14.2 Growth and Decay

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: compound interest, depreciation and repeated percentage change. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Tell the difference between simple and compound interest
• Use a multiplier to the power n for repeated percentage increase or decrease
• Solve problems about compound interest, depreciation and population growth
• Find how many years it takes for a value to reach a target

Key words

• Compound interest: Interest added each year is included in the next year’s calculation, so you earn interest on interest.
• Depreciation: A fall in value over time, for example a car losing value each year.
• Growth: An amount getting bigger by the same percentage each time period.
• Decay: An amount getting smaller by the same percentage each time period.

Simple and compound interest

With simple interest you earn the same amount every year. With compound interest the interest is added on, and next year’s interest is worked out on the new, bigger amount.

• 1; Simple: £2000 at 4% 2000 + 80 = £2080; Compound: £2000 at 4% 2000 × 1.04 = £2080
• 2; Simple: £2000 at 4% 2080 + 80 = £2160; Compound: £2000 at 4% 2080 × 1.04 = £2163.20
• 3; Simple: £2000 at 4% 2160 + 80 = £2240; Compound: £2000 at 4% 2163.20 × 1.04 = £2249.73

Using a multiplier to the power n

Instead of working year by year, multiply by the same multiplier n times.

Key fact: Amount after n years = original × (multiplier)ⁿ Increase of r%: multiplier = 1 + r/100. Decrease of r%: multiplier = 1 − r/100.

Worked example: compound interest

£2000 is invested at 4% compound interest. Find the value after 3 years and the interest earned.
Multiplier = 1.04. 2000 × 1.04³ = 2249.728 = £2249.73 (to the nearest penny).
Interest = 2249.73 − 2000 = £249.73.

Answer: £2249.73; interest £249.73

Depreciation

Values that fall by a percentage each year use a multiplier less than 1. The value gets smaller but never reaches zero.

Worked example: a car

A car costs £15 000. It loses 12% of its value each year. Find its value after 3 years.
Multiplier = 1 − 0.12 = 0.88. 15 000 × 0.88³ = 10 222.08.

Answer: £10 222.08

Growth and decay in real life

Worked example: bacteria

A sample has 500 bacteria. The number grows by 30% every hour. How many are there after 4 hours?
500 × 1.3⁴ = 1428.05

Answer: about 1428 bacteria

Worked example: how many years?

A boat is worth £8000 and loses 15% of its value each year. After how many years is it first worth less than £5000?
Keep multiplying by 0.85: 1 year: £6800; 2 years: £5780; 3 years: £4913.
£4913 is less than £5000.

Answer: after 3 years

On a calculator, type 8000, press =, then type × 0.85 and press = again and again. Count the presses.

Common mistakes

• Using simple interest when the question says compound (or the other way round).
• Using 0.15 as the multiplier for a 15% decrease. It should be 0.85.
• Thinking that 10% off for 5 years is 50% off altogether. 0.9⁵ = 0.59…, so it is about 41% off.
• Rounding in the middle of a calculation. Keep the full value and round only at the end.

Revision questions

Try these, then check your answers below each one.

Question 1. What is the multiplier for a 6% decrease? (Choose from: 0.06, 0.94, 1.06, 0.6.)

Answer: 0.94

Working: 94% = 0.94.

Question 2. £2500 is invested at 2% per year compound interest. Work out its value after 3 years.

Answer: 2653.02 £

Working: 2500 × 1.02³ = 2653.02.

Question 3. A car costs £12 000. It loses 15% of its value each year. Work out its value after 2 years.

Answer: 8670 £

Working: 12 000 × 0.85² = £8670.

Related lessons

• Previous lesson: 14.1 Percentages
• Next lesson: 14.3 Compound Measures
• In this chapter: 14.4 Distance, Speed and Time
• In this chapter: 14.5 Direct and Inverse Proportion

Continue in the full course

This lesson is section 14.2 (EGCMF1402) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.