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Key Stage 3 maths · Number

KS3 fractions, decimals and percentages: a guide for parents

In Years 7, 8 and 9 pupils learn to treat fractions, decimals and percentages as three ways of writing the same thing. They convert between fractions and terminating decimals, express one quantity as a percentage of another, work with percentage change and percentages over 100%, and use fractions and percentages as multipliers that act on an amount.

Fractions, decimals and percentages, explained

Pupils work interchangeably with terminating decimals and their matching fractions. To change a fraction to a decimal, divide the numerator by the denominator: 3 ÷ 8 = 0.375, and 7/2 = 7 ÷ 2 = 3.5. To go the other way, read the decimal using place value and simplify. 0.375 is 375 thousandths, which is 375/1000, and dividing the top and bottom by 125 gives 3/8. A terminating decimal is one that stops. Some fractions, such as 1/3, give decimals that go on for ever (0.333…). Turning recurring decimals back into fractions comes later, for pupils working at the higher level of GCSE, so at this stage the focus is on decimals that end.

A percentage means “parts per hundred”. 35% is 35/100, which simplifies to 7/20 and is 0.35 as a decimal. Pupils interpret percentages as fractions and as decimals, and express one quantity as a percentage of another by writing it as a fraction and converting. If a pupil scores 18 out of 24, then 18/24 = 3/4 = 75%. Percentages make comparison easier: 18 out of 24 (75%) is a better result than 21 out of 30 (70%). Pupils also work with percentages greater than 100%. 100% is the whole amount, 150% is one and a half times it, and 250% of £40 is £100.

A percentage change is a multiplication in disguise. To increase an amount by 15%, you want 100% + 15% = 115% of it, which is the same as multiplying by 1.15. To decrease by 35%, you keep 100% − 35% = 65%, which means multiplying by 0.65. Seeing it this way, pupils work out the new amount in one step, work backwards to the original amount, and find a percentage change using change ÷ original. A price that goes from £8 to £10 has gone up by £2, and 2/8 = 25%, so it is a 25% increase. Because percentages are always of a particular starting amount, a rise and a fall of the same percentage don’t cancel each other out.

Fractions and percentages also act as operators, meaning they do something to a quantity. 3/5 of 45 is 45 ÷ 5 × 3 = 27. 40% of 65 is 4 × 6.5 = 26, since 10% of 65 is 6.5. These ideas link directly to ratio, proportion and algebra later in the course. Pupils choose whichever form makes the calculation easier, such as 25% as a quarter or 0.2 as a fifth.

Words your child will hear

Terminating decimal
A decimal that stops after a certain number of digits, such as 0.375.
Percentage
A number of parts per hundred. 35% means 35 out of every 100.
Equivalent
Worth the same amount. 3/4, 0.75 and 75% are equivalent.
Percentage change
An increase or decrease expressed as a percentage of the original amount.
Multiplier
The number you multiply by to apply a change in one step. A 15% increase has multiplier 1.15.
Original value
The amount before a percentage change was applied.
Operator
A fraction or percentage used to act on an amount, such as 3/5 of 45.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

  • Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8)
  • Define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100%
  • Interpret fractions and percentages as operators

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A jacket costs £64. In a sale it is reduced by 35%. A week later the shop raises the sale price by 35%. What does the jacket cost at each stage, and is it back to £64?

  1. A 35% reduction leaves 100% − 35% = 65% to pay. As a decimal that is a multiplier of 0.65.
  2. Work out 65% of £64 by building it from 10%: 10% is £6.40, so 60% is 6 times that, and 5% is half of 10%.
    10% = £6.40
60% = £38.40
5%  = £3.20
65% = £41.60
  3. Now increase £41.60 by 35%. The multiplier is 1.35, so add 35% of £41.60 to £41.60.
    10% = £4.16
30% = £12.48
5%  = £2.08
35% = £14.56
  4. Add the increase to the sale price.
    £41.60 + £14.56 = £56.16
  5. The price is not back to £64. The second 35% was taken from a smaller amount. As a check, multiply the two multipliers: 0.65 × 1.35 = 0.8775, and 64 × 0.8775 = £56.16. That is 87.75% of the original, so the jacket is 12.25% cheaper than before the sale.

The sale price is £41.60 and the price after the increase is £56.16, which is not £64.

Common mistakes and how to help

Adding the percentage back to the sale price to find the original price

A coat costs £52 after a 20% discount. Pupils work out 20% of 52 (£10.40) and add it on to get £62.40. But the 20% was taken from the original price, not from £52.

What helps: The £52 is 80% of the original, so divide by 0.8: 52 ÷ 0.8 = £65. Check by taking 20% off £65, which is £13, leaving £52.

Writing a fraction as a decimal by putting a point between the numbers, so 1/4 becomes 1.4

A fraction has two numbers and a decimal point looks like a way of separating two numbers.

What helps: Remember that a fraction is a division. 1 ÷ 4 = 0.25, and 3/8 is 3 ÷ 8 = 0.375. Money can help: a quarter of £1 is 25p.

Believing a percentage can’t be more than 100%

In primary school 100% is “everything”, so anything bigger seems impossible.

What helps: Explain that 100% means the whole of the starting amount, and more than 100% means more than that. A price at 150% of last year’s is one and a half times as much.

Finding 5% by dividing by 5

Pupils know that 10% is a tenth, found by dividing by 10, and wrongly extend the pattern. Dividing by 5 actually gives 20%.

What helps: Divide by 10 to get 10%, then halve it to get 5%. Dividing by 100 gives 1%, and any percentage can be built from these pieces.

How to help at home

  • Sale-price detective. When you see “30% off” or “a quarter off”, ask your child to work out the new price and which offer is better. For a £45 item, 30% off gives £31.50 and a quarter off gives £33.75.
  • Matching sets. Make cards showing a fraction, its decimal and its percentage: 3/4, 0.75 and 75%, then 1/8, 0.125 and 12.5%. Shuffle and play pairs or sort them into triples. Go on to include fractions such as 7/20 that need working out.
  • Fractions and percentages of amounts. Splitting a bill, leaving a tip or working out a deposit all use fractions and percentages as operators. Ask “what is 3/5 of £45?” or “what is 40% of £65?” and let them choose a method.
  • Which is the better score? Compare 18 out of 24 with 21 out of 30 by converting both to percentages. Do this with sports results, quiz scores or test marks, anywhere a fair comparison is needed.
  • Percentages over 100. Look for statements such as “prices are 120% of last year’s” or “the recipe makes 150% of the usual amount”. Ask your child what they mean in plain English, and how to work out the new amount.

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