Common Core Math

In geometry, you will various math concepts like Angles, Lines, Shapes, Area and Perimeter, etc. Today, we will discuss completely the concept named Angles and its types. Based on measurements, there are various types of angles. Usually, an angle is measured in degrees and it is one of the core concepts of geometry in Maths. Are you excited to learn deeply about the topic called Angle? Then, refer to the below modules thoroughly and know what is an angle, what are the Types of Angles, definitions, figures, and some solved examples.

What is an Angle?

An Angle is a geometrical shape formed when two rays join with a common end-point. “side and vertex’ are the two components of an angle. Angles are classified based on their measures.

Parts of Angle

• Vertex – Point where the arms meet.
• Arms – Two straight line segments form a vertex.
• Angle – When a ray turn about its endpoint, the measure of its rotation between its initial and final position is called Angle.

If these two ray joins in various fashions to form a various type of angles in maths. Let’s, start learning what are the different types of angles and their definitions and figures.

Classification of Angles

In nature, there are several types of angles that exist. Each and every angle of them hold great value in our everyday living standards.

Basically, Angles are classified on the basis of:

• Magnitude
• Rotation

Types of Angles Based on Magnitude

In maths, mainly, there are 6 types of angles on the basis of direction. And also, all these six angle types are commonly used in geometry. The names of different angles types are as follows:

• Acute Angle
• Obtuse Angle
• Right Angle
• Straight Angle
• Reflex Angle
• Complete Angle

The below image illustrates specific types of angles based on magnitude:

1. Acute Angle:

An acute angle is an angle that lies between 0 degrees to 90 degrees. In other words, an angle is less than 90° is called an acute angle.

Illustration:

∠XYZ is greater than 0° but less than 90° so, this is an acute angle type.

2. Obtuse Angle:

Opposite of the Acute angle is called Obtuse Angle. In other words, the angle that lies between 90 degrees and 180 degrees is known as the obtuse angle.

Illustration:

3. Right Angle:

An angle that measures at exactly 90 degrees is called a right angle. Basically, it forms when two lines are perpendicular to each other. See the below-illustrated figure of a right angle.

4. Straight Angle:

An angle that measures 180° is known as Straight angle. The following figure illustrated the straight angle.

5. Reflex Angle:

The angle that lies between 180 degrees and 360 degrees is called a reflex angle. If you want to calculate the reflex angle then you must require an acute angle. The below figure illustrates the reflex angle.

6. Complete Angle:

An angle measured 360 degrees is called a Complete Angle. 1 Revolution is equal to 360° and the illustration of the complete angle is as shown in the below figure:

Solved Examples on Types of Angles

1. The sum of three angles is (x+6), (x -4), and (x + 8) forms a right angle. Find the value of x.

Solution:

⇒ (x+6) + (x-4) + (x+8) = 90

⇒ 3x + 10 = 90

⇒ 3x = 80

x = 26

Therefore, the value of x is 26 degrees.

2. A certain angle is such that, two times the sum of its size and 70° is 90°. What is the name of this angle?

Solution:

The angle be x°

⇒ 2(x + 70°) = 90°

⇒ 2x + 140° = 90°

⇒ 2x = 50°

x = 25°

The angle is 25°

Since 25° is less than 90°, so the type of the angle is an Acute angle.

Are you looking for help on how to convert a decimal fraction to a fraction number? Don’t Fret as you will find easy methods to convert from decimal to fraction here. Before, diving let’s learn about Decimals, Fractions Definitions. To Convert a Decimal to a Fraction place the decimal number over its place value. For better understanding, we even listed Solved Problems on Decimal Fraction to Fraction Number Conversions here. You can easily convert from decimals to fractions and no calculators are needed.

Decimal Definition

Decimal Numbers are the Numbers that have base 10 in Computer Science. However, in Mathematics Decimal Number is a number that has a decimal point in between digits. In Other Words, we can say that decimals are fractions that have denominator 10 or multiples of 10.

Example: 2.35, 6.78, 8.79 are decimals

Fraction Definition

A fraction is a part of a whole number and is represented as a ratio of two numbers a/b in which a, b are integers and b≠0. The two numbers are namely numerator and denominator. There are different types of fractions namely proper, improper, mixed fractions. we can perform all basic operations on the fractions.

Example: \(\frac { 1 }{ 3 } \), \(\frac { 3 }{ 4 } \) are fractions

How to Convert a Decimal to Fraction?

Learn the Steps to Convert Decimal to Fraction here. Follow the below-listed procedure to change between Decimals to Fractions easily. They are in the following fashion

• Firstly, write the fraction with the decimal number as the numerator and with 1 in the denominator.
• Remove the decimal places by multiplication. Firstly, count how many places are there right to the decimal. Let Suppose there are x places then you need to multiply both the numerator and denominator with 10^x
• Reduce the fraction to the lowest form by dividing both the numerator and denominator of the fraction with GCF.

Steps to Convert a Repeating Decimal to Fraction

Converting a regular Decimal to Fraction is an easy method. But, converting a recurring or repeating decimal fraction is a bit tedious and can be confusing. Let us learn how to convert a repeating decimal to a fraction by considering few examples.

Step 1: Let us assume the decimal number as X

Step 2: Count the number of trailing or repeating digits. If there are x digits multiply with 10^x and consider it as the 2nd equation.

Step 3: Subtract Equation (1) from (2) and Solve for X

Step 4: Reduce the obtained fraction to the lowest form by dividing both the numerator and denominator with their GCF. The obtained fraction is the converted value of the repeating decimal given.

Decimal to Fraction Table

Below is the list of decimal values converted to fractions that you might find useful during your calculations. They are in the following fashion

Decimal to Fraction Conversion Examples

1. Convert 2.25 to fraction?

Solution:

Step 1: To change 2.25 to fraction firstly write the numerator part with a decimal number leaving the denominator part with 1.

Step 2: Count the number of decimal places to the right of the decimal point. Since give decimal value has 2 digits next to the decimal point multiply with 10² both the numerator and denominator.

= \(\frac { (2.25*100) }{ (1*100) } \)

= \(\frac { 225 }{ 100 } \)

Step 3: Reduce the obtained fraction in the earlier step to its lowest form by dividing them with GCF. GCF(225, 100) = 25

i.e. \(\frac { 225÷25 }{ 100÷25 } \)

= \(\frac { 9 }{ 4 } \)

Therefore, 2.25 converted to fraction form is \(\frac { 9 }{ 4 } \)

2. Convert 101.1 to fraction?

Solution:

Given Decimal value is 101.1

Step 1: Place the given decimal value in the numerator of the fraction and place 1 in the denominator.

Step 2: Count the number of digits after the decimal point. Since the given decimal value has only 1 digit multiply both the numerator and denominator with 10. i.e. \(\frac { 101.1*10 }{ 1*10 } \) = \(\frac { 1011 }{ 10 } \)

Step 3:  The above fraction can’t be reduced further since the GCF is 1.

Therefore, 2.25 converted to fraction form is \(\frac {1011 }{ 10 } \)

FAQs on Decimal to Fraction

1. What is a Decimal?

Decima Number is defined as a number whose whole number part and fraction part is separated by a decimal point(dot).

2. What are the types of Decimals?

There are two different types of Decimals

• Terminating Decimals or Non-Recurring Decimals
• Non-Terminating or Recurring Decimals

3. What is a Fraction?

A fraction is a numerical value that is a part of a whole. It is evaluated by dividing a whole into a number of parts.

4. How do I Convert a Decimal to a Fraction?

To Convert a Decimal to a Fraction, place the decimal number over its place value.

