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This page sets out the method for percentages 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 Percentages topic page.
When you see a percentage on Paper 1, break it into 10%s, 5%s and 1%s.
When you have a calculator, turn the percentage into a decimal and multiply once.
When you see 'what percentage of' or 'express … as a percentage of …', divide then multiply by 100.
30% means 30 out of 100, or 30/100.
As a decimal that is 0.3.
10% of an amount is the amount ÷ 10.
5% is half of 10%. 20% is two lots of 10%.
1% is the amount ÷ 100.
Build an awkward percentage from 10% and 1%.
30% of an amount is the amount × 0.3.
Use the multiplier when a calculator is allowed.
150% is the whole amount plus another half.
The result is larger than the amount you started with.
50% should be half of the amount. 10% should be one tenth of it.
If 10% of 80 comes out as 40, you have found half by mistake. 10% of 80 is 8.
There are two reliable ways to find a percentage of an amount. On a non-calculator paper, build the percentage from 10%, 5% and 1%. On a calculator paper, multiply by the decimal multiplier. Both ways must give the same answer.
Suppose you want 35% of an amount. 10% is the amount divided by 10, 5% is half of that, and 1% is the amount divided by 100. Then 35% = 3 × 10% + 5%. The bar here is a picture of 35 parts out of 100, which is why the multiplier is 0.35.
To write one quantity as a percentage of another, divide the part by the whole and multiply by 100. The whole is the original group, not the part.
Method: 1. Turn the percentage into a multiplier by dividing by 100. 2. Or find 10%, 5% and 1% and build the percentage up. 3. Multiply the amount by the multiplier. 4. For money, round to the nearest penny at the end.
Tip: If 10% looks wrong, stop. 10% of £80 is £8, not £0.80 and not £80.
An increase keeps the original amount and adds a percentage of it. The multiplier is more than 1. A 4% increase multiplies by 1.04. A decrease multiplier is less than 1. A 4% decrease multiplies by 0.96. Multiplying by 0.04 finds only the change, not the new total.
Percentage change compares the change with the starting amount: dfracchangeoriginal × 100. If a price goes up, the change is new minus original and the answer is an increase. If it goes down, original minus new is the fall, and the answer is a decrease.
Two changes do not cancel just because the percentages match. Up 10% then down 10% multiplies by 1.1 × 0.9 = 0.99, a 1% fall. Multiply the multipliers.
Method: 1. Write the multiplier: 1 + p/100 or 1 - p/100. 2. Multiply the amount by that multiplier to get the new amount. 3. For a change you are asked to find, divide the change by the original. 4. Multiply by 100 and say increase or decrease.
Tip: Write the multiplier down before you use it. A correct multiplier shows the method clearly, even if the final button press slips.
A reverse percentage starts from the amount after the change and works back to the original. You need the multiplier that was used, then you divide by it. Adding the percentage back on is not the same, because that percentage was of the original, not of the new amount.
After a 12% discount, the customer paid 88% of the original price. The multiplier is 0.88. Divide the sale price by 0.88. After a 20% increase, the new amount is 120% of the original, so divide by 1.2, not by 0.2 and not by 0.8.
Check by going forwards. If the original was £200 and the change was 12% off, then £200 - £24 = £176. That should match the sale price.
Method: 1. Decide whether the change was an increase or a decrease. 2. Write the multiplier that produces the amount you were given. 3. Divide the given amount by that multiplier. 4. Check by applying the percentage forwards.
Tip: The phrase "after a reduction of" or "including a 20% rise" means the number in the question is the new amount, not the original.
Simple interest adds the same percentage of the original amount every year. Compound interest adds a percentage of the latest total, so the amount you earn interest on grows. The quick way is one multiplier raised to a power: starting amount × (multiplier)^n, where n is the number of years.
Depreciation is the same structure with a multiplier below 1. A car that loses 18% of its value each year is multiplied by 0.82 each year. After two years the value is the original × 0.82^2. It is not the original minus 36%, because the second 18% is taken from a smaller value.
