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Percentage changes

Percentage increase and decrease, and an introduction to reverse percentages.

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Prices go up. Sales take money off. Plants grow. A percentage change tells you how much something goes up or down, as a percentage of what it was before. You will use the building blocks from Lesson 3.7 (50%, 25%, 10%, 5% and 1%). In this lesson you will learn how to:

  • increase an amount (make it bigger) and decrease it (make it smaller) by a percentage
  • work out sale prices, like 30% off
  • use the new percentage: 30% off leaves 70%
  • work backwards to find the amount before a change
  • spot a trap: 10% up and then 10% down does not take you back to the start
Dark blue: whole numbers Blue: decimals, the parts of a whole Red dot: the decimal point

1. Going up and going down

Key idea The amount you start with is 100%. To increase, find the change and add it on. To decrease, find the change and take it away.

Increase 40 by 30%

4444444444444100% = 40+30% = 12

10% of 40 is 4, so 30% is 3 blocks of 4, which is 12. Add it on: 40 + 12 = 52.

Decrease 40 by 30%

4444444444100% = 4070% left = 28−30% = 12

30% of 40 is 12 again. Take it away: 40 − 12 = 28.

Words to look out for

  • Increase words: rise, grow, go up, a pay rise, more, extra.
  • Decrease words: fall, drop, go down, reduce, a discount, a sale, “30% off”.

2. Method 1: find the change, then add or take away

Key idea Two jobs. First find the change with your building blocks. Then add it on or take it away.

Example: increase 80 by 15%

  1. Find 10%:80 ÷ 10 = 8
  2. 5% is half of 10%:half of 8 = 4
  3. The change is 15% = 10% + 5%:8 + 4 = 12
  4. It is an increase, so add it on:80 + 12 = 92
Answer80 increased by 15% is 92

Sale price: 30% off a £60 coat

Coat£60£4230% off

10% of £60 is £6, so 30% is £18. Take it off: £60 − £18 = £42.

With pence: increase £18 by 15%

10% of £18 is £1.80, and 5% is half of that, £0.90. So 15% is £2.70.

Add it on: £18 + £2.70 = £20.70.

Watch out The change is not the answer. 15% of 80 is 12, but 80 increased by 15% is 80 + 12 = 92.

3. Method 2: the new percentage

Key idea Start at 100%. A 30% decrease leaves 100% − 30% = 70%. A 25% increase makes 100% + 25% = 125%. Then find that percentage in one go.

A 30% decrease leaves 70%

0%50%100%150%−30%70%

So 40 decreased by 30% is the same as 70% of 40. Find 10% first: 40 ÷ 10 = 4, so 70% is 4 × 7 = 28, the same answer as section 1.

Decrease 60 by 75%

100% − 75% = 25% is left. 25% of 60 is a quarter of 60, which is 15.

Method 1 works too: 75% of 60 is 45, and 60 − 45 = 15. It is just longer.

Everything is 90% off!

A £50 game is 90% off. You pay 100% − 90% = 10% of the price.

10% of £50 is £5.

Key tip Use whichever method is quicker. A big decrease, like 75% or 90% off, is much quicker with the new percentage.

5. Working backwards: reverse percentages

Key idea Sometimes you know the amount after the change and need the amount before. The amount before is 100%. Find what percentage you know, find one block, then build back up to 100%.

30% of a number is 12. What is the number?

444444444430% = 12100% = 40

30% is 3 blocks of 10%, so one block is 12 ÷ 3 = 4. The number is all 10 blocks: 4 × 10 = 40.

A plant grows by 25% to 50 cm. How tall was it before?

1010101010125% = 50 cm100% = 40 cm

50 cm is 100% + 25% = 125%: 5 blocks of 25%. One block is 50 ÷ 5 = 10. Before, it was 4 blocks: 40 cm.

Example: after 20% off, a coat costs £48. What did it cost before?

£6£6£6£6£6£6£6£6£6£680% = £48−20%100% = £60
  1. What percentage is £48?100% − 20% = 80%
  2. 80% is 8 blocks of 10%, so one block is:£48 ÷ 8 = £6
  3. The price before is 100%, all 10 blocks:£6 × 10 = £60
  4. Check: 20% of £60 is £12, and£60 − £12 = £48 ✓
AnswerThe coat cost £60 before
Watch out Do not take 20% of £48 and add it back on. That gives £48 + £9.60 = £57.60, which is wrong: the 20% was 20% of the price before, not of £48.

6. A trap: up and then down again

Key tip Going up 10% and then down 10% does not take you back to the start. The second 10% is 10% of a different amount.

Start with 200. Up 10%, then down 10%.

  1. Up 10%: 10% of 200 is 20.200 + 20 = 220
  2. Down 10%: 10% of 220 is 22.220 − 22 = 198

More came off (22) than went on (20), so you end up below where you started.

Two discounts are not the same as one big one

£100 with 20% off is £80. Then 10% off £80 takes £8 more: £72.

30% off £100 in one go is £70. So 20% off and then 10% off is not 30% off.

The mistakes that cost marks

  1. Stopping at the change. Giving 12 as 80 increased by 15%. 12 is only the change. Add it on: 80 + 12 = 92.
  2. Adding the percentage as a number. Working out 80 increased by 15% as 80 + 15 = 95. 15% of 80 is 12, not 15. Find the percentage first, then add it.
  3. Going the wrong way. Adding on for “30% off”. “Off”, “fall” and “discount” mean take away.
  4. Working backwards from the new amount. Adding 20% of £48 back on to undo 20% off. £48 is 80% of the price before. One 10% block is £6, so 100% is £60.
  5. Thinking up and down cancel out. Saying 200 up 10% and then down 10% is 200 again. The second 10% is 10% of 220: 220 − 22 = 198.
  6. Money with one decimal place. Writing £20.7. write £20.70. Pence always take two places.

Quick checklist

Tick each one when you can do it without help.

Download

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Year 4 practice paper PDF · 11+ style, with answer sheet and answers Year 5 practice paper PDF · 11+ style, with answer sheet and answers Year 6 practice paper PDF · 11+ style, with answer sheet and answers

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