Conversion Of Minutes Into Seconds:  Wondering how to calculate the conversion of minutes into seconds in a simple way? Then, you have stepped on the correct page. Here, we have curated the definitions, formula, process of converting min to sec with solved problems. Students can easily convert minutes into seconds within no time by referring to this article. Also, have a glance at the Math Conversion Chart for getting knowledge about length, mass, capacity, conversions, etc. Let’s get into this article and ace up your preparation about conversion between min and sec.

What is Minute?

The definition of a minute is a unit of time equal to 1/60 of an hour or 60 seconds. A Minute is separated into seconds and multiplied into hours. The short form of Minutes is ‘min’. For instance, 1 Minute can be written as 1 min.

What is Second?

The second is called the base unit of time. Seconds are classified into milliseconds and multiplied into minutes. Scientists defined one minute to be 60 seconds. The second is the SI base unit for a time in the metric system. The shortened of the is ‘sec’. For example, 1 Second can be written as 1 sec.

Minutes to Seconds Conversion Formula

In terms of math, the conversion formula below is the correct way to calculate minutes to seconds conversion. The simple formula to convert min to sec is as follows:

How to Convert Minutes(min) to Seconds(sec)

To convert minutes unit of time into seconds unit of time, all you need to perform is multiple the time value by 60. Hence, one minute is equal to 60 seconds. Use the simple Minute to Second conversion formula and substitute the given values in the formula and find out the conversion of minutes into seconds easily.

[number of ] mins x 60 = [number of ] secs

To understand the process of converting minute to second, we have listed out some worked-out examples on minutes to seconds conversion in the below module. Have a look at them and practice well to grasp the concept of conversion of min to sec.

Solved Examples of Min to Sec Conversion

1. Convert 7 minutes to seconds using the conversion formula?

Solution:

Given Minutes = 7

Now, convert them into seconds by using the conversion formula,

The conversion formula for minutes to seconds is Seconds = Minutes x 60 s/min

Substitute the minutes into the formula and calculate the seconds,

Seconds = 7 x 60 = 420 sec

Therefore, 7 minutes = 420 seconds. 

2. How many seconds are there in 4 minutes?

Solution:

Since we know that, 1 minute = 60 seconds, we can make use of this information to solve:

4 minutes x 60 seconds / 1 minute = 4 x 60 sec = 240 sec. 

3. Convert 25 minutes 15 seconds into seconds?

Solution:

As we know, 1 minute = 60 seconds

25 minutes 15 seconds = (25 × 60) seconds + 15 seconds

= 1500 seconds + 15 seconds

= 1515 seconds.

For better learnings about time conversions like Conversion Of Hours Into Seconds visit our website thoroughly.

Minute to Second Conversion Table

FAQs on Conversion Of Minutes Into Seconds

1. How many seconds does 1 minute have?

In one minute, there are 60 seconds. In short, a second is 1/60 of a minute.

2. What is 75 minutes as seconds?

75 minutes = 4500 seconds.

3. Why we use minutes and seconds units?

Minutes and Seconds units are used to measure the time.

The Perimeter of a Square is the length that the boundary covers. You can obtain the Perimeter of a Square by adding all the sides together. Refer to the entire article and learn about Perimeter of a Square Definition, Formula, Derivation, Solved Examples, etc.

A Square is a type of rectangle in which the adjacent sides are equal. In other words, we can frame the Definition as Square has all sides of equal length. Refer to the Properties of Square here along with Solved Example Questions in the later sections.

(i) All the angles in a square are the same and equal 90º.

(ii) All the sides of a square are equal.

What is the Perimeter of a Square?

The Perimeter of any closed geometrical shape is defined as the distance around the object. Usually, the Perimeter of a Square is found by summing all four sides together. Since the square has equal sides thus perimeter will be 4 times the side i.e., 4*Side

The Formula for Perimeter of a Square(P) = 4 × Side

Derivation of Perimeter of a Square

The Perimeter of a Square is defined as the length of the boundary of the object.

The Formula to Calculate the Perimeter of a Square = Sum of Lengths of all 4 Sides

Here Length of Each Side = a units

Perimeter = Side +Side + Side+Side

= a+a+a+a

= 4a units

where a, is the length of a side of square.

Solved Perimeter of Square Questions

Question 1.

Find the Perimeter of a Square whose side is 4 cm?

Solution:

We know the formula to calculate Perimeter of a Square = 4*Side

Given Length of the Side = 4 cm

Substitute the Side Length in the formula of Perimeter

Perimeter of a Square = 4*4

= 16 cm

Therefore, the Perimeter of a Square is 16 cm.

Question 2.

Calculate the Perimeter of a Square if its Side is 12cm?

Solution:

The formula for Perimeter of a Square = 4*Side

Given Length of the Side = 12 cm

Substitute the Side Length in the Formula of Perimeter we have

Perimeter of a Square = 4*12 cm

= 48 cm

Therefore, the Perimeter of a Square is 48 cm.

Question 3.

If the Perimeter of a Square is 84 cm find its Side?

Solution:

Perimeter of a Square = 84 cm

The Formula for Perimeter of a Square =4*Side

4*Side = 84 cm

Side = 84/4

= 21 cm

Therefore, the Length of the Side is 21 cm.

FAQs on Perimeter of a Square

1. What is the Perimeter of a Square?

Perimeter of a Square is the total length of the boundary of squares.

2. How to find the Perimeter of a Square?

To find the Perimeter of a Square add all the sides. There are 4 Sides in a Square and Sum of all 4 Sides gives the Perimeter.

3. What is the Formula for Perimeter of a Square?

The Formula for Perimeter of a Square is P = Side+Side+Side+Side i.e. 4*Side

Have a look at the Properties of Perfect Squares and know how to solve problems on Perfect Square concepts. You can easily learn to solve problems on your own once you get a complete grip on the Perfect Square Concept. We have clearly mentioned every property of a perfect square along with examples. Check out the examples for a better understanding. All concepts available on Square are given on our website for free of cost and you can prepare anytime and anywhere.

Different Properties of Perfect Squares

See different properties of perfect squares given below.

Property 1:
Numbers those end with 2, 3, 7, or 8 will never a perfect square. Also, all the numbers ending in 1, 4, 5, 6, 9, 0 are not square numbers.
Examples:
The numbers 20, 32, 73, 167, 298 end in 0, 2, 3, 7, 8 respectively.
So, none of them is a perfect square.

Property 2:
A number that ends with an odd number of zeros is never a perfect square.
Examples:
The numbers 150, 3000, 700000 end in one zero, three zeros, and five zeros respectively.
So, none of them is a perfect square.

Property 3:
The square of an even number is always even.
Examples:
12² = 144, 14² = 196, 16² = 256, 18² = 324, etc.

Property 4:
The square of an odd number is always odd.
Examples:
11² = 121, 13² = 169, 15² = 225, 17² = 289, 19² = 361, etc. All the numbers are odd.

Property 5:
The square of a proper fraction is smaller than the fraction.
Examples:
(3/4)² = (3/4 × 3/4) = 9/16 and 9/16 < 3/4, since (16 × 3) < (9 × 4).

