Percentage Change and Reverse Percentages

Percentages appear on every paper of OCR GCSE Mathematics (J560), from a quick "find $30\%$30% of an amount" to multi-step reverse-percentage problems on Higher. They are also one of the most useful pieces of mathematics in everyday life, behind every sale, interest rate, tax and statistic you meet. This lesson covers increasing and decreasing by a percentage using multipliers, expressing one quantity as a percentage of another, calculating percentage change (profit, loss and error), and the often-misunderstood reverse percentage, where you work back from a final amount to the original. The unifying idea throughout is the multiplier — a single decimal that captures a percentage change and lets you apply, combine and reverse changes with confidence.

This lesson builds AO1 fluency with multipliers, AO2 reasoning when you justify using the original amount as the base, and AO3 problem-solving when a percentage is one step inside a longer calculation.

Key Vocabulary

Term	Meaning
Percentage	A number out of $100$100 (per cent means "per hundred")
Multiplier	The decimal you multiply by to apply a percentage change
Percentage increase	The amount goes up by a stated percentage
Percentage decrease	The amount goes down by a stated percentage
Percentage change	The change written as a percentage of the original
Reverse percentage	Finding the original amount before a percentage change

Increasing and Decreasing with Multipliers

The fastest, most reliable method for percentage change is the multiplier. A multiplier is a single decimal that performs the whole change in one step, so there is no separate "find the percentage then add or subtract" stage where errors creep in. The idea is that the starting amount is $100\%$100% of itself, written as the decimal $1$1. An increase adds to that $100\%$100%, and a decrease takes away from it.

• To increase by $r\%$r%, multiply by $1 + \dfrac{r}{100}$1+100r​.
• To decrease by $r\%$r%, multiply by $1 - \dfrac{r}{100}$1−100r​.

So a $20\%$20% increase multiplies by $1.20$1.20 (because $100\% + 20\% = 120\% = 1.20$100%+20%=120%=1.20), and a $15\%$15% decrease multiplies by $0.85$0.85 (because $100\% - 15\% = 85\% = 0.85$100%−15%=85%=0.85). Writing the multiplier down clearly is also worth a method mark in the exam, even if your later arithmetic slips.

Change	Multiplier
Increase by $20\%$20%	$1.2$1.2
Increase by $4.5\%$4.5%	$1.045$1.045
Decrease by $15\%$15%	$0.85$0.85
Decrease by $7\%$7%	$0.93$0.93
Decrease by $2.5\%$2.5%	$0.975$0.975

Worked Example 1

A coat costs £80. In a sale the price is reduced by $25\%$25%. Work out the sale price.

A $25\%$25% decrease has multiplier $1 - 0.25 = 0.75$1−0.25=0.75.

Sale price $= 80 \times 0.75 = 60$=80×0.75=60, i.e. £60.

Answer: £60.

Worked Example 2

A train season ticket costs £1,240. The price rises by $6\%$6%. Work out the new price.

A $6\%$6% increase has multiplier $1.06$1.06.

New price $= 1{,}240 \times 1.06 = 1{,}314.40$=1,240×1.06=1,314.40, i.e. £1,314.40.

Answer: £1,314.40.

Worked Example 3

A population of $45{,}000$45,000 falls by $12\%$12%. Work out the new population.

Multiplier for a $12\%$12% decrease $= 0.88$=0.88.

New population $= 45{,}000 \times 0.88 = 39{,}600$=45,000×0.88=39,600.

Answer: $39{,}600$39,600.

Common error: finding $12\%$12% and forgetting to subtract it. The multiplier method does the whole calculation in one step and avoids this.

Worked Example 3b

A restaurant adds a $12.5\%$12.5% service charge to a bill of £48. Work out the total amount to pay.

A $12.5\%$12.5% increase has multiplier $1 + 0.125 = 1.125$1+0.125=1.125.

Total $= 48 \times 1.125 = 54$=48×1.125=54, i.e. £54.

Answer: £54.

Common error: working out $12.5\%$12.5% (which is £6) but then forgetting to add it to the £48. The multiplier $1.125$1.125 builds the "add it on" into a single step.

Without a Calculator (Paper 1)

On the non-calculator paper, build the percentage you need from easy "building-block" percentages: $10\%$10% (divide by $10$10), $5\%$5% (half of $10\%$10%) and $1\%$1% (divide by $100$100). Almost any percentage can be assembled from these. For example, $35\% = 30\% + 5\%$35%=30%+5%, and $17\% = 10\% + 5\% + 1\% + 1\%$17%=10%+5%+1%+1%. This "ladder" method is reliable because it uses only halving and simple multiplication, avoiding awkward decimal multipliers by hand.

Worked Example 4

Increase $240$240 by $35\%$35% without a calculator.

$10\%$10% of $240 = 24$240=24, so $30\% = 3 \times 24 = 72$30%=3×24=72. $5\% = \dfrac{24}{2} = 12$5%=224​=12. So $35\% = 72 + 12 = 84$35%=72+12=84.

New amount $= 240 + 84 = 324$=240+84=324.

