Class 6 Fractions

Fraction is a part of whole and it is denoted as ^p⁄_q. 'p' is known as numerator and 'q' is known as denominator. Denominator tells us about into how many equal parts the whole is divided. Numerator tells us how many parts are considered of the whole.

Type of Fractions

Mainly there are five types of fractions.

1. Proper fractions

2. Improper fractions

3. Mixed fractions

4. Like fractions

5. Unlike fractions

Proper Fractions

Fractions in which denominator is greater than numerator are known as proper fraction.

Example. ²⁄₃, ³⁄₅, ⁴⁄₇ (Here all the numerators are less than denominators)

Improper Fractions

Fractions in which numerator is greater than or equal to the denominator are known as improper fractions.

Example. ³⁄₂, ⁴⁄₃, ⁹⁄₇

Mixed Fractions

A number with a whole number part and a fractional part is known as mixed fraction.

Example. 1 ²⁄₃, 1¹⁄₂, 1²⁄₃

Like Fractions

Fractions having same denominators are known as like fractions.

Example. ²⁄₅, ³⁄₅, ⁴⁄₅

Unlike Fractions

Example. ²⁄₃, ³⁄₅, ⁴⁄₇

Conversion of an Improper Fraction to Mixed Fraction

Let's consider an improper fraction i.e. ⁷⁄₅, now divide 7 by 5 as shown below.

D − Divisor

Q − Quotient

R − Remainder

7/5 = Q^R⁄_D = 1²⁄₅

Conversion of a Mixed Fraction to Improper Fraction

W^N⁄_D = ^((WxD)+N)⁄_D

W = Whole number

N = Numerator

D = Denominator

Let's see some examples.

Example 1. Convert 2²⁄₅ into improper fraction.

Solution. 2²⁄₅ = ^((2x5)+2)⁄₅ = ^((10)+2)⁄₅ = ¹²⁄₅

Example 2. Convert 1²⁄₃ into improper fraction.

Solution. 1²⁄₃ = ^((1x3)+2)⁄₃ = ⁵⁄₃

Equivalent Fraction

All fractions that have the same value are known as equivalent fraction. Let's see some examples of equivalent fractions.

¹⁄₂ = ²⁄₄ = ⁴⁄₈ = ⁸⁄₁₆ = ...

²⁄₃ = ⁴⁄₆ = ⁸⁄₁₂ = ¹⁶⁄₂₄ = ...

Addition of Like Fractions

If the denominators of two or more fractions are same, then we add the numerators and keep the denominators as is. Let's see some examples.

Example 1. Add ²⁄₅ and ⁴⁄₅.

Solutions. ²⁄₅ + ⁴⁄₅ = ⁽²⁺⁴⁾⁄₅ = ²⁄₅

Example 2. Add ²⁄₉ and ⁵⁄₉.

Solutions. ²⁄₉ + ⁵⁄₉ = ⁽²⁺⁵⁾⁄₉ = ⁷⁄₉

Addition of Unlike Fractions

If the denominators of two fractions are different, then we find the LCM of the denominators. Rewrite each fraction with LCM as common denominators and add the fractions as like fractions. Let's see some examples.

Examples 1. Add ²⁄₅ and ³⁄₇.

Solution. LCM of 5 and 7 is 35.

²⁄₅ = ¹⁴⁄₃₅ and ³⁄₇ = ¹⁵⁄₃₅

Now add ¹⁴⁄₃₅ and ¹⁵⁄₃₅
= ¹⁴⁄₃₅ + ¹⁵⁄₃₅
= ^(14+15)⁄₃₅ = ²⁹⁄₃₅

Examples 2. Add ⁵⁄₁₂, ⁷⁄₁₆ and 9/24.

Solution. The LCM of 12, 16 and 24 is 48.

⁵⁄₁₂ = ²⁰⁄₄₈ ⁷⁄₁₆ = ²¹⁄₄₈ ⁹⁄₂₄ = ¹⁸⁄₄₈

²⁰⁄₄₈ + ²¹⁄₄₈ + ¹⁸⁄₄₈

= ^(20+21+18)⁄₄₈ = ⁵⁹⁄₄₈ = 1¹¹⁄₄₈

Subtraction of Like Fractions

If the denominators of two fractions are same, then we subtract the numerators and keep the denominators as is. Let's see some examples.

Example 1. Subtract ³⁄₇ from ⁵⁄₇.