Property 6:
For every natural number n, we have
(n + 1)² – n² = (n + 1 + n)(n + 1 – n) = {(n + 1) + n}.
Therefore, {(n + 1)² – n²} = {(n + 1) + n}.
Examples:
(i) {1 + 3 + 5 + 7 + 9} = sum of first 5 odd numbers = 5²
(ii) {1 + 3 + 5 + 7 + 9 + 11 + 13 + 15} = sum of first 8 odd numbers = 8²

Property 7:
For every natural number n, we can write as the sum of the first n odd numbers = n²
Examples:
(i) {1 + 3 + 5 + 7 + 9} = sum of first 5 odd numbers = 5²
(ii) {1 + 3 + 5 + 7 + 9 + 11 + 13 + 15} = sum of first 8 odd numbers = 8²

Property 8 (Pythagorean Triplets):
Three natural numbers m, n, p is said to form a Pythagorean triplet (m, n, p) if (m² + n²) = p².
Note:
For every natural number m > 1, we have (2m, m² – 1, m² + 1) as a Pythagorean triplet.
Examples:
(i) Putting m = 6 in (2m, m² – 1, m² + 1) we get (12, 35, 37) as a Pythagorean triplet.
(ii) Putting m = 7 in (2m, m² – 1, m² + 1) we get (14, 48, 50) as a Pythagorean triplet.

Solved Examples on Properties of Perfect Squares

1. Without adding, find the sum (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19).

Solution:
(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) = sum of first 10 odd numbers
Find the square of the 10 to get the answer.
= (10)² = 100

100 is the sum of the given numbers 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19.

2. Express 81 as the sum of nine odd numbers.

Solution:
81 = 9² = sum of first nine odd numbers
Let us write the first nine odd numbers and add them naturally.
= (1 + 3 + 5 + 7 + 9 + 11 + 13) = 81.

3. Find the Pythagorean triplet whose smallest member is 4.

Solution:
For every natural number m > 1. (2m, m² – 1, m² + 1) is a Pythagorean triplet.
Putting 2m = 8, i.e., m = 4, we get the triplet (8, 15, 17).

The final answer is (8, 15, 17).

A Perfect Square is formed by squaring a whole number. For example 1² = 1; 2² = 4; 3² = 9; 4² = 16; 5² = 25 and so on. Thus 1, 4, 9, 16, 25, etc., are perfect squares. Learn complete information regarding the perfect square and how to find it in this article. We have given examples and also their explanations to understand it easily. Therefore, it is now your part to begin practice and get a complete grip on the concept.

Examples:
1 = 1²; 4 = 2²; 9 = 3²; 16 = 4²; 25 = 5² and so on. Here 1, 4, 9, 16, 25, etc., are perfect squares.

How to Find a Perfect Square or Square Number?

A perfect square number is defined as the product of pairs of equal factors. Or it can also express as grouped in pairs of equal factors.

1. Find out if the following numbers are perfect squares?
(i) 169
(ii) 512
(iii) 64

Solution:
(i) Given number is 169
Find the prime factors of the given number 169.
The prime factors of 169 are 13 and 13.
Grouping the factors into the pairs of equal factors.
(13 × 13)
Factors of the 169 are 13 × 13.

Therefore, 169 is a perfect square.

(ii) Given number is 512
Find the prime factors of the given number 512.
The prime factors of 169 are 8, 8, and 8.
Grouping the factors into the pairs of equal factors.
(8 × 8) × 8
Factors of the 169 are 8 × 8 × 8.
8 is not grouped in pairs of equal factors.

Therefore, 512 is not a perfect square.

(iii) Given number is 64
Find the prime factors of the given number 64.
The prime factors of 64 are 8, and 8.
Grouping the factors into the pairs of equal factors.
(8 × 8)
Factors of the 169 are 8 × 8.

Therefore, 64 is a perfect square.

2. Is 16 a perfect square? If so, find the number whose square is 16.

Solution:
Given number is 16
Find the prime factors of the given number 16.
The prime factors of 16 are 2, 2, 2, and 2.
Grouping the factors into the pairs of equal factors.
(2 × 2) ×(2 × 2)
Therefore, 16 is a perfect square.
Take one number from each group and multiply them to find the number whose square is 16.
2 × 2 = 4.

4 is the number whose square is 16.

3. Is 576 a perfect square? If so, find the number whose square is 576.

Solution:
Given number is 576
Find the prime factors of the given number 16.
The prime factors of 16 are 2, 2, 3, 3, 4, and 4.
Grouping the factors into the pairs of equal factors.
(2 × 2) × (3 × 3) × (4 × 4)
Therefore, 576 is a perfect square.
Take one number from each group and multiply them to find the number whose square is 576.
2 × 3 × 4 = 24.

24 is the number whose square is 576.

4. Show that 288 is not a perfect square.

Solution:
Given number is 288
Find the prime factors of the given number 288.
The prime factors of 16 are 2, 3, 3, 4, and 4.
Grouping the factors into the pairs of equal factors.
2 × (3 × 3) × (4 × 4)
2 is not grouped in pairs of equal factors.

Therefore, 288 is not a perfect square.

5. Find the smallest number by which 100 must be multiplied to make it a perfect square?

Solution:
The given number is 100.
Find the prime factors of the given number 100.
The prime factors of 100 are 5, 5, and 4.
Grouping the factors into the pairs of equal factors.
4 × (5 × 5)
4 is not grouped in pairs of equal factors.
Therefore, by multiplying 4 to 100, we make 100 as a perfect square.

4 is the smallest number by which 100 must be multiplied to make it a perfect square

6. Find the smallest number by which 180 must be divided so as to get a perfect square.

Solution:
The given number is 180.
Find the prime factors of the given number 180.
The prime factors of 180 are 2, 2, 3, 3, and 5.
Grouping the factors into the pairs of equal factors.
5 × (3 × 3) × (2 × 2)
5 is not grouped in pairs of equal factors.
Therefore, by dividing 5 by 180, we make 180 a perfect square.

5 is the smallest number by which 180 must be divided so as to get a perfect square.

A square of a number is calculated by multiplying a number by itself twice. Geometrically, a square is a two-dimensional plane that has equal sides.
Area of a square = Side × Side
Square number = a × a = a²

1, 4, 9, 16, 25, 36, 49, 64, etc. are some of the examples of the number for a square of a number. If S is a number that formed by multiplying a by two times, then S is called the square of a number. For example, 16 is a number then it can write as 4 . 4 where 4 is the natural number and 16 is the square of a number. Also, 42 is a number and it is the multiplication of 7 and 6. However, 42 is not considered as a square of a number. Square numbers are also treated as perfect square numbers.

List of Square Concepts

We have given a list of Square Concepts and their concerned links for you. Click on the required link and learn the entire topic easily.

• Perfect Square or Square Number
• Properties of Perfect Squares

Solved Examples on Square of a Number

Check the below examples to understand which numbers are called squares of a number.

• 2² = (2 × 2) = 4. Therefore, we can say that the square of 2 is 4.
• 3² = (3 × 3) = 9. Therefore, we can say that the square of 3 is 9.
• 4² = (4 × 4) = 16. Therefore, we can say that the square of 4 is 16.
• 5² = (5 × 5) = 25. Therefore, we can say that the square of 5 is 25.
• 6² = (6 × 6) = 36. Therefore, we can say that the square of 6 is 36.
• 7² = (7 × 7) = 49. Therefore, we can say that the square of 7 is 49.
• 8² = (8 × 8) = 64. Therefore, we can say that the square of 8 is 64.
• 9² = (9 × 9) = 81. Therefore, we can say that the square of 9 is 81.
• 10² = (10 × 10) = 100. Therefore, we can say that the square of 10 is 100.
• 11² = (11 × 11) = 121. Therefore, we can say that the square of 11 is 121.
• 12² = (12 × 12) = 144. Therefore, we can say that the square of 12 is 144.
• 13² = (13 × 13) = 169. Therefore, we can say that the square of 13 is 169.
• 14² = (14 × 14) = 196. Therefore, we can say that the square of 14 is 196.
• 15² = (15 × 15) = 225. Therefore, we can say that the square of 15 is 225.

