Cancelling common factors

The idea

When multiplying fractions, the numerator and denominator can first be divided by their common factors before multiplying them; this is called cancelling the common factors, and cancelling all of them gives the product in its lowest form.

Big numbers, small answer

Multiply 12/7 × 5/24. Brahmagupta’s rule gives one fraction: (12 × 5)/(7 × 24). That is 60/168, which is not in lowest form.

Which numbers in (12 × 5)/(7 × 24) share a common factor?

• 12 and 24
• 5 and 7
• 12 and 5

Cancel before you multiply

The fraction is made of tops and bottoms. Tap a top number and a bottom number. If they share a factor, divide both by it.

Divide out the common factors first

12 on top and 24 at the bottom have the common factor 12. A fraction does not change when its numerator and denominator are divided by the same number. So divide first, then multiply the small numbers.

When multiplying fractions, the numerator and denominator can first be divided by their common factors before multiplying. This is called cancelling the common factors. Cancelling all of them gives the product in its lowest form.

Notes

When multiplying fractions, the numerator and denominator can first be divided by their common factors before multiplying. This is called cancelling the common factors. Cancelling all of them gives the product in its lowest form.

Check yourself

Cancel first, then multiply: 8/9 × 3/4. Type a fraction like 5/7.

Answer: 2/3

8 and 4 ÷ 4 give 2 and 1. 3 and 9 ÷ 3 give 1 and 3. (2 × 1)/(3 × 1) = 2/3

In 6/7 × 5/12, which two numbers can be cancelled?

Cancel first, then multiply: 9/10 × 5/6. Type a fraction like 5/7.

Answer: 3/4

9 and 6 ÷ 3 give 3 and 2. 5 and 10 ÷ 5 give 1 and 2. (3 × 1)/(2 × 2) = 3/4

In 6/5 × 7/3, Meera cancels the 6 and the 3 (both divided by 3). Is that allowed?

• 6 on top and 12 at the bottom, both divided by 6 — correct. Yes! 6 is a factor of both, and one is on top and one at the bottom.
• 5 on top and 12 at the bottom, both divided by 5. 5 does not divide 12. These two numbers have no common factor except 1.
• 6 and 5, both on top, divided by 5. Both are numerators. Cancelling needs one number on top and one at the bottom.
• No. Only the numerator and denominator of the same fraction can cancel.. After multiplying, all the tops are in one numerator and all the bottoms in one denominator. Any top and bottom can cancel.
• No. Cancelling is only for adding fractions.. Cancelling common factors is a way to multiply fractions with smaller numbers.
• Yes. The product is one fraction, (6 × 7)/(5 × 3). A top number and a bottom number can cancel, whichever fraction they came from. — correct. Yes! 6 × 7 over 5 × 3 becomes (2 × 7)/(5 × 1) = 14/5.