _____________
5b. _____________
5c. _____________
5d. _____________

Answer:
5a. Yes
5b. No
5c. Yes
5d. No

Explanation:
1/2 = 0.5 × 100/100 = 50/100 = 50%
4/5 = 0.8 × 100/100 = 80/100 = 80%
3/8 = 0.375 × 100/100 = 37.5/100 = 37.5%
2/10 = 0.2 × 100/100 = 20/100 = 20%

Question 6.
The school orchestra has 25 woodwind instruments, 15 percussion instruments, 30 string instruments, and 30 brass instruments. Select the portion of the instruments that are percussion. Mark all that apply.
Options:
a. 15%
b. 1.5
c. \(\frac{3}{20}\)
d. 0.15

Answer:
a. 15%
c. \(\frac{3}{20}\)
d. 0.15

Explanation:
25 + 15 + 30 + 30 = 100
15 percussion instruments = 15/100 = 15% = 0.15

Question 7.
For a science project, \(\frac{3}{4}\) of the students chose to make a poster and 0.25 of the students wrote a report. Rosa said that more students made a poster than wrote a report. Do you agree with Rosa? Use numbers and words to support your answer
Type below:
_____________

Answer:
Yes, because 3/4 is equal to 0.75 and 0.75 > 0.25
Or 0.25 is equal to 1/4, and 1/4 < 3/4

Question 8.
Select other ways to write 0.875. Mark all that apply.
Options:
a. 875%
b. 87.5%
c. \(\frac{7}{8}\)
d. \(\frac{875}{100}\)

Answer:
c. \(\frac{7}{8}\)

Explanation:
0.875 = 8.75/100 = 8.75%

Page No. 309

Question 9.
There are 88 marbles in a bin and 25% of the marbles are red.
There are _____________ red marbles in the bin.

Answer:
There are 22 red marbles in the bin.

Explanation:
88 × 25% = 88 × 25/100 = 22

Question 10.
Harrison has 30 CDs in his music collection. If 40% of the CDs are country music and 30% are pop music, how many CDs are other types of music?
_____ CDs

Answer:
9 CDs

Explanation:
Harrison has 30 CDs in his music collection. If 40% of the CDs are country music and 30% are pop music,
40 + 30 = 70
100 – 70 = 30%
30 × 30/100 = 9

Question 11.
For numbers 11a–11b, choose <, >, or =.
11a. 30% of 90 Ο 35% of 80
11b. 25% of 16 Ο 20% of 25
30% of 90 _____ 35% of 80
25% of 16 _____ 20% of 25

Answer:
30% of 90 < 35% of 80
25% of 16 < 20% of 25

Explanation:
30% of 90 = 30/100 × 90 = 27
35% of 80 = 35/100 × 80 = 28
30% of 90 < 35% of 80
25% of 16 = 25/100 × 16 = 4
20% of 25 = 20/100 × 25 = 5
25% of 16 < 20% of 25

Question 12.
There were 200 people who voted at the town council meeting. Of these people, 40% voted for building a new basketball court in the park. How many people voted against building the new basketball court? Use numbers and words to explain your answer.
Type below:
_____________

Answer:
There were 200 people who voted at the town council meeting. Of these people, 40% voted for building a new basketball court in the park.
100 – 40% = 60%
200 × 60/100 = 120 people

Page No. 310

Question 13.
James and Sarah went out to lunch. The price of lunch for both of them was $20. They tipped their server 20% of that amount. How much did each person pay if they shared the price of lunch and the tip equally?
$ _____

Answer:
$12

Explanation:
James and Sarah went out to lunch. The price of lunch for both of them was $20. They tipped their server 20% of that amount.
20% of 20 = 20/100 × 20 = 4
20 + 4 = 24
24/2 = 12
$12

Question 14.
A sandwich shop has 30 stores and 60% of the stores are in California. The rest of the stores are in Nevada.
Part A
How many stores are in California and how many are in Nevada?
Type below:
_____________

Answer:
30 × 60/100 = 18 stores in California
30 – 18 = 12 stores in Nevada

Question 14.
Part B
The shop opens 10 new stores. Some are in California, and some are in Nevada. Complete the table.

Type below:
_____________

Answer:

Explanation:
100 – 45 = 55%
55% of 40 = 55/100 × 40 = 22
45% of 40 = 45/100 × 40 = 18

Question 15.
Juanita has saved 35% of the money that she needs to buy a new bicycle. If she has saved $63, how much money does the bicycle cost? Use numbers and words to explain your answer
$ _____

Answer:
$180

Explanation:
Juanita has saved 35% of the money that she needs to buy a new bicycle. If she has saved $63,
35/100 = 7/20
7/20 × 9/9 = 63/180
The bicycle cost is $180

Page No. 311

Question 16.
For 16a–16d, choose Yes or No to indicate whether the statement is correct.
16a. 12 is 20% of 60.
16b. 24 is 50% of 48.
16c. 14 is 75% of 20.
16d. 9 is 30% of 30.
16a. _____________
16b. _____________
16c. _____________
16d. _____________

Answer:
16a. Yes
16b. Yes
16c. No
16d. Yes

Explanation:
20% of 60 = 20/100 × 60 = 12
50% of 48 = 50/100 × 48 = 24
75% of 20 = 75/100 × 20 = 15
30% of 30 = 30/100 × 30 = 9

Question 17.
Heather and her family are going to the grand opening of a new amusement park. There is a special price on tickets this weekend. Tickets cost $56 each. This is 70% of the cost of a regular price ticket
Part A
What is the cost of a regular price ticket? Show your work.
$ _____

Answer:
$80

Explanation:
70/100 = 56/s
s = 56 × 100/70 = 80

Question 17.
Part B
Heather’s mom says that they would save more than $100 if they buy 4 tickets for their family on opening weekend. Do you agree or disagree with Heather’s mom? Use numbers and words to support your answer. If her statement is incorrect, explain the correct way to solve it.
Type below:
_____________

Answer:
80 × 4 = 320
56 × 4 = 224
320 – 224 = 96
$96

Question 18.
Elise said that 0.2 equals 2%. Use words and numbers to explain her mistake.
Type below:
_____________

Answer:
0.2 × 100/100 = 20/100 = 2%

Page No. 312

Question 19.
Write 18% as a fraction.
\(\frac{□}{□}\)

Answer:
\(\frac{9}{50}\)

Explanation:
18% = 18/100 = 9/50

Question 20.
Noah wants to put a variety of fish in his new fish tank. His tank is large enough to hold a maximum of 70 fish.
Part A
Complete the table.

Type below:
_____________

Answer:

Explanation:
70 × 20/100 = 14
70 × 40/100 = 28
70 × 30/100 = 21

Question 20.
Part B
Has Noah put the maximum number of fish in his tank? Use numbers and words to explain how you know. If he has not put the maximum number of fish in the tank, how many more fish could he put in the tank?
Type below:
_____________

Answer:
No, since 20% + 40% + 30% = 90%, he can add 10% in the tank.

Conclusion:

Test your knowledge by solving the problems from Go Math Grade 6 Answer Key Chapter 5 Model Percents. Get the solutions for Mid Chapter Checkpoint and Review Test along with the exercise problems in Go Math Grade 6 Chapter 5 Model Percents Solution Key. Quick learning and best practice come in a single hand with our Go Math Grade 6 Solution Key Chapter 5 Model Percents @ ccssmathanswers.com

Students of Grade 8 can get a detailed explanation for all the questions in Go Math Answer Key Chapter 8 Solving Systems of Linear Equations. In addition to the exercise problems we also provide the solutions for the review test. So, go through all the answers and explanations provided by the math experts in Go Math Grade 8 Chapter 8 Solving Systems of Linear Equations Answer Key. Our aim is to provide easy and simple tricks to solve the problems in Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations.

Download Go Math Grade 8 Chapter 8 Solving Systems of Linear Equations Answer Key Pdf

Students who are interested to secure the highest marks in the exams are suggested to download the Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations pdf. All the solutions are provided in the pdf format as per the list of the chapters provided in the latest edition. Hence refer to Go Math 8th Grade Solution Key to learning the easy way of maths practice. Check the list of the topics covered in Chapter 8 Solving Systems of Linear Equations from the following section.

Lesson 1: Solving Systems of Linear Equations by Graphing

• Solving Systems of Linear Equations by Graphing – Page No. 232
• Solving Systems of Linear Equations by Graphing – Page No. 233
• Solving Systems of Linear Equations by Graphing Lesson Check – Page No. 234

Lesson 2: Solving Systems by Substitution

• Solving Systems by Substitution – Page No. 240
• Solving Systems by Substitution – Page No. 241
• Solving Systems by Substitution Lesson Check – Page No. 242

Lesson 3: Solving Systems by Elimination

• Solving Systems by Elimination – Page No. 248
• Solving Systems by Elimination – Page No. 249
• Solving Systems by Elimination Lesson Check – Page No. 250

Lesson 4: Solving Systems by Elimination with Multiplication

• Solving Systems by Elimination with Multiplication – Page No. 256
• Solving Systems by Elimination with Multiplication – Page No. 257
• Solving Systems by Elimination with Multiplication Lesson Check – Page No. 258

Lesson 5: Solving Solving Special Systems

• Solving Solving Special Systems – Page No. 262
• Solving Solving Special Systems – Page No. 263
• Solving Solving Special Systems Lesson Check – Page No. 264

Model Quiz

• Model Quiz – Page No. 265

Review

• Mixed Review – Page No. 266

Guided Practice – Solving Systems of Linear Equations by Graphing – Page No. 232

Solve each system by graphing.

