Multiplying & Dividing Fractions

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: 23 × 34

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

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You may cancel before you multiply to keep the numbers small: in 23 × 34 the 3 on top and the 3 underneath cancel, leaving 21 × 14 = 24 = 12. 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 ab is ba, and a number × its reciprocal = 1.

Worked ExampleWrite down the reciprocal of 35, of 7 and of 113

1
35 → 53.
Simply swap top and bottom.
2
7 = 71 → 17.
A whole number has a hidden denominator of 1.
3
113 = 43 → 34.
Convert the mixed number first, then flip.

3Dividing — flip and multiply

To divide by a fraction, multiply by its reciprocal.
a ÷ bc = a × cb

Worked ExampleCalculate: 23 ÷ 34

1
Flip the second fraction: 34 becomes 43.
Dividing by a fraction is multiplying by its reciprocal.
2
Change ÷ to ×: 23 × 43.
Only the fraction after the ÷ sign gets flipped.
3
Multiply straight across: 89.
8 and 9 share no factor, so it is in lowest terms.

Worked ExampleCalculate: 3 ÷ 14 (a whole number divided by a fraction)

1
Write 3 as 31, then flip the 14 to 41.
2
31 × 41 = 12.
3
Answer: 12 — and it makes sense: there are 12 quarters in 3 wholes.
Dividing by a fraction smaller than 1 gives a bigger answer.

4Mixed numbers

Worked ExampleCalculate: 112 × 223

1
Convert both: 112 = 32 and 223 = 83.
Never multiply the whole parts separately — convert first.
2
Multiply: (3 × 8)/(2 × 3) = 246.
3
246 = 4.
Simplify the answer fully.

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
12 × 35
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(1×3)/(2×5) = 310
2
23 × 34
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612 = 12
3
45 × 10
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405 = 8
4
Write down the reciprocal of 27.
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72
5
Write down the reciprocal of 5.
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15
6
34 ÷ 12
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34 × 21 = 64 = 32 = 112
7
23 ÷ 49
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23 × 94 = 1812 = 32 = 112
8
5 ÷ 13
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5 × 3 = 15
9
114 × 25
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54 × 25 = 1020 = 12
10
How many 18 pieces are there in 2 whole pizzas?
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2 ÷ 18 = 2 × 8 = 16 pieces