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GCSE Percentage Revision Notes

Foundation and HigherRevision Notes

Topics labelled Foundation appear on both tiers. Topics labelled Higher are examined on the Higher tier only.

1. Percentage of an amountFoundation

Use the multiplier method: multiply the amount by the decimal equivalent of the percentage.

Worked example

Find 35% of 240.

Multiplier = 35 ÷ 100 = 0.35

240 × 0.35 = 84

2. Percentage increase and decreaseFoundation

For an increase, the multiplier is greater than 1. For a decrease, the multiplier is less than 1.

Increase

Increase £320 by 15%.

Multiplier = 1 + 0.15 = 1.15

£320 × 1.15 = £368

Decrease

Decrease £320 by 15%.

Multiplier = 1 - 0.15 = 0.85

£320 × 0.85 = £272

3. Expressing as a percentageFoundation

(part ÷ whole) × 100

Worked example

A student scores 63 out of 90. What percentage is that?

(63 ÷ 90) × 100 = 0.7 × 100 = 70%

4. Percentage changeFoundation

Percentage change = (change ÷ original) × 100

Worked example

A jacket was £85, now costs £102. What is the percentage increase?

Change = £102 - £85 = £17

Percentage change = (17 ÷ 85) × 100 = 20%

5. Reverse percentagesHigher

A reverse percentage question gives you the new value after a percentage change and asks for the original value. Divide by the multiplier.

Original = new value ÷ multiplier

Worked example

After a 20% increase, a price is £144. Find the original price.

Multiplier for a 20% increase = 1.20

Original = 144 ÷ 1.20 = £120

Common mistake

Do NOT subtract 20% from £144. That gives £144 - £28.80 = £115.20, which is wrong because you would be subtracting 20% of the new (larger) value, not 20% of the original.

6. Simple interestFoundation

Simple interest is calculated on the original amount only. The interest does not grow year on year.

Interest = P × r × t ÷ 100

where P = principal, r = rate per year (%), t = number of years

Worked example

£500 is invested at 3% simple interest for 4 years. How much interest is earned?

Interest = 500 × 3 × 4 ÷ 100 = £60

Total amount = £500 + £60 = £560

7. Compound interestHigher

Compound interest is calculated on the original amount plus all accumulated interest. The total grows faster than simple interest.

A = P(1 + r/100)n

A = final amount, P = principal, r = annual rate (%), n = number of years

Worked example

£2,000 is invested at 5% compound interest for 3 years. Find the total amount.

A = 2000 × (1 + 5/100)3 = 2000 × 1.053

1.053 = 1.157625

A = 2000 × 1.157625 = £2,315.25

8. DepreciationHigher

Depreciation is a repeated percentage decrease in value. Use the same formula as compound interest, but with a multiplier less than 1.

Worked example

A car is worth £15,000. It depreciates at 12% per year for 4 years. Find its value after 4 years.

Multiplier = 1 - 0.12 = 0.88

A = 15000 × 0.884 = 15000 × 0.59969536

A = £8,995.43 (to the nearest penny)

9. Repeated and combined percentage changesHigher

When two percentage changes happen one after the other, multiply the multipliers together to find the overall multiplier.

Worked example

A price increases by 20%, then decreases by 20%. What is the overall percentage change?

Combined multiplier = 1.20 × 0.80 = 0.96

0.96 means the final price is 96% of the original, so there is an overall 4% decrease.

This result surprises many students. The two 20% changes do not cancel out, because each percentage is applied to a different base value.

10. Percentage profit and lossFoundationHigher

Percentage profit and loss are always calculated as a fraction of the cost price, not the selling price.

% profit = (profit ÷ cost price) × 100

% loss = (loss ÷ cost price) × 100

Worked example

A trader buys an item for £40 and sells it for £52. What is the percentage profit?

Profit = £52 - £40 = £12

% profit = (12 ÷ 40) × 100 = 30%

11. Practice questions

Difficulty is indicated after each question. Attempt all questions before checking your answers.

  1. Find 42% of 350. [Foundation]
  2. A television costs £480. It is reduced by 35% in a sale. Find the sale price. [Foundation]
  3. A shop buys a lamp for £18 and sells it for £27. Find the percentage profit. [Foundation]
  4. A house is valued at £180,000. Over one year its value falls to £162,000. Find the percentage decrease. [Foundation]
  5. £3,500 is invested at 2.5% simple interest per year. How much interest is earned in 6 years? [Foundation]
  6. After a 30% increase, a price is £195. Find the original price. [Higher]
  7. £5,000 is invested at 4% compound interest per year. Find the total amount after 5 years. Give your answer to the nearest penny. [Higher]
  8. A motorbike is bought for £8,400. It depreciates by 18% per year. Find its value after 3 years, to the nearest pound. [Higher]
  9. A price increases by 10% and then increases by a further 10%. Find the overall percentage increase. [Higher]
  10. A shop has a sale. All prices are reduced by 20%, then reduced by a further 15%. What single percentage discount is this equivalent to? [Higher]

Answers

  1. 350 × 0.42 = 147
  2. 480 × 0.65 = £312
  3. Profit = £9. % profit = (9 ÷ 18) × 100 = 50%
  4. Change = £18,000. % decrease = (18000 ÷ 180000) × 100 = 10%
  5. Interest = 3500 × 2.5 × 6 ÷ 100 = £525
  6. Original = 195 ÷ 1.30 = £150
  7. A = 5000 × 1.045 = 5000 × 1.2166529... = £6,083.26
  8. A = 8400 × 0.823 = 8400 × 0.551368 = £4,632 (to nearest £)
  9. Combined multiplier = 1.10 × 1.10 = 1.21, so overall 21% increase
  10. Combined multiplier = 0.80 × 0.85 = 0.68, so overall 32% discount

Related resources

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