Mathematics
Multiplying Fractions Practice Questions, Cancelling Before You Multiply
Twelve questions on fraction times fraction, fraction times whole number and mixed numbers, with cross-cancelling done before the multiplication, not after.
The short answer
To multiply fractions, multiply the numerators together and the denominators together — no common denominator is needed. Cancel any shared factor across the diagonal first: 4/9 × 3/8 becomes 1/3 × 1/2 = 1/6. Mixed numbers must be converted to improper fractions before you start.
The questions
- Question 1Foundation2 marks
Work out 2/3 × 4/5.
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8/15
- Nothing cancels: 2 and 5 share no factor, and 4 and 3 share none.
- Multiply the numerators: 2 × 4 = 8.
- Multiply the denominators: 3 × 5 = 15.
- 8/15, already in lowest terms.
- Question 2Foundation2 marks
Work out 2/5 × 20.
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8
- Write 20 as 20/1.
- 20 and 5 cancel by 5, leaving 2/1 × 4/1.
- 2 × 4 = 8.
- Or in words: 20 ÷ 5 = 4, then 4 × 2 = 8.
- Question 3Foundation2 marks
Work out 3/8 × 4, giving your answer as a mixed number.
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1 1/2
- Write 4 as 4/1. Only the numerator is multiplied by 4, not the denominator.
- 4 and 8 cancel by 4, leaving 3/2 × 1/1.
- 3 × 1 = 3 and 2 × 1 = 2, giving 3/2.
- 3/2 = 1 1/2.
- Question 4Core3 marks
Work out 4/9 × 3/8, cancelling first.
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1/6
- 4 and 8 share a factor of 4: they become 1 and 2.
- 3 and 9 share a factor of 3: they become 1 and 3.
- The calculation is now 1/3 × 1/2.
- 1 × 1 = 1 and 3 × 2 = 6, giving 1/6.
- Multiplying first would have given 12/72, which needs dividing by 12.
- Question 5Core3 marks
Work out 5/6 × 9/10.
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3/4
- 5 and 10 cancel by 5: they become 1 and 2.
- 9 and 6 cancel by 3: they become 3 and 2.
- The calculation is now 1/2 × 3/2.
- 1 × 3 = 3 and 2 × 2 = 4, giving 3/4.
- Question 6Core3 marks
Work out 7/12 × 8/21.
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2/9
- 7 and 21 cancel by 7: they become 1 and 3.
- 8 and 12 cancel by 4: they become 2 and 3.
- The calculation is now 1/3 × 2/3.
- 1 × 2 = 2 and 3 × 3 = 9, giving 2/9.
- Without cancelling this is 56/252, which needs dividing by 28.
- Question 7Core2 marks
Work out 2/3 of 45.
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30
- "Of" means multiply: 2/3 × 45.
- Divide by the denominator first to keep the numbers whole: 45 ÷ 3 = 15.
- Multiply by the numerator: 15 × 2 = 30.
- Check the other order: 45 × 2 = 90, and 90 ÷ 3 = 30.
- Question 8Stretch4 marks
Work out 1 2/3 × 2 1/4.
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3 3/4
- Convert both to improper fractions: 1 2/3 = 5/3 and 2 1/4 = 9/4.
- 9 and 3 cancel by 3: they become 3 and 1.
- The calculation is 5/1 × 3/4.
- 5 × 3 = 15 and 1 × 4 = 4, giving 15/4.
- 15/4 = 3 3/4.
- Question 9Stretch4 marks
Work out 3 1/5 × 1 7/8.
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6
- Convert: 3 1/5 = 16/5 and 1 7/8 = 15/8.
- 16 and 8 cancel by 8: they become 2 and 1.
- 15 and 5 cancel by 5: they become 3 and 1.
- The calculation is 2/1 × 3/1 = 6.
- Multiplying without cancelling gives 240/40, which is the same answer by a longer road.
- Question 10Stretch4 marks
Work out (3/4)² × 8/9.
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1/2
- (3/4)² squares both parts: 9/16.
- The calculation is 9/16 × 8/9.
- 9 cancels with 9, leaving 1 and 1.
- 8 and 16 cancel by 8, leaving 1 and 2.
- 1/2 × 1/1 = 1/2.
- Question 11Stretch4 marks
A recipe for 4 people uses 3/4 kg of flour. How much flour is needed for 10 people?
