Percentage Increase and Decrease: Edexcel GCSE Foundation maths revision notes

Keep the original amount in mind

A percentage means "out of 100". The original amount is always 100\%. An increase adds a percentage of it on. A decrease takes a percentage of it away.

In the exam you are usually asked to work out a new price, a pay rise or a sale price, for 2 or 3 marks. Longer questions, worth 4 or 5 marks, have two changes one after the other, or ask you to compare two amounts and decide which is bigger.

There are two good methods. Without a calculator, find the percentage of the amount, then add it on or take it off. With a calculator, use a multiplier, which finds the new amount in one step.

Worked example 1: Increasing by finding the extra amount

A weekly bus pass costs £65. The price increases by 20\%. Work out the new price of the bus pass.

10\% of £65 is 65\div10=6.5, which is £6.50.
20\% is two lots of 10\%, so the increase is 6.50\times2=13, which is £13.
The question asks for the new price, so £13 is not the answer on its own.
Your turn. Add the increase to the original price. What is the new price, in pounds?

Your answer

65+13=78
Answer: £78

Worked example 2: Decreasing by finding the reduction

A coat costs £120. In a sale, the price is reduced by 15\%. Work out the sale price of the coat.

10\% of £120 is £12, and 5\% is half of that, which is £6.
So 15\% of £120 is 12+6=18, which is £18.
Your turn. A reduction is taken off. What is the sale price, in pounds?

Your answer

120-18=102
Check: you pay 100\%-15\%=85\% of the price, and 85\% of £120 is £102.
Answer: £102

Use a multiplier

A multiplier is the number you multiply the original amount by to get the new amount in one step. Work out what percentage of the original you end up with, then divide it by 100.

Increase by 20\%: 100\%+20\%=120\%, so multiply by 1.2
Decrease by 15\%: 100\%-15\%=85\%, so multiply by 0.85
Increase by 3\%: 100\%+3\%=103\%, so multiply by 1.03, not 1.3

Multiplying by 0.15 finds only the 15\% itself. To find what is left after a 15\% decrease, multiply by 0.85.

Worked example 3: Using a multiplier for an increase

A school has 1350 students. Next year, the number of students increases by 6\%. Work out the number of students next year.

The new number is 100\%+6\%=106\% of the old number.
Your turn. Write 106\% as a decimal. This is the multiplier.

Your answer

106\div100=1.06
1350\times1.06=1431
Check: 6\% of 1350 is 81, and 1350+81=1431
Answer: 1431 students

Worked example 4: A sale price to the nearest penny

A pair of trainers costs £34.99. In a sale, the price is reduced by 35\%. Work out the sale price. Give your answer to the nearest penny.

A 35\% decrease leaves 100\%-35\%=65\% of the price.
So the multiplier is 65\div100=0.65
Your turn. Work out 34.99\times0.65. Give your answer to the nearest penny.

Your answer

34.99\times0.65=22.7435, which is £22.74 to the nearest penny.
Use the full calculator answer and round only at the end. Money needs two decimal places.
Answer: £22.74

One change after another

When one change follows another, the second percentage is a percentage of the new amount, not of the original amount. Do the changes one at a time, in order.

You cannot just add or subtract the percentages. A 10\% rise and then a 10\% fall does not take you back to the start: £100 goes up to £110, then down by £11 to £99.

With multipliers, multiply by each one in turn: 100\times1.1\times0.9=99

Worked example 5: Two changes one after the other

A concert ticket costs £45. The price increases by 20\%. Later, the new price is reduced by 10\%. Work out the final price of the ticket.

20\% of £45 is 9, so the new price is 45+9=54, which is £54.
The reduction is 10\% of £54, the new price, not 10\% of £45.
Your turn. What is 10\% of £54, in pounds?

Your answer

54\div10=5.4, which is £5.40.
54-5.40=48.60
Check with multipliers: 45\times1.2\times0.9=48.6
Answer: £48.60

Compare and decide

Some questions ask you to decide, such as "Which is cheaper?" or "Does she have enough money?". Work out each final amount in full. Then write a sentence that gives your answer and uses your numbers.

A bigger percentage does not always mean a bigger amount, because the percentages may be of different starting amounts.

If a fixed charge, such as delivery, is added after a discount, take the percentage off the price first. Then add the charge, which is not reduced.

Worked example 6: Comparing two pay rises

Kim earns £420 a week. She gets a 5\% pay rise. Lee earns £380 a week. He gets a 15\% pay rise. Who earns more each week after the pay rises, and by how much?

Kim: 10\% of £420 is £42, so 5\% is 42\div2=21, which is £21.
Kim's new pay is 420+21=441, which is £441.
Lee: 10\% of £380 is £38 and 5\% is £19, so 15\% is 38+19=57, which is £57.
Your turn. What is Lee's new weekly pay, in pounds?

Your answer

380+57=437, so Lee's new pay is £437.
Compare the new amounts: 441-437=4
Answer: Kim earns £4 a week more than Lee.

Writing a change as a percentage

Some questions give you the amount before and after, and ask for the percentage increase or decrease. For example, a gym membership rises from £50 to £56 a month. The change is 56-50=6, which is £6.

Divide the change by the original amount, then multiply by 100:

\text{percentage change}=\frac{\text{change}}{\text{original amount}}\times100

Here, \frac{6}{50}\times100=12, so the increase is 12\%.

The original amount is the one before the change, not the new amount. If there were two changes, it is the amount just before the change you are asked about.

Common mistakes

Stopping at the change. When the question asks for the new amount, add the increase on or take the decrease off.
Using the wrong multiplier. A 3\% increase is \times1.03, not \times1.3, and a 15\% decrease is \times0.85, not \times0.15.
Taking the second percentage of the original amount. Work out the amount after the first change, then take the next percentage of that.
Building up a percentage in lots of small pieces on a calculator paper. Use a multiplier and write the calculation down.
Not giving a decision. When the question asks which is bigger or cheaper, say which, using your final amounts.
Dividing by the new amount. To find a percentage change, divide the change by the original amount.