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Maths · Multiplicative reasoning

Growth and decay

Use multipliers to solve repeated percentage change problems, including compound interest, depreciation and exponential growth and decay.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    What is the multiplier for a 15% increase?

    Show answerHide answer

    1.15

  2. 2

    What is the multiplier for a 20% decrease?

    Show answerHide answer

    0.8

  3. 3

    Work out \(1.1^2\).

    Show answerHide answer

    1.21

  4. 4

    Increase £200 by 10%.

    Show answerHide answer

    £220

  5. 5

    What is \(0.5^3\)?

    Show answerHide answer

    0.125

Learning Objectives

  1. 1Use multipliers for repeated percentage change.
  2. 2Calculate compound interest and depreciation.
  3. 3Use the formula \(\text{final} = \text{initial} \times \text{multiplier}^n\).
  4. 4Solve growth and decay problems in context.

THE KEY IDEA

For repeated percentage change, multiply by the multiplier once for each period.

\(\text{final amount} = \text{original amount} \times (\text{multiplier})^n\)

Multipliers

Write the multiplier as a decimal.

  • Increase by 5%

    Multiplier: 1.05. Type: Growth

  • Increase by 20%

    Multiplier: 1.2. Type: Growth

  • Decrease by 15%

    Multiplier: 0.85. Type: Decay

  • Decrease by 40%

    Multiplier: 0.6. Type: Decay

  • Halves each time

    Multiplier: 0.5. Type: Decay

  • Doubles each time

    Multiplier: 2. Type: Growth

Compound Interest

£5000 is invested at 3% compound interest per year. Work out its value after 4 years.

Show the solutionHide the solution
  1. 1 Multiplier for a 3% increase \(1.03\)
  2. 2 Apply it 4 times \(5000 \times 1.03^4\)
  3. 3 Work it out \(5000 \times 1.12550881 = 5627.54\)

Answer£5627.54

Depreciation

A car worth £12 000 loses 15% of its value each year. Find its value after 3 years.

Show the solutionHide the solution
  1. 1 Multiplier for a 15% decrease \(0.85\)
  2. 2 Apply it 3 times \(12\,000 \times 0.85^3\)
  3. 3 Work it out \(12\,000 \times 0.614125\)

Answer£7369.50

Exponential Growth

A culture of 500 bacteria grows by 20% every hour. Find the number of bacteria after 6 hours, to the nearest whole number.

Show the solutionHide the solution
  1. 1 Multiplier \(1.2\)
  2. 2 Apply it 6 times \(500 \times 1.2^6\)
  3. 3 Work it out \(500 \times 2.985984 = 1492.99\)

Answer1493 bacteria

Halving Each Time

A radioactive sample has mass 80 g. Its mass halves every 10 days. Find its mass after 30 days.

Show the solutionHide the solution
  1. 1 30 days is three halvings \(30 \div 10 = 3\)
  2. 2 Multiplier \(0.5\)
  3. 3 Apply it 3 times \(80 \times 0.5^3 = 80 \times 0.125\)

Answer10 g

Growth or Decay?

Growth

  • The multiplier is greater than 1.
  • Examples: interest, population increase, inflation.
  • The graph curves upwards.

Decay

  • The multiplier is between 0 and 1.
  • Examples: depreciation, radioactive decay, cooling.
  • The graph curves down and flattens towards zero.

Working Backwards and Solving for Time

Sometimes the unknown is the starting amount or the number of periods.

  • Original value

    Divide by the multiplier: if £441 is after 2 years at 5%, the start is \(441 \div 1.05^2 = £400\).

  • Number of periods

    Try values of n in \(\text{initial} \times \text{multiplier}^n\) until you reach the target.

  • Use the formula

    Write the equation first, then substitute.

Double Your Money

An investment of £1000 earns 6% compound interest per year. By trying different numbers of years, find how many complete years it takes for the value to reach at least £2000.

1. Use a table of values.

2. Check the final year carefully.

A good answer shows: \(1000 \times 1.06^{11} = 1898.30\) and \(1000 \times 1.06^{12} = 2012.20\). It takes 12 years.

Can I...?

  1. 1Find a multiplier from a percentage change.
  2. 2Use a power for repeated change.
  3. 3Calculate compound interest.
  4. 4Calculate depreciation.
  5. 5Solve exponential growth problems.
  6. 6Solve exponential decay problems.
  7. 7Find an original value.
  8. 8Find the number of periods by trial.

Summary & Exam Focus

  • Growth multiplier above 1; decay multiplier below 1.
  • Final \(=\) initial \(\times\) multiplier\(^n\).
  • Compound interest and depreciation use the same idea.
  • To reverse a change, divide by the multiplier.

