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Lesson overview

Ratio and proportion — finance and growth

Two quantities are in direct proportion when one is a constant multiple of the other: $y = kx$ . Two quantities are in inverse proportion when their product is constant: $y = \frac{k}{x}$ . In finance, growth or decay with a constant percentage uses the formula: Final value $= \text{Initial value} \times (\text{multiplier})^n$ where the multiplier is (1 + rate) for growth and (1 − rate) for decay. Ratio divides a total in given proportions: if the ratio is $a:b$ , then the parts are $\frac{a}{a+b} \times \text{total}$ and $\frac{b}{a+b} \times \text{total}$ .

Direct proportion

Apples cost £0.50 each. Complete the table for cost against number.

Number of apples	1	3	5	10
Cost (£)	0.50	1.50	2.50	5.00

Direct proportion: as one quantity increases, the other increases at a constant rate. The constant is the multiplier $k$ .

Inverse proportion

A fixed area of 120 ${cm}^{2}$ is divided with length 12 cm. If length increases to 15 cm, what is the new width?

Inverse proportion: as one quantity increases, the other decreases. Their product remains constant.

Dividing in a ratio

Divide £400 in the ratio 3:2.

Divide the total by the sum of ratio parts, then multiply each part by its ratio number.

Compound growth with percentage multiplier

£500 is invested at 4% compound interest per year for 3 years. What is the final amount?

For compound growth, multiply by (1 + rate) for each time period. Use exponents for multiple periods.

Worked example — compound growth investment

A savings account starts with £2000. It earns 3% compound interest per year. How much will be in the account after 5 years?

Identify the initial amount: £2000.
Identify the annual interest rate: 3% = 0.03.
Calculate the multiplier: 1 + 0.03 = 1.03.
Identify the number of years: 5.
Apply the compound formula: Final amount = £2000 × ${1.03}^{5}$ .
Calculate ${1.03}^{5}$ ≈ 1.1593.
Multiply: £2000 × 1.1593 ≈ £2318.55.
The account will contain approximately £2318.55 after 5 years.

Try it

Use ratio and percentage methods carefully. For growth or decay, identify the multiplier and how many time periods. For dividing in ratios, use the sum of the parts.

Question 1

If y is directly proportional to x and y = 12 when x = 3, find y when x = 8.

💡 First find k: y = kx, so 12 = 3k, thus k = 4. Then y = 4 × 8 = 32.

Question 2

If y is inversely proportional to x and y = 20 when x = 2, find y when x = 5.

💡 First find k: y = k/x, so 20 = k/2, thus k = 40. Then y = $\frac{40}{5}$ = 8.

Question 3

Divide £180 in the ratio 2:1.

💡 Total parts: 2+1=3. First share: ( $\frac{2}{3}$ )×£180 = £120.

Question 4

£1000 is invested at 5% compound interest per year. How much after 2 years?

💡 $Use 1000 × {1.05}^{2}.$

Question 5

A population of 50,000 decreases by 8% per year. Approximately how many after 3 years?

💡 Multiplier = 0.92. Calculate 50,000 × ${0.92}^{3}$ .

Common mistakes

Watch for these when working through the lesson.

Confusing direct and inverse proportion. Direct: y = kx (both increase together). Inverse: y = k/x (one increases, other decreases).
Using the wrong multiplier for compound growth. Remember: 5% growth means multiply by 1.05, not 0.05.
Adding percentages instead of multiplying by the multiplier in compound problems.
Incorrectly dividing a ratio by adding instead of multiplying by the fractions.

Direct and inverse proportion, finance and growth

Lesson overview

Worked example — compound growth investment

Try it

Common mistakes

Related topics

Independent practice will plug in here