Fractions, decimals and percentages
Fractions of amounts, equivalent fractions, decimals and percentages: what each question asks, the quick way to answer it, and the traps the wrong answers set.
Chapter 1
What are fractions questions?
A fraction is part of a whole that’s been cut into equal parts. 5/8 means: cut the whole into 8 equal parts, and take 5 of them.
The bottom number, the denominator. How many equal parts the whole is cut into. Denominator and down both start with d.
The top number, the numerator. How many of those parts you take.
Most questions ask for a fraction of an amount, like this one. The rest swap a fraction for one the same size, compare fractions, or turn them into decimals and percentages. This page covers them all.
5/8 of 72 = ?
The options
- Option 1: 27
- Option 2: 36
- Option 3: 45
- Option 4: 9
A Maths question from Rush, level 2.
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Equal parts, always
Eighths only means something if all 8 parts are the same size. If they aren’t, they aren’t eighths.
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Same amount, new numbers
These are equivalent fractions: the same size, written differently. The one with the smallest numbers, 1/2, is the simplest form.
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Three ways to write it
Decimals and percentages are fractions too. Per cent means out of 100, so 50% is 50/100, which is a half.
Chapter 2
Solve it in 4 steps.
Here’s the question, one step at a time.
The question
5/8 of 72 = ?
The options
- Option 1: 27
- Option 2: 36
- Option 3: 45
- Option 4: 9
A Maths question from Rush, level 2.
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Step 1
Size it up.
Is the fraction more or less than a half? 5 is more than half of 8, so 5/8 is more than a half. Half of 72 is 36, so the answer is bigger than 36. That already rules out 9, 27 and 36.
More than half: bigger than 36.
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Step 2
Divide by the denominator (the bottom number).
The bottom number says how many equal parts: 8. So one part is 72 ÷ 8 = 9.
72 ÷ 8 = 9 in each part.
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Step 3
Multiply by the numerator (the top number).
The top number says how many parts you take: 5. Five parts of 9 is 9 × 5 = 45.
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Step 4
Check it adds up.
The 3 parts you didn’t take make 27. And 45 + 27 = 72, the whole amount. It adds up, so 45 is right.
45 + 27 = 72.
Three mistakes this question sets
| The mistake | Do this instead |
|---|---|
| Stopping at one part. 9 is one eighth: 72 ÷ 8. That’s only half the job. | Multiply by the numerator. You want 5 parts, not 1. |
| Giving the part left. 27 is the 3 parts you didn’t take: three eighths, not five. | Check which part the question asks for. Here it’s the 5 you take. |
| Taking a half. 36 is half of 72. Five eighths is close to a half, but more. | Size it up first: more than a half means bigger than 36. |
So it’s 45. Five parts of 9. And 45 is more than half of 72, just as step 1 said.
Remember: Size it up. Bottom, then top.
Chapter 3
Everything else Rush asks.
Each card is one kind of question: the rule, one worked example, and the trap. They all use the same idea: a fraction is equal parts.
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- 10% of 160 = 16
- 30% = 48
- 5% = 8
- 35% = 48 + 8 = 56
Percentages: find 10% first
Divide by 10 for 10%, then build.
35% of 160 is three lots of 10% (48) plus half of 10% (8): 56. Trap: 16 and 48 are steps on the way, not the answer.
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=
× 4, top and bottom
Missing numbers
Do the same to the top and bottom. Multiply, never add.
3/5 = ?/20. The bottom went from 5 to 20, which is × 4, so the top goes × 4 too: 12. Trap: 18, from adding the 15 the bottom gained.
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=
÷ 6, top and bottom
Simplest form
Divide both by the biggest number that goes into both.
6 goes into 18 and 24, so 18/24 is 3/4. Trap: halving once to 9/12 and stopping. 3 still goes into both, so keep going.
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=
Improper fractions and mixed numbers
The bottom number never changes.
34/9 is an improper fraction: the numerator is bigger than the denominator. How many 9s in 34? Three (27), with 7 left, so 3 7/9. Trap: putting the 7 over 34. Going back: 3 × 9 + 7 = 34, so 3 7/9 is 34/9.
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=<=
Both in twelfths
Which is bigger?
Same top? The smaller bottom wins. Otherwise, match the bottoms.
2/3 beats 2/5, because thirds are bigger parts than fifths. Which is biggest: 2/3, 3/4 or 7/12? Turn them all into twelfths: 8/12, 9/12 and 7/12, so 3/4 wins. Trap: 7/12 has the biggest numbers, but it’s the smallest.
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+=+=
Adding and taking away
Match the bottoms, then add the tops.
You can only add parts the same size. 3/5 is 6/10, so 3/5 + 3/10 = 9/10. Trap: adding the bottoms. 6/15 is just 2/5, smaller than you started with.
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÷ 2 =
Times and share
Times: the top grows. Share: the bottom grows.
2/7 × 5 = 10/7: five lots of two sevenths. 1/3 ÷ 2 = 1/6: half of a third is a sixth, because each part gets smaller. Fraction times fraction: top × top, bottom × bottom (1/2 × 5/6 = 5/12).