Square of a Negative Number

The square of a negative number always a positive number.

• (-2)² = ((-2) × (-2)) = 4. Therefore, we can say that the square of (-2) is 4.
• (-3)² = ((-3) × (-3)) = 9. Therefore, we can say that the square of (-3) is 9.
• (-4)² = ((-4) × (-4)) = 16. Therefore, we can say that the square of (-4) is 16.
• (-5)² = ((-5) × (-5)) = 25. Therefore, we can say that the square of (-5) is 25.
• (-6)² = ((-6) × (-6)) = 36. Therefore, we can say that the square of (-6) is 36.
• (-7)² = ((-7) × (-7)) = 49. Therefore, we can say that the square of (-7) is 49.
• (-8)² = ((-8) × (-8)) = 64. Therefore, we can say that the square of (-8) is 64.
• (-9)² = ((-9) × (-9)) = 81. Therefore, we can say that the square of (-9) is 81.
• (-10)² = ((-10) × (-10)) = 100. Therefore, we can say that the square of (-10) is 100.
• (-11)² = ((-11) × (-11)) = 121. Therefore, we can say that the square of (-11) is 121.
• (-12)² = ((-12) × (-12)) = 144. Therefore, we can say that the square of (-12) is 144.
• (-13)² = ((-13) × (-13)) = 169. Therefore, we can say that the square of (-13) is 169.
• (-14)² = ((-14) × (-14)) = 196. Therefore, we can say that the square of (-14) is 196.
• (-15)² = ((-15) × (-15)) = 225. Therefore, we can say that the square of (-15) is 225.

What is the Square of a number?

A number is multiplied by itself to form a square of a number. Thus, the number with exponent 2 is called the square number.
Example:
\(\frac { 3 }{ 7 } \) × \(\frac { 3 }{ 7 } \) = (\(\frac { 3 }{ 7 } \))² = \(\frac { 9 }{ 49 } \)
Here \(\frac { 9 }{ 49 } \) is the square of \(\frac { 3 }{ 7 } \).
0.2 × 0.2 = (0.2)² = 0.04
Here 0.04 is the square of 0.2.

Odd and Even Square numbers

• Square of an even number is always even, i.e, (2n)² = 4n².
• Square of an odd numbers is always odd, i.e, (2n + 1) = 4(n² + n) + 1.
• Since every odd square is of the form 4n + 1, the odd numbers that are of the form 4n + 3 are not square numbers.

Properties of Square Numbers

Check out the properties of Square Numbers given below to completely understand the Square concept.

1. If the numbers 2, 3, 7, or 8 present in the unit’s place, then the number will not become a perfect square. Therefore, the numbers that end with 2, 3, 7, or 8 will never become a perfect square.
2. The number ends with even zeros becomes perfect squares. Also, the numbers with an odd number of zeros will never become a perfect square.
3. Square of even numbers always an even number and square of odd numbers always an odd number.
4. If the natural numbers that are more than one are squared, then it should be either of multiple of 3 or more than the multiple of 3 by 1.
5. Also, if the natural numbers that are more than one are squared, then it should be either of multiple of 4 or more than the multiple of 4 by 1.
6. If the unit’s digit of the square of a number is equal to the unit’s digit of the square of the digit at the unit’s place of the given natural number.
7. If there are n natural numbers, say x and y such that x² = 2y².
8. For every natural number n, we can write it as (n + 1)² – n² = ( n + 1) + n.
9. For any natural number, say”n” which is greater than 1, we can say that (2n, n² – 1, n²+ 1) should be a Pythagorean triplet.
10. If a number n is squared, it equals the sum of first n odd natural numbers.

Example:

Simply the equation by adding \(\frac { 3 }{ 8 } \) and \(\frac { 5 }{ 12 } \)?

Solution:

As given in the question, \(\frac { 3 }{ 8 } \) + \(\frac { 5 }{ 12 } \) are the fractions.

Now find the LCM of 8 and 12, we get

LCM of (8, 12) = 2 * 2 * 2 * 3 = 24

Now multiply the fractions to get the denominator values equal to 24, such that

= \(\frac { (3 * 3) }{ (8 * 3) } \) + \(\frac { (5 * 2) }{(12 * 2) } \)

= \(\frac { 9 }{ 24 } \) + \(\frac { 10 }{ 24 } \)

= \(\frac { 19 }{ 24 } \)

Confused between various fraction types? If yes, then check here for the important type of fraction i.e., equivalent fraction. Definition of Equivalent Fractions is here. Check rules, methods, and formulae of Equivalent Fractions. Refer step by step procedure to know the problems of equal fractions. Follow the important points and example problems to know in-depth of equivalent problems. Check the below sections to know the detailed description of equivalent fractions and their rules.

What are Equivalent Fractions?

Equivalent Fractions or equal fractions are the fractions that have different numerators and denominators but gives the same value. For example, the value of both the fractions \(\frac { 4 }{ 8 } \) and \(\frac { 3 }{ 6} \) is equal to \(\frac { 1 }{ 2 } \). Hence, both the values are the same they are equivalent in nature. This equivalent fraction represents a similar proportion of the whole.

To define the equivalent fractions, suppose that \(\frac { a }{ b } \) and \(\frac { c }{ d } \) are 2 fractions. After simplification of the given fractions, both results in equal fractions suppose e/f which are equal to each other.

Why do fractions have the same values in spite of having a different number?

For the above question, the answer is the denominator and numerator are not co-prime numbers. This fraction has a common multiple that gives the same value therefore they have a common multiple, which on division gives exactly the same value.

Example:

= \(\frac { 1 }{ 2 } \)

= \(\frac { 4 }{ 8 } \)

In the above given example, it is clearly shown that the fractions have different denominators and numerators.

Dividing both denominator and numerator by their common factor, we have:

= \(\frac { 2 }{ 2 } \) ÷ \(\frac { 4 }{ 2 } \)

= \(\frac { 1 }{ 2 } \)

In a similar way, on simplifying \(\frac { 4 }{ 8 } \) we get

= \(\frac { 4 }{ 4} \) ÷ \(\frac { 8 }{ 4 } \)

= \(\frac { 1 }{ 2 } \)

Therefore both the fractions have an equal value \(\frac { 1 }{ 2 } \)

How to Find Equal Fractions?

Equal fractions are actually similar because when we divide or multiply both the denominator and numerator by the same number, the fraction value doesn’t change. On simplifying the value of the equivalent fractions, the value will be the same.

Example:

Simplify the fraction \(\frac { 1 }{ 5 } \)

On multiplying denominator and numerator with 2, the result will be = \(\frac { 2 }{ 10 } \)

On multiplying denominator and numerator with 3, the result will be = \(\frac { 3 }{ 15 } \)

Multiplying denominator and numerator with 4, the result will be = \(\frac { 4 }{ 20 } \)

Thus, we can conclude from the above simplification as,

\(\frac { 1 }{ 5 } \) = \(\frac { 2 }{ 10 } \) = \(\frac { 3}{ 15 } \) = \(\frac { 4 }{ 20 } \)

We can only divide or multiply by similar numbers to get an equal or equivalent fraction and not subtraction or addition. Simplification has to be done where both the denominator and numerator should be whole numbers.

Key Points to Remember

• Equal Fractions or equivalent fractions will look different, but they both have the same values.
• You can easily divide or multiply to find an equivalent fraction.
• The functions of addition and subtraction do not work for similar fractions or equivalent fractions.
• If you divide or multiply with the top part of the fraction, you must also do the same for the denominator.
• Use the rule of cross multiplication, to determine if both the fractions are equivalent.