Question 1.
\(\left\{\begin{array}{l}y=3 x-4 \\y=x+2\end{array}\right.\)

Type below:
______________

Answer:

Explanation:
y = 3x – 4
y = x + 2
The solution of thr linear system of equations is the intersection point of the two equations.
(3, 5) is the solution of the system of equations.
If x = 3, y = 3(3) – 4 = 9 – 4 = 5; y = 3 + 2 = 5
5 = 5; True

Question 2.
\(\left\{\begin{array}{l}x-3 y=2 \\-3x+9y=-6\end{array}\right.\)

Type below:
______________

Answer:

Infinitely many solutions

Explanation:
x – 3y = 2
-3x + 9y = -6
x – 3y – x = -x + 2
-3y = -x + 2
y = 1/3 . x – 2/3
-3x + 9y + 3x = 3x – 6
9y = 3x – 6
y = 3/9 . x – 6/9
y = 1/3 . x – 2/3
The solution of the linear system of equations is the intersection of the two equations.
Infinitely many solutions

Question 3.
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points. The maximum number of points possible on the test is 100.
a. Write an equation in slope-intercept form to represent the number of questions on the test.

Type below:
______________

Answer:
y = -x + 15

Explanation:
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points.
x + y = 15
x + y – x = -x + 15
y = -x + 15

Question 3.
b. Write an equation in slope-intercept form to represent the total number of points on the test.
Type below:
______________

Answer:
y = -1/2 . x + 10

Explanation:
The total number of points on test is 100
5x + 10y = 100
5x + 10y – 5x = -5x + 100
10y = -5x + 100
y = -5/10 . x + 100/10
y = -1/2 . x + 10

Question 3.
c. Graph the solutions of both equations.
Type below:
______________

Answer:

Question 3.
d. Use your graph to tell how many of each question type are on the test.
_________ spelling questions
_________ vocabulary questions

Answer:
10 spelling questions
5 vocabulary questions

ESSENTIAL QUESTION CHECK-IN

Question 4.
When you graph a system of linear equations, why does the intersection of the two lines represent the solution of the system?
Type below:
______________

Answer:
To solve a system of linear equations means finding the solutions that satisfy all the equations of that system. When we graph a system of linear equations, the intersection point lies on the line of each equation, which means that satisfies all the equations. Therefore, it is considered to be the solution to that system.

Solving Systems of Linear Equations by Graphing – Page No. 233

Question 5.
Vocabulary
A_________________ is a set of equations that have the same variables.
______________

Answer:
system of equations

Explanation:
A system of equations is a set of equations that have the same variables.

Question 6.
Eight friends started a business. They will wear either a baseball cap or a shirt imprinted with their logo while working. They want to spend exactly $36 on the shirts and caps. Shirts cost $6 each and caps cost $3 each.
a. Write a system of equations to describe the situation. Let x represent the number of shirts and let y represent the number of caps.
______________

Answer:
6x + 3y = 36

Explanation:
The sum of caps and shirts is 8. The total cost of caps and shirts is $36.
x + y = 8
6x + 3y = 36

Question 6.
b. Graph the system. What is the solution and what does it represent?

Type below:
______________

Answer:
The solution is (4, 4)

Explanation:
x + y – x = -x + 8
y = -x + 8
6x + 3y – 6x = -6x + 36
3y = -6x + 36
y = -6/2 . x + 36/3
y = -2x + 12
(4, 4). They should order 4 shirts and 4 caps.

Question 7.
Multistep The table shows the cost for bowling at two bowling alleys.

a. Write a system of equations, with one equation describing the cost to bowl at Bowl-o-Rama and the other describing the cost to bowl at Bowling Pinz. For each equation, let x represent the number of games played and let y represent the total cost.
Type below:
______________

Answer:
y = 2.5x + 2
y = 2x + 4

Explanation:
Cost at Bowl-o-Rama => y = 2.5x + 2
Cost at Bowling Pinz => y = 2x + 4

Question 7.
b. Graph the system. What is the solution and what does it represent?

Type below:
______________

Answer:

Explanation:
The solution of the linear system of equations is the intersection of the two equations.
(4, 12)
When 4 games are played, the total cost is $12.

Solving Systems of Linear Equations by Graphing – Page No. 234

Question 8.
Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week. Tony runs 3 miles per week and increases his distance by 2 miles each week. In how many weeks will Jeremy and Tony be running the same distance? What will that distance be?
Type below:
______________

Answer:
After 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles.

Explanation:
Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week.
y = x + 7
Tony runs 3 miles per week and increases his distance by 2 miles each week.
y = 2x + 3

The solution of the system of linear equation is (4, 11) which means that after 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles.

Question 9.
Critical Thinking Write a real-world situation that could be represented by the system of equations shown below.
\(\left\{\begin{array}{l}y=4 x+10 \\y=3x+15\end{array}\right.\)
Type below:
______________

Answer:
The entry fee of the first gym is $10 and for every hour that you spend there, you pay an extra $4. If we denote with x the number of hours that somebody spends at the gym and with y the total cost is
y = 4x + 10
The entry fee of the second gym is $15 and for every hour that you spend there, you pay an extra $3. If we denote with x the number of hours that somebody spends at the gym and with y the total cost is
y = 3x + 15
y = 4x + 10
y = 3x + 15

FOCUS ON HIGHER ORDER THINKING

Question 10.
Multistep The table shows two options provided by a high-speed Internet provider.

a. In how many months will the total cost of both options be the same? What will that cost be?
________ months
$ ________

Answer:
5 months
$ 200

Explanation:
Let y be the total cost after x month
y = 30x + 50
Let y be the total cost after x month
y = 40x
Substitute y = 40x in y = 30x + 50
40x = 30x + 50
40x – 30x = 50
10x = 50
x = 50/10
x = 5
The total cost of both options will be the same after 5 months. Total cost would be y = 40(5) = $200.

Question 10.
b. If you plan to cancel your Internet service after 9 months, which is the cheaper option? Explain.
______________

Answer:
When x = 9 months
y = 30(9) + 50 = $320
y = 40(9) = $360
$320 < $360
Option 1 is cheaper as the total cost is less for option 1

Question 11.
Draw Conclusions How many solutions does the system formed by x − y = 3 and ay − ax + 3a = 0 have for a nonzero number a? Explain.
Type below:
______________

Answer:
x – y = 3
ay – ax + 3a =0
ay – ax + 3a – 3a = 0 – 3a
ay – ax = – 3a
a(y – x) = -3a
y – x = -3
x – y = 3
Both equations are the same. The system of linear equations have infinitely many solutions.

Guided Practice – Solving Systems by Substitution – Page No. 240

Solve each system of linear equations by substitution.

Question 1.
\(\left\{\begin{array}{l}3x-2y=9 \\y=2x-7\end{array}\right.\)
x = ________
y = ________

Answer:
x = 5
y = 3

Explanation:
\(\left\{\begin{array}{l}3x-2y=9 \\y=2x-7\end{array}\right.\)
Substitute 2x – 7 in 3x – 2y = 9
3x – 2(2x – 7) = 9
3x – 4x + 14 = 9
-x + 14 = 9
-x + 14 – 14 = 9 – 14
-x = -5
x = -5/-1 = 5
y = 2(5) – 7 = 3
Solution is (5, 3)

Question 2.
\(\left\{\begin{array}{l}y=x-4 \\2x+y=5\end{array}\right.\)
x = ________
y = ________

Answer:
x = 3
y = -1

Explanation:
\(\left\{\begin{array}{l}y=x-4 \\2x+y=5\end{array}\right.\)
2x + x – 4 = 5
3x – 4 = 5
3x – 4 + 4 = 5 + 4
3x = 9
x = 9/3 = 3
y = 3 – 4 = -1
The solution is (3, -1)

Question 3.
\(\left\{\begin{array}{l}x+4y=6 \\y=-x+3\end{array}\right.\)
x = ________
y = ________

Answer:
x = 2
y = 1

Explanation:
\(\left\{\begin{array}{l}x+4y=6 \\y=-x+3\end{array}\right.\)
Substitute y = -x + 3 in x + 4y = 6
x + 4(-x + 3) = 6
x – 4x + 12 = 6
-3x + 12 = 6
-3x + 12 – 12 = 6 – 12
-3x = -6
x = -6/-3 = 2
y = -2 + 3 = 1
The solution is (2, 1)

Question 4.
\(\left\{\begin{array}{l}x+2y=6 \\x-y=3\end{array}\right.\)
x = ________
y = ________

Answer:
x = 4
y = 1

Explanation:
\(\left\{\begin{array}{l}x+2y=6 \\x-y=3\end{array}\right.\)
y = x – 3
Substitute y = x – 3 in x + 2y = 6
x + 2(x – 3) = 6
x + 2x – 6 = 6
3x = 12
x = 12/3
x = 4
4 – y = 3
-y = 3 – 4
-y = -1
y = 1
The solution is (4, 1)

Solve each system. Estimate the solution first.

Question 5.
\(\left\{\begin{array}{l}6x+y=4 \\x-4y=19\end{array}\right.\)
Estimate ______________
Solution ______________
Type below:
______________

Answer:
Estimate (2, -5)
Solution (1.4, -4.4)

Explanation:
\(\left\{\begin{array}{l}6x+y=4 \\x-4y=19\end{array}\right.\)
Let’s find the estimation by graphing the equations
Estimate: (2, -5)

x = 4y + 19
6(4y + 19) + y = 4
24y + 114 + y = 4
25y + 114 = 4
25y = 4 – 114
25y = -110
y = -110/25
y = -4.4
x + 4(-4.4) = 19
x + 17.6 = 19
x = 19 – 17.6
x = 1.4
The solution is (1.4, -4.4)

Question 6.
\(\left\{\begin{array}{l}x+2y=8 \\3x+2y=6\end{array}\right.\)
Estimate ______________
Solution ______________
Type below:
______________

Answer:
Estimate (-1, 5)
Solution (-1, 4.5)

Explanation:
\(\left\{\begin{array}{l}x+2y=8 \\3x+2y=6\end{array}\right.\)
Let’s find the estimation by graphing the equations
Estimate: (-1, 5)

x = -2y + 8
Substitute the equation x = -2y + 8 in 3x + 2y = 6
3(-2y + 8) + 2y = 6
-6y + 24 + 2y = 6
-4y = 6 – 24
-4y = -18
y = -18/-4
y = 4.5
x + 2(4.5) = 8
x + 9 = 8
x = 8 – 9
x = -1
The solution is (-1, 4.5)

Question 7.
\(\left\{\begin{array}{l}3x+y=4 \\5x-y=22\end{array}\right.\)
Estimate ______________
Solution ______________
Type below:
______________

