Percentage change: how to work out an increase or decrease

How to work out percentage increase

To work out a percentage increase, find how much the amount went up, divide that by the original amount, then multiply by 100. Always divide by the amount you started with.

Worked example: a price goes up from £40 to £46.

1. Find the change: 46 − 40 = 6.
2. Divide by the original: 6 ÷ 40 = 0.15.
3. Multiply by 100: 0.15 × 100 = 15.

So the price went up by 15%. A comparison bar shows the change as the gap between the old and new amounts:

The same steps answer "how do you find the percentage of growth?". If a town grows from 2,500 people to 2,800, the change is 2,800 − 2,500 = 300. Then 300 ÷ 2,500 = 0.12, so the town has grown by 12%.

How to work out percentage decrease

To work out a percentage decrease, find how much the amount went down, divide that by the original amount, then multiply by 100. It's the same method as an increase.

Worked example: a bike is reduced from £80 to £68.

1. Find the change: 80 − 68 = 12.
2. Divide by the original: 12 ÷ 80 = 0.15.
3. Multiply by 100: 0.15 × 100 = 15.

So the price fell by 15%. Working out the "percent off" in a sale is a percentage decrease.

The percentage change formula

The percentage change formula is:

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

If the answer is positive, it's an increase. If it's negative, it's a decrease. The percentage increase formula and the percentage decrease formula are this same formula: for a decrease, you can take the new amount from the original instead, so the answer comes out positive.

The answer can be more than 100%. The national curriculum asks secondary pupils to work with percentages greater than 100%. If a plant grows from 12 cm to 30 cm, the change is 30 − 12 = 18. Then 18 ÷ 12 = 1.5, so it has grown by 150%. A 100% increase means the amount has doubled.

Using multipliers for percentage increase and decrease

A multiplier turns a percentage change into a single multiplication. The national curriculum describes this as interpreting percentage changes "multiplicatively", and it's quick on a calculator.

The original amount is 100%. A 15% increase takes it to 115%, which is 1.15 as a decimal. A 15% decrease leaves 85%, which is 0.85.

• To increase 80 by 15%: 1.15 × 80 = 92.
• To decrease 80 by 15%: 0.85 × 80 = 68.

Change	Percentage of the original	Multiplier
increase by 5%	105%	1.05
increase by 20%	120%	1.2
increase by 3%	103%	1.03
decrease by 10%	90%	0.9
decrease by 25%	75%	0.75
decrease by 3%	97%	0.97

Multipliers work backwards too. Divide the new amount by the original to find the multiplier, then read off the change: 46 ÷ 40 = 1.15, which is 115% of the original, so a 15% increase. And 68 ÷ 80 = 0.85, which is 85%, so a 15% decrease.

If your child is unsure about turning percentages into decimals, fractions, decimals and percentages and multiplying decimals cover that step.

Reverse percentages: finding the original amount

A reverse percentage question gives you the amount after a change and asks for the amount before it. Divide by the multiplier. The national curriculum calls these "original value problems". They're in the key stage 3 programme of study, and in the GCSE content that every student is taught, so they come up on both GCSE tiers.

Worked example: after 20% off, a coat costs £60. What was the price before the sale?

1. After 20% off, £60 is 80% of the original price, so the multiplier is 0.8.
2. Divide by the multiplier: 60 ÷ 0.8 = 75.
3. Check: 0.8 × 75 = 60.

So the coat cost £75. Without a calculator, find 1% first: if 80% is 60, then 1% is 60 ÷ 80 = 0.75, and 100% is 0.75 × 100 = 75.

Worked example: after a 15% rise, a price is £46. What was it before? The multiplier is 1.15, and 46 ÷ 1.15 = 40, so it was £40.

The common wrong answer to the coat question is £72, from adding 20% of £60 back on. That doesn't work, because the 20% was taken off the original price, not off £60.

How to get a percentage difference

To get a percentage difference between two numbers in school maths, treat it as a percentage change from one to the other, and say which one you're comparing with. The national curriculum calls this comparing two quantities using percentages.

The direction matters. Going from 40 to 50 is a change of 10, and 10 ÷ 40 = 0.25, so it's a 25% increase. Going from 50 to 40 is the same change, but 10 ÷ 50 = 0.2, so it's a 20% decrease.

Repeated changes and interest

Some percentage change questions change an amount more than once. Simple interest, in key stage 3, adds the same amount each year. £200 saved at 3% simple interest earns 3% of 200 = 6 a year, so £24 over 4 years, because 6 × 4 = 24.

Compound interest, at GCSE, takes the percentage of the new amount each time, so you multiply by the multiplier again and again: £200 at 3% compound interest for 4 years is 200 × 1.03⁴, which is about £225.10.

Repeated changes catch people out. A 10% rise followed by a 10% fall doesn't get back to the start. From 100, 1.1 × 100 = 110, then 0.9 × 110 = 99.

Common mistakes with percentage change

• Dividing by the new amount. The change always goes over the original. From £40 to £46 is 6 ÷ 40, not 6 ÷ 46.
• Using the new amount, not the change. 46 ÷ 40 gives 1.15, the multiplier, not a 15% change on its own. Take away 1 (or 100%) to get the change.
• The wrong multiplier for a decrease. A 15% decrease is 0.85, not 0.15. Multiplying by 0.15 gives the amount taken off, not what's left.
• Adding the percentage back on in reverse percentages. Divide by the multiplier instead.
• Thinking changes cancel out. Up 10% then down 10% is a 1% loss overall.

How to help at home

• Use real prices. When something in a shop is reduced, ask what percentage has come off, then check it with the multiplier.
• Ask for an estimate first. From £40 to £46 is a bit more than 10%, since 10% of 40 is 4. That catches answers like 1.5%.
• Talk about the direction. Ask "what did it start at?" before anything else. That's the number you divide by.
• Practise the basic percentages. Secure 10%, 1% and percentages of amounts first. How to work out percentages covers that primary groundwork, with the bar model as a way to draw it.

When do children learn percentage change?

Percentage change is secondary school maths. At primary, children find percentages of amounts and use them to compare, finishing in Year 6 maths. Percentage increase and decrease, multipliers, original value problems and simple interest are all in the key stage 3 programme of study, which covers Years 7 to 9 as one key stage: schools decide the order, so ask your child's school for its scheme of work if you want to know when it's coming. Compound interest follows at GCSE. Percentage change also links to ratio and proportion. The KS3 maths guide covers the rest of the key stage.

Percentage change questions to try

1. A price goes up from £50 to £56. What is the percentage increase?
2. A car's value falls from £8,000 to £6,000. What is the percentage decrease?
3. What multiplier increases an amount by 8%?
4. Decrease 240 by 35%.
5. After a 25% increase, a bill is £150. What was it before?
6. A plant grows from 15 cm to 36 cm. What is the percentage increase?

Answers

1. 12% (the change is 6, and 6 ÷ 50 = 0.12)
2. 25% (the change is 2,000, and 2,000 ÷ 8,000 = 0.25)
3. 1.08
4. 156 (0.65 × 240 = 156)
5. £120 (150 ÷ 1.25 = 120)
6. 140% (the change is 21, and 21 ÷ 15 = 1.4)