Order of multiplication

The idea

The order of multiplying fractions does not matter, a/b × c/d = c/d × a/b, just as the area of a rectangle stays the same when its length and breadth are interchanged.

Swap the two fractions

The tortoise walks 1/4 km in an hour. In half an hour it walks 1/2 × 1/4 = 1/8 km.

What is 1/4 × 1/2, with the two fractions swapped?

• 1/8
• 1/6
• 3/4

Turn the rectangle

A rectangle has a length and a breadth. Its area is length × breadth. Tap Turn it and watch the small squares.

Same rectangle, same area

Turning a rectangle swaps its length and breadth, but its area stays the same. So the product of two fractions does not depend on which comes first.

Brahmagupta’s formula shows it too. The tops are a × c or c × a, and the bottoms are b × d or d × b. Whole numbers give the same either way round.

The order of multiplying fractions does not matter: a/b × c/d = c/d × a/b. The area of a rectangle stays the same when its length and breadth are interchanged.

• One way | Swapped | Both give
• 1/2 × 1/4 | 1/4 × 1/2 | 1/8
• 2/3 × 3/5 | 3/5 × 2/3 | 2/5
• 3/4 × 5/6 | 5/6 × 3/4 | 5/8

Notes

The order of multiplying fractions does not matter: a/b × c/d = c/d × a/b. The area of a rectangle stays the same when its length and breadth are interchanged.

Check yourself

You know that 5/7 × 3/8 = 15/56. What is 3/8 × 5/7? Type a fraction like 2/9.

Answer: 15/56

3/8 × 5/7 = (3 × 5)/(8 × 7) = 15/56, the same as 5/7 × 3/8.

Which statement is true?

A rectangle is 3/5 m long and 2/3 m broad. Its area is 2/5 square metres. It is turned so that it is 2/3 m long and 3/5 m broad. What is its area now, in square metres? Type a fraction like 3/7.

Answer: 2/5

The area is the same: 2/3 × 3/5 = 2/5 square metres.

Aarav says: “3/4 × 5 and 5 × 3/4 are different, because the fraction comes first in one and second in the other.” Is he right?

• 2/5 × 7/3 = 7/3 × 2/5 — correct. Yes! Swapping the two fractions does not change the product.
• 2/5 × 7/3 = 5/2 × 7/3. Here the first fraction was turned upside down. That changes the product: it is not the same as swapping the order.
• 2/5 × 7/3 = 2/7 × 5/3. Here the 5 and the 7 traded places, which breaks the fractions apart. Whole fractions trade places: 7/3 × 2/5.
• Yes. The fraction must always come first.. There is no such rule. Swapping the numbers gives the same product.
• Yes. One of them is 15/4 and the other is 5/12.. Check: 5 × 3/4 = 5/1 × 3/4 = 15/4. It is not 5/12.
• No. Both are 15/4. The order does not matter. — correct. Yes! 3/4 × 5 = 15/4 and 5 × 3/4 = 15/4.