How to Determine Whether Two Fractions are Equivalent?

If you are seeking help on the concept of Ratio and Proportion you can always make use of Worked out Problems on Ratio and Proportion. All the Problems are explained with straightforward description making it easy for you to understand the concept. Solve different problems on Ratio and Proportion available here firstly on your own and cross-check your solutions.

You can find Ratio and Proportion Questions related to Simplification of Ratios, Comparison of Ratios, Arranging Ratios in Ascending Order, Descending Order, Word Problems on Ratio and Proportion, etc. Sample Problems on Ratio and Proportion will help you get a good grip on the concept and its fundamentals too in no time.

Ratio and Proportion Problems and Solutions

1. Two numbers are in the ratio 4 : 5. If the sum of numbers is 72, find the numbers?

Solution:

Let the numbers be 4x and 5x

Since Sum of the Numbers is 72 we have the equation as such

4x+5x = 72

9x= 72

x = 72/9

= 8

Substitute the value of x to obtain the numbers

4x = 4*8  32

5x = 5*8 = 40

Therefore, the Numbers are 32 and 40.

2. If x : y = 3 : 2, find the value of (2x + 4y) : (x + 5y)?

Solution:

We know x:y = 3:2

we can rewrite it as

x/y = 3/2

Given equation (2x + 4y) : (x + 5y)

we can rewrite it as

(2x + 4y)/(x + 5y)

Dividing Numerator and Denominator with y we have the equation as follows

= (2(x/y)+4(y/y))/((x/y)+5(y/y))

Since we know the value of x/y substitute it in the above equation

= (2(1/2)+4(1))/((1/2)+5(1))

= (1+4)/(1/2+5)

= 5/(11/2)

= 10/11

Therefore, value of (2x + 4y) : (x + 5y) is 10/11.

3. The average age of three boys is 36 years and their ages are in the proportion 5 : 6 : 7. Find the age of the youngest boy?

Solution:

From the ratio 5:6:7, the ages of boys are 5x, 6x, 7x

Given Average Age of Boys = 36

5x+6x+7x = 36

18x = 36

x = 2

Age of Youngest Boy = 5x

= 5*2

= 10 years

Therefore, the Age of the Youngest Boy is 10 Years.

4. If 3A = 4B = 5C, find A : B : C?

Solution:

Let us assume a constant k

3A=4B=5C = k

equating them we have

3A= k, 4B = k, 5C = k

A = k/3, B = k/4, C = k/5…….(1)

Finding LCM for the obtained values 3, 4, 5

LCM(3, 4, 5) = 60

Multiplying with 60 the eqn (1) we get the Ratio as Follows

Ratio of A:B:C is 20:15:12

5. What must be added to each term of the ratio 3 : 2, so that it may become equal to 5 : 4?

Solution:

Let the Number to be added be x then (3+x):(2+x) = 5:4

(3+x)/(2+x) = 5/4

(3+x)4 = 5(2+x)

12+4x= 10+5x

12-10 = 5x-4x

x =2

To make the ratio 3:2 to 4:5 you need to add 2.

6. The length of the ribbon was originally 33 cm. It was reduced in the ratio 3:2. What is its length now?

Solution:

Length of Ribbon = 33 cm

Let the Original Length be 3x

Reduced Length be 2x

But 3x = 33 cm

x = 33 cm/3

= 11 cm

Reduced Length = 2x

= 2*11

= 22 cm

Therefore, the Length of the Ribbon is 22 Cm.

7. The ratio of the number of boys and girls is 5 : 3. If there are 15 girls in a class, find the number of boys in the class and the total number of students in the class?

Solution:

Given Ratio of Boys to Girls is 5:3

There are 15 Girls in the Class

Boys/Girls = 5/3

Boys/15 = 5/3

Boys = (5*15)/3

= 25

Number of Students in Class = Boys +Girls

= 25+15

= 40

Therefore, there are 25 Boys and 40 Students in the Class.

8. Find the third proportional of 10 and 20?

Solution:

Let us consider the Third Proportional of 10 and 20 be x

10, 20, and x are in Proportion

10:20 = 20:x

Product of Means = Product of Extremes

20*20 = 10*x

400 =10x

x = 400/10

= 40

Third Proportional of 10, 20 is 40

9. The first, second, and third terms of the proportion are 40, 36, 35. Find the fourth term?

Solution:

Let us consider the fourth term be x

40, 36, 35, x

Product of Means = Product of Extremes

36*35 = 40*x

x = (36*35)/40

= 31.5

Fourth Proportional of 40, 36, 35 is 31.5

10. Arrange the following ratios in Ascending Order

3:2, 4:3, 5 : 6, 1 : 4

Solution:

Given Ratios are 3/2, 4/3, 5/6 and 1/4

Finding the LCM of 2, 3, 6, 4 we get 12

Express the given ratios in terms of common denominator we get

3/2 = (3*6/2*6) = 18/12

4/3 = (4*4/3*4) = 16/12

5/6 = (5*2/6*2) = 10/12

1/4 = (1*3/4*3) = 3/12

Clearly, 3/12<10/12<16/12<18/12

Therefore, 1:4 <5:6<4:3<3:2

Practice Test on Ratio and Proportion helps students to get knowledge on different levels. The Ratio and Proportion Questions and Answers provided range from beginner, medium, hard levels. Practice the Questions here and get to know how to solve different problems asked. All the Ratio and Proportion Word Problems covered are as per the latest syllabus. Master the topic of Ratio and Proportion by practicing the Problems on a consistent basis and score better grades in your exam.

Ratio and Proportion Questions and Answers

1. The ratio of monthly income to the savings in a family is 5 : 3 If the savings be $6000, find the income and the expenses?

Solution:

Let us assume the Income be 5x

whereas savings be 3x

Given Savings = $6000

3x = 6000

x = 6000/3

= 2000

Income = 5x

= 5*2000

= $10,000

Expenses = Income – Savings

= 5x – 3x

= 2x

= 2*2000

= $4000

Therefore, Income and Expenses are $10,000 and $ 4000.

2. Two numbers are in the ratio 7: 4. If 3 is subtracted from each of them, the ratio becomes 5 : 2. Find the numbers?

Solution:

Let us consider the number be x

so 7x:4x

If 3 is subtracted the ratio becomes 5:2 then we have

7x-3:4x-3 = 5:2

equating them ad solving we get the values as

7x-3/4x-3 = 5/2

(7x-3)2 = 5(4x-3)

14x-6 = 20x-15

-6+15 = 20x-14x

9 = 6x

6x = 9

x = 9/6

= 3/2

Therefore the numbers are 7(3/2) and 4(3/2)

= 21/2, 6

3. Two numbers are in the ratio 3 : 5. If their sum is 720, find the numbers?

Solution:

Let us consider the number be x

Therefore two numbers become 3x:5x

Since their Sum = 720

3x+5x = 720

8x = 720

x = 720/8

= 90

Numbers are 3x and 4x

Thus, they become 3(90) and 4(90) i.e. 270 and 360.

4. A sum of money is divided among Rohan and Anand in the ratio 4 : 6. If Anand’s share is $600, find the total money?

Solution:

Let the money be x

Rohan and Anand’s Share = 4x:6x

Anand’s Share = $600

6x = $600

x = $100

Rohan’s Share = 4x

= 4*100

= $400

Total Money = Rohan’s Share + Anand’s Share

= $400+$600

= $1000

Therefore, the Sum of Money is $1000

5. The difference between the two numbers is 33 and the ratio between them is 5 : 2. Find the numbers?

Solution:

Let the number be x

From the given data

we have 5x-2x = 33

3x = 33

x = 11

Numbers are 5x, 2x

thus, they become 5*11 and 2*11

= 55, 22

Therefore, the numbers are 55 and 22.