Solution. ⁵⁄₇ − ³⁄₇ = ⁽⁵⁻³⁾⁄₇ = ²⁄₇

Example 2. Subtract ⁹⁄₂₄ from ¹³⁄₂₄

Solution. ¹³⁄₂₄ − ⁹⁄₂₄ = ⁽¹³⁻⁹⁾⁄₂₄ = ⁴⁄₂₄ = ¹⁄₆

Subtraction of Unlike Fractions

If the denominators of two fractions are different, then we find the LCM of the denominators. Rewrite each fraction with LCM as common denominators and subtract the fractions as like fractions. Let's see some examples.

Example 1. Subtract ¹⁄₃ from ⁴⁄₅

Solution. LCM of 3 and 5 is 15.

¹⁄₃ can be written as ^(1x5)⁄_(3x5) = ⁵⁄₁₅

⁴⁄₅ can be written as ^(4x3)⁄_(5x3) = ¹²⁄₁₅

¹²⁄₁₅ − ⁵⁄₁₅ = ⁽¹²⁻⁵⁾⁄₁₅ = ⁷⁄₁₅

Example 2. Simplify 2¹⁄₃ − 1³⁄₄₈

Solution. 2¹⁄₃ − 1³⁄₄₈

First, we must convert the mixed fractions into improper fractions.

⁷⁄₃ − ⁵¹⁄₄₈ (LCM of 3 and 48 is 48)

= ¹¹²⁄₄₈ − ⁵¹⁄₄₈ = ^(112−51)⁄₄₈ = ⁶¹⁄₄₈

Example 3. Simplify 2⁵⁄₁₂ + 2⁵⁄₉ − 1²⁄₃

Solution. 2⁵⁄₁₂ + 2⁵⁄₉ - 1²⁄₃

First, we must convert the mixed fractions into improper fractions.

²⁹⁄₁₂ + ²³⁄₉ − ⁵⁄₃ (LCM of 12, 9 and 3 is 36)

= ^(29x3)⁄_(12x3) + ^(23x4)⁄_(9x4) − ^(5x12)⁄_(3x12)

= ⁸⁷⁄₃₆ + ⁹²⁄₃₆ − ⁶⁰⁄₃₆ = ^(87+92−60)⁄₃₆

= ¹¹⁹⁄₃₆

Multiplication of Fraction with a Whole Number

To get the product of a fraction and a whole number multiply the numerator of the fraction with the whole number. Let's see some examples.

Example 1. Multiply 3 by ²⁄₇.

Solution. 3 x ²⁄₇ = ^(3x2)⁄₇ = ⁶⁄₇

Example 2. Multiply 25 by ³⁄₅.

Solution. 25 x ³⁄₅ = ^(25*3)⁄₅ = ⁷⁵⁄₅ = 15

Multiplication of Fractions

To get the product of two fractions numerators are multiplied and the denominators are multiplied. Let's see some examples.

Example 1. Multiply ³⁄₅ and ²⁄₇.

Solution. ³⁄₅ x 2/7 = (3x2)/(5x7) = 6/35

Example 2. Multiply ²⁄₅, ³⁄₄ and ²⁄₃.

Solution. ²⁄₅ x ³⁄₄ x ²⁄₃ = ^(2x3x2)⁄_(5x4x3) = ¹²⁄₆₀ = ¹⁄₅

1. When any fraction is multiplied by 1, the product is fraction itself.

2. When any fraction is multiplied by 0, the product is 0.

Reciprocal of Fraction

In any fraction, if we interchange the numerator and the denominator, we get the reciprocal of that fraction. Let's see some examples.

Example 1. Find the reciprocal of ²⁄₅.

Solution. Reciprocal of ²⁄₅ is ⁵⁄₂.

Example 2. Find the reciprocal of ¹⁄₅.

Solution. Reciprocal of ¹⁄₅ is 5.

Division of Fractions

Division is nothing but multiplying a fraction with the reciprocal of another fraction. Let's see some examples.

Example 1. ²⁄₅ ÷ ³⁄₅.

Solution. Reciprocal of ³⁄₅ is ⁵⁄₃.
²⁄₅ x ³⁄₅ = ²⁄₃

Example 2. ⁶⁄₃₅ ÷ ²⁄₅.

Solution. Reciprocal of ²⁄₅ is ⁵⁄₂.

⁶⁄₃₅ x ⁵⁄₂ = 3/7

1. When any fraction is divided by 1, the result is fraction itself.

2. When 0 is divide by any fraction, then the result is 0.

3. When any fraction is divided by itself, then the result is 1.

4. When any fraction is multiplied by it's reciprocal, then the result is 1.