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1 7/8 kg
- 10 people is 10/4 of the recipe, which is 2 1/2 times.
- 3/4 × 10/4: 10 and 4 cancel by 2, giving 3/4 × 5/2.
- 3 × 5 = 15 and 4 × 2 = 8, so 15/8 kg.
- 15/8 = 1 7/8 kg.
- Check: one person needs 3/16 kg, and 10 × 3/16 = 30/16 = 15/8.
- Question 12Stretch4 marks
5/8 of a class of 32 are girls, and 2/5 of the girls wear glasses. How many girls wear glasses?
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8
- Girls: 5/8 × 32. Cancel 32 with 8 to get 4, so 5 × 4 = 20 girls.
- Girls with glasses: 2/5 × 20.
- Cancel 20 with 5 to get 4, so 2 × 4 = 8.
- Eight girls wear glasses.
- Check in one step: 2/5 × 5/8 × 32 = 1/4 × 32 = 8.
Where these go wrong
- Finding a common denominator before multiplying, which is the addition procedure applied to the wrong operation.
- Multiplying mixed numbers part by part, so 1 2/3 × 2 1/4 becomes 2 2/12 instead of 15/4.
- Multiplying a fraction by a whole number by scaling both parts, turning 3/8 × 4 into 12/32.
- Cancelling two numerators against each other, or two denominators, instead of cancelling across the diagonal.
- Squaring only the numerator in (3/4)², giving 9/4 rather than 9/16.
Why this is the easy operation and addition is the hard one
Adding fractions needs a common denominator because you can only add pieces of the same size. Multiplying does not, because it is not counting pieces at all — 2/3 × 4/5 asks for two thirds of four fifths, and the answer is a new size of piece: fifteenths.
That is why the rule is so short. Multiply across the top, multiply across the bottom, and the denominators multiply to give the size of the new piece: 3 × 5 = 15. Pupils who find a common denominator first have imported the addition procedure, and while their answer often survives it, the extra work is entirely wasted.
Cancel before multiplying, not after
7/12 × 8/21 can be multiplied out to 56/252 and then simplified, which means finding that 28 divides both. Or the 7 and the 21 can be cancelled by 7, and the 8 and the 12 by 4, leaving 1/3 × 2/3 = 2/9 with no simplifying at the end.
The cancelling goes diagonally: any numerator may cancel with any denominator, because all the numerators end up on top and all the denominators end up underneath. What is not allowed is cancelling two numerators against each other, or two denominators.
The saving grows with the numbers. In 3 1/5 × 1 7/8 the improper fractions are 16/5 and 15/8; cancelling 16 with 8 and 15 with 5 leaves 2 × 3 = 6, an answer reached without multiplying anything larger than six.
"Of" means multiply
Two thirds of 45 is 2/3 × 45, and 5/8 of a class of 32 is 5/8 × 32. Recognising "of" as multiplication is what turns a worded question into an arithmetic one, and it is also what makes percentages of amounts the same skill later on.
For a fraction of a whole number, divide by the denominator and multiply by the numerator, in whichever order keeps the numbers whole. 2/3 of 45 is easier as 45 ÷ 3 = 15, then 15 × 2 = 30, than as 90 ÷ 3.
Common questions
Why does multiplying by a fraction make the answer smaller?
Because multiplying by a number below 1 is taking a part of something. 2/3 × 45 asks for two thirds of 45, which must be less than 45. The expectation that multiplication always enlarges comes from years of whole numbers, and naming it out loud usually settles it faster than another worked example.
Is cancelling first compulsory?
No. Multiplying straight out and simplifying afterwards reaches the same answer and earns the same marks. Cancelling first is worth the habit because it keeps the numbers small: 7/12 × 8/21 is 1/3 × 2/3 if you cancel, and 56/252 if you do not, and spotting that 28 divides 56 and 252 is the harder job.
Can mixed numbers be multiplied without converting them?
Only by expanding all four products, which is more work and where most errors come from. Converting 1 2/3 to 5/3 takes one line and makes the rest mechanical. The attractive shortcut — multiplying whole by whole and fraction by fraction — is simply wrong, and it is worth showing a pupil once with numbers where the error is obvious.
When do children learn to multiply fractions?
In England, multiplying pairs of proper fractions and writing the answer in its simplest form is a Year 6 expectation, with mixed numbers and harder cancelling coming in Year 7 and 8. The stretch questions here sit in that secondary range rather than at primary level.
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