Dividing fractions

Division as multiplication

We already know that 12 ÷ 4 = 3. The same fact can be asked the other way round: what number times 4 gives 12? Writing every division as a missing-number multiplication is the trick that makes dividing fractions easy. In 12 ÷ 4 = 3, 12 is the dividend, 4 the divisor and 3 the quotient.

The reciprocal

What is 1 ÷ 2/3? Rewrite it as 2/3 × ? = 1. If the 2 and the 3 could cancel each other we would be left with 1, and 2/3 × 3/2 does exactly that. So 1 ÷ 2/3 = 3/2. The fraction 3/2, with numerator and denominator swapped, is the reciprocal of 2/3, and any fraction multiplied by its reciprocal gives 1. Now 3 ÷ 2/3 asks for 3 times as much: 2/3 × 3/2 × 3 = 3, so 3 ÷ 2/3 = 9/2. In the same way, 1/5 ÷ 1/2 = 2 × 1/5 = 2/5.

The rule

Look at 2/3 ÷ 3/5, that is, 3/5 × ? = 2/3. The reciprocal of the divisor, 5/3, turns 3/5 into 1, and multiplying by 2/3 then gives 2/3, so the quotient is 5/3 × 2/3 = 10/9. Every division follows the same two steps: find the reciprocal of the divisor, then multiply it by the dividend.

a/b ÷ c/d = a/b × d/c = (a × d)/(b × c).

Brahmagupta first stated this rule in general form in his Brahmasphutasiddhanta (628 CE), and Bhaskara II later described it as multiplying by the reciprocal.

Is the quotient bigger?

6 ÷ 3 = 2 is smaller than 6, but 6 ÷ 1/4 = 24 is much bigger: six wholes contain 24 quarters. Likewise 1/8 ÷ 1/4 = 1/2, bigger than 1/8. Since dividend = divisor × quotient, the multiplication rules settle it: a divisor between 0 and 1 makes the quotient larger than the dividend, and a divisor above 1 makes it smaller. Comparing the dividend with the divisor tells us about the quotient too: it is more than 1 exactly when the dividend is bigger than the divisor.

Using division in problems

Leena makes 5 cups of tea with 1/4 litre of milk. Each cup has 1/4 ÷ 5 of a litre: the reciprocal of 5 is 1/5, so each cup gets 1/4 × 1/5 = 1/20 litre. Other problems follow the same pattern: 8 m of lace used 1/4 m at a time decorates 8 ÷ 1/4 = 32 bags; 1/2 m of ribbon shared by 8 badges gives 1/2 ÷ 8 = 1/16 m each; and 5 kg of flour at 1/6 kg a loaf makes 5 ÷ 1/6 = 30 loaves. The key step is deciding which quantity is being shared or measured, and which is doing the dividing.

Why it matters

Division of fractions answers ‘how many fit?’ and ‘how much each?’ questions in cooking, sharing, measuring and building. How multiplication works is in multiplying fractions, and old Indian puzzles that use division, such as Baudhayana’s bricks and the four fountains, are in fraction problems from Indian mathematics. Answers to every question are in the NCERT solutions for Chapter 8; see also the full Chapter 8 notes.