Answer:
Estimate (3, -6)
Solution (3.25, -5.75)

Explanation:
\(\left\{\begin{array}{l}3x+y=4 \\5x-y=22\end{array}\right.\)
Find the Estimation using graphing the equations.
Estimate: (3, -6)

y = -3x + 4
Substitute y = -3x + 4 in 5x – y = 22
5x – (-3x + 4) = 22
5x + 3x -4 = 22
8x = 26
x = 26/8
x = 3.25
3(3.25) + y = 4
9.75 + y = 4
y = 4 – 9.75
y = -5.75
The solution is (3.25, -5.75)

Question 8.
\(\left\{\begin{array}{l}2x+7y=2 \\x+y=-1\end{array}\right.\)
Estimate ______________
Solution ______________
Type below:
______________

Answer:
Estimate (-2, 1)
Solution (-1.8, 0.8)

Explanation:
\(\left\{\begin{array}{l}2x+7y=2 \\x+y=-1\end{array}\right.\)
Find the Estimation using graphing the equations.
Estimate: (-2, 1)

y = -x -1
Substitute y = -x – 1 in 2x + 7y = 2
2x + 7(-x – 1) = 2
2x – 7x -7 = 2
-5x = 2 + 7
-5x = 9
x = -9/5
x = -1.8
-1.8 + y = -1
y = -1 + 1.8
y = 0.8
The solution is (-1.8, 0.8)

Question 9.
Adult tickets to Space City amusement park cost x dollars. Children’s tickets cost y dollars. The Henson family bought 3 adult and 1 child tickets for $163. The Garcia family bought 2 adult and 3 child tickets for $174.
a. Write equations to represent the Hensons’ cost and the Garcias’ cost.
Hensons’ cost: ________________
Garcias’ cost:__________________
Type below:
______________

Answer:
Hensons’ cost: 3x + y = 163
Garcias’ cost: 2x + 3y = 174

Explanation:
Henson’s cost
3x + y = 163
Garcia’s cost
2x + 3y = 174

Question 9.
b. Solve the system.
adult ticket price: $ _________
Garcias’ cost: $ _________

Answer:
adult ticket price: $ 45
Garcias’ cost: $ 28

Explanation:
y = -3x + 163
Substitute y = -3x + 163 in 2x + 3y = 174
2x + 3(-3x + 163) = 174
2x -9x + 489 = 174
-7x = -315
x = -315/-7 = 45
3(45) + y = 163
135 + y = 163
y = 163 – 135
y = 28
adult ticket price: $ 45
Garcias’ cost: $ 28

ESSENTIAL QUESTION CHECK-IN

Question 10.
How can you decide which variable to solve for first when you are solving a linear system by substitution?
Type below:
______________

Answer:
The variable with the unit coefficient should be solved first when solving a linear system by substitution.

8.2 Independent Practice – Solving Systems by Substitution – Page No. 241

Question 11.
Check for Reasonableness Zach solves the system
\(\left\{\begin{array}{l}x+y=-3 \\x-y=1\end{array}\right.\)
and finds the solution (1, -2). Use a graph to explain whether Zach’s solution is reasonable.

Type below:
______________

Answer:

Explanation:
\(\left\{\begin{array}{l}x+y=-3 \\x-y=1\end{array}\right.\)
The x coordinate of the solution is negative, hence Zach’s solution is not reasonable.

Represent Real-World Problems Angelo bought apples and bananas at the fruit stand. He bought 20 pieces of fruit and spent $11.50. Apples cost $0.50 and bananas cost $0.75 each.
a. Write a system of equations to model the problem. (Hint: One equation will represent the number of pieces of fruit. A second equation will represent the money spent on the fruit.)

Type below:
______________

Answer:
x + y = 20
0.5x + 0.75y = 11.5

Explanation:
x + y = 20
0.5x + 0.75y = 11.5
where c is the number of Apples and y is the number of Bananas.

Question 12.
b. Solve the system algebraically. Tell how many apples and bananas Angelo bought.
________ apples
________ bananas

Answer:
14 apples
6 bananas

Explanation:
y = -x + 20
Substitute y = -x + 20 in 0.5x + 0.75y = 11.5
0.5x + 0.75(-x + 20) = 11.5
0.5x – 0.75x + 15 = 11.5
-0.25x + 15 = 11.5
-0.25x = 11.5 – 15
-0.25x = -3.5
x = -3.5/-0.25
x = 14
14 + y = 20
y = 6
Angelo bought 14 apples and 6 bananas.

Question 13.
Represent Real-World Problems A jar contains n nickels and d dimes. There is a total of 200 coins in the jar. The value of the coins is $14.00. How many nickels and how many dimes are in the jar?
________ nickels
________ dimes

Answer:
120 nickels
80 dimes

Explanation:
A jar contains n nickels and d dimes. There is a total of 200 coins in the jar. The value of the coins is $14.00.
$14 = 1400 cents
n + d = 200
5n + 10d = 1400
d = -n + 200
5n + 10(-n + 200) = 1400
5n – 10n + 2000 = 1400
-5n = -600
n = -600/-5
n = 120
120 + d = 200
d = 200 – 120
d = 80
There are 120 nickles and 80 dimes in the jar.

Question 14.
Multistep The graph shows a triangle formed by the x-axis, the line 3x−2y=0, and the line x+2y=10. Follow these steps to find the area of the triangle.
a. Find the coordinates of point A by solving the system
\(\left\{\begin{array}{l}3x-2y=0 \\x-2y=10\end{array}\right.\)
Point A: ____________________

Type below:
______________

Answer:
Point A: (2.5, 3.75)Coordinate of A is (2.5, 3.75)

Explanation:
\(\left\{\begin{array}{l}3x-2y=0 \\x-2y=10\end{array}\right.\)
x = -2y + 10
Substitute x = -2y + 10 in 3x – 2y = 0
3(-2y + 10) -2y = 0
-6y + 30 – 2y = 0
-8y = -30
y = -30/-8 = 3.75
x + 2(3.75) = 10
x + 7.5 = 10
x = 10 – 7.5
x = 2.5
Coordinate of A is (2.5, 3.75)

Question 14.
b. Use the coordinates of point A to find the height of the triangle.
height:__________________
height: \(\frac{□}{□}\) units

Answer:
height: 3.75
height: \(\frac{15}{4}\) units

Explanation:
Height of the triangle is the y coordinate of A
Height = 3.75

Question 14.
c. What is the length of the base of the triangle?
base:________________
base: ______ units

Answer:
base: 10 units

Explanation:
Length of the base = 10

Question 14.
d. What is the area of the triangle?
A = ______ \(\frac{□}{□}\) square units

Answer:
A = 18.75 square units
A = 18 \(\frac{3}{4}\) square units

Explanation:
Area of the triangle = 1/2 . Height . Base
Area = 1/2 . 3.75 . 10 = 18.75

Solving Systems by Substitution – Page No. 242

Question 15.
Jed is graphing the design for a kite on a coordinate grid. The four vertices of the kite are at A(−\(\frac{4}{3}\), \(\frac{2}{3}\)), B(\(\frac{14}{3}\), −\(\frac{4}{3}\)), C(\(\frac{14}{3}\), −\(\frac{16}{3}\)), and D(\(\frac{2}{3}\), −\(\frac{16}{3}\)). One kite strut will connect points A and C. The other will connect points B and D. Find the point where the struts cross.

Type below:
______________

Answer:
The struts cross as (8/3, 10/3)

Explanation:
1. From AC
Slope = (y2 – y1)/(x2 – x1) = [(-16/3)-(2/3)] ÷ [(14/3) – (-4/3)] = (-18/3) ÷ (18/3) = -1
y = mx + b
2/3 = -1(-4/3) + b
2/3 = 4/3 + b
1. From BD
Slope = (y2 – y1)/(x2 – x1) = [(-16/3)-(-4/3)] ÷ [(2/3) – (144/3)] = (-12/3) ÷ (-12/3) = 1
y = mx + b
-4/3 = 1(14/3) + b
-4/3 = 14/3 + b
-18/3 = b
-6 = b
y = mx + b
y = x -6
3. y = -x -2/3
y = x – 6
4. y = -x – 2/3
x – 6 = -x – 2/3
x = -x – 2/3 + 6
x = – x + 16/3
2x = 16/3
x = 16/6
x = 8/3
then y = x – 6
y = 8/3 – 18/3
y = -10/3
The struts cross as (8/3, 10/3)

FOCUS ON HIGHER ORDER THINKING

Question 16.
Analyze Relationships Consider the system
\(\left\{\begin{array}{l}6x-3y=15 \\x+3y=-8\end{array}\right.\)
Describe three different substitution methods that can be used to solve this system. Then solve the system.
Type below:
______________

Answer:
(1, -3) is the answer.

Explanation:
As there are three different substitution methods, we can write
Solve for y in the first equation, then substitute that value into the second equation.
Solve for x in the second equation, then substitute that value into the first equation.
Solve either equation for 3y, then substitute that value into the other equation.
From the Second method,
x + 3y = -8
x = -3y – 8
6x – 3y = 15
6 (-3y – 8) -3y = 15
-18y – 48 -3y = 15
-21y – 48 = 15
-21y = 63
y = -3
x + 3y = -8
x + 3(-3) = -8
x – 9 = -8
x = 1
(1, -3) is the answer.

Question 17.
Communicate Mathematical Ideas Explain the advantages, if any, that solving a system of linear equations by substitution has over solving the same system by graphing.
Type below:
______________

Answer:
The advantage of solving a system of linear equations by graphing is that it is relatively easy to do and requires very little algebra.

Question 18.
Persevere in Problem Solving Create a system of equations of the form
\(\left\{\begin{array}{l}Ax+By=C \\Dx+Ey=F\end{array}\right.\)
that has (7, −2) as its solution. Explain how you found the system.
Type below:
______________

Answer:
x + y = 5
x – y = 9
solves in :
x = (5+9)/2 = 7
y = 5-9)/2 = -2
A=1, B=2, C= 5
D=1, E= -1, F=9
x = 7
y = -2
IS a system (even if it is a trivial one) of equations so this answer would be acceptable.
The target for a system is to find it SOLUTION SET and not to conclude with x=a and y=b

Guided Practice – Solving Systems by Elimination – Page No. 248

Question 1.
Solve the system
\(\left\{\begin{array}{l}4x+3y=1 \\x-3y=-11\end{array}\right.\)
by adding.

