Multiplying Fractions

The simplest fraction operation

Many students expect multiplying fractions to be the hardest of the four operations, but it is actually the easiest. Unlike adding and subtracting, you do not need a common denominator. You simply multiply straight across. By the end of this lesson you will be able to multiply any two fractions, simplify the result, use a shortcut called cancelling, and handle whole and mixed numbers.

A quick reminder: a fraction's numerator (top) counts the pieces and its denominator (bottom) gives the size of each piece. If you need to review the basics, see our Introduction to Fractions lesson.

The core rule

Rule: To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Then simplify if you can.

In symbols: a/b × c/d = (a × c)/(b × d).

Example 1 — Work out 2/3 × 4/5.

1. Multiply the numerators: 2 × 4 = 8.
2. Multiply the denominators: 3 × 5 = 15.
3. So 2/3 × 4/5 = 8/15.
4. Check for simplifying: 8 and 15 share no common factor except 1, so 8/15 is already in lowest terms.

Example 2 — Work out 3/4 × 2/9.

1. Numerators: 3 × 2 = 6.
2. Denominators: 4 × 9 = 36.
3. So the answer is 6/36.
4. Simplify: the greatest common factor of 6 and 36 is 6, so 6/36 = 1/6.

What "of" really means

A key idea: in mathematics, the word "of" means multiply. Finding 1/2 of 1/2 is the same as 1/2 × 1/2.

Example 3 — What is 1/2 of 1/2?

Picture half a chocolate bar. Now take half of that half. You are left with a quarter of the whole bar. And the rule agrees: 1/2 × 1/2 = (1×1)/(2×2) = 1/4. This is why multiplying by a fraction smaller than 1 makes the answer smaller — you are taking only a part of something.

Cancelling: the smart shortcut

When the numbers are larger, you can cancel common factors before multiplying. This means dividing a numerator and a denominator (from either fraction) by a common factor. It keeps the numbers small and often leaves your answer already simplified.

Rule: Before multiplying, simplify any numerator with any denominator that share a common factor.

Example 4 — Work out 3/4 × 8/9 using cancelling.

1. Look at the 3 (top) and the 9 (bottom). They share a factor of 3: 3 ÷ 3 = 1, and 9 ÷ 3 = 3.
2. Look at the 8 (top) and the 4 (bottom). They share a factor of 4: 8 ÷ 4 = 2, and 4 ÷ 4 = 1.
3. The problem becomes 1/1 × 2/3.
4. Multiply: (1 × 2)/(1 × 3) = 2/3.

To check, multiply without cancelling: 3/4 × 8/9 = 24/36, and 24/36 simplifies to 2/3. Same answer, but cancelling avoided the big numbers.

Multiplying by a whole number

Any whole number can be written as a fraction by putting it over 1. For example, 6 = 6/1.

Example 5 — Work out 1/2 × 6.

1. Write 6 as 6/1.
2. Multiply across: (1 × 6)/(2 × 1) = 6/2.
3. Simplify: 6/2 = 3.

This matches common sense: half of 6 is 3.

Example 6 — Work out 2/3 × 9.

1. Write 9 as 9/1: 2/3 × 9/1.
2. Cancel 3 (bottom) with 9 (top): 9 ÷ 3 = 3, 3 ÷ 3 = 1, giving 2/1 × 3/1.
3. Multiply: (2 × 3)/(1 × 1) = 6.

Multiplying mixed numbers

You cannot multiply mixed numbers as they stand. First turn each one into an improper fraction, then multiply, then convert back if you wish.

Example 7 — Work out 1 and 1/2 × 2 and 2/3.

1. Convert each mixed number to an improper fraction:
2. 1 and 1/2 = (1×2 + 1)/2 = 3/2
3. 2 and 2/3 = (2×3 + 2)/3 = 8/3
4. Cancel: the 8 (top) and 2 (bottom) share a factor of 2, giving 3/1 × 4/3.
5. Cancel again: the 3 (top) and 3 (bottom) cancel, giving 1/1 × 4/1.
6. Multiply: 4.

So 1 and 1/2 × 2 and 2/3 = 4.

A reference table

Problem	Multiply across	Simplify	Answer
1/2 × 1/3	(1×1)/(2×3)	1/6	1/6
2/3 × 3/5	(2×3)/(3×5) = 6/15	÷3	2/5
3/4 × 2/9	6/36	÷6	1/6
4 × 3/8	12/8	÷4	3/2 = 1 and 1/2
2 and 1/2 × 4	5/2 × 4/1 = 20/2	÷2	10

Why you multiply straight across

It is worth understanding the why so the rule is not just memorised. Take 2/3 × 4/5. The 4/5 tells you to take four fifths of something. Picture a square split into 5 equal columns (fifths) and 3 equal rows (thirds). The whole square is divided into 3 × 5 = 15 small equal rectangles — that is where the denominator 15 comes from. Now shade 2 of the 3 rows and 4 of the 5 columns. The overlap, the part shaded both ways, is 2 × 4 = 8 of those small rectangles — that is where the numerator 8 comes from. So 2/3 × 4/5 = 8/15. The picture shows exactly why you multiply tops together and bottoms together.

A practice activity

Use grid paper or draw a rectangle.

1. Draw a rectangle and divide it into 4 columns (quarters) and 3 rows (thirds), making 12 small boxes.
2. To model 3/4 × 1/3, shade 3 of the 4 columns one colour, and 1 of the 3 rows another colour.
3. Count the boxes shaded both colours: there are 3. Out of 12 boxes, that is 3/12 = 1/4.
4. Check with the rule: 3/4 × 1/3 = 3/12 = 1/4. The drawing matches.

Then try these on paper (answers below): (a) 1/3 × 3/5, (b) 5/6 × 2/3, (c) 3/8 × 4, (d) 2 and 1/4 × 2/3.

Answers: (a) 3/15 = 1/5, (b) 10/18 = 5/9, (c) 12/8 = 3/2 = 1 and 1/2, (d) 9/4 × 2/3 = 18/12 = 3/2 = 1 and 1/2.

Where this leads

Multiplying fractions is the gateway to Dividing Fractions, which uses multiplication with a clever flip, and to working with Equivalent Fractions and proportions. The single most important habit is this: multiply straight across, then simplify — no common denominator required.