Ratio, proportion and rates of change: set up, solve and interpret the answers in growth and decay problems, including compound interest (Foundation tier)

The idea

This is a core Foundation tier topic and essential knowledge for Higher tier. Repeated percentage change is handled with a decimal multiplier. An increase of 5% uses a multiplier of 1.05; a decrease of 12% uses 0.88. This multiplier is applied for each period, so for n periods, it is raised to the power n. For example, £2000 invested at 3% compound interest for four years becomes 2000 × 1.03⁴ = £2251.02. The interest earned is £251.02. Decay problems use the same method with a multiplier between 0 and 1. A car worth £8000 that loses a fifth (20%) of its value each year is worth 8000 × 0.8³ = £4096 after three years. Always check if the question asks for the final amount or the total change, and round money to two decimal places.

What earns the marks

• A correct multiplier for the percentage change, e.g., 1.03 for a 3% increase or 0.8 for a 20% decrease.
• Raising the multiplier to the correct power, matching the number of time periods in the question.
• A correct final value after the full period, before any final subtraction if only the change is required.
• The final answer correctly rounded as requested, typically to the nearest penny for monetary contexts.

Common mistakes and how to put them right

• Calculating one period of interest and multiplying it by the number of periods (this is simple interest, not compound).

• Using a multiplier of 0.2 for a decrease of 20% when it should be 1 - 0.2 = 0.8.

• Multiplying the initial value by the number of years instead of raising the multiplier to the power of the number of years.

• Giving the total final amount when the question asked only for the interest earned, or vice versa.

Exam tips

• Write down the multiplier before using the calculator, then check it makes sense: above 1 for growth, below 1 for decay.
• Use the power key (e.g., xʸ) on your calculator instead of multiplying repeatedly; it is faster and more accurate.
• Keep the full calculator value on screen until the very end of the calculation, then round just once as the final step.