# Division of a Fraction by a Whole Number

This topic would discuss on dividing a fraction by a whole number. Whole numbers are numbers from 0 to infinity i.e. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9……} They are not represented in numerator and denominator form. However, for the purpose of division and multiplication of the whole number has to be represented in numerator and denominator form. For that we have to place 1 in the denominator of a whole number. However, in division the general method used is:

Fraction ÷ fraction / Whole number) = fraction × reciprocal of the fraction/ Whole number

Then after that the steps are same as multiplication of fraction with whole number.

Here are few examples to show division of a fraction by whole number:

1. 7 2/10  ÷ 36

⟹ 72/10 ÷ 36;  [Changing mixed fraction into improper fraction]

⟹ 72/10 ÷ 36/1; [Placing 1 in the denominator]

⟹ 72/10 × 1/36;  [Reciprocal of the whole number]

⟹ (72 × 1)/(10 × 36); [Product of numerator and denominator]

⟹ (72 ÷ 72)/(360 ÷ 72);   [Change into lowest term]

⟹ 1/5

Explanation:

The fraction is in mixed form so, in the first step it is changed into improper fraction. Then the whole number is changed into fraction by placing 1 in the denominator. In the next step the fraction is multiplied with the reciprocal or multiplicative inverse of the whole number. That is Product of the numerator/ Product of the denominator. Then the answer is changed to lowest terms to arrive at the final answer.

2. 9/23 ÷ 108

⟹ 9/23 ÷ 108/1; [Placing 1 in the denominator]

⟹ 9/23 × 1/108;  [Reciprocal of the whole number]

⟹ (9 × 1)/(23 × 108); [Product of numerator and denominator]

⟹ (9 ÷ 9)/(2484 ÷ 9) [Change into lowest term]

⟹ 1/276

Explanation:

In this example the whole number is changed into fraction by placing 1 in the denominator. In the next step the fraction is multiplied with the reciprocal or multiplicative inverse of the whole number. That is Product of the numerator/ Product of the denominator. Then the answer is changed to lowest terms to arrive at the final answer

3. 9 1/7 ÷ 64

⟹ 64/7 ÷ 64;  [Changing mixed fraction into improper fraction]

⟹ 64/7 ÷ 64/1; [Placing 1 in the denominator]

⟹ 64/7 × 1/64;  [Reciprocal of the whole number]

⟹ (64 ×1)/(7 ×64); [Product of numerator and denominator]

⟹ (64 ÷64)/(448 ÷64);   [Change into lowest term]

⟹ 1/7

Explanation:

The fraction is in mixed form so, in the first step it is changed into improper fraction. Then the whole number is changed into fraction by placing 1 in the denominator. In the next step the fraction is multiplied with the reciprocal or multiplicative inverse of the whole number. That is Product of the numerator/ Product of the denominator. Then the answer is changed to lowest terms to arrive at the final answer.

The above examples show the division of fraction by whole number where the fraction is multiplied by the reciprocal or the multiplicative inverse of the whole number.

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