Brahmagupta’s formula for dividing fractions

The idea

To divide by a fraction, find the reciprocal of the divisor and multiply it by the dividend: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c).

Dividing is multiplying by something

Earlier, 2/3 ÷ 3/5 was the same as 3/5 × ? = 2/3, and the answer was 5/3 × 2/3. The divisor 3/5 seems to have turned into its reciprocal.

So 2/3 ÷ 3/5 is 2/3 × ?. What goes in the box?

• 5/3
• 3/5
• 2/5

Flip it and multiply

Divide in three moves. 1. Flip the divisor. 2. Change ÷ into ×. 3. Multiply the tops and multiply the bottoms. Pick the right numbers to multiply.

Brahmagupta’s formula

To divide by a fraction, multiply by its reciprocal. Brahmagupta wrote this rule in 628 CE.

To divide fractions, find the reciprocal of the divisor and multiply it by the dividend: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c)

• Division | Times the reciprocal | Answer
• 2/3 ÷ 3/5 | 2/3 × 5/3 | 10/9
• 1/4 ÷ 5 | 1/4 × 1/5 | 1/20
• 6 ÷ 1/4 | 6 × 4 | 24

Notes

To divide fractions, find the reciprocal of the divisor and multiply it by the dividend: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c)

Check yourself

What is 4/5 ÷ 2/3? Type a fraction like 7/9.

Answer: 6/5

4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10 = 6/5.

Which is the right way to work out 3/7 ÷ 2/5?

Leena pours 1/4 litre of milk equally into 5 cups. How much milk is in each cup, in litres? That is 1/4 ÷ 5. Type a fraction like 3/8.

Answer: 1/20

1/4 ÷ 5 = 1/4 × 1/5 = 1/20 litre.

An area of 15/2 square units is covered with square bricks of area 1/25 square unit each. How many bricks are needed? That is 15/2 ÷ 1/25 Type a fraction like 7/2.

Answer: 375/2

15/2 ÷ 1/25 = 15/2 × 25 = 375/2, which is 187 and a half bricks.

• 3/7 × 5/2 — correct. Yes! The dividend stays. The divisor 2/5 is flipped to 5/2.
• 3/7 × 2/5. The sign was changed to × but the divisor was not flipped. That is 3/7 of 2/5, which is 6/35.
• 7/3 × 2/5. The dividend was flipped instead of the divisor. Only the divisor is flipped.