Multiplying Fractions

The easiest of the four operations. No common denominator needed.

The rule

a/b × c/d = (a × c) / (b × d)

Multiply tops, multiply bottoms. Then simplify.

Why it's easier than add/subtract

Because you do not need a common denominator. The denominators just multiply together.

That's the whole reason fraction multiplication feels mechanical.

Worked examples

Example A — straight up

2/3 × 4/5
= (2 × 4) / (3 × 5)
= 8/15        (8 and 15 share no factor; lowest terms)

Example B — simplify before you multiply (recommended)

You can cancel a factor in any numerator with any denominator before multiplying. The numbers stay smaller.

3/8 × 4/9

3 and 9 share a factor of 3       3 ÷ 3 = 1, 9 ÷ 3 = 3
4 and 8 share a factor of 4       4 ÷ 4 = 1, 8 ÷ 4 = 2

= (1/2) × (1/3)
= 1/6

This is the same as multiplying first and then simplifying, but with much smaller numbers along the way.

Example C — answer greater than 1

5/4 × 6/5
5 and 5 cancel
4 and 6 share factor 2 → 4 ÷ 2 = 2, 6 ÷ 2 = 3
= (1/2) × (3/1)
= 3/2          (improper fraction is fine)

Example D — multiplying by a whole number

A whole number n is the fraction n/1.

3/7 × 2 = 3/7 × 2/1 = 6/7

When the answer is a whole number

It can happen.

2/3 × 9/4
2 and 4 share factor 2 → 1 and 2
3 and 9 share factor 3 → 1 and 3
= (1/1) × (3/2)
= 3/2

(In this one we still got an improper fraction, but for example 6/5 × 5/6 = 30/30 = 1.)

Try these (answers below)

1. 1/2 × 3/4
2. 2/5 × 5/6
3. 3/8 × 4/9
4. 7/3 × 6/7

1. 3/8
2. 1/3
3. 1/6
4. 2 (or 2/1)