Percentages

Mathematik GCSE Updated Wed 7 Oct 2026

A percentage is a number out of 100. The quickest route through nearly every percentage question is a multiplier: a single decimal that does the increase or decrease in one step, and that you can divide by to work backwards.

Part 1 of 3: Learn it

In short

Find a multiplier: a 12% increase is × 1.12, a 12% decrease is × 0.88.
Percentage change = change ÷ original × 100.
Reverse percentage: divide the new amount by the multiplier to get the original.

Where this is in your specification

Spec points: DfE N12 and R9 (repeated percentage change and compound interest: R16)

Board	Topic: Fractions, decimals and percentages
DfE content	N8, N10, N11, N12, R9
AQA 8300	Number; Ratio, proportion and rates of change
Edexcel 1MA1	Number; Ratio, proportion and rates of change
OCR J560	2 Fractions, decimals and percentages
Eduqas C300	HN8, HN10 to HN12, HR10
Cambridge IGCSE 0580	E1.4, E1.13
National 5 C847 75	Numerical skills: fractions, percentages, appreciation and depreciation

Percentages, decimals and fractions

Percentage	Decimal	Fraction
50%	0.5	1/2
25%	0.25	1/4
20%	0.2	1/5
12.5%	0.125	1/8
5%	0.05	1/20

To change a percentage to a decimal, divide by 100. Without a calculator, build answers from 10% (divide by 10) and 1% (divide by 100): 35% of 260 is 3 × 26 + 13 = 91.

Multipliers

Start from 100%. An increase of r% gives (100 + r)%, a decrease gives (100 − r)%. Write that as a decimal and multiply once.

Increase by 8%: × 1.08
Decrease by 30%: × 0.7
Increase by 2.5%: × 1.025

Percentage change

percentage change = (new − original) ÷ original × 100

Always divide by the original value, not the new one. A negative answer is a decrease. The same idea gives percentage profit and percentage error.

Reverse percentages

If you know the amount after a change and want the amount before, divide by the multiplier. After a 15% rise the price is 115% of the original, so divide by 1.15.

Simple interest adds the same amount every year. Compound interest multiplies by the same multiplier every year: £P after n years at r% is P × (1 + r/100)ⁿ.

Quick check

Write down the multiplier for a 7% decrease.

Show the answer

100 − 7 = 93%, so × 0.93.

Part 2 of 3: See it worked

Worked examples

Example 1

A train ticket cost £250 last year and £215 this year. Find the percentage change.

Change = 215 − 250 = −35
−35 ÷ 250 = −0.14
−0.14 × 100 = −14%

Answer: A 14% decrease.

Example 2

In a sale everything is reduced by 20%. A jacket costs £68 in the sale. What was its original price?

Sale price = 80% of the original, multiplier 0.8
Original = 68 ÷ 0.8
= 85

Answer: £85.

Example 3

£2400 is invested at 3% compound interest per year. How much is it worth after 4 years, to the nearest penny?

Multiplier for each year: 1.03
2400 × 1.03⁴ = 2400 × 1.12550881
= 2701.2211...

Answer: £2701.22.

Common mistakes

Working out a reverse percentage by taking the percentage off the new price. 20% of £68 is not the amount that was taken off.
Dividing by the new value in a percentage change.
Using 1.3 for a 3% increase. 3% is 0.03, so the multiplier is 1.03.
Using simple interest when the question says compound.

Quick check

A coat costs £126 after a 5% price rise. What did it cost before?

Show the answer

126 ÷ 1.05 = £120.

Part 3 of 3: Test yourself

Check yourself

Answer each one in your head or on paper first, then open it to check.

Write down the multiplier for a 7% decrease.

100 − 7 = 93%, so × 0.93.

A coat costs £126 after a 5% price rise. What did it cost before?

126 ÷ 1.05 = £120.

A plant grows from 40 cm to 46 cm. Find the percentage increase.

6 ÷ 40 × 100 = 15%.

Jobs that use this

Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).

Diese Lernzettel sind auf Englisch, weil sie britischen Prüfungslehrplänen folgen.

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