How to Add Subtract Multiply and Divide Fractions?

Introduction

The previous chapter Fractions, Types, Shaded Diagrams & Real Life Applications discussed about what a fraction is and what the various types of fractions are. This chapter, discusses the arithmetic operations like addition, subtraction, multiplication and division on fractions.

Addition of fractions

1. Add like fractions

In addition of fractions of like fractions, only numerators are added and keep the value of denominator as it is.

Example

Example 1: Add ⁵₈ and ⁶₈
As the denominator is 8 and add numerator of both fractions only
= ⁵₈ + ⁶₈
= ^{5 + 6}₈ = ¹¹₈

Example 2: Add ¹₄, ²₄ and ⁶₄
Here, as denominator is 4 for all fractions, so add numerators of all fractions only.
= ¹₄ + ²₄ + ⁶₄
= ^{1 + 2 + 6}₄ = ⁹₄

2. Add unlike fractions

To add unlike fractions, convert unlike fractions into like fractions by taking LCM of their denominators.

Example

Example 1: Add ³₄ and ⁵₆
Step 1: Take LCM of denominators of fractions. ³₄ and ⁵₆. LCM of 4, 6
Step 2: Multiply both fractions by numbers to make their denominators same equal to LCM 12.
³₄ × ³₃ = ⁹₁₂
⁵₆ × ²₂ = ¹⁰₁₂
Step 3: Add the like fractions ⁹₁₂ and ¹⁰₁₂.
= ⁹₁₂ + ¹⁰₁₂
= ^{9 + 10}₁₂ = ¹⁹₁₂
∴³₄ + ⁵₆ = ¹⁹₁₂

Example 2: Add ²₃ , ⁴₆ and ¹₄
Step 1: Take LCM of denominators of fractions ²₃ , ⁴₆ and ¹₄ LCM of 3, 6, 4
Step 2: Multiply all fractions by numbers to make their denominators same equal to LCM 12.
²₃ × ⁴₄ = ⁸₁₂
⁴₆ × ²₂ = ⁸₁₂
¹₄ × ³₃ = ³₁₂
Step 3: Add the like fractions ⁸₁₂ , ⁸₁₂ and ³₁₂
= ⁸₁₂ + ⁸₁₂ + ³₁₂
= ^{8 + 8 + 3}₁₂ = ¹⁹₁₂
∴ ²₃ + ⁴₆ + ¹₄ = ¹⁹₁₂

Subtraction of fractions

1. Subtract like fractions

In subtraction of like fractions, only numerators are subtracted and keep the value of denominator as it is.

Example

Example 1: Subtract ⁸₉ from ¹⁰₉
As these are like fractions and denominator is 9 for both fractions.
So, subtract numerator of both fractions only.
¹⁰₉ – ⁸₉
^{10 – 8}₉ = ²₉

Example 2: Subtract ¹¹₁₂ from ¹⁵₁₂
Here, the two fractions are like fractions and their denominator is 12.
So, subtract numerators of both fractions only.
= ¹⁵₁₂ – ¹¹₁₂
^{15 – 11}₁₂ = ⁴₁₂ = ¹₃

2. Subtract unlike fractions

To subtract the unlike fractions, convert unlike fractions into like fractions by taking LCM of their denominators.

Example

Subtract ⁶₄ from ⁸₃
Step 1: Take LCM of denominators of fractions.
LCM of 4, 3
LCM of 4 and 3 is 12
Step 2: Convert unlike fractions into like fractions by multiplying a number which makes denominator equal to LCM 12.
⁸₃ × ⁴₄ = ³²₁₂
⁶₄ × ³₃ = ¹⁸₁₂
Step 3: Subtract ¹⁸₁₂ from ³²₁₂
= ³²₁₂ – ¹⁸₁₂
= ^{32 – 18}₁₂ = ¹⁴₁₂ = ⁷₆
∴ ⁸₃ – ⁶₄ = ¹⁴₁₂

Multiplication of fractions

1. Multiply a fraction with whole number

To multiply a fraction with whole number, multiply only numerator by the given whole number and keep the denominator same. Then reduce it to its the lowest term.

Example

Example 1: ⁷₅ × 2
Here, ⁷₅ is a fraction and 2 is a whole number.
2 is written as ²₁
= ⁷₅ × ²₁
= ^{7 × 2}_{5 × 1} = ¹⁴₅

Example 2: ⁶₅ × 4
= ⁶₅ × ⁴₁
= ^{6 × 4}_{4 × 1} = ²⁴₅

2. Multiply a fraction by fraction

To multiply a fraction by another fraction, multiply their corresponding numerators and denominators. Then reduce the obtained fraction into its the lowest form.