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0.7=0.70>0.69
Decimals and percentages
Make the bottom 100.
3/20 = 15/100 = 15% = 0.15. To compare a mix, turn them all into percentages: 0.7 is 70%, 0.69 is 69%. Trap: 0.69 looks bigger because it’s longer. It isn’t.
Learn these by heart
They come up again and again, and knowing them saves the working. 1/8 is half of 1/4, so it’s half of 25%: 12.5%.
- 0.550%
- 0.2525%
- 0.7575%
- 0.220%
- 0.110%
- 0.12512.5%
- 0.333…33⅓%
- 0.055%
Chapter 4: In the exam
A fraction of what’s left.
The exam’s longer questions take a fraction, then a fraction of what’s left. Rush doesn’t ask these, but the bar makes them easy.
The questionSam has 60 stickers. He gives 1/3 to his sister, then 1/4 of what’s left to his friend. How many does Sam have now?
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Step 1
Take the first fraction.
A third of 60 is 60 ÷ 3 = 20, so his sister gets 20. Sam has 40 left.
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Step 2
Draw a new bar for what’s left.
The quarter is of the 40 he has now, not the 60 he started with. 40 ÷ 4 = 10, so his friend gets 10.
A new bar: just the 40.
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Step 3
Count what’s left.
Three parts of 10 are left: 30. Check: 20 + 10 + 30 = 60.
Two mistakes this question sets
| The mistake | Do this instead |
|---|---|
| Taking the quarter of the whole. A quarter of 60 is 15, and 60 − 20 − 15 = 25. But the quarter is of what’s left. | Read “of what’s left” and draw a new bar for it. |
| Stopping after one step. 40 is what Sam has after his sister, before his friend. | Count the steps in the question. Two gifts means two bars. |
So it’s 30. A quarter of the 40 he had left, not of the 60 he started with.
Remember: Of what’s left? Draw a new bar.
Other ways the exam asks it
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What fraction is shaded?
Count the shaded squares (6) and all the squares (16): 6/16, which simplifies to 3/8. Trap: comparing shaded with white (6/10) instead of with the whole.
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Pieces that aren’t equal
Four pieces, one shaded, but it isn’t 1/4: the pieces aren’t the same size. The shaded one would fit into the whole 8 times, so it’s 1/8.
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How many don’t?
30 children, and 2/5 walk to school. How many don’t? 30 ÷ 5 = 6, so 12 walk and 18 don’t. Trap: 12 is the ones who do. Read what it asks for twice.
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Working backwards
2/5 of a number is 18. What’s the number? Two parts make 18, so one part is 9, and all 5 parts are 45. Share by the top, then multiply by the bottom.
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40%0.40.25
Which is NOT equal?
Which is not equal to 2/5? Turn each into a percentage: 4/10, 6/15 and 0.4 are all 40%. 0.25 is 25%: it just copies the 2 and the 5.
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80p of 400p = 20%
Different units
What percentage of £4 is 80p? Make the units the same first: £4 is 400p. 80/400 = 20/100 = 20%. Trap: 80%, from forgetting that £4 is 400p.
Percentage increase and decrease
Find the percentage, then take it off or add it on.
A £45 coat has 20% off in a sale. 10% is £4.50, so 20% is £9, and the price is £45 − £9 = £36. Trap: £9, the money off, not the new price. A 20% rise adds it on instead: £54.
- 10% of £45 = £4.50
- 20% = £9
- £45 − £9 = £36
Chapter 5
Try it yourself.
Six real questions from Rush, easy to hard, 60 seconds on the clock.
Use the steps
- Size it up.
- Divide by the denominator (the bottom number).
- Multiply by the numerator (the top number).
- Check it adds up.
Remember: Size it up. Bottom, then top.
For parents.
What parents ask us about fractions, decimals and percentages.
Questions parents ask
How do you find a fraction of an amount?
Divide by the bottom number, then multiply by the top. For 3/5 of 35: 35 ÷ 5 = 7 is one fifth, and three fifths is 21. Size it up first: three fifths is more than a half, so the answer must be more than 17½.
Which fractions, decimals and percentages should you know by heart?
Halves, quarters, fifths, tenths and eighths: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/10 = 0.1 = 10% and 1/8 = 0.125 = 12.5%. Learn 1/3 too, which is about 0.33 or 33⅓%. With these, most conversions need no working at all.
How do you simplify a fraction?
Divide the top and bottom by the biggest number that goes into both. For 20/25 that's 5, giving 4/5. If you divide by a smaller number first, see if it divides again: 12/18 halves to 6/9, which still divides by 3, to 2/3.
How do you add fractions with different denominators?
Change them so the bottoms match, then add the tops. If one bottom goes into the other, only one fraction changes: 1/2 + 3/8 = 4/8 + 3/8 = 7/8. If not, find a number both go into: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
Rush: 11+ Practice & KS2 SATsPractise fractions, decimals and percentages in Rush.
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