Multiplying & Dividing Fractions — Grade 8 Maths (CAPS)

Multiplying straight across, the reciprocal, and why dividing by a fraction means multiplying by its flip — including mixed numbers and dividing whole numbers by fractions, with worked examples, a worksheet with memo and a quiz.

Here is the good news: multiplying and dividing fractions is easier than adding them, because you do not need a common denominator at all.

1Multiplying — straight across

Multiplying fractions as areas: two thirds of three quarters covers 6 of the 12 blocks, which is a half.

top × top, bottom × bottom — then simplify.

Worked ExampleCalculate: 2/3 × 3/4

1
Multiply the tops: 2 × 3 = 6.
2
Multiply the bottoms: 3 × 4 = 12.
3
6/12 simplifies to 1/2.
Divide top and bottom by the HCF, 6.

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You may cancel before you multiply to keep the numbers small: in 2/3 × 3/4 the 3 on top and the 3 underneath cancel, leaving 2/1 × 1/4 = 2/4 = 1/2. Same answer, easier arithmetic.

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In maths, 'of' means multiply. So '½ of ¾' is ½ × ¾ = ⅜. Learners often add instead — read the word carefully.

2The reciprocal

The reciprocal of a fraction is that fraction turned upside down.
The reciprocal of a/b is b/a (a, b ≠ 0), and a number × its reciprocal = 1.

Worked ExampleWrite down the reciprocal of 3/5, of 7 and of 11/3

1
3/5 → 5/3.
Simply swap top and bottom.
2
7 = 7/1 → 1/7.
A whole number has a hidden denominator of 1.
3
11/3 = 4/3 → 3/4.
Convert the mixed number first, then flip.

3Dividing — flip and multiply

To divide by a fraction, multiply by its reciprocal.
a ÷ b/c = a × c/b (b, c ≠ 0)

Worked ExampleCalculate: 2/3 ÷ 3/4

1
Flip the second fraction: 3/4 becomes 4/3.
Dividing by a fraction is multiplying by its reciprocal.
2
Change ÷ to ×: 2/3 × 4/3.
Only the fraction after the ÷ sign gets flipped.
3
Multiply straight across: 8/9.
8 and 9 share no factor, so it is in lowest terms.

Worked ExampleCalculate: 3 ÷ 1/4 (a whole number divided by a fraction)

1
Write 3 as 3/1, then flip the 1/4 to 4/1.
2
3/1 × 4/1 = 12.
3
Answer: 12 — and it makes sense: there are 12 quarters in 3 wholes.
Dividing a positive number by a fraction between 0 and 1 gives a bigger answer.

4Mixed numbers

Worked ExampleCalculate: 11/2 × 22/3

1
Convert both: 11/2 = 3/2 and 22/3 = 8/3.
Never multiply the whole parts separately — convert first.
2
Multiply: (3 × 8)/(2 × 3) = 24/6.
3
24/6 = 4.
Simplify the answer fully.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
1/2 × 3/5
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(1 × 3)/(2 × 5) = 3/10
2
2/3 × 3/4
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6/12 = 1/2
3
4/5 × 10
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40/5 = 8
4
Write down the reciprocal of 2/7.
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7/2
5
Write down the reciprocal of 5.
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1/5
6
3/4 ÷ 1/2
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3/4 × 2/1 = 6/4 = 3/2 = 11/2
7
2/3 ÷ 4/9
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2/3 × 9/4 = 18/12 = 3/2 = 11/2
8
5 ÷ 1/3
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5 × 3 = 15
9
11/4 × 2/5
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5/4 × 2/5 = 10/20 = 1/2
10
How many 1/8 pieces are there in 2 whole pizzas?
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2 ÷ 1/8 = 2 × 8 = 16 pieces