Exam focus

A car is worth £12 000. It loses 15% of its value each year. Work out its value after 3 years. (3 marks) (3 marks)

Write the multiplier, then use the power. Do not calculate 15% three times separately.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Multiplier
The number you multiply by to increase or decrease.
Compound interest
Interest paid on the original amount and on interest already added.
Depreciation
The fall in value of an item over time.
Exponential growth
Growth by a constant multiplier greater than 1 each period.
Exponential decay
Decay by a constant multiplier between 0 and 1 each period.
Original value
The amount at the start.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Calculator 3 marks

    £5000 is invested at 3% compound interest per year. Work out the value of the investment after 4 years. Give your answer to the nearest penny.

    Show answerHide answer

    Model answer

    \(5000 \times 1.03^4 = 5627.54\).

    Mark scheme

    • Multiplier 1.03 — M1
    • \(5000 \times 1.03^4\) — M1
    • 5627.54 — A1
  2. Question 2 Calculator 3 marks

    A car is worth £12 000. Its value falls by 15% each year. Work out its value after 3 years.

    Show answerHide answer

    Model answer

    \(12\,000 \times 0.85^3 = 7369.50\), so £7369.50.

    Mark scheme

    • Multiplier 0.85 — M1
    • \(12\,000 \times 0.85^3\) — M1
    • 7369.50 — A1
  3. Question 3 Calculator 3 marks

    A culture has 500 bacteria. The number increases by 20% every hour. Work out the number of bacteria after 6 hours. Give your answer to the nearest whole number.

    Show answerHide answer

    Model answer

    \(500 \times 1.2^6 = 1492.99\), so 1493.

    Mark scheme

    • Multiplier 1.2 — M1
    • \(500 \times 1.2^6\) — M1
    • 1493 — A1
  4. Question 4 Calculator 4 marks

    The graph shows the number of bacteria in a culture, \(N\), after \(t\) hours. The graph is a curve. (a) Use the graph to estimate the number of bacteria after 3.5 hours. (b) Use the graph to estimate the time when there are 200 bacteria.

    An exponential growth curve of bacteria starting at 100 and reaching about 299 after 6 hours.
    Show answerHide answer

    Model answer

    (a) About 190 (accept 185 to 195). (b) About 3.8 hours (accept 3.7 to 3.9).

    Mark scheme

    • (a) Reading at \(t = 3.5\) — M1
    • (a) 190 (185 to 195) — A1
    • (b) Reading at \(N = 200\) — M1
    • (b) 3.8 (3.7 to 3.9) — A1
  5. Question 5 Non-calculator 3 marks

    A radioactive sample has a mass of 80 g. Its mass halves every 10 days. Work out its mass after 30 days.

    Show answerHide answer

    Model answer

    30 days is 3 halvings, so \(80 \times 0.5^3 = 10\) g.

    Mark scheme

    • 3 halvings — M1
    • \(80 \times \dfrac{1}{8}\) — M1
    • 10 — A1
  6. Question 6 Calculator 3 marks

    The population of a town is 2000. It decreases by 5% each year. Work out the population after 10 years. Give your answer to the nearest whole number.

    Show answerHide answer

    Model answer

    \(2000 \times 0.95^{10} = 1197.47\), so 1197.

    Mark scheme

    • Multiplier 0.95 — M1
    • \(2000 \times 0.95^{10}\) — M1
    • 1197 — A1

Quick check

  1. What is the multiplier for a 12% decrease?

    1. A0.12
    2. B0.88
    3. C1.12
    4. D1.88
    Show answerHide answer

    B: 0.88

    \(100\% - 12\% = 88\% = 0.88\).

  2. £200 grows at 10% per year for 2 years. What is it worth?

    1. A£220
    2. B£240
    3. C£242
    4. D£400
    Show answerHide answer

    C: £242

    \(200 \times 1.1^2 = 242\).

  3. A value of £600 falls by 50% each year. What is it worth after 2 years?

    1. A£150
    2. B£300
    3. C£0
    4. D£450
    Show answerHide answer

    A: £150

    \(600 \times 0.5^2 = 150\).

  4. To find the original value after a 5% rise gave £420, you...

    1. AMultiply by 1.05
    2. BSubtract 5%
    3. CDivide by 0.95
    4. DDivide by 1.05
    Show answerHide answer

    D: Divide by 1.05

    Divide by the multiplier: \(420 \div 1.05 = 400\).

  5. Which multiplier shows exponential decay?

    1. A1.03
    2. B0.97
    3. C2
    4. D1.5
    Show answerHide answer

    B: 0.97

    A multiplier between 0 and 1.

  6. How does the graph of compound interest compare with simple interest?

    1. AIt is a straight line
    2. BIt is below simple interest
    3. CIt curves upwards
    4. DIt is the same
    Show answerHide answer

    C: It curves upwards

    Compound interest curves above the straight line of simple interest.

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