Type below:
______________

Answer:
4x + 3y = 1
x – 3y = -11
Add the above two equations
4x + 3y = 1
+(x – 3y = -11)
Add to eliminate the variable y
5x + 0y = -10
Simplify and solve for x
5x = -10
Divide both sided by 5
x = -10/5 = -2
Substitute into one of the original equations and solve for y.
4(-2) + 3y = 1
-8 + 3y = 1
3y = 9
y = 9/3 = 3
So, (-2, 3) is the solution of the system.

Solve each system of equations by adding or subtracting.

Question 2.
\(\left\{\begin{array}{l}x+2y=-2 \\-3x+2y=-10\end{array}\right.\)
x = ________
y = ________

Answer:
x = 2
y = -2

Explanation:
\(\left\{\begin{array}{l}x+2y=-2 \\-3x+2y=-10\end{array}\right.\)
Subtract the equations
x + 2y = -2
-(-3x + 2y = -10)
y is eliminated as it has reversed coefficients. Solve for x
x + 2y + 3x – 2y = -2 + 10
4x = 8
x = 8/4 = 2
Substituting x in either of the equation to find y
2 + 2y = -2
2 + 2y -2 = -2 -2
2y = -4
y = -4/2 = -2
(2, -2) is the answer.

Question 3.
\(\left\{\begin{array}{l}3x+y=23 \\3x-2y=8\end{array}\right.\)
(________ , ________)

Answer:
(6, 5)

Explanation:
\(\left\{\begin{array}{l}3x+y=23 \\3x-2y=8\end{array}\right.\)
Subtract the equations
3x + y = 23
-(3x – 2y = 8)
x is eliminated as it has reversed coefficients. Solve for y
3x + y – 3x + 2y = 23 – 8
3y = 15
y = 15/3 = 5
Substituting y in either of the equation to find x
3x + 5 = 23
3x + 5 – 5 = 23 – 5
3x = 18
x = 18/3 = 6
Solution is (6, 5)

Question 4.
\(\left\{\begin{array}{l}-4x-5y=7 \\3x+5y=-14\end{array}\right.\)
(________ , ________)

Answer:
(7, -7)

Explanation:
\(\left\{\begin{array}{l}-4x-5y=7 \\3x+5y=-14\end{array}\right.\)
Add the equations
-4x – 5y = 7
+(3x + 5y = -14)
y is eliminated as it has reversed coefficients. Solve for x
-4x -5y +3x + 5y = 7 -14
-x = -7
x = -7/-1 = 7
Substituting x in either of the equation to find y
3(7) + 5y = -14
21 + 5y -21 = -14 -21
5y = -35
y = -35/5 = -7
The answer is (7, -7)

Question 5.
\(\left\{\begin{array}{l}x-2y=-19 \\5x+2y=1\end{array}\right.\)
(________ , ________)

Answer:
(-3, 8)

Explanation:
\(\left\{\begin{array}{l}x-2y=-19 \\5x+2y=1\end{array}\right.\)
Add the equations
x – 2y = -19
+(5x + 2y = 1)
y is eliminated as it has reversed coefficients. Solve for x
x – 2y + 5x + 2y = -19 + 1
6x = -18
x = -18/6 = -3
Substituting x in either of the equation to find y
-3 -2y = -19
-3 -2y + 3 = -19 + 3
-2y = -16
y = -16/-2 = 8
The answer is (-3, 8)

Question 6.
\(\left\{\begin{array}{l}3x+4y=18 \\-2x+4y=8\end{array}\right.\)
(________ , ________)

Answer:
(2, 3)

Explanation:
\(\left\{\begin{array}{l}3x+4y=18 \\-2x+4y=8\end{array}\right.\)
Subtract the equations
3x + 4y = 18
-(-2x + 4y = 8)
y is eliminated as it has reversed coefficients. Solve for x
3x + 4y + 2x – 4y = 18 – 8
5x = 10
x = 10/5 = 2
Substituting x in either of the equation to find y
3(2) + 4y = 18
6 + 4y – 6 = 18 – 6
4y = 12
y = 12/4 =3
Solution is (2, 3)

Question 7.
\(\left\{\begin{array}{l}-5x+7y=11 \\-5x+3y=19\end{array}\right.\)
(________ , ________)

Answer:
(-5, -2)

Explanation:
\(\left\{\begin{array}{l}-5x+7y=11 \\-5x+3y=19\end{array}\right.\)
Subtract the equations
-5x + 7y = 11
-(-5x + 3y = 19)
x is eliminated as it has reversed coefficients. Solve for y
-5x + 7y + 5x – 3y = 11 – 19
4y = -8
y = -8/4 = -2
Substituting y in either of the equation to find x
-5x + 7(-2) = 11
-5x -14 + 14 = 11 + 14
-5x = 25
x = 25/-5 = -5
Solution is (-5, -2)

Question 8.
The Green River Freeway has a minimum and a maximum speed limit. Tony drove for 2 hours at the minimum speed limit and 3.5 hours at the maximum limit, a distance of 355 miles. Rae drove 2 hours at the minimum speed limit and 3 hours at the maximum limit, a distance of 320 miles. What are the two speed limits?
a. Write equatios to represent Tony’s distance and Rae’s distance.
Type below:
______________

Answer:
Tony’s distance: 2x + 3.5y = 355
Rae’s distance: 2x + 3y = 320
where x is the minimum speed and y is the maximum speed.

Question 8.
b. Solve the system.
minimum speed limit:______________
maximum speed limit______________
minimum speed limit: ________ mi/h
maximum speed limit: ________ mi/h

Answer:
minimum speed limit:55
maximum speed limit70
minimum speed limit: 55mi/h
maximum speed limit: 70mi/h

Explanation:
Subtract the equations
2x + 3.5y = 355
-(2x + 3y = 320)
x is eliminated as it has reversed coefficients. Solve for y
2x + 3.5y – 2x – 3y = 355 – 320
0.5y = 35
y = 35/0.5 = 70
Substituting y in either of the equation to find x
2x + 3(70) = 320
2x + 210 – 210 = 320 – 210
2x = 110
x = 110/2 = 55
Minimum speed limit: 55 miles per hour
Maximum speed limit: 70 miles per hour

ESSENTIAL QUESTION CHECK-IN

Question 9.
Can you use addition or subtraction to solve any system? Explain.
________

Answer:
No. One of the variables should have the same coefficient in order to add or subtract the system.

8.3 Independent Practice – Solving Systems by Elimination – Page No. 249

Question 10.
Represent Real-World Problems Marta bought new fish for her home aquarium. She bought 3 guppies and 2 platies for a total of $13.95. Hank also bought guppies and platies for his aquarium. He bought 3 guppies and 4 platies for a total of $18.33. Find the price of a guppy and the price of a platy.

Guppy: $ ________
Platy: $ ________

Answer:
Guppy: $ 3.19
Platy: $ 2.19

Explanation:
3x + 2y = 13.95
3x + 4y = 18.33
where x is the unit price of guppy and y is the unit price of platy
Subtract the equations
3x + 2y = 13.95
-(3x + 4y = 18.33)
x is eliminated as it has reversed coefficients. Solve for y
3x + 2y – 3x – 4y = 13.95 – 18.33
-2y = -4.38
y = -4.38/-2 = 2.19
Substituting y in either of the equation to find x
3x + 2(2.19) = 13.95
3x + 4.38 – 4.38 = 13.95 – 4.38
3x = 9.57
x = 9.57/3 = 3.19
The price of a guppy is $3.19 and price of a platy is $2.19

Question 11.
Represent Real-World Problems The rule for the number of fish in a home aquarium is 1 gallon of water for each inch of fish length. Marta’s aquarium holds 13 gallons and Hank’s aquarium holds 17 gallons. Based on the number of fish they bought in Exercise 10, how long is a guppy and how long is a platy?
Length of a guppy = ________ inches
Length of a platy = ________ inches

Answer:
Length of a guppy = 3 inches
Length of a platy = 2 inches

Explanation:
3x + 2y = 13
3x + 4y = 17
where x is the length of guppy and y is the length of a platy
Subtract the equations
3x + 2y = 13
-(3x + 4y = 17)
x is eliminated as it has reversed coefficients. Solve for y
3x + 2y – 3x – 4y = 13 – 17
-2y = -4
y = -4/-2 = 2
Substituting y in either of the equation to find x
3x + 2(2) = 13
3x + 4 – 4 = 13 – 4
3x = 9
x = 9/3 = 3
The length of a guppy is 3 inches and price of a platy is 2 inches

Question 12.
Line m passes through the points (6, 1) and (2, -3). Line n passes through the points (2, 3) and (5, -6). Find the point of intersection of these lines.
Type below:
________________

Answer:
The intersection of these lines is (3.5, -1.5)

Explanation:
Find the slope of line m = (y2 – y1)/(x2 – x1) where (x2, y2) = (2, -3) and (x1, y1) = (6, 1)
Slope = (-3 -1)/(2 – 6) = -4/-4 = 1
Substitute the value of m and any of the given ordered pair (x, y) in point-slope form of equation: y – y1 = m(x – x1)
y – 1 = 1(x – 6)
y – 1 = x – 6
y = x – 6 + 1
x – y = 5
Find the slope of line n = (y2 – y1)/(x2 – x1) where (x2, y2) = (5, -6) and (x1, y1) = (2, 3)
Slope = (-6 -3)/(5 – 2) = -9/3 = -3
Substitute the value of m and any of the given ordered pair (x, y) in point-slope form of equation: y – y1 = m(x – x1)
y – 3 = -3(x – 2)
y – 3 = -3x + 6
y = -3x + 6 + 3
3x + y = 9
Add the equations
x – y = 5
+(3x + y = 9)
y is eliminated as it has reversed coefficients. Solve for x
x – y + 3x + y = 5 + 9
4x = 14
x = 14/4 = 3.5
Substituting x in either of the equation to find y
3.5 – y = 5
3.5 – y – 3.5 = 5 – 3.5
-y = 1.5
y = -1.5
The intersection of these lines is (3.5, -1.5)