6.  The ages of A and B are in the ratio 3 : 6. Four years later, the sum of their ages is 53. Find their present ages?

Solution:

Let the Present Ages be 3x and 6x

After four Years Age of A and B Becomes 3x+4 and 6x+4

We know sum of their ages after 4 years = 53

3x+4+6x+4 = 53

9x+8 = 53

9x = 53-8

9x = 45

x =5

Present Ages of A and B is 3x and 6x

thus 3*5 and 6*5 i.e. 15 and 30

Therefore, the Present Ages of A and B are 15 and 30.

7. If 3A = 4B = 5C, find the ratio of A : B : C?

Solution:

Let us assume that 3A = 4B = 5C = k

Equating them we have A = k/3, B = k/4, C = k/5

Therefore, Ratio becomes = k/3:k/4:k/5

LCM of 3, 4, 5 is 60

Thus expressing them in terms of least common multiple we have

A:B:C = 20:15:12

Therefore, Ratio of A:B:C is 20:15:12

8. A certain sum of money is divided among a, b, c in the ratio 3:4:5. of a share is $300, find the share of b and c?

Solution:

Let us consider the sum of money as x

Since it is shared among the ratio of 3:4:5 we have 3x:4x:5x

We know a’s share is 3x = $300

x =$100

Share of b = 4x

= 4*100

= $400

Share of C = 5x

= 5*100

= $500

9. Divide $900 among A, B, C in the ratio 3: 4 ∶ 5?

Solution:

Let us assume the total money as x

Since the sum is to be shared among A, B, C in the ratio of 3:4:5 we have

3x+4x+5x = $900

12x = $900

x = $900/12

=$75

Share of A = 3x

= 3*75

= $225

Share of B = 4x

= 4*75

= $300

Share of C = 5x

= 5*75

= $375

10. Find the first term, if second, third, and fourth terms are 21, 80, 120?

Solution:

Let the Terms be a, a+d, a+2d, a+3d

Given Second Term = 21

a+d = 21

Third Term = 80

a+2d = 80

Fourth Term = 120

a+3d = 120

Using the Eliminating Method

a+d = 21

a+2d = 80

_______

Subtracting them we get the value of d as

-d = -59

d= 59

Substitute the value of d in any of the terms

a+d = 21

a+59 =21

a =21-59

= -38

The word problems on profit and loss are solved here to get the basic idea of how to use the formulae of profit and loss in terms of cost price and selling price. We have explained the entire concept of profit and loss and various formulae when C.P, S.P, Profit %, or Loss % is given. Solve Different Questions on Profit and Loss available here to test your grip on the fundamentals of the concept.

Profit or Gain

If the selling price of an item is more than the cost price of the same item, then it is said to be gain (or) profit i.e. S.P. > C.P.

Net profit= S.P. – C.P.

Loss

If the selling price of an item is less than the cost price of the same item, then it is said to be a loss i.e. S.P.  < C.P.

Net loss = C.P. – S.P.

Profit and Loss Word Problems with Answers

Question 1:

A laptop was brought for $ 80,000 and sold at a loss of $ 5000. Find the selling price.

Solution:

Given data:

The cost price of the laptop is $ 80,000

Loss = $ 5000

We know that,

Loss = C.P. – S.P.

$ 5000 =$ 80,000 – S.P.

S.P. = $ 80,000 – $ 5000

S.P. = $ 75,000

Therefore, the selling price of the laptop is $ 75,000.

Question 2:

Abhi sold his water purifier for $ 4000, at a loss of $ 300. Find the cost price of the water purifier.

Solution:

Given Data:

The selling price of water purifier = $ 4000

Loss = $ 300

We know that, Loss = C.P. – S.P.

From this, we can note that,

Cost price = loss + selling price

= $ 300 + $ 4000

= $ 4300

Hence, the cost price of a water purifier is $ 4300.

Question 3:

Deepika sold her gold necklace for $ 60,000 at a profit of $ 10,000. Find the cost price of the gold necklace.

Solution:

Given data

The selling price of gold necklace = $ 60,000

Gained a profit = $ 10,000

From the formula

Gain = Selling price (S.P.) – Cost price (C.P.)

We get,

Cost price (C.P.) = Selling price (S.P.) – Gain

= $ 60,000 – $ 10,000

= $ 50,000.

Hence, the cost price (C.P.) of the gold necklace is $ 50,000.

Question 4:

Karthik buys a watch for $ 6000 and sells it at a gain of 5⅓ %. For how much does he sell it?

Solution:

Given Data:

Cost price (C.P.) of watch = $ 6000

Gain = 5⅓% = 16/3 %

We know that

Gain% = ((S.P. – C.P.)/C.P. *100) %

From above,

S.P. = [{(100 + gain %) /100) * C.P.]

= $ [{(100 + 16/3)/100} * 6000]

= ${(103.33/100) * 6000]

= $ 6199.8

Hence, karthik sells his watch at an amount of $ 6199.8.

Question 5:

Siva ram bought an old bike for $ 15000 and spends $ 2000 on repairs. If he sells the bike for $ 21150, what is his gain percentage?

Solution:

Given data:

Cost price (C.P.) of bike = $ 15000

Repair cost = $ 2000

Total Cost Price = Original Price of the Bike + Repair Cost

= $ 15000+$2000

= $17000

Selling price (S.P.) of bike = $ 21150

As the selling price (S.P.) is more than the cost price (C.P.) of the bike then it is said to be in gain

Therefore, Gain = Selling price (S.P.) – Cost Price (C.P.)

= $ 21150 – $ 17000

= $ 4150.

Gain % = ((S.P. – C.P.)/C.P. *100) %

= $ (4150/17000 * 100) %

= 24. 41%

Hence, He got a 24.41% gain on his bike.

Question 6:

If the selling price of an object is doubled, the profit of the object triples. Find the profit percentage.

Solution:

Given data:

Let the cost price of the object be $ ‘a’

And selling price of the object be $ ‘b’

According to the question, the profit is tripled and selling price is doubled hence

Profit = $ 3(b – a)

Profit = S.P. – C.P.

3(b – a) = 2b –a

3b – 3a = 2b – a

Therefore, b = 2a.

Profit =$ b – a

= 2a –a

= $ a.

Profit% = ((S.P. – C.P.)/C.P. *100) %

= (a/a * 100) %

= 100%.

Question 7:

The percentage profit earned by selling an article for $ 3000 is equal to the percentage loss incurred by selling the same article for $ 2500. At what price should the article be sold to make a 20% profit?

Solution:

The above question says that the % profit earned by selling the article is equal to the % loss incurred

by the same article.

Given data:

Let the C.P. of the article be ‘P’

We know that

Profit % = ((S.P. – C.P.)/C.P. *100) %

And loss % = ((C.P. – S.P.)/C.P. *100) %

((3000 – p)/p * 100) = ((p – 2500)/p * 100)

2p = 7500

P = $ 3750.

Calculating selling price at a profit of 20%

Profit % = ((S.P. – C.P.)/C.P. *100) %

From above,

S.P. = $ [{(100 + gain %) /100) * C.P.]

= $ ((100 + 20)/100) * 3750)

= $ 4500.

Question 8.

If mangoes are bought at prices ranging from $ 300 to $ 450 are sold at prices ranging from $ 400 to $ 525, what is the greatest possible profit that might be made in selling ten mangoes?