Example

Example 1: Multiply fraction ⁷₅ by ³₄
Here, multiply numerators 7 and 3. Also, multiply their denominators 5 and 4.
⁷₅ × ³₄
= ^{7 × 3}_{5 × 4} = ²¹₂₀

Example 2: Multiply fractions ²₅ , ⁴₆ and ³₂
²₅ × ³₂ ×
multiply numerators 2, 4 and 3 and also multiply denominators 5, 6 and 2.
= ^{2 × 4 × 3}_{5 × 6 ×
2} = ²⁴₆₀
Now ²⁴₆₀ should be reduced to its the lowest term by dividing with the common factor 12.
²⁴₆₀ = ²₅

Division of fractions

To divide a fraction with another fraction, first the division is changed into multiplication by changing the division sign into multiplication and taking reciprocal of the second fraction.
Let’s learn it by following examples.

Example

Example 1: Divide ²₃ by ⁵₇
= ²₃ ÷ ⁵₇
= ²₃ × ⁷₅
= ^{2 × 7}_{3 × 5} = ¹⁴₁₅

Example 2: Divide fraction ⁴₅ by whole number 6
= ⁴₅ ÷ ⁶₁
= ⁴₅ × ¹₆
= ^{4 × 1}_{5 × 6} = ⁴₃₀
⁴₃₀. should be reduced into its the lowest term by dividing with common factor 2
= ²₁₅

Example 3: Divide 5 by ⁶₇
= 5 ÷ ⁶₇
= ^{5 × 7}_{1 × 6}
= ³⁵₆

Frequently Asked Questions

1) How to add fractions?

Step I: Add the numerator of fractions.
Step II: Keep the value of common denominator as it is.
Step III: Value of fraction = ^{Sum of all numerators}_{Value of common denominator}

2) How to add unlike fractions?

Step I: Convert the unlike fractions into like fractions by taking LCM of all denominators.
Step II: Add the numerator of each equivalent fraction.
Step III: Value of fraction = ^{Sum of numerators}_{Denominator of equivalent fractions}

3) How to subtract like fractions?

Step I: Subtract the numerator of each fraction.
Step II: Keep the value of common denominator as it is.
Step III: Value of fraction = ^{Difference of all numerators}_{Value of common denominator}

4) How to subtract unlike fractions?

Step I: Convert the unlike fractions into like fractions by taking LCM of all denominators. Step II: Subtract the numerator of each equivalent fraction.
Step III: Value of fraction = ^{Difference of all numerators}_{Denominator of equivalent fractions}

Solved Examples

1) Add ¹₄ , ⁷₄ and ¹¹₄.

As the denominator of of all fractions is same, then add only numerators of all fractions. ^{1 + 7 + 11}₄ = ¹⁹₄

2) Add ³₄ , ⁵₃ and ⁹₂.

Here, denominators of all fractions are different.
So, first take LCM of all denominators.
LCM of 4, 3, 2
LCM of 4, 3 and 2 is 12.
Now, make denominator as 12 for all fractions.
³₄ × ³₃ = ⁹₁₂
⁵₃ × ⁴₄ = ²⁰₁₂
⁹₂ × ⁶₆ = ⁵⁴₁₂
Add the fractions with same denominator of 12
= ⁹₁₂ + ²⁰₁₂ + ⁵⁴₁₂
^{9 + 20 + 54}₁₂
⁸³₁₂

3) Simplify the fraction ⁵₃ + ³₄ - ³₂

Take LCM of all denominators 3, 4 and 2.
LCM of 3, 4, 2
LCM of 3, 4 and 2 = 12
Make denominator as 12 for all fractions.
⁵₃ × ⁴₄ = ²⁰₁₂
³₄ × ³₃ = ⁹₁₂
³₂ × ⁶₆ = ¹⁸₁₂
Add the fractions
= ²⁰₁₂ + ⁹₁₂ - ¹⁸₁₂
^{20 + 9 - 18}₁₂
^{29 - 18}₁₂
¹¹₁₂

4) Solve ³₂ - 2¹₃ - 2¹₅

= ³₂ - ⁷₃ - ¹¹₅
Take LCM of all denominators 2, 3 and 5.
LCM of 2, 3, 5
LCM of 2, 3 and 5= 30
Make denominator as 30 for all fractions.
³₂ × ¹⁵₁₅ = ⁴⁵₃₀
⁷₃ × ¹⁰₁₀ = ⁷⁰₃₀
¹¹₅ × ⁶₆ = ⁶⁶₃₀
³₂ - ⁷₃ - ¹¹₅ can be written as
⁴⁵₃₀ - ⁷⁰₃₀ - ⁶⁶₃₀
^{45 - 70 - 66}₃₀
^{45 - 136}₃₀
- ⁹¹₃₀