Ratio and proportion.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Ratio and proportion

Multiplicative reasoning

Lesson 4 of 4

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Multiplicative reasoning

Lesson 4 of 4

Answer each one, then check.

1

4 pens cost £3.20. How much does 1 pen cost?

2

If y = 3x and x = 5, what is y?

3

Solve 12 = k/4.

4

Work out 48 ÷ 6.

5

What does "directly proportional" mean?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Multiplicative reasoning

Lesson 4 of 4

1

4 pens cost £3.20. How much does 1 pen cost?

£0.80

2

If y = 3x and x = 5, what is y?

15

3

Solve 12 = k/4.

k = 48

4

Work out 48 ÷ 6.

8

5

What does "directly proportional" mean?

Both increase in the same ratio

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Multiplicative reasoning

Lesson 4 of 4

1

Solve direct proportion problems with y = kx.

2

Solve inverse proportion problems with y = k/x.

3

Recognise proportion from a table or graph.

4

Use y ∝ x², y ∝ x³ and y ∝ √x (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Proportion Graphs

Multiplicative reasoning

Lesson 4 of 4

The shape of the graph tells you the type of proportion.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Direct and Inverse Proportion

Multiplicative reasoning

Lesson 4 of 4

DIRECT PROPORTION

• As x increases, y increases in the same ratio.

• y = kx, so y/x is constant.

• The graph is a straight line through the origin.

INVERSE PROPORTION

• As x increases, y decreases in the opposite ratio.

• y = k/x, so xy is constant.

• The graph is a curve that never touches the axes.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Direct Proportion

Multiplicative reasoning

Lesson 4 of 4

y is directly proportional to x. When x = 7, y = 21. Find y when x = 10.

1

Write y = kx

21 = k × 7

2

Find k

k = 3

3

Substitute x = 10

y = 3 × 10

ANSWER

y = 30

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Inverse Proportion

Multiplicative reasoning

Lesson 4 of 4

y is inversely proportional to x. When x = 4, y = 12. Find y when x = 6.

1

Write y = k/x

12 = k/4

2

Find k

k = 48

3

Substitute x = 6

y = 48/6

ANSWER

y = 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Worker Problem

Multiplicative reasoning

Lesson 4 of 4

5 workers take 12 days to build a wall. How long would 8 workers take, working at the same rate?

1

More workers means fewer days: inverse proportion

Workers × days is constant

2

Constant

5 × 12 = 60

3

Days for 8 workers

60 ÷ 8

ANSWER

7.5 days

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

HIGHER TIER

Other Proportions

Squares, cubes and square roots.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Proportional to a Square

Multiplicative reasoning

Lesson 4 of 4

y is directly proportional to x². When x = 2, y = 20. Find y when x = 3.

1

Write y = kx²

20 = k × 4

2

Find k

k = 5

3

Substitute x = 3

y = 5 × 9

ANSWER

y = 45

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Proportional to a Square Root

Multiplicative reasoning

Lesson 4 of 4

y is directly proportional to √x. When x = 4, y = 6. Find y when x = 25.

1

Write y = k√x

6 = k × 2

2

Find k

k = 3

3

Substitute x = 25

y = 3 × 5

ANSWER

y = 15

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Types of Proportion

Multiplicative reasoning

Lesson 4 of 4

Learn the equation for each phrase.

Statement

Equation

y is proportional to x

y = kx

y is inversely proportional to x

y = k/x

y is proportional to x²

y = kx²

y is proportional to x³

y = kx³

y is proportional to √x

y = k√x

y is inversely proportional to x²

y = k/x²

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Multiplicative reasoning

Lesson 4 of 4

Direct proportion

Two quantities that increase together in the same ratio.

Inverse proportion

As one quantity increases the other decreases in the opposite ratio.

Constant of proportionality

The number k in a proportion equation.

Proportional to

Written y ∝ x.

Unitary method

Finding the value of one unit first.

Rate

How much of one quantity per unit of another.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Multiplicative reasoning

Lesson 4 of 4

YOUR TASK

Which Type?

12 minutes

For each situation decide whether it is direct or inverse proportion, and write an equation with a constant k. (a) The cost of buying n identical books (b) the time t to fill a pool with p identical pumps (c) the area A of a circle with radius r.

1

Say what happens when one doubles.

2

Write the equation.

WHAT A GOOD ANSWER SHOWS

(a) Direct: C = kn. (b) Inverse: t = k/p. (c) A = kr² (with k = π).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Multiplicative reasoning

Lesson 4 of 4

Solve a direct proportion problem.

Solve an inverse proportion problem.

Find the constant k.

Write y = kx and y = k/x.

Recognise proportion from a graph.

Use y = kx² (Higher).

Use y = k√x (Higher).

Explain the type of proportion.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Direct: y = kx. Inverse: y = k/x.

Find k from the first pair of values.

Higher: y = kx², y = kx³, y = k√x.

Check your answer makes sense.

EXAM FOCUS

y is directly proportional to x². When x = 2, y = 20. Find the value of y when x = 3. (3 marks)

Write the equation with k first, use the given pair to find k, then substitute the new value.

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