Learn | Grasp Maths

Lesson overview

Number — percentage and growth

When a quantity changes by a percentage repeatedly, we use multipliers. A percentage increase of 10% means multiply by 1.10 (the multiplier). A decrease of 20% means multiply by 0.80. For compound change over multiple periods, we multiply by the multiplier each time. For example, a £1000 investment growing by 5% per year for 3 years is $1000 \times 1.05^3 = 1157.63$ pounds. To reverse a percentage change, divide by the multiplier or use the inverse multiplier.

Finding the multiplier

A house price increases by 8%. What is the multiplier?

RuleFor an increase of $p\%$ , multiplier = $1 + \frac{p}{100}$

The new value is the old value plus the increase. Always start with 1 (the original 100%), then add the percentage as a decimal.

Compound percentage change with powers

An investment of £5000 grows by 3% per year for 4 years. What is the final value?

Compound growthUse $\text{Final} = \text{Initial} \times \text{(multiplier)}^n$ where $n$ is the number of periods

Each year, multiply by 1.03. After 4 years, we've multiplied by 1.03 four times, so we use the power ${1.03}^{4}$ .

Percentage decrease — the multiplier is less than 1

A car depreciates by 15% each year. What is the multiplier?

RuleFor a decrease of $p\%$ , multiplier = $1 - \frac{p}{100}$

The new value is the old value minus the decrease. For a 15% drop, keep 85%, so the multiplier is 0.85.

Worked example — reverse percentage problem

After a 20% increase, the price of a laptop is £960. What was the original price?

Check£800 × 1.20 = £960 ✓

A 20% increase means multiply by 1.20.
We know the final value is £960, so: Original × 1.20 = 960.
To find the original, divide by 1.20: Original = 960 ÷ 1.20 = £800.
Check your answer: £800 × 1.20 = £960. Correct!

Try it

Use multipliers to model each percentage change. Show your working clearly.

Question 1

A quantity increases by 25%. What is the multiplier?

💡 Multiplier = 1 + 0.25 = 1.25.

Question 2

A value decreases by 30%. What is the multiplier?

💡 Multiplier = 1 - 0.30 = 0.70.

Question 3

£500 grows by 4% per year for 2 years. Which calculation gives the final amount?

💡 Compound growth uses: Final = Initial × (multiplier)^n.

Question 4

A population of 1200 bacteria multiplies by a factor of 1.5 each day. How many bacteria are there after 3 days?

💡 $1200 \times 1.5^3 = 1200 \times 3.375 = 6075$ .

Question 5

After a 10% discount, a book costs £13.50. What was the original price?

💡 Original × 0.90 = 13.50, so Original = 13.50 ÷ 0.90 = £15.

Common mistakes

Watch for these when working through the lesson.

Using the percentage as the multiplier: for a 5% increase, the multiplier is 1.05, not 0.05.
Multiplying by the percentage in separate time periods instead of using a power: after 3 years at 2%, use ${1.02}^{3}$ , not 1.02 + 1.02 + 1.02.
Forgetting to divide when reversing a change: if the new price is P and the multiplier is M, the old price is P ÷ M, not P × M.

Repeated percentage change and compound growth

Lesson overview

Worked example — reverse percentage problem

Try it

Common mistakes

Related topics

Independent practice will plug in here