What you'll learn
This topic covers percentages — finding a percentage of an amount, percentage change, and reverse percentages. In this guide you will learn how to calculate percentages with and without a calculator, how to increase and decrease using multipliers, how to find percentage change, and how to work backwards to an original value. Percentages appear constantly in money, statistics and everyday life.
Key terms and definitions
Percentage — a number out of 100 (% means "per hundred").
Multiplier — a decimal used to increase or decrease by a percentage.
Percentage change — the change as a percentage of the original.
Reverse percentage — finding the original amount before a percentage change.
Original (or 100%) value — the starting amount that a percentage refers to.
Core concepts
Percentage of an amount
To find a percentage of an amount, convert the percentage to a decimal and multiply, or build it from 10% and 1%. For example, 15% of 80 = 0.15 × 80 = 12. Without a calculator, 10% of 80 = 8 and 5% = 4, so 15% = 12.
Increase and decrease with multipliers
A multiplier does an increase or decrease in one step. To increase by 20%, multiply by 1.2; to decrease by 20%, multiply by 0.8. The multiplier is 1 plus or minus the percentage as a decimal. This is efficient and reduces errors.
Percentage change
Percentage change = (change ÷ original) × 100. Work out the actual increase or decrease, divide by the original amount (not the new one), and multiply by 100. This covers profit, loss and growth.
Reverse percentages
A reverse percentage finds the original value before a change. If a price is £60 after a 20% increase, that £60 is 120% of the original, so divide: 60 ÷ 1.2 = £50. Always identify what percentage the known amount represents.
Repeated change and compound interest
For repeated percentage changes (e.g. compound interest), apply the multiplier once per period, or raise it to a power: an amount growing 5% per year for 3 years is multiplied by 1.05³.
Worked examples
Example 1: Percentage of an amount
Find 35% of 240.
0.35 × 240 = 84.
Example 2: Percentage change
A price rises from £40 to £50. Find the percentage change.
(10 ÷ 40) × 100 = 25% increase.
Example 3: Reverse percentage
After a 25% increase, a value is 75. Find the original.
75 ÷ 1.25 = 60.
Common mistakes and how to avoid them
Dividing by the new value in percentage change. Use the original.
Wrong multiplier. Increase uses 1 + rate; decrease uses 1 − rate.
Treating reverse percentages as a simple decrease. Divide by the multiplier instead.
Forgetting compounding. Repeated change uses powers of the multiplier.
Mixing up percentage points and percentages. They are not the same.
Exam technique for Percentages
Convert to a decimal to find a percentage of an amount.
Use multipliers for increase, decrease and compound change.
Divide the change by the original for percentage change.
Identify the known percentage for reverse problems, then divide.
Raise the multiplier to a power for repeated change.
Quick revision summary
To find a percentage of an amount, convert to a decimal and multiply (15% of 80 = 0.15 × 80 = 12), or build from 10% and 1% without a calculator. Use a multiplier for change: increase by 20% → ×1.2, decrease by 20% → ×0.8 (1 plus or minus the rate as a decimal). Percentage change = (change ÷ original) × 100, always dividing by the original amount. A reverse percentage finds the original: if £60 is after a 20% rise it is 120% of the original, so 60 ÷ 1.2 = £50. For repeated change such as compound interest, apply the multiplier once per period or raise it to a power (1.05³ for 5% over 3 years). The common errors are dividing by the new value, using the wrong multiplier, treating reverse problems as a simple decrease, and ignoring compounding. Convert to decimals, use multipliers, divide change by the original, and identify the known percentage for reverse problems.