Question 13.
Represent Real-World Problems Two cars got an oil change at the same auto shop. The shop charges customers for each quart of oil plus a flat fee for labor. The oil change for one car required 5 quarts of oil and cost $22.45. The oil change for the other car required 7 quarts of oil and cost $25.45. How much is the labor fee and how much is each quart of oil?
Labor fee: $ ________
Quart of oil: $ ________

Answer:
Labor fee: $ 14.95
Quart of oil: $ 1.5

Explanation:
5x + y = 22.45
7x + y = 25.45
where x is the unit cost of quarts of oil and y is the flat fee for labor
Subtract the equations
5x + y = 22.45
-(7x + y = 25.45)
y is eliminated as it has reversed coefficients. Solve for x
5x + y – 7x – y = 22.45 – 25.45
-2x = -3
x = -3/-2 = 1.5
Substituting x in either of the equation to find y
5(1.5) + y = 22.45
7.5 + y – 7.5 = 22.45 – 7.5
y = 14.95
Labor fee is $14.95 and unit cost of quart of oil is $1.5

Question 14.
Represent Real-World Problems A sales manager noticed that the number of units sold for two T-shirt styles, style A and style B, was the same during June and July. In June, total sales were $2779 for the two styles, with A selling for $15.95 per shirt and B selling for $22.95 per shirt. In July, total sales for the two styles were $2385.10, with A selling at the same price and B selling at a discount of 22% off the June price. How many T-shirts of each style were sold in June and July combined?
________ T-shirts of style A and style B were sold in June and July.

Answer:
15.95x + 22.95y = 2779
15.95x + 17.9y = 2385.10
where x is number of style A shirt and y is the number of style B shirt
In July, the price of style B shirt is 22% of the price of style B shirt in June, hence 0.78(22.95) = 17.90
Subtract the equations
15.95x + 22.95y = 2779
-(15.95x + 17.9y = 2385.10)
x is eliminated as it has reversed coefficients. Solve for y
15.95x + 22.95 – 15.95x – 17.9y = 2779 – 2385.10
5.05y = 393.9
y = 393.9/5.05 = 78
Substituting y in either of the equation to find x
15.95x +22.95(78) = 2779
15.95x + 1790.1 – 1790.1 = 2779 – 1790.1
15.95x = 988.9
x = 988.9/15.95 = 62
The number of style A T shirt sold in June is 62.
Since the number of T-shirts sold in both numbers is the same, the total number = 2. 62 = 124.
The number of style B T-shirts sold in June is 78.
Since the number of T-shirts sold in both numbers is the same, the total number = 2. 78 = 156.

Question 15.
Represent Real-World Problems Adult tickets to a basketball game cost $5. Student tickets cost $1. A total of $2,874 was collected on the sale of 1,246 tickets. How many of each type of ticket were sold?
img 14
________ adult tickets
________ student tickets

Answer:
407 adult tickets
839 student tickets

Explanation:
x + y = 1246
5x + y = 2874
where x is the number of adult tickets sold and y is the number of student tickets sold.
Subtract the equations
x + y = 1246
-(5x + y = 2874)
y is eliminated as it has reversed coefficients. Solve for x
x + y – 5x – y = 1246 – 2874
-4x = -1628
x = -1628/-4 = 407
Substituting x in either of the equation to find y
407 + y = 1246
407 + y – 407 = 1246 – 407
y = 839
The number of adult tickets sold is 407 and student tickets sold is 839.

FOCUS ON HIGHER ORDER THINKING – Solving Systems by Elimination – Page No. 250

Question 16.
Communicate Mathematical Ideas Is it possible to solve the system
\(\left\{\begin{array}{l}3x-2y=10 \\x+2y=6\end{array}\right.\)
by using substitution? If so, explain how. Which method, substitution or elimination, is more efficient? Why?
________

Answer:
The system can be solved by substitution as x in equation 2 can be isolated.
3x – 2y = 10
x + 2y = 6
Solve the equation for x in the equation.
x = -2y + 6
Substitute the expression for x in the other equation and solve.
3(-2y + 6) -2y = 10
-6y + 18 – 2y = 10
-8y + 18 = 10
-8y = -8
y = -8/-8 = 1
Substitute the values of y into one of the equations and solve for the other variable x.
x + 2(1) = 6
x = 4
The solution is (4, 1)
As the cofficient if variable y is opposite, it will be eliminated and solved for x in less number of steps.
Elimination would be more efficient.

Question 17.
Jenny used substitution to solve the system
\(\left\{\begin{array}{l}2x+y=8 \\x-y=1\end{array}\right.\). Her solution is shown below.
Step 1: y = -2x + 8               Solve the first equation for y.
Step 2: 2x + (-2x + 8) = 8     Substitute the value of y in an original equation.
Step 3: 2x – 2x + 8 = 8          Use the Distributive Property.
Step 4: 8 = 8                         Simplify.
a. Explain the Error Explain the error Jenny made. Describe how to correct it.
Type below:
______________

Answer:
2x + y = 8
x – y = 1
Rewritten equation should be substituted in the other original equation
Error is that Jenny solved for y in the first equation and substitute it in the original equation.
x – (-2x + 8) = 1
3x – 8 = 1
3x = 9
x = 9/3 = 3
x = 3

Question 17.
b. Communicate Mathematical Ideas Would adding the equations have been a better method for solving the system? If so, explain why.
________

Answer:
Yes

Explanation:
As the coefficient, if variable y is the opposite, it will be eliminated and solved for x in less number of steps.

Guided Practice – Solving Systems by Elimination with Multiplication – Page No. 256

Question 1.
Solve the system
\(\left\{\begin{array}{l}3x-y=8 \\-2x+4y=-12\end{array}\right.\)
by multiplying and adding.

Type below:
______________

Answer:
\(\left\{\begin{array}{l}3x-y=8 \\-2x+4y=-12\end{array}\right.\)
Multiply each term in the first equation by 4 to get opposite coefficients for the y-terms.
4(3x – y = 8)
12x – 4y = 32
Add the second equation to the new equation
12x – 4y = 32
+(-2x + 4y = -12)
Add to eliminate the variable y
10x = 20
Divide both sides by 10
x = 20/10 = 2
Substitue into one of the original equations and solve for y
y = 3(2) – 8 = -1
S0, (2, -2)is the solution of the system.

Solve each system of equations by multiplying first.

Question 2.
\(\left\{\begin{array}{l}x+4y=2 \\2x+5y=7\end{array}\right.\)
(________ , ________ )

Answer:
(6, -1)

Explanation:
x + 4y = 2
2x + 5y = 7
To eliminate x terms, multiply the 2nd equation by 2
2(x + 4y = 2)
2x + 8y = 4
Subtract the equations
2x + 8y = 4
-(2x + 5y = 7)
x is eliminated as it has reversed coefficients. Solve for y
2x + 8y – 2x – 5y = 4 – 7
3y = -3
y = -3/3 = -1
Substituting y in either of the equation to find x
x + 4(-1) = 2
x – 4 + 4 = 2 + 4
x = 6
Solution: (6, -1)

Question 3.
\(\left\{\begin{array}{l}3x+y=-1 \\2x+3y=18\end{array}\right.\)
(________ , ________ )

Answer:
(-3, 8)

Explanation:
\(\left\{\begin{array}{l}3x+y=-1 \\2x+3y=18\end{array}\right.\)
To eliminate y terms, multiply the 1st equation by 3
3(3x + y = -1)
9x + 3y = -3
Subtract the equations
9x + 3y = -3
-(2x + 3y = 18)
y is eliminated as it has reversed coefficients. Solve for x
9x + 3y – 2x – 3y = -3 -18
7x = -21
x = -21/7
x = -3
Substituting x in either of the equation to find y
3(-3) + y = -1
-9 + y + 9 = -1 + 9
y = 8
Solution: (-3, 8)

Question 4.
\(\left\{\begin{array}{l}2x+8y=21 \\6x-4y=14\end{array}\right.\)
Type below:
______________

Answer:
The soultion is (3.5, 1.75)

Explanation:
\(\left\{\begin{array}{l}2x+8y=21 \\6x-4y=14\end{array}\right.\)
To eliminate y terms, multiply the 2nd equation by 2
2(6x – 4y = 14)
2x + 8y = 21
Add the equations
2x + 8y = 21
+(12x – 8y = 28)
y is eliminated it has reversed coefficients. Solve for x
2x + 8y + 12x – 8y = 21 + 28
14x = 49
x = 49/14 = 3.5
Substituting x in either of the equation to find y
6(3.5) – 4y = 14
21 – 4y – 21 = 14 – 21
-4y = -7
y = -7/-4 = 1.75
The soultion is (3.5, 1.75)

Question 5.
\(\left\{\begin{array}{l}2x+y=3 \\-x+3y=-12\end{array}\right.\)
(________ , ________ )

Answer:

Explanation:
\(\left\{\begin{array}{l}2x+y=3 \\-x+3y=-12\end{array}\right.\)
To eliminate x terms, multiply the 2nd equation by 2
2(-x + 3y = -12)
-2x + 6y = -24
Add the equations
2x + y = 3
+(-2x + 6y = -24)
x is eliminated it has reversed coefficients. Solve for y
2x + y – 2x + 6y = 3 – 24
7y = -21
y = -21/7 = -3
Substituting y in either of the equation to find x
-x + 3(-3) = -12
-x -9 + 9 = -12 + 9
-x = -3
x = 3
The soultion is (3, -3)

Question 6.
\(\left\{\begin{array}{l}6x+5y=19 \\2x+3y=5\end{array}\right.\)
(________ , ________ )

Answer:
The soultion is (4, -1)