For money, keep the full power and round to the nearest penny at the end. An overall percentage loss still divides by the original price.
Method: 1. Write the yearly multiplier from the rate. 2. Raise it to the power of the number of years. 3. Multiply by the starting amount. 4. Round money to the nearest penny if that is what the question wants.
Tip: A power of 1 is one year. A power of 3 is three years. Do not multiply the percentage by the number of years and call it compound.
On a non-calculator paper, build any percentage from pieces you can find in your head. Start with 10%, which is the amount divided by 10. Then 5% is half of that and 1% is a tenth of it. For 55% of 640 you find 10% = 64, then 50% = 320, then 5% = 32, and add: 320 + 32 = 352.
Some percentages are better as fractions. 50% = 1/2, 25% = 1/4, 12.5% = 1/8 and 75% = 3/4. Taking 12.5% of 64 is just 64 ÷ 8 = 8. Write each building step on its own line. The marks are for the build-up, so a correct answer with no working is risky when the question says "show your working".
Method: 1. Find 10% by dividing by 10, and write it down. 2. Halve 10% for 5%, or divide by 10 again for 1%. 3. Add or subtract the pieces to build the percentage you need. 4. Check against a fraction: for example 25% should be a quarter of the amount.
Tip: Remember 12.5% = 1/8 and 37.5% = 3/8. They come up often in non-calculator questions.
To write one amount as a percentage of another, make a fraction with the part on top and the whole on the bottom, then multiply by 100. If 48 of 150 students travel by bus, the percentage is 48/150 × 100 = 32%. Make sure the whole is the right total. Real questions give you a table and you must decide whether the whole is all students or just one group.
Percentages are the fair way to compare groups of different size. A test score of 36 out of 40 is 90% and 41 out of 50 is 82%, so the first is better even though the second has a higher mark. A decimal such as 0.07 is 7%, and a fraction like 3/8 is 37.5%. Convert to a decimal first if the fraction does not have a denominator of 100.
Method: 1. Identify the part and the whole. 2. Write the fraction part over whole. 3. Convert to a decimal and multiply by 100, or scale to a denominator of 100. 4. Write the percentage sign and compare with other percentages if the question asks.
VAT at 20% is added to a price before VAT. The multiplier is 1.2, so a bicycle at £340 plus VAT costs 340 × 1.2 = £408. If you are given the price including VAT, divide by 1.2 to get back to the price before VAT. A price of £690 including VAT was 690 ÷ 1.2 = £575 before VAT. Taking 20% off 690 is wrong because 20% was added to the smaller amount.
Simple interest is the same amount every year. It is calculated from the original investment only. Interest of 3% per year on £800 for 4 years is 800 × 0.03 × 4 = £96, and the total is £896. A deposit is a percentage of the full price, paid first. Keep money to two decimal places, and label each amount in your working.
Method: 1. Decide whether the percentage is added to the price (VAT), taken from it, or is interest. 2. Use a multiplier such as 1.2 for VAT, or find the percentage and add or subtract it. 3. For simple interest, multiply the yearly interest by the number of years. 4. Round money to two decimal places and write the unit.
The hardest percentage questions change the numbers part way through. Tom has 30 stamps and 60% are British, so 18 are British. If he sells 8 British stamps, he has 22 stamps and 10 are British. If he then buys 3 British stamps, he has 25 stamps and 13 are British, which is 52%.
Notice that the total changes at every step, and the new percentage is always based on the new total. Keep a small table of "British" and "Total" and update both columns after each event. Other typical multi-step questions ask for a percentage of what is left, or combine two groups, for example boys and girls who walk, to find an overall percentage. Work in whole numbers wherever you can.
Method: 1. Write the starting numbers for the part and the total. 2. Update both the part and the total for each event. 3. Convert to a percentage only at the end, using the final part over the final total. 4. Check that the percentage is sensible compared with where you started.
Tip: Draw a two-column table, Part and Total, and update it line by line.
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