Percentage Change

This page is about comparing two amounts using a percentage. You might write one amount as a percentage of another, such as 9 out of 36, which is 25\%. Or you might be told an amount before and after a change, and work out the change as a percentage of the amount you started with.

In the exam you are usually asked to write one number as a percentage of another, for 2 marks, or to work out a percentage increase, decrease, profit or loss, for 2 or 3 marks. Longer questions, worth 4 or 5 marks, make you work out a total cost and the total money taken before you find the percentage profit, or ask you to compare two percentage changes.

Every question on this page uses the same idea: make a fraction with the amount you are comparing with on the bottom, then multiply by 100.

To write one amount as a percentage of another, first write it as a fraction: the part over the whole. Then multiply by 100.

\text{percentage}=\frac{\text{part}}{\text{whole}}\times100

For example, 12 as a percentage of 48 is \frac{12}{48}\times100=25, so 25\%. The whole is the amount after the word "of".

Without a calculator, make the bottom of the fraction 100. Multiply or divide the top and the bottom by the same number. So \frac{9}{20}=\frac{45}{100}, which is 45\%. On a calculator paper, divide and then multiply by 100: 9\div20\times100=45

Both amounts must be in the same units before you divide. To write 30 minutes as a percentage of 2 hours, change 2 hours to 120 minutes. Then \frac{30}{120}\times100=25, so it is 25\%.

Read whether the question asks about the part you are given or the rest. If 35 of the 50 seats on a bus are taken, \frac{35}{50}\times100=70, so 70\% are taken. The rest, 100\%-70\%=30\%, are empty.

If the amount is bigger than the one you compare it with, the answer is more than 100\%. For example, 90 as a percentage of 75 is \frac{90}{75}\times100=120, so 120\%.

When an amount changes, first work out the change: the bigger amount subtract the smaller one. For example, a price rises from £80 to £92. The change is 92-80=12, which is £12.

Then write the change as a percentage of the original amount, the amount before the change.

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

Here, \frac{12}{80}\times100=15, so the increase is 15\%.

A decrease works in the same way: the change is how much it went down, and you still divide by the amount before the change. On a calculator you can also divide the new amount by the original: 92\div80=1.15, which is 115\%, so the increase is 115\%-100\%=15\%.

A bigger change in pounds does not always mean a bigger percentage change. Work out each percentage change from its own original amount, then compare the percentages, not the changes.

If a question asks how many times bigger one percentage is than another, divide one by the other. For example, 18\% is 18\div6=3 times 6\%.

For an overall change after two changes, compare the final amount with the very first amount. You can't add the two percentages. A price that goes from £80 up to £100, then down to £90, has risen by 25\% and then fallen by 10\%. Overall it has risen by £10, and \frac{10}{80}\times100=12.5, so the overall increase is 12.5\%, not 15\%.

The cost is what is paid at the start. The profit is the money taken subtract the cost. If the money taken is less than the cost, the difference is a loss.

Percentage profit and percentage loss are percentages of the cost, because the cost is the original amount.

\text{percentage profit}=\frac{\text{profit}}{\text{cost}}\times100

For example, a phone bought for £250 and sold for £190 makes a loss of 250-190=60, which is £60. The percentage loss is \frac{60}{250}\times100=24, so 24\%. The loss of £60 is not 60\%.

When there are lots of items, work out two totals first: the total cost and the total money taken. Use the same units for both, so change pence to pounds. Items that are not sold still count in the cost.

• Giving the change as the percentage. A loss of £24 is not 24\%: divide the change by the original amount, then multiply by 100.
• Dividing by the new amount or the selling price. Divide by the amount before the change; for a percentage profit, divide by the cost.
• Stopping at the decimal. 0.6 is not the answer: multiply by 100 to get 60\%.
• Comparing the changes instead of the percentages. Work out each percentage change from its own original amount, then compare.
• Leaving out some of the cost. Include every item bought, even the ones that are not sold, and any extra charges.
• Mixing pounds and pence. Change everything to pounds, or everything to pence, before you divide.