Explanation:
\(\left\{\begin{array}{l}6x+5y=19 \\2x+3y=5\end{array}\right.\)
To eliminate x terms, multiply the 2nd equation by 3
3(2x + 3y = 5)
6x + 9y = 15
Subtract the equations
6x + 5y = 19
-(6x + 9y = 15)
x is eliminated it has reversed coefficients. Solve for y
6x + 5y – 6x – 9y = 19 – 15
-4y = 4
y = 4/-4 = -1
Substituting y in either of the equation to find x
2x + 3(-1) = 5
2x – 3 + 3 = 5 + 3
2x = 8
x = 8/2 = 4
The soultion is (4, -1)

Question 7.
\(\left\{\begin{array}{l}2x+5y=16 \\-4x+3y=20\end{array}\right.\)
(________ , ________ )

Answer:
The soultion is (-2, 4)

Explanation:
\(\left\{\begin{array}{l}2x+5y=16 \\-4x+3y=20\end{array}\right.\)
To eliminate x terms, multiply the 1st equation by 2
2(2x + 5y = 16)
4x + 10y = 32
Add the equations
4x + 10y = 32
+(-4x + 3y = 20)
x is eliminated it has reversed coefficients. Solve for y
10y + 3y = 32 + 20
13y = 52
y = 52/13 = 4
Substituting y in either of the equation to find x
2x + 5(4) = 16
2x + 20 – 20 = 16 – 20
2x = -4
x = -4/2 = -2
The soultion is (-2, 4)

Question 8.
Bryce spent $5.26 on some apples priced at $0.64 each and some pears priced at $0.45 each. At another store he could have bought the same number of apples at $0.32 each and the same number of pears at $0.39 each, for a total cost of $3.62. How many apples and how many pears did Bryce buy?
a. Write equations to represent Bryce’s expenditures at each store
First store: _____________
Second store: _____________
Type below:
_____________

Answer:
First store: 0.64x + 0.45y = 5.26
Second store: 0.32x + 0.39y = 3.62

Explanation:
First store = 0.64x + 0.45y = 5.26
Second store = 0.32x + 0.39y = 3.62
where x is the number of apples and y is the number of pears.

Question 8.
b. Solve the system.
Number of apples: _______
Number of pears: _______

Answer:
Number of apples: 4
Number of pears: 6

Explanation:
First store = 0.64x + 0.45y = 5.26
Second store = 0.32x + 0.39y = 3.62
Multiply by 100
64x + 45y = 526
32x + 39y = 362
To eliminate x terms, multiply the 2nd equation by 2
2(32x + 39y = 362)
64x + 45y = 526
Subtract the equations
64x + 45y = 526
-(64x + 78y = 724)
x is eliminated it has reversed coefficients. Solve for y
64x + 45y – 64x – 78y = 526 – 724
-33y = -198
y = -198/-33 = 6
Substituting y in either of the equation to find x
32x + 39(6) = 362
32x + 234 – 234 = 362 – 234
32x = 128
x = 128/32 = 4
He bought 4 apples and 6 pears.

ESSENTIAL QUESTION CHECK-IN

Question 9.
When solving a system by multiplying and then adding or subtracting, how do you decide whether to add or subtract?
Type below:
_____________

Answer:
If the variable with the same coefficient but reversed sign, we add and if they have the same sign, we subtract.

Solving Systems by Elimination with Multiplication – Page No. 257

Question 10.
Explain the Error Gwen used elimination with multiplication to solve the system
\(\left\{\begin{array}{l}2x+6y=3 \\x-3y=-1\end{array}\right.\)
Her work to find x is shown. Explain her error. Then solve the system.
2(x − 3y) = -1
2x − 6y = -1
+2x + 6y = 3
_____________
4x + 0y = 2
x = \(\frac{1}{2}\)
Type below:
____________

Answer:
2x + 6y = 3
x – 3y = -1
To eliminate x terms, multiply the 2nd equation by 2
2(x – 3y = -1)
2x – 6y = -2
Error is the Gnew did not multiply the entire expression with 2.
Add the equations
2x + 6y = 3
+(2x – 6y = -2)
y is eliminated it has reversed coefficients. Solve for x
2x + 6y + 2x – 6y = 3 – 2
4x = 1
x = 1/4
Substituting x in either of the equation to find y
x – 3y = -1
1/4 – 3y – 1/4 = -1 -1/4
-3y = -5/4
y = -5/4(-3) = 5/12

Question 11.
Represent Real-World Problems At Raging River Sports, polyester-fill sleeping bags sell for $79. Down-fill sleeping bags sell for $149. In one week the store sold 14 sleeping bags for $1,456.
a. Let x represent the number of polyester-fill bags sold and let y represent the number of down-fill bags sold. Write a system of equations you can solve to find the number of each type sold.

Type below:
____________

Answer:
x + y = 14
79x + 149y = 1456
where x is the polyster-fill bags and y is the number of down-fill bags

Question 11.
b. Explain how you can solve the system for y by multiplying and subtracting.
Type below:
____________

Answer:
x + y = 14
79x + 149y = 1456
Multiply the second equation by 79. Subtract the new equation from the first equation and solve the resulting equation for y.

Question 11.
c. Explain how you can solve the system for y using substitution.
Type below:
____________

Answer:
Solve the second equation for x. Substitute the expression for x , in the first equation and solve the resulting equation for y.

Question 11.
d. How many of each type of bag were sold?
_______ polyester-fill
_______ down-fill

Answer:
9 polyester-fill
5 down-fill

Explanation:
x + y = 14
79x + 149y = 1456
To eliminate x terms, multiply the 2nd equation by 2
79(x + y = 14)
79x + 149y = 1456
Subtract the equations
79x + 79y = 1106
-(79x + 149y = 1456)
x is eliminated it has reversed coefficients. Solve for y
79x + 79y – 79x – 149y = 1106 – 1456
-70y = -350
y = -350/-70 = 5
Substituting y in either of the equation to find x
x + 5 = 14
x = 14 – 5
x = 9
There were 9 polyster-fill bags and 5 down-fill bags sold.

Question 12.
Twice a number plus twice a second number is 310. The difference between the numbers is 55. Find the numbers by writing and solving a system of equations. Explain how you solved the system.
x = _______
y = _______

Answer:
x = 105
y = 50

Explanation:
2x + 2y = 310
x – y = 55
To eliminate y terms, multiply the 2nd equation by 2
2(x – y = 55)
2x – 2y = 110
Add the equations
2x + 2y = 310
+ (2x – 2y = 110)
y is eliminated it has reversed coefficients. Solve for x
2x + 2y + 2x – 2y = 310 + 110
4x = 420
x = 420/4 = 105
Substituting x in either of the equation to find y
105 – y = 55
y = 105 – 55
y = 50
The solution is (105, 50)

Solving Systems by Elimination with Multiplication – Page No. 258

Question 13.
Represent Real-World Problems A farm stand sells apple pies and jars of applesauce. The table shows the number of apples needed to make a pie and a jar of applesauce. Yesterday, the farm picked 169 Granny Smith apples and 95 Red Delicious apples. How many pies and jars of applesauce can the farm make if every apple is used?

_______ pies
_______ jars of applesauce

Answer:
21 pies
16 jars of applesauce

Explanation:
5x + 4y = 169
3x + 2y = 95
where x is the number of apples needed for pie and y is the number of apples for jar of applesauce
To eliminate y terms, multiply the 2nd equation by 2
2(3x + 2y = 95)
6x + 4y = 190
Subtract the equations
5x + 4y = 169
– (6x + 4y = 190)
y is eliminated it has reversed coefficients. Solve for x
5x + 4y – 6x – 4y = 169 – 190
-x = -21
x = -21/-1 = 21
Substituting x in either of the equation to find y
5(21) + 4y = 169
105 + 4y – 105 = 169 – 105
4y = 64
y = 64/4 = 16
The number of apples needed for pie is 21 and the number of apples for jar of applesauce is 16.

FOCUS ON HIGHER ORDER THINKING

Question 14.
Make a Conjecture Lena tried to solve a system of linear equations algebraically and in the process found the equation 5 = 9. Lena thought something was wrong, so she graphed the equations and found that they were parallel lines. Explain what Lena’s graph and equation could mean.
Type below:
____________

Answer:
Lena’s graph is a parallel line which means the graph does not intersect each other, hence they have no solutions. Equation 5 = 9 means variables are eliminated and this statement is not true. This linear system has no solution.

Question 15.
Consider the system
\(\left\{\begin{array}{l}2x+3y=6 \\3x+7y=-1\end{array}\right.\)
a. Communicate Mathematical Ideas Describe how to solve the system by multiplying the first equation by a constant and subtracting. Why would this method be less than ideal?
Type below:
____________

Answer:
Multiplying the first equation by a constant and subtracting
2x + 3y = 6
3x + 7y = -1
Multiply the first equation by 1.5 and subtract. This would be less than ideal because you would introduce decimals into the solution process.

Question 15.
b. Draw Conclusions Is it possible to solve the system by multiplying both equations by integer constants? If so, explain how.
Type below:
____________

Answer:
Yes

Explanation:
Multiply the first equation by 3 and the second equation by 2. Both x-term coefficients would be 6. Solve by eliminating the x-terms using subtraction.

Question 15.
c. Use your answer from part b to solve the system.
(_______ , _______)

Answer:
(9, -4)

Explanation:
2x + 3y = 6
3x + 7y = -1
Multiply the first equation by 3 and the second equation by 2.
3(2x + 3y = 6)
2(3x + 7y = -1)
Subtract the equations
6x + 9y = 18
-(6x + 14y = -2)
x is eliminated it has reversed coefficients. Solve for y
6x + 9y – 6x – 14y = 18 + 2
-5y = 20
y = 20/-5 = -4
Substituting y in either of the equation to find x
2x + 3(-4) = 6
2x = 18
x = 18/2 = 9
The solution is (9, -4)

Guided Practice – Solving Solving Special Systems – Page No. 262

Use the graph to solve each system of linear equations

Question 1.
A. \(\left\{\begin{array}{l}4x-2y=-6 \\2x-y=4\end{array}\right.\)
B. \(\left\{\begin{array}{l}4x-2y=-6 \\x+y=6\end{array}\right.\)
C. \(\left\{\begin{array}{l}2x-y=4 \\6x-3y=-12\end{array}\right.\)
STEP 1 Decide if the graphs of the equations in each system intersect, are parallel, or are the same line.

