Percentages: revision notes and practice

What you will learn

• Convert percentages into fractions and decimal multipliers.
• Calculate a percentage of an amount and a percentage change.
• Recover an original amount after an increase or decrease.
• Explain repeated changes and distinguish simple from compound growth.

Key rules

Find a percentage

p% of an amount = amount × p/100. For example, 15% uses the multiplier 0.15.

Change an amount

For a p% increase multiply by 1 + p/100. For a p% decrease multiply by 1 − p/100.

Work backwards

Original amount = final amount ÷ change multiplier. Round money to the nearest penny only at the end.

Worked examples

Example 1: Find 15% of 240.

1. Write 15% as 0.15.
2. Calculate 240 × 0.15 = 36.

36

Example 2: A price is £72 after a 20% increase. Find the original price.

1. A 20% increase uses the multiplier 1.20.
2. The final price is 120% of the original.
3. Divide: 72 ÷ 1.20 = 60.

£60.00

A practice question

What is 15% of 240?

1. Percentages describe parts of a whole

Percent means out of 100. Therefore 35% = 35/100 = 0.35. A percentage compares an amount with a stated whole, which is treated as 100%. Identifying that whole is more important than memorising a calculator button.

Percentages can be greater than 100%. If one quantity is 150% of another, it is one and a half times as large. Do not confuse 150% of an amount with an increase of 150%: the latter adds 1.5 times the original to the original, producing 250% altogether.

2. Find a percentage of an amount

Convert the percentage to a decimal or fraction, then multiply. To find 18% of 250, calculate 0.18 × 250 = 45. The percentage acts as a multiplier applied to the amount; it is not a number to add to it.

Without a calculator, build useful percentages from easy parts. Ten percent of 80 is 8, five percent is half of that, or 4, and one percent is 0.8. Thus 16% of 80 is 8 + 4 + 0.8 = 12.8. Choose combinations that match the percentage you need.

p% of A = A × p/100

3. Use a multiplier for a change

After a 12% increase, the final amount is 100% + 12% = 112% of the original. Use the multiplier 1.12. A £50 price therefore becomes £56. After a 12% decrease, 88% remains, so use 0.88 and obtain £44.

This one-step method includes the original amount and the change together. It is also valid to calculate the percentage first, then add or subtract it. Before calculating, predict whether the answer should be larger or smaller than the starting value; that checks whether your multiplier has the correct direction.

Increase: new = original × (1 + p/100). Decrease: new = original × (1 − p/100).

4. Work backwards to the original

A reverse percentage question gives the amount after a change. Write the relationship before rearranging. If a coat costs £68 after a 15% reduction, final = original × 0.85. Divide £68 by 0.85 to get the original price, £80.

Subtracting 15% of £68 would calculate a new reduction from the sale price; it would not undo the earlier change. With an increase, use the same inverse idea: a value of 132 after a 10% rise came from 132 ÷ 1.10 = 120. For a 100% decrease, the final value is zero and the original cannot be recovered by division.

Original = final ÷ change multiplier, when the multiplier is non-zero

5. Calculate the percentage change

First find the numerical difference, then compare it with the original amount. A price rising from £40 to £46 changes by £6. Since 6/40 = 0.15, the percentage increase is 15%. Using 46 as the denominator would answer a different comparison.

For a decrease from 80 to 68, the drop is 12, giving 12/80 × 100 = 15%. State whether the change is an increase or decrease. The standard percentage-change calculation needs a non-zero original value; a change from zero cannot be described by dividing by the original.

Percentage change = (change in amount ÷ original amount) × 100%

6. Combine successive changes

Each change applies to the amount at that stage. An increase of 20% followed by a decrease of 20% uses 1.20 × 0.80 = 0.96, so the final amount is 96% of the original: an overall decrease of 4%. Equal percentages in opposite directions do not necessarily cancel.

Apply the multipliers in sequence, or multiply them together before applying the result to the original. A £100 item reduced by 10%, then by another 10%, costs 100 × 0.9 × 0.9 = £81. Two 10% discounts produce a 19% overall reduction, rather than 20%.

7. Recognise simple and compound growth

With simple interest, each year's interest is calculated from the initial principal. At 4% simple interest, £200 earns £8 each year, so after three years the total is £224. This is a mathematical model; actual financial products may use different terms.

With annual compounding at a fixed rate, each year's growth is applied to the updated balance. The same £200 at 4% compound growth for three years gives 200 × 1.04³ = £224.9728, or £224.97 when rounded at the end. Repeated percentage decrease uses a multiplier below 1 in the same way.

Fixed compound growth: final = original × multiplierⁿ

8. Compare fairly and round at the end

To express one amount as a percentage of another, divide by the comparison whole and multiply by 100. If 18 out of 24 questions are correct, the accuracy is 18/24 × 100 = 75%. Keep the same whole when comparing two results.

A rise from 20% to 25% is 5 percentage points, but a 25% relative increase because 5/20 × 100 = 25%. Keep full calculator accuracy during reverse or repeated calculations. Round money to the nearest penny at the final step, unless the question specifies a different rounding rule.

Check your understanding

How to earn the marks

• Underline which amount represents 100%.
• Write the multiplier before a forward or reverse change.
• Use the original amount as the denominator for percentage change.
• Keep unrounded intermediate values and include the final unit.

A common mistake

Decreasing the final price by 20% does not undo a 20% increase; the percentages use different starting amounts.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Mathematics — N12, R9

  Percentage calculations, multipliers and reverse percentages are available at both tiers. Full growth and decay coverage is not included.

• Edexcel GCSE Mathematics — N12, R9

  Percentage calculations, multipliers and reverse percentages are available at both tiers. Full growth and decay coverage is not included.

• OCR GCSE Mathematics — 2.03

  Percentage calculations, multipliers and reverse percentages are available at both tiers. Full growth and decay coverage is not included.

Keep your next step small

Try five questions, read the feedback, and revisit one thing you found difficult. The app tracks a 20-answer practice goal separately from accuracy.

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