Solution:

The question says that the mangoes are bought at a certain range and sold at a certain range. It says to find the greatest profit on selling ten mangoes.

Given data:

Cost price (C.P.) of mangoes ranging from = $ 300 – $ 450

Selling price (S.P.) of mangoes ranging from = $ 400 – $ 525

Considering,

Least cost price (C.P.) for ten mangoes = $ 300*10

= $ 3000

Greatest selling price (S.P.) of mangoes = $ 525*10

= $ 5250

Profit = S.P. – C.P.

= $ 5250 – $ 3000

= $ 2250.

Hence, the required profit obtained is $ 2250 on selling mangoes.

Question 9:

On selling 18 toys at $ 800, there is a loss equal to the cost price of 7 toys. The cost price of the toys is?

Solution:

Given data:

Let the cost price (C.P.) of the toys be ‘m’

Given, on selling 18 toys at $ 800, there is a loss equal to the cost price of 7 toys

According to the question, the equation is written as:

18m – 800 = 7m

Solving the above equation

We get m = $ 32

Therefore, the cost price of the toy is $ 32.

Before, going for examples on simplification, know the rules and methods. Check all the best possible methods to solve the simplification of expressions. Refer to all the formulae, rules, and methods for the simplification of integers. You can go through the entire article to know more on what are the rules to be followed while simplifying expressions. Practice the Simplification Questions available and cross-check your answers here to know where you stand in your preparation level.

Rules of Simplification

While solving the Examples on Simplification keep the below pointers in mind. They are as follows

1. Remove Brackets – If two or more signs occur in the expression, then convert them into one. The brackets will be square, round and curly braces.
2. Group the values into one group i.e., either positive or negative.
3. Whenever there are two integers, the result will give the sign of the greater value.

Examples on Simplification

Question 1.

At 135 feet below sea level, a submarine has started. It dives 239 feet before rising 307 feet. Find the exact depth of the submarine at which it is spent currently?

Solution:

As per the given question,

At 135 feet below sea level, a submarine has started

The level at which it dives = 239 feet

The rise of the sea level =307 feet

To find, the current depth of the submarine, we write the equation as

-135+(-239)+307

=-374+307

=-67

Therefore, the current depth of the submarine is 67 feet below sea level.

The final solution is -67 feet.

Question 2:

Jenny purchases a credit card from a local retailer. She begins with a $200 balance. She then makes the following purchases: lamp $8, rug $63, vacuum $39. After this shopping trip, she loads $147 on her card, then spends $113 on groceries. Express each transaction as an integer, then determine the new balance on the prepaid credit card?

Solution:

As per the given question,

Jenny begins with a balance = $200

She purchases lamp = $8

She purchases rug = $63

She purchases vacuum = $39

She loads on her card = $133

As she has the balance of $200, it is positive, and also she loads $147 to her card, therefore it is also positive. The remaining values are negative.

The new balance on the prepaid credit card = 200 +(-8) + (-63) + (-39) + 147 + (-113)

= 347 + (-223)

=124

Therefore, the new balance on the prepaid credit card = $124

Hence, the final solution = $124

Question 3:

Simplify: 37 – [5 + {28 – (19 – 7)}]

Solution:

Step 1: Innermost grouping symbols removal is the first step to simplify the expression. Consider {28-(19-7)}, to remove the brackets we subtract the equation {28-(19-7)}, here comes as {28-12}
The simplified expression is 37 – [5 + {28 – 12}]
Step 2: Follow the same procedure to remove parentheses in {28-12}, we have to subtract the equation {28-12}, here comes the final equation as
37 – [5 + 16]
Step 3: Now, the equation is further simplified and it contains only square brackets, we should perform all the set of operations within two brackets.
37 – 21
Step 4: In the final step, we subtract the values and the final result will be 16.
Thus the final solution will be 16.

Question 4:

Simplify the equation 15 – (-5) {4 – 7 – 3} ÷ [3{5 + (-3) x (-6)}]

Solution:

Step 1: Innermost grouping symbols removal is the first step to simplify the expression. Consider {5+(-3)x(-6)}, to remove the brackets we multiply the equation {5+18}, here comes the final expression as

15 – (-5) {4 – 7 – 3} ÷[3 {5 + 18}]

Step 2: Follow the same procedure to remove parentheses in {4-7-3}, we have multiplied the equation {4-7-3}, here comes the final result as 4-4

After performing 2 steps, the result equation will be

15 – (-5) x 0 ÷[3 {5 + 18}]

Step 3: Next simplification must be the addition of 5 and 18. The result will be

15 – (-5) x 0 ÷ 3 x 23

Step 4: As the brackets of two sets are removed. Therefore, change the negative signs and rewrite the equation.

15 + (5 x 0) ÷ 3 x 23

Step 5: As all other brackets are removed and the expression contains only and braces. Perform all the operations that are possible within the brackets.

= 15 – (-5) x 0 ÷ 69

Step 6: On further simplification, the result will be

15 – (-5) x 0

Step 7:

Therefore, the final solution is 15

Question 5:

The temperature of the fridge compartment is set at 8 degrees C. The freezer compartment is set at -10 degrees C. What is the difference between the temperature settings?

Solution:

As per the given question,

The temperature of fridge compartment = 8 degrees C

Temperature of freezer compartment = -10 degree C

Difference between temperature = 8-(-10)

=8+10

=18 degrees C

Question 6:

Rekha climbs up 5 stairs every second and then climbs down 2 stairs over the next second. How many seconds will she take to climb 60 steps?

Solution:

As per the given question,

Stairs climbed up by Rekha = 5 in 1 sec

Stairs climbed down by Rekha = 2 in 1 sec

Climbing down is considered as a decrease in value, hence it is negative.

Therefore, stairs climbed = 5+(-2) in 2 secs

Steps climbed = 3 stairs in 2 secs

Thus, the time to climb 1 step in time = 2/3 secs

Thus, the time taken to climb 60 steps = 2 x 60/3

= 40sec

Hence, the final solution is 40 sec.

Question 7:

I start with integer (-8), Add (-12) to it, subtract 10 from the result. Divide the result by (+3) and multiply the answer by (-2). What do you get?

Solution:

Step 1: As given in the question, the integer value is -8

Step 2: Adding -12 to the integer value

i.e., -8-12=-20

Step 3: Subtracting 10 from the value = -20-(10) = -30

Step 4: Dividing by +3

=(-30)/(3) = -10

Step 5: Multiply the answer by -2

(-10) x (-2) = 20

Therefore, the final solution is 20

Question 8:

Arnav has $20. He spent $8 on Monday. He got $5 as pocket money on Tuesday. He gave a $7 loan to his friend on Wednesday. He ate ice cream for $10 on Thursday. He received a reward of $5 on Friday. He repaid the loan of $7 on Saturday. How much money did Arnav totally have on Sunday?

Solution:

As per the given question,

Arnav has $20

He spent on Monday = $8

He got pocket money on Tuesday = $5

He gave a loan to a friend on Wednesday = $7

He ate ice cream on Thursday = $10

Received a reward on Friday = $5

His friend repay the loan on Saturday = $7

Money Arnav has on Sunday = 20+(-8)+5(-7)+(-10)+5+7

=20+5+5+7+(-8)+(-7)+(-10)

=37-8-7-10

=37-25

=12

Therefore, the money Arnav has on Sunday = $12

Question 9:

A Hiker is descending 152m in 8 minutes. What will be his elevation in half an hour?

Solution:

As given in the question,

In 8 mins Hiker descends 152cm.

