1MA1 · R9, N12Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 2–5

Percentages

"Per cent" means "out of 100". Percentages are everywhere — discounts, pay rises, interest rates, exam scores — and one tool handles almost all of them: the $\textbf{multiplier}$ . Increasing by 15% is the same as multiplying by 1.15, and once you think this way, even reverse percentages become one-line calculations.

50 min Ratio & Proportion

The big picture

A percentage change multiplies the original amount. An increase of $r\%$ multiplies by $1 + \tfrac{r}{100}$ ; a decrease multiplies by $1 - \tfrac{r}{100}$ . Undoing a change divides by the same multiplier — which is why reverse percentage questions ("the sale price is £68 after 20% off; what was it before?") are solved by division, not by adding 20% back.

Edexcel sets percentages on every paper, often combined with money, ratio or compound interest, and reverse percentages are a classic source of lost marks.

What you'll learn

Find a percentage of an amount with and without a calculator
Increase and decrease amounts by a percentage using multipliers
Express a change as a percentage of the original amount
Solve reverse percentage problems

Percentages of amounts

Without a calculator, build from easy percentages: 10% is one tenth, 1% is one hundredth, 5% is half of 10%. So 35% of 80 is $3 \times 8 + 4 = 28$ .

With a calculator, convert to a decimal: 35% of 80 is $0.35 \times 80 = 28$ .

To write one quantity as a percentage of another, divide and multiply by 100: 18 out of 24 is $\tfrac{18}{24} \times 100 = 75\%$ .

Your turnSame method, new numbers — try it before moving on.

Work out 35% of 480.

Your answer

Show worked solution

Solution

110% is 48, so 30% is 144.
25% is 24: $144 + 24 = 168$ .

Answer: $168$

Multipliers for increase and decrease

Increase by $r\%$ : multiply by $1 + \tfrac{r}{100}$ . A 15% increase is $\times 1.15$ .

Decrease by $r\%$ : multiply by $1 - \tfrac{r}{100}$ . A 15% decrease is $\times 0.85$ .

Successive changes multiply: a 10% increase followed by a 10% decrease is $\times 1.1 \times 0.9 = \times 0.99$ , an overall 1% decrease.

\text{new} = \text{original} \times \left(1 \pm \tfrac{r}{100}\right)

Percentage increase or decrease.

Example 1A jacket costs £240. Its price rises by 15%. Find the new price.

1Multiplier: 1.15.

240 \times 1.15

Answer£276

Your turnSame method, new numbers — try it before moving on.

Decrease £85 by 12%.

Your answer

Show worked solution

Solution

1Multiplier $0.88$ .
2 $85 \times 0.88 = 74.80$ .

Answer: £74.80

Why this works

A 10% rise followed by a 10% fall does not return you to the start, because the fall is 10% of a larger amount. Multipliers make this obvious: $1.1 \times 0.9 \neq 1$ .

Percentage change

Percentage change $= \dfrac{\text{change}}{\text{original}} \times 100$ .

Always divide by the $\textbf{original}$ value, not the new one.

Percentage profit and loss use the same formula, with the cost price as the original.

\%\ \text{change} = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100

Percentage change.

Example 1A share price rises from 45p to 54p. Find the percentage increase.

1Change: 9p.

\dfrac{9}{45} \times 100

Answer20%

Your turnSame method, new numbers — try it before moving on.

A value falls from 250 to 215. What is the percentage decrease?

Your answer

Show worked solution

Solution

1Decrease $= 35$ .
2 $\frac{35}{250} \times 100 = 14\%$ .

Answer: $14\%$

Reverse percentages

If you know the value $\textit{after}$ a percentage change, divide by the multiplier to find the original.

After a 20% discount, the sale price is 80% of the original. If that is £68, the original is $68 \div 0.8$ .

Do not find 20% of the sale price and add it on — that uses the wrong 100%.

Example 1In a sale, all prices are reduced by 20%. A coat costs £68 in the sale. What was its original price?

1Sale price

=

original

\times 0.8

2Original

= 68 \div 0.8

Answer£85

Your turnSame method, new numbers — try it before moving on.

After a 25% increase, a price is £75. What was the original price?

Your answer

Show worked solution

Solution

1Multiplier $1.25$ .
2 $75 \div 1.25 = 60$ .

Answer: £60

Tip — Check reverse percentages by working forwards: £85 reduced by 20% is £68 ✓. Adding 20% to £68 would give £81.60, which is wrong.

Think like an examiner

Write the multiplier you are using.
Divide by the original amount for percentage change.
For reverse percentages, show the division by the multiplier.
Give money answers to 2 decimal places.

Percentages

\times\left(1 + \tfrac{r}{100}\right)

Increase by r%.

\times\left(1 - \tfrac{r}{100}\right)

Decrease by r%.

\text{original} = \tfrac{\text{new}}{\text{multiplier}}

Reverse percentage.

Watch out for these

Dividing by the new value for percentage change.

Divide by the original value.

Finding an original price by adding the percentage to the new price.

Divide the new price by the multiplier.

Using 0.15 as the multiplier for a 15% decrease.

The multiplier is 0.85.

Adding successive percentage changes: +10% then −10% is 0%.

Multiply:

1.1 \times 0.9 = 0.99

, a 1% decrease.

Stretch yourself

A shop increases a price by 25%, then later reduces the new price by 20%. A customer says the final price is the same as the original. Are they right? Then find the single percentage change for an increase of 30% followed by a decrease of 30%.

Hint — Multiply the multipliers.

Reveal solution

SolutionFirst:

1.25 \times 0.8 = 1

, so yes, the final price equals the original. Second:

1.3 \times 0.7 = 0.91

, which is a 9% decrease overall. The two changes of 30% do not cancel, because the decrease is applied to a larger amount.

\blacksquare

Questions students ask

Why use multipliers?

They turn every percentage change into one multiplication, handle repeated changes easily, and make reverse percentages simple division.

Can a percentage be more than 100?

Yes. A 150% increase multiplies by 2.5, and a quantity can be 250% of another.

How do I find the original after an increase?

Divide by the increase multiplier. After a 12% rise to £560, the original is

560 \div 1.12 =

£500.

Key takeaways

Find percentages from 10% and 1%, or by converting to decimals.
Increase: $\times (1 + r/100)$ . Decrease: $\times (1 - r/100)$ .
Successive changes multiply.
Percentage change divides by the original.
Reverse percentages divide by the multiplier.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

Work out 15% of £60 without a calculator.[2]
Show mark scheme
- M110% = £6 and 5% = £3
- A1£9
Answer: £9

Question 2Grade 42 marks

A price including 20% VAT is £96. Work out the price before VAT.[2]
Show mark scheme
- M1 $96 \div 1.2$
- A1£80
Answer: £80

Question 3Grade 53 marks

Shop A sells a jacket for £45 with $\frac{1}{3}$ off. Shop B sells the same jacket for £42 with 30% off.

Which shop is cheaper? Show your working.[3]
Show mark scheme
- M1Shop A: £30
- M1Shop B: $42 \times 0.7 = £29.40$
- C1Shop B is cheaper, with both prices

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

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