System A: The graphs __________
System B: The graphs __________
System C: The graphs __________

Answer:
System A: The graphs are parallel
System B: The graphs are intersecting
System C: The graphs are  the same line

Explanation:
System A: 4x – 2y = -6
2x – y = 4
System B: 4x – 2y = -6
x + y = 6
System C: 2x – y = 4
6x – 3y = 12

Question 1.
STEP 2 Decide how many points the graphs have in common.
a. Intersecting lines have _______________ point(s) in common.
b. Parallel lines have _______________ point(s) in common.
c. The same lines have ___________ point(s) in common.
a. __________
b. __________
c. __________

Answer:
a. Intersecting lines have one point(s) in common.
b. Parallel lines have no point(s) in common.
c. The same lines have infinitely many points (s) in common.

Explanation:
From the graphs,
Intersecting lines have one point(s) in common
Parallel lines have no point(s) in common
The same lines have infinitely many points (s) in common

Question 1.
STEP 3 Solve each system.
System A has __________ points in common, so it has __________ solution.
System B has __________ point in common. That point is the solution, __________.
System C has __________ points in common. ________ ordered pairs on the line will make both equations true.
Type below:
___________

Answer:
System A has no points in common, so it has no solution. System B has one point in common. That point is the solution, (1,5). System C has an infinite number of points in common. All ordered pairs on the line will make both equations true.

Explanation:
Number of solutions for each system
System A has no points in common, so it has no solution. System B has one point in common. That point is the solution, (1,5). System C has an infinite number of points in common. All ordered pairs on the line will make both equations true.

Solve each system. Tell how many solutions each system has.

Question 2.
\(\left\{\begin{array}{l}x-3y=4 \\-5x+15y=-20\end{array}\right.\)
___________

Answer:
infinitely many solutions

Explanation:
x – 3y = 4
-5x + 15y = -20
To eliminate y terms, multiply the 1st equation by 5
5(x – 3y = 4)
5x – 15y = 20
Add the equations
5x – 15y = 20
+(-5x + 15y = -20)
x and y is eliminated as it has reversed coefficients.
5x – 15y – 5x + 15y = 20 – 20
0 = 0
The statement is true, hence the solution has infinitely many solutions.

Question 3.
\(\left\{\begin{array}{l}6x+2y=-4 \\3x+y=4\end{array}\right.\)
___________

Answer:
no solution

Explanation:
6x + 2y = -4
3x + y = 4
To eliminate y terms, multiply the 2nd equation by 5
2(3x + y = 4)
6x + 2y = 8
Subtract the equations
6x + 2y = -4
-(6x + 2y = 8)
x and y is eliminated as it has reversed coefficients.
6x + 2y – 6x – 2y = -4 -8
0 = -12
The statement is false, hence the solution has no solution.

Question 4.
\(\left\{\begin{array}{l}6x-2y=-10 \\3x+4y=-25\end{array}\right.\)
___________

Answer:
one solution

Explanation:
6x – 2y = -10
3x + 4y = -25
To eliminate y terms, multiply the 1st equation by 2
2(6x – 2y = -10)
12x – 4y = -20
Add the equations
12x – 4y = -20
+(3x + 4y = -25)
y is eliminated as it has reversed coefficients. Solve for x.
12x – 4y + 3x + 4y = -20 – 25
15x = -45
x = -45/15 = -3
Substitute x in any one of the original equations and solve for y
3(-3) + 4y = -25
-9 + 4y + 9 = -25 + 9
4y = -16
y = -16/4
y = -4
There is one solution, (-3, -4)

ESSENTIAL QUESTION CHECK-IN

Question 5.
When you solve a system of equations algebraically, how can you tell whether the system has zero, one, or an infinite number of solutions?
Type below:
___________

Answer:
When x and y are eliminated and the statement is true, the system has infinitely many solutions.
When x and y are eliminated and the statement is false, the system has no solutions.
When the system has one solution by solving, the system has one solution.

8.5 Independent Practice – Solving Solving Special Systems – Page No. 263

Solve each system by graphing. Check your answer algebraically.

Question 6.
\(\left\{\begin{array}{l}-2x+6y=12 \\x-3y=3\end{array}\right.\)

Solution: ______________
___________

Answer:
\(\left\{\begin{array}{l}-2x+6y=12 \\x-3y=3\end{array}\right.\)
Graph the equations on same coordinate plane
No solution as equations are parallel

To eliminate y terms, multiply the 2nd equation by 2
2(x – 3y = 3)
2x – 6y = 6
Add the equations
-2x + 6y = 12
2x – 6y = 6
x and y is eliminated as it has reversed coefficients.
-2x + 6y + 2x – 6y = 12 + 6
0 = 18
The statement is false, hence the system has no solution.

Question 7.
\(\left\{\begin{array}{l}15x+5y=5 \\3x+y=1\end{array}\right.\)

Solution: ______________
___________

Answer:
\(\left\{\begin{array}{l}15x+5y=5 \\3x+y=1\end{array}\right.\)
Graph the equations on same coordinate plane

Infinitely many solutions as equations are overlapping
To eliminate y terms, multiply the 2nd equation by 5
5(3x + y = 1)
15x + 5y = 5
Subtarct the equations
15x + 5y = 5
-(15x + 5y = 5)
x and y is eliminated as it has reversed coefficients.
15x + 5y -15x – 5y = 5 – 5
0 = 0
The statement is true, hence the system has infinitely many solutions.

For Exs. 8–

14, state the number of solutions for each system of linear equations

Question 8.
a system whose graphs have the same slope but different y-intercepts
___________

Answer:
No solutions

Explanation:
Equations are parallel
No solutions

Question 9.
a system whose graphs have the same y-intercepts but different slopes
___________

Answer:
One solution

Explanation:
Equations are intersecting
One solution

Question 10.
a system whose graphs have the same y-intercepts and the same slopes
___________

Answer:
Infinitely many solutions

Explanation:
Equations are overlapping
Infinitely many solutions

Question 11.
a system whose graphs have different y-intercepts and different slopes
___________

Answer:
One solution

Explanation:
Equations are intersecting
One solution

Question 12.
the system
\(\left\{\begin{array}{l}y=2 \\y=-3\end{array}\right.\)
___________

Answer:
No solutions

Explanation:
Equations are parallel
No solutions

Question 13.
the system
\(\left\{\begin{array}{l}y=2 \\y=-3\end{array}\right.\)
___________

Answer:
One solution

Explanation:
Equations are intersecting
One solution

Question 14.
the system whose graphs were drawn using these tables of values:

___________

Answer:
No solutions

Explanation:
Equations are parallel The slope is the same for both equations but the y-intercept is different.
No solutions

Question 15.
Draw Conclusions The graph of a linear system appears in a textbook. You can see that the lines do not intersect on the graph, but also they do not appear to be parallel. Can you conclude that the system has no solution? Explain.
___________

Answer:

No; although the lines do not intersect on the graph, they intersect at a point that is not on the graph. To prove that a system has no solution, you must do so algebraically

Solving Solving Special Systems – Page No. 264

Question 16.
Represent Real-World Problems Two school groups go to a roller skating rink. One group pays $243 for 36 admissions and 21 skate rentals. The other group pays $81 for 12 admissions and 7 skate rentals. Let x represent the cost of admission and let y represent the cost of a skate rental. Is there enough information to find values for x and y? Explain.

___________

Answer:
36x + 21y = 243
12x + 7y = 81
where x is the cost of admission and y is the cost of stake rentals.
Although the information can be used to develop a system of linear equation, where each equation has two variables when the system is solved, the number of solutions is infinite, Mee the values of x and y cannot be determined.
No

Question 17.
Represent Real-World Problems Juan and Tory are practicing for a track meet. They start their practice runs at the same point, but Tory starts 1 minute after Juan. Both run at a speed of 704 feet per minute. Does Tory catch up to Juan? Explain.
___________

Answer:
No; Both Juan and Tory-run at the same rate, so the lines representing the distances each has run are parallel. There is no solution to the system

FOCUS ON HIGHER ORDER THINKING

Question 18.
Justify Reasoning A linear system with no solution consists of the equation y = 4x − 3 and a second equation of the form y = mx + b. What can you say about the values of m and b? Explain your reasoning.
Type below:
___________

Answer:
y = 4x – 3
y = mx + b
Since the system has no solutions, the two equations are parallel. The value of the slope, m would be the same i.e. 4. The value of y-intercept, b can be any number except -3 as b is different for parallel lines.

Question 19.
Justify Reasoning A linear system with infinitely many solutions consists of the equation 3x + 5 = 8 and a second equation of the form Ax + By = C. What can you say about the values of A, B, and C? Explain your reasoning.
Type below:
___________

Answer:
3x + 5 = 8
Ax + By = C
Since the system has infinitely many solutions, the values of A, B, and C must all be the same multiple of 3, 5, and 8, respectively. The two equations represent a single line, so the coefficients and constants of one equation must be a multiple of the other.

Question 20.
Draw Conclusions Both the points (2, -2) and (4, -4) are solutions of a system of linear equations. What conclusions can you make about the equations and their graphs?
Type below:
___________

Answer:
If a system has more than one solution, the equations represent the same line and have infinitely many solutions.

Ready to Go On? – Model Quiz – Page No. 265

8.1 Solving Systems of Linear Equations by Graphing

Solve each system by graphing.

Question 1.
\(\left\{\begin{array}{l}y=x-1 \\y=2x-3\end{array}\right.\)

(________ , ________)

Answer:
(2, 1)

Explanation:
y = x – 1
y = 2x – 3
Graph the equations on the same coordinate plane

The solution of the system is the point of intersection
The solution is (2, 1)

Question 2.
\(\left\{\begin{array}{l}x+2y=1 \\-x+y=2\end{array}\right.\)

(________ , ________)

Answer:
(-1, 1)

Explanation:
x + 2y = 1
-x + y = 2
Graph the equations on same coordinate plane

The solution of the system is the point of intersection
The solution is (-1, 1)

8.2 Solving Systems by Substitution

Solve each system of equations by substitution.