In 1 min Hiker descends = 152/8 = 19m

Therefore, In 30min Hiker descends = 19 x 30 = 570m

Hence, the final solution is 570m

Question 10:

In an exam, the student gets +4 for each correct answer and -2 for each wrong answer. Rohith’s final score is 68 marks and he attempted 25 questions correctly. How many marks did he lose for wrong answers?

Solution:

Rohith’s final score = 68 marks

Rohith attempted correctly = 25 questions

Let the number of wrong answers be a

Marks for correct answers = 25 x 4 = 100

Marks for wrong answers = a x (-2) = -2a

Total wrong answers = 100 +(-2a) = 68

100-2a=68

-2a=68-100

-2a=-32

a=32/2

a=16

Therefore, Rohith lost 16 marks for wrong answers.

Removal of brackets involves step by step process. We are providing a detailed procedure and various rules to follow while removing the brackets. Also, check the various methods to remove different types of brackets. Rewrite the expression by simplifying it with the help of the brackets removal function. Know the process of writing in equivalent forms by removing the brackets. Refer to the definitions of “expanding” or “removing” brackets.

How to Remove Brackets from an Equation?

There are various steps involved in bracket removal. In the further sections, we will see how the equations can be written in equivalent forms by removing the braces. This process is called “removing” or “expanding” brackets.

The actual procedure to remove the brackets is to multiply the term which is outside the brackets with the term which is inside the brackets. This is also called the law of distribution.

Steps to Brackets Removal

In Order to take off brackets, you need to follow the step by step guidelines listed below and they are as follows

Step 1:

Check for the given question or the expression whether it contains a vinculum(The horizontal line which is used in the mathematical notation for the desired purpose) or not. If there is a vinculum, then perform the operations in it. Or else go to step 2

Step 2:

Now, go for the equation and check for the innermost bracket and then perform the operation within that bracket.

Step 3:

To perform step 3, there are various rules to be followed.

Rule 1: If the equation is preceded by addition or plus sign, then remove it by writing the terms as mentioned in the equation.

Rule 2: If the equation is preceded by a negative or minus sign, then change the negative signs within it to positive or vice versa.

Rule 3: The indication of multiplication is there will be no sign between a grouping symbol and a number.

Rule 4: If there is any number before the brackets, then we multiply that number inside the brackets with the number which is outside the brackets.

Step 4:

Check for the next innermost bracket and perform the calculations or operations within it. Remove the next innermost bracket by using Step III rules. Follow this process until all the brackets are removed.

Removing Brackets by FOIL Method

Brackets can be easily removed and rules can be remembered by using the FOIL Method. Whenever multiplication is necessary outside the brackets, multiply each of the terms in the first bracket by each of the terms in the second bracket.

To avoid confusion, we apply the FOIL Method and make the calculations simple further.

In FOIL, F indicates First, O indicates Outside, I indicates Inside, L indicates Last

First: Multiply the initial terms of each bracket i.e., the first term of the first bracket with the first term in the second bracket.

Outside: Outside indicates multiplying the 2 outside terms i.e., the first term in the first bracket with the second term in the second bracket.

Inside: Multiplication of 2 inside terms inside Inside in FOIL Method i.e., the second term in the first bracket with the first term in the second bracket.

Last: Last indicates multiplication of last terms from both the brackets i.e., the second term in the first bracket with the second term in the second bracket.

Example:

(x+5)(x+10)

= (x+5)x + (x+5)10

x²+ 5x + 10x + 50

= x²+15x + 50

Removing Brackets Questions

Consider an algebraic expression which contains parentheses or round brackets ( ), square brackets [ ], curly brackets { }.

5{[4(y-4)+15] – [2(5y-3)+1]}

Step 1:

Innermost grouping symbols removal is the first step to simplify the expression. Consider (y-4), to remove the brackets we multiply the equation (y-4) by 4, here comes as 4y-16.

Step 2:

Follow the same procedure to remove parentheses in 2(5y-3), we have multiplied the equation (5y-3) by 2, here comes the final equation as 10y-6

After performing 2 steps, the result equation will be

5{[4y-16+15] – [10y-6+1]}

Step 3:

Now, the equation is further simplified and it contains only square brackets and curly brackets, we should perform all the set of operations within two brackets.

5{[4y-16+15] – [10y-6+1]}

=5{[4y-1]-[10y-5]}

Step 4:

As the brackets of two sets are removed. Therefore, the negative sign before the 2nd set implies for all the terms present in the 2nd set. Therefore, it is multiplied by -1

5{[4y-1]-[10y-5]}

=5{4y-1-10y+5}

Step 5:

As all other brackets are removed and the expression contains only curly braces. Perform all the operations that are possible within the brackets.

5{4y-1-10y+5}

=5{-6y+4}

Step 6:

To the open curly braces, apply the distributive law.

5{-6y+4}

=-30y+20

Example 2:

Simplify the equation 95 – [144 ÷ (12 x 12) – (-4) – {3 – 17 – 10}]

Solution:

Step 1:

Innermost grouping symbols removal is the first step to simplify the expression. Consider {3-17-10}, to remove the brackets we subtract the equation {3-17-10}, here comes as {3-7}

Step 2:

Follow the same procedure to remove parentheses in (12 x 12), we have multiplied the equation (12 x12), here comes the final equation as 144

After performing 2 steps, the result equation will be

95 – [144 ÷ 144 – (-4) – {3-7}]

Step 3: 

Now, the equation is further simplified and it contains only square brackets and curly brackets, we should perform all the set of operations within two brackets.

95 – [1 – (-4) – (-4)]

Step 4:

As the brackets of two sets are removed. Therefore, change the negative signs and rewrite the equation.

= 95 – [1 + 4 + 4]

Step 5:

As all other brackets are removed and the expression contains only square braces. Perform all the operations that are possible within the brackets.

= 95 – 9

Step 6:

Therefore, after subtraction the final solution is

=86

Example 3:

Simplify the equation 197 – [1/9{42 + (56 – 8 + 9)} +108]

Solution:

Step 1:

Innermost grouping symbols removal is the first step to simplify the expression. Consider {56-8+9}, to remove the brackets we subtract the equation {56-8+9}, here comes as {56-17}

= 197 – [1/9 {42 + (56 – 17)} + 108]

Step 2:

Follow the same procedure to remove parentheses in {42 + (56 – 17)}, we have subtracted the equation (56-17), here comes the final equation as {42 + 39}

After performing 2 steps, the result equation will be

197 – [1/9 {42 + 39} + 108]

Step 3:

Now, the equation is further simplified and it contains only square brackets and round brackets, we should perform all the set of operations within two brackets.

197 – [(81/9) + 108]

Step 4:

Now, simplify the equation (81/9), then the result will be 9. The final equation will be

197 – [9 + 108]

Step 5:

As all other brackets are removed and the expression contains only square braces. Perform all the operations that are possible within the brackets.

= 197 – 117

Step 6:

To get the final result, subtract 117 from 197, therefore the final result will be 80

How to Expand Brackets?

There are few different methods to expand brackets and simplify expressions. We have explained each of them in detail and even took expanding brackets examples for explaining the entire process. They are as such

1. Multiplying two bracketed terms together

If we have a situation to multiply two bracketed terms, then we multiply each term in the first bracket with each term in the second bracket. These types of multiplications lead to quadratic expressions.

Example:

(x+5)(x+10)

=(x+5)x + (x+5)10

2. Dealing with nested brackets

These are the collection of expressions nested in various sets of brackets.

Example:

Simplify -{5x-(11y-3x)-[5y-(3x-6y)]}

-[5x-(11y-3x)-[5y-3x+6y]}

-{5x-11y+3x-5y+3x-6y}

-{11x-22y}

-11x+22y