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This page sets out the method for percentage change in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Percentage change topic page.
When you see 'increase by' or 'decrease by', build the multiplier and multiply once.
When you have no calculator and the percentage is friendly, find the percentage of the amount and adjust.
When you are given an old value and a new value and asked for the percentage change.
Increase by 20% means multiply by 1.2.
Decrease by 20% means multiply by 0.8. Do not use 1.2 for a decrease.
The question may ask for the increase itself, or for the price after the increase.
Read which one is wanted.
Change ÷ original × 100 gives the percentage change.
The original is the starting amount, not the final one.
This topic is a single change.
Repeated change, year after year, comes later.
Give money answers to the nearest penny, with two decimal places, such as £7.50.
Do not round to the nearest pound unless the question asks you to.
An increase is larger than the start. A decrease is smaller.
If a decrease came out larger, the multiplier was above 1.
An increase of p% multiplies by 1 + p/100. For 15% that is 1 + 0.15 = 1.15. The 1 keeps the original amount. The 0.15 is the increase. 80 × 1.15 = 92. You can find the increase on its own: 10% of 80 is 8 and 5% is 4, so the increase is 12, and 80 + 12 = 92.
Adding 15 onto 80 gives 95, which treats the percentage as a raw number of units. That is only correct when the original amount is 100. Read what the question asks. The increase is 12. The new amount is 92. A price after the rise is 92. The rise itself is 12.
The multiplier must be above 1 for an increase. 0.15 × 80 = 12 finds only the extra, and stopping there leaves out the original 80. If an 'increase' comes out smaller than the start, the multiplier was below 1. Check 92 > 80 before you move on.
Method: 1. Add the percentage as a decimal to 1 to get the multiplier. 2. Multiply the original amount by that multiplier. 3. The increase itself is the new amount minus the start.
A decrease of p% multiplies by 1 - p/100. For 20% that is 1 - 0.20 = 0.8. You keep 80% of the original. 50 × 0.8 = 40. The amount removed is 10, which is 20% of 50. Using 1.2 would raise 50 to 60, the opposite direction.
A 10% decrease uses 0.9, not 1.1 and not 0.1. The 0.1 finds the piece that is removed: 10% of 50 is 5. The new amount is 50 - 5 = 45, or 50 × 0.9 = 45. Subtract from 1 to build the multiplier. Do not subtract from 0.
The new amount must be smaller than the start. 40 < 50 agrees with a decrease. Give money answers to the nearest penny, with two decimal places, such as £7.50. Here 40 is already exact. A decrease that comes out larger means the increase multiplier was used.
Method: 1. Subtract the percentage as a decimal from 1. 2. Multiply the original amount by that multiplier. 3. A decrease finishes below the original amount.
Subtract to find the change, divide by the original, then multiply by 100. From 45 to 54, the change is 54 - 45 = 9. The original is 45, not 54. 9/45 = 0.2, and 0.2 × 100 = 20. The score increased by 20%. Check forwards: 45 × 1.2 = 54.
Dividing by the new amount gives a different percentage. 9/54 × 100 is not 20. That answers '9 is what percentage of 54', which is not the change from the start. The original is always the denominator for a percentage change, whether the value went up or down.
A fall gives a negative change if you keep new minus old. From 50 to 40, (40 - 50)/50 × 100 = -20. That is a decrease of 20%. The multiplier check is 50 × 0.8 = 40. State the direction in words if the question asks for an increase or a decrease rather than a signed number.
Method: 1. Subtract to find the change, and keep the original as the denominator. 2. Divide the change by the original. 3. Multiply by 100 to turn that into a percentage.
Apply one multiplier, then the other. A price of pounds 80 rises by 25% and then falls by 20%. The multipliers are 1.25 and 0.8. Their product is 1.25 × 0.8 = 1, so the final price is pounds 80, the same as the start.
Check it in stages. 80 × 1.25 = 100, and 100 × 0.8 = 80. The rise adds pounds 20. The fall removes 20% of the new price, which is pounds 20. These two percentages were chosen so the second change undoes the first. Equal percentages, such as +10% then -10%, do not do that.
Always multiply the multipliers and compare the product with 1. A product above 1 is an overall increase. A product below 1 is an overall decrease. A product of exactly 1 means the final amount matches the start. Here 1.25 × 0.8 = 1.
Method: 1. Write a multiplier for each change. 2. Multiply those multipliers together. 3. Multiply the starting price by the combined multiplier.
Profit and loss questions are percentage change in disguise. Profit is the selling price minus the cost price, and loss is the cost price minus the selling price. The percentage is found with the same rule as always: change divided by the original, times 100. The original is the cost price, because that is where the trader started.
For example, a trader buys a watch for £48 and sells it for £60. The profit is £12 and the percentage profit is 12/48 × 100 = 25%. A common error is to divide by the selling price. Another is to forget that the cost may be a total for several items, in which case the income is the number sold times the price each.
Method: 1. Work out the total cost and the total income. 2. Find the profit or loss as the difference between them. 3. Divide the profit or loss by the cost price. 4. Multiply by 100 and write the answer as a percentage profit or loss.
Tip: The original amount is the one before the change. For profit, that is the cost price.
A reverse percentage question gives you the amount after a change and asks for the original. After a 15% increase, the new price is 115% of the original, so the multiplier is 1.15. To get back, divide by the multiplier: 207 ÷ 1.15 = 180. The original is not found by taking 15% off 207, because 15% of 207 is a different amount from 15% of the original.
For a decrease the multiplier is below 1. If a bike is £276 after a 20% reduction, then 276 is 80% of the original and the original is 276 ÷ 0.8 = 345. Check by applying the change forwards: 345 × 0.8 must equal 276. If several changes happen, multiply all the multipliers together first, then divide once.
Method: 1. Write the multiplier for the change, such as 1.15 or 0.8. 2. Write "new = original × multiplier". 3. Divide the new amount by the multiplier to find the original. 4. Check by multiplying your answer by the multiplier.
"The length falls by 15% and the width falls by 10%, so the area falls by 25%" is a classic wrong claim. Each percentage applies to a different quantity, and the area depends on both. Multiply the multipliers: 0.85 × 0.9 = 0.765. That is 76.5% of the original area, so the area falls by 23.5%.
The same idea explains why a rise of 10% followed by a fall of 10% does not return to the start. The second change is applied to a bigger amount. On £200 the rise gives £220 and the fall gives £198. Exam questions ask "Is she correct?" and expect a full calculation with a conclusion, not a bare yes or no.
Method: 1. Write the multiplier for each change. 2. Multiply the multipliers together. 3. Convert the final multiplier into a percentage change. 4. Compare with the claim and write a sentence that gives your verdict.
In a sale question you may be asked whether a customer has enough money. Work out the sale price with a multiplier, then compare it with the amount the customer has. Two reductions in a row, such as 10% then a further 20%, are two multipliers: 0.9 and then 0.8. State clearly whether the price is more or less than the budget.
Read the accuracy instruction at the end. Money is given to the nearest penny, so keep two decimal places. A percentage change is often asked "to 1 decimal place" or "to 3 significant figures". Do not round in the middle of the working, only at the end. When there is no calculator, use 10%, 5% and 1% to build up a percentage, then add or subtract.
Method: 1. Convert each change into a multiplier. 2. Apply the multipliers one after the other, or multiply them together. 3. Compare the final value with the amount the question uses for comparison. 4. Round only at the end, and give units or a percentage sign.
Tip: On a non-calculator paper, 15% = 10% + 5% and 35% = 30% + 5% are quick builds.
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