Question 3.
\(\left\{\begin{array}{l}y=2x \\x+y=-9\end{array}\right.\)
(________ , ________)

Answer:
(-3, -6)

Explanation:
y = 2x
x + y = -9
Substitute y from equation 1 in the other equation.
x + 2x = -9
3x = -9
x = -9/3
x = -3
Then, y = 2(-3) = -6
The Solution is (-3, -6)

Question 4.
\(\left\{\begin{array}{l}3x-2y=11 \\x+2y=9\end{array}\right.\)
(________ , ________)

Answer:
(5, 2)

Explanation:
3x – 2y = 11
x + 2y = 9
Solve for x in equation 2
x = – 2y + 9
Substitute x from equation 2 in the other equation
3(-2y + 9) – 2y = 11
-6y + 27 -2y = 11
-8y = -16
y = -16/-8 = 2
Substitute y in any of the equations to find x
x + 2(2) = 9
x + 4 – 4 = 9 – 4
x = 5
The solution is (5, 2)

8.3 Solving Systems by Elimination

Solve each system of equations by adding or subtracting.

Question 5.
\(\left\{\begin{array}{l}3x+y=9 \\2x+y=5\end{array}\right.\)
(________ , ________)

Answer:
(4, -3)

Explanation:
\(\left\{\begin{array}{l}3x+y=9 \\2x+y=5\end{array}\right.\)
Subtract the equations
3x + y = 9
-(2x + y = 5)
y is eliminated as it has reversed coefficients. Solve for x
3x + y – 2x – y = 9 – 5
x = 4
Substituting x in either of the equation to find y
2(4) + y = 5
8 + y – 8 = 5 – 8
y = -3
The solution is (4, -3)

Question 6.
\(\left\{\begin{array}{l}-x-2y=4 \\3x+2y=4\end{array}\right.\)
(________ , ________)

Answer:
(4, -4)

Explanation:
\(\left\{\begin{array}{l}-x-2y=4 \\3x+2y=4\end{array}\right.\)
Add the equations
-x – 2y = 4
+(3x + 2y = 4)
y is eliminated as it has reversed coefficients. Solve for x
-x – 2y + 3x + 2y = 4 + 4
2x = 8
x = 8/2 = 4
Substituting x in either of the equation to find y
3(4) + 2y = 4
12 + 2y – 12 = 4 – 12
2y = -8
y = -8/2 = -4
The solution is (4, -4)

8.4 Solving Systems by Elimination with Multiplication

Solve each system of equations by multiplying first.

Question 7.
\(\left\{\begin{array}{l}x+3y=-2 \\3x+4y=-1\end{array}\right.\)
(________ , ________)

Answer:
(1, -1)

Explanation:
\(\left\{\begin{array}{l}x+3y=-2 \\3x+4y=-1\end{array}\right.\)
Subtract the equations
3x + 9y = -6
-(3x + 4y = -1)
x is eliminated as it has reversed coefficients. Solve for y
3x + 9y – 3x – 4y = -6 + 1
5y = -5
y = -5/5
y = -1
Substituting y in either of the equation to find x
x + 3(-1) = -2
x – 3 = -2
x = -2 + 3
x = 1
The solution is (1, -1)

Question 8.
\(\left\{\begin{array}{l}2x+8y=22 \\3x-2y=5\end{array}\right.\)
(________ , ________)

Answer:
(3, 2)

Explanation:
\(\left\{\begin{array}{l}2x+8y=22 \\3x-2y=5\end{array}\right.\)
Multiply equation 2 by 4 so that y can be eliminated
4(3x – 2y = 5)
12x – 8y = 20
Add the equations
2x + 8y = 22
+(12x – 8y = 20)
y is eliminated as it has reversed coefficients. Solve for x
2x + 8y + 12x – 8y = 22 + 20
14x = 42
x = 42/14
x = 3
Substituting y in either of the equation to find x
2(3) + 8y = 22
6 + 8y = 22
8y = 22 – 6
8y = 16
y = 16/8
y = 2
The solution is (3, 2)

8.5 Solving Special Systems

Solve each system. Tell how many solutions each system has.

Question 9.
\(\left\{\begin{array}{l}-2x+8y=5 \\x-4y=-3\end{array}\right.\)
_____________

Answer:
no solution

Explanation:
\(\left\{\begin{array}{l}-2x+8y=5 \\x-4y=-3\end{array}\right.\)
Multiply equation 2 by 2 so that y can be eliminated
2(x – 4y = -3)
2x – 8y = -6
Add the equations
-2x + 8y = 5
+(2x – 8y = -6)
x and y is eliminated
-2x + 8y + 2x – 8y = 5 – 6
0 = -1
The statement is false. Hence, the system has no solution.

Question 10.
\(\left\{\begin{array}{l}6x+18y=-12 \\x+3y=-2\end{array}\right.\)
_____________

Answer:
infinitely many solutions

Explanation:
\(\left\{\begin{array}{l}6x+18y=-12 \\x+3y=-2\end{array}\right.\)
Multiply equation 2 by 6 so that x can be eliminated
6(x + 3y = -2)
6x + 18y = -12
Subtract the equations
6x + 18y = -12
-(6x + 18y = -12)
x and y is eliminated
6x + 18y -6x -18y = -12 + 12
0 = 0
The statement is true. Hence, the system has infinitely many solutions.

ESSENTIAL QUESTION

Question 11.
What are the possible solutions to a system of linear equations, and what do they represent graphically?
Type below:
___________

Answer:
System of linear equations can have no solution, which is represented by parallel lines; one solution, which is represented by intersecting lines; and infinitely many solutions, which is represented by overlapping lines.

Selected Response – Mixed Review – Page No. 266

Question 1.
The graph of which equation is shown?

Options:
A. y = −2x + 2
B. y = −x + 2
C. y = 2x + 2
D. y = 2x + 1

Answer:
C. y = 2x + 2

Explanation:
Option A and B are eliminated as the slope of the graph is 2.
Option D is eliminated as the y-intercept from the graph should be 2.
Option C is the equation of the graph

Question 2.
Which best describes the solutions to the system
\(\left\{\begin{array}{l}x+y=-4 \\-2x-2y=0\end{array}\right.\)
Options:
A. one solution
B. no solution
C. infinitely many
D. (0, 0)

Answer:
B. no solution

Explanation:
\(\left\{\begin{array}{l}x+y=-4 \\-2x-2y=0\end{array}\right.\)
Multply equation 1 by 2 so that x can be eliminated
2(x + y = -4)
2x + 2y = -8
Add the equations
2x + 2y = -8
-2x – 2y = 0
x and y is eliminated
2x + 2y – 2x -2y = -8 + 0
0 = -8
The statement is false. Hence, the system has no solution.

Question 3.
Which of the following represents 0.000056023 written in scientific notation?
Options:
A. 5.6023 × 10⁵
B. 5.6023 × 10⁴
C. 5.6023 × 10^-4
D. 5.6023 × 10^-5

Answer:
D. 5.6023 × 10^-5

Explanation:
Move the decimal 5 points right to get the equation.
D. 5.6023 × 10^-5

Question 4.
Which is the solution to
\(\left\{\begin{array}{l}2x-y=1 \\4x+y=11\end{array}\right.\)
Options:
A. (2, 3)
B. (3, 2)
C. (-2, 3)
D. (3, -2)

Answer:
A. (2, 3)

Explanation:
\(\left\{\begin{array}{l}2x-y=1 \\4x+y=11\end{array}\right.\)
Add the equations
2x – y = 1
4x + y = 11
y is eliminated as it has reversed coefficients. Solve for x.
2x – y + 4x + y = 1 + 11
6x = 12
x = 12/6 = 2
Substituting x in either of the equation to find y
4(2) + y = 11
8 + y = 11
y = 11 – 8
y = 3
The solution is (2, 3)

Question 5.
Which expression can you substitute in the indicated equation to solve
\(\left\{\begin{array}{l}3x-y=5 \\x+2y=4\end{array}\right.\)
Options:
A. 2y – 4 for x in 3x – y = 5
B. 4 – x for y in 3x – y = 5
C. 3x – 5 for y in 3x – y = 5
D. 3x – 5 for y in x + 2y = 4

Answer:
D. 3x – 5 for y in x + 2y = 4

Explanation:
\(\left\{\begin{array}{l}3x-y=5 \\x+2y=4\end{array}\right.\)
Solve for y in equation 1
y = 3x – 5
Substitute in other equation x + 2y = 4

Question 6.
What is the solution to the system of linear equations shown on the graph?

Options:
A. -1
B. -2
C. (-1, -2)
D. (-2, -1)

Answer:
C. (-1, -2)

Explanation:
The point of intersection is (-1, -2), which is the solution of the system

Question 7.
Which step could you use to start solving
\(\left\{\begin{array}{l}x-6y=8 \\2x-5y=3\end{array}\right.\)
Options:
A. Add 2x – 5y = 3 to x – 6y = 8.
B. Multiply x – 6y = 8 by 2 and add it to 2x – 5y = 3.
C. Multiply x – 6y = 8 by 2 and subtract it from 2x – 5y = 3.
D. Substitute x = 6y – 8 for x in 2x – 5y = 3.

Answer:
C. Multiply x – 6y = 8 by 2 and subtract it from 2x – 5y = 3.

Explanation:
x – 6y = 8
2x – 5y = 3
Multiply the 1st equation by 2 so that the coefficient of variable x is the same in both equations
Subtract the equations as x has the same sign.

Mini-Task

Question 8.
A hot-air balloon begins rising from the ground at 4 meters per second at the same time a parachutist’s chute opens at a height of 200 meters. The parachutist descends at 6 meters per second.
a. Define the variables and write a system that represents the situation.
Type below:
_____________

Answer:
y represents the distance from the ground and x represents the time in seconds
y = 4x
y = -6x + 200

Question 8.
b. Find the solution. What does it mean?
Type below:
_____________

Answer:
Substitute y from the equation 1 in the equation 2
4x = -6x + 200
4x + 6x = -6x + 200 + 6x
10x = 200
x = 200/10 = 20
Substitute x in any one of the equations and solve for x
y = 4(20) = 80
The solution is (20, 80)
The ballon and parachute meets after 20sec at 80m from the ground.

Final Words:

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