Percentages

Find percentages of amounts, percentage change, reverse percentages and successive changes.

Also known as: percentage change, reverse percentages, percent increase.

SAT · Problem-Solving and Data Analysis GCSE Maths · Ratio, proportion and rates of change IGCSE Maths · Number

Key ideas

"Percent" means "out of 100": $15\% = 0.15$ .
Percentage change $= \frac{\text{change}}{\text{original}} \times 100$ .
For reverse percentages, divide by the multiplier, never subtract the percentage from the new value.
Successive percentage changes multiply; they do not add.

Percentages as multipliers

To find $15\%$ of $240$ , multiply by $0.15$ : $0.15 \times 240 = 36$ .

A $25\%$ discount leaves $75\%$ of the price, so multiply by $0.75$ . A $20\%$ increase gives $120\%$ , so multiply by $1.2$ .

Percentage change

From $250$ to $290$ is a change of $40$ : $\frac{40}{250} \times 100 = 16\%.$ Always divide by the original value.

Reverse percentages and repeated change

If a price is $96$ after a $20\%$ increase, the original is $96 \div 1.2 = 80$ .

Successive changes multiply: a $10\%$ increase followed by a $10\%$ decrease gives $1.1 \times 0.9 = 0.99$ , an overall $1\%$ decrease, not "no change".

Worked examples

Example 1 · Reverse percentage

After a 20% increase, a price is 96. What was the original price?

Step 1.The new price is $120\%$ of the original: multiplier $1.2$ .
Step 2. $96 \div 1.2 = 80$ .

Answer: 80

Example 2 · Percent of a percent

$p$ is $30\%$ of $q$ , and $q$ is $40\%$ of $r$ . What percent of $r$ is $p$ ?

Step 1. $p = 0.3q$ and $q = 0.4r$ .
Step 2. $p = 0.3 \times 0.4r = 0.12r$ .

Answer: $12\%$

Common mistakes

Finding the percentage change relative to the new value instead of the original.
Subtracting 20% of the new price to reverse a 20% increase.
Adding successive percentage changes.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1PercentagesEasier

What is $15\%$ of $240$ ?

Question 2PercentagesEasier

A jacket priced at $80$ is reduced by $25\%$ . What is the new price?