Answer: $324$324.

Worked Example 4b

Work out $12.5\%$12.5% of £64 without a calculator.

A neat fact is that $12.5\% = \dfrac{1}{8}$12.5%=81​. Dividing by $8$8 is the same as halving three times: $64 \div 2 = 32$64÷2=32, $32 \div 2 = 16$32÷2=16, $16 \div 2 = 8$16÷2=8.

So $12.5\%$12.5% of $64$64 is $8$8, i.e. £8.

Answer: £8.

Common error: treating $12.5\%$12.5% as "about $12\%$12%" and estimating. Recognising the exact fraction $\dfrac{1}{8}$81​ gives an exact, quick answer on Paper 1.

Expressing One Quantity as a Percentage of Another

A very common question type asks you to write one quantity as a percentage of another — for example, a test mark out of a total, or the proportion of a class who walk to school. To write a part as a percentage of a whole, divide the part by the whole and multiply by $100$100:

$\text{percentage} = \dfrac{\text{part}}{\text{whole}} \times 100.$percentage=wholepart​×100.

Make sure the two quantities are in the same units before dividing — comparing pence with pounds, for instance, would give a nonsensical answer.

Worked Example 5

In a test, Maya scores $54$54 out of $72$72. Work out her score as a percentage.

$\dfrac{54}{72} \times 100 = 0.75 \times 100 = 75\%.$7254​×100=0.75×100=75%.

Answer: $75\%$75%.

Worked Example 5b

A football team won $18$18 of its $24$24 matches. Work out the percentage of matches the team won.

$\dfrac{18}{24} \times 100 = 0.75 \times 100 = 75\%.$2418​×100=0.75×100=75%.

Answer: $75\%$75%. Notice this is the same calculation as Maya's test score — "$18$18 out of $24$24" and "$54$54 out of $72$72" both simplify to the fraction $\dfrac{3}{4}$43​, which is $75\%$75%.

Percentage Change

So far you have been applying a percentage change. The reverse task is to measure a change that has already happened and express it as a percentage — for instance, working out the percentage profit a shop made, or the percentage by which a population fell. The single most important formula in this lesson is:

$\text{percentage change} = \dfrac{\text{change}}{\text{original}} \times 100.$percentage change=originalchange​×100.

This works for increases (profit) and decreases (loss). The "change" is the difference between the new and original amounts, and the base is always the original — not the new value. This is the rule students most often get wrong, so it is worth saying twice: divide by the value you started with.

Worked Example 6

A shopkeeper buys a bike for £150 and sells it for £189. Work out the percentage profit.

Change $= 189 - 150 = 39$=189−150=39, i.e. a £39 profit.

$\text{percentage profit} = \dfrac{39}{150} \times 100 = 26\%.$percentage profit=15039​×100=26%.

Answer: $26\%$26% profit.

Worked Example 7

A car bought for £18,000 is sold a year later for £14,400. Work out the percentage loss.

Change $= 18{,}000 - 14{,}400 = 3{,}600$=18,000−14,400=3,600, i.e. a £3,600 loss.

$\text{percentage loss} = \dfrac{3{,}600}{18{,}000} \times 100 = 20\%.$percentage loss=18,0003,600​×100=20%.

Answer: $20\%$20% loss.

Common error: dividing by the new value ($14{,}400$14,400). Percentage change is always measured against the original.

Worked Example 7b

A scientist records a plant's height as $40$40 cm in week 1 and $46$46 cm in week 2. Work out the percentage increase in height.

Change $= 46 - 40 = 6$=46−40=6 cm.

$\text{percentage increase} = \dfrac{6}{40} \times 100 = 15\%.$percentage increase=406​×100=15%.

Answer: $15\%$15% increase.

Worked Example 7c (Percentage error)

A length is measured as $9.6$9.6 cm, but the true length is $10$10 cm. Work out the percentage error.

Percentage error is a percentage change measured against the true value:

$\text{percentage error} = \dfrac{|10 - 9.6|}{10} \times 100 = \dfrac{0.4}{10} \times 100 = 4\%.$percentage error=10∣10−9.6∣​×100=100.4​×100=4%.

Answer: $4\%$4%. This is exactly the percentage-change formula, with the "true" value playing the role of the original.

Reverse Percentages

A reverse percentage gives you the amount after a change and asks for the original. These are among the most commonly misunderstood questions on the whole paper, because the natural-but-wrong instinct is to take a percentage of the final figure and adjust it. That fails because the percentage in the question was applied to the original, which is exactly the number you are trying to find. The correct, dependable method is to think about what multiplier turned the original into the final amount, and then reverse that multiplication by dividing. If a $20\%$20% increase multiplied the original by $1.2$1.2, then dividing the final amount by $1.2$1.2 undoes it and returns the original. The method: identify the multiplier, then divide the final amount by it.

Worked Example 8

After a $20\%$20% increase, a phone bill is £54. Work out the original bill.

A $20\%$20% increase has multiplier $1.2$1.2, so $54$54 represents the original $\times 1.2$×1.2.