Growth and decay.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Growth and decay

Multiplicative reasoning

Lesson 1 of 4

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Multiplicative reasoning

Lesson 1 of 4

Answer each one, then check.

1

What is the multiplier for a 15% increase?

2

What is the multiplier for a 20% decrease?

3

Work out 1.1².

4

Increase £200 by 10%.

5

What is 0.5³?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Multiplicative reasoning

Lesson 1 of 4

1

What is the multiplier for a 15% increase?

1.15

2

What is the multiplier for a 20% decrease?

0.8

3

Work out 1.1².

1.21

4

Increase £200 by 10%.

£220

5

What is 0.5³?

0.125

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Multiplicative reasoning

Lesson 1 of 4

1

Use multipliers for repeated percentage change.

2

Calculate compound interest and depreciation.

3

Use the formula final = initial × multiplierⁿ.

4

Solve growth and decay problems in context.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Multiplicative reasoning

Lesson 1 of 4

THE KEY IDEA

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

final amount = original amount × (multiplier)ⁿ

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Multipliers

Multiplicative reasoning

Lesson 1 of 4

Write the multiplier as a decimal.

Change

Multiplier

Type

Increase by 5%

1.05

Growth

Increase by 20%

1.2

Growth

Decrease by 15%

0.85

Decay

Decrease by 40%

0.6

Decay

Halves each time

0.5

Decay

Doubles each time

2

Growth

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Compound Interest

Multiplicative reasoning

Lesson 1 of 4

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

1

Multiplier for a 3% increase

1.03

2

Apply it 4 times

5000 × 1.03⁴

3

Work it out

5000 × 1.12550881 = 5627.54

ANSWER

£5627.54

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Compound Growth Beats Simple Interest

Multiplicative reasoning

Lesson 1 of 4

Compound interest earns interest on interest, so its graph curves upwards.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Depreciation

Multiplicative reasoning

Lesson 1 of 4

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

1

Multiplier for a 15% decrease

0.85

2

Apply it 3 times

12 000 × 0.85³

3

Work it out

12 000 × 0.614125

ANSWER

£7369.50

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exponential Growth

Multiplicative reasoning

Lesson 1 of 4

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

1

Multiplier

1.2

2

Apply it 6 times

500 × 1.2⁶

3

Work it out

500 × 2.985984 = 1492.99

ANSWER

1493 bacteria

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Halving Each Time

Multiplicative reasoning

Lesson 1 of 4

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

1

30 days is three halvings

30 ÷ 10 = 3

2

Multiplier

0.5

3

Apply it 3 times

80 × 0.5³ = 80 × 0.125

ANSWER

10 g

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Growth or Decay?

Multiplicative reasoning

Lesson 1 of 4

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Working Backwards and Solving for Time

Multiplicative reasoning

Lesson 1 of 4

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 ÷ 1.05² = £400.

Number of periods

Try values of n in initial × multiplierⁿ until you reach the target.

Use the formula

Write the equation first, then substitute.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Multiplicative reasoning

Lesson 1 of 4

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Multiplicative reasoning

Lesson 1 of 4

YOUR TASK

Double Your Money

12 minutes

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.

WHAT A GOOD ANSWER SHOWS

1000 × 1.06¹¹ = 1898.30 and 1000 × 1.06¹² = 2012.20. It takes 12 years.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Multiplicative reasoning

Lesson 1 of 4

Find a multiplier from a percentage change.

Use a power for repeated change.

Calculate compound interest.

Calculate depreciation.

Solve exponential growth problems.

Solve exponential decay problems.

Find an original value.

Find the number of periods by trial.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Growth multiplier above 1; decay multiplier below 1.

Final = initial × multiplierⁿ.

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)

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

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