Percentage Increase and Decrease

Grade 3

Finding a percentage of an amount

The core skill everything else builds on: finding a percentage of a number. Percentage just means "out of 100", so to find, say, 30% of 240 you divide by 100 to get 1%, then multiply by 30.

percentage of amount = (percentage ÷ 100) × amount
EXAMPLE
Find 35% of £420.
1% of £420 = £420 ÷ 100 = £4.20. Then 35% = £4.20 × 35 = £147.

Increasing and decreasing by a percentage

To increase by a percentage, find that percentage of the amount and add it on. To decrease by a percentage, find it and subtract.

EXAMPLE
A coat costs £80. It is reduced by 25%. What is the sale price?
25% of £80 = £80 ÷ 100 × 25 = £20. Sale price = £80 − £20 = £60.

The multiplier method — faster every time

There is a slicker way that GCSE questions reward: the multiplier method. Instead of finding the percentage and then adding or subtracting, you combine both steps into a single multiplication.

For an increase of r%, multiply by (1 + r/100). For a decrease of r%, multiply by (1 − r/100). So +20% → ×1.2; −15% → ×0.85.
EXAMPLE
A phone costs £360. VAT of 20% is added. What is the price including VAT?
Multiplier for +20% is 1.20. Price = £360 × 1.20 = £432.

The multiplier method is especially useful when you need to apply repeated percentage changes — just multiply the multipliers together.

Watch out

A common mistake is to find the percentage of the new amount instead of the original. Always apply the percentage to the value you started with before the change.
Nova
Try this one

A car depreciates in value by 20% each year.

At the end of three years, the car is worth £5120.

(a) Show that the original price of the car was £10 000. [3]

(b) By what overall percentage did the car depreciate over the three years? [2]

Nova's hint:

Part (a) uses reverse compound percentage: undo each year's 20% decrease by dividing by the multiplier three times (or divide by the multiplier cubed).

Ask yourself: 'What single multiplier represents one year's 20% decrease, and how do I raise a multiplier to a power of 3?' For example, two 10% decreases in succession use multiplier \( 0.9^2 \).

Part (b) compares the total change to the original using \( \frac{\text{change}}{\text{original}} \times 100 \).

Try to Solve it with Nova?