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SavannahYear 8
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MathsRatio, proportion and rates of change30 min★★★ difficulty

Direct and inverse proportion

Two quantities can grow together or trade off against each other. Deciding which relationship applies before calculating prevents nearly every error in this topic.

Part of your national curriculum
  • Ratio, proportion and rates of change: Solve problems involving direct and inverse proportion, including graphical and algebraic representations

Lesson overview

What you'll learn in this lesson

Solve problems involving direct and inverse proportion, including graphical and algebraic representations

Key learning points

  • • Direct proportion
  • • Inverse proportion
  • • Recognising which is which
  • • Modelling assumptions

This lesson at a glance

  • 30 minutes
  • 17 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

problemsinvolvingdirectinverseproportion

Scroll down — the lesson carries on below

Part 1 of 17 · Discover6%
1

Watch & discover

Part 1 of 17

Watch first

Direct and inverse proportion

An English teaching video on this topic from The Organic Chemistry Tutor. Watch it, then scroll down for the lesson.

In a nutshell

Solve problems involving direct and inverse proportion, including graphical and algebraic representations

2

Learning cycle

Part 2 of 17

Learning cycle 1 of 2

Part 1 · Direct proportion

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 17

Learn

Direct proportion

If y is directly proportional to x, then y = kx and the graph is a straight line through the origin. Doubling x doubles y. If 6 books cost £27, then k = 4.5 and 10 books cost £45. The unitary method, finding the value of one, is the same idea.

4

Explore the idea

Part 4 of 17

Learn

Inverse proportion

If y is inversely proportional to x, then y = k/x and the product xy is constant. The graph is a curve that never touches the axes. If 4 workers take 9 hours, the total work is 36 worker-hours, so 6 workers take 6 hours, assuming they work at the same rate.

5

Quick check

Part 5 of 17

Quick check

If 6 books cost £27, 10 books cost

6

Quick check

Part 6 of 17

Quick check

y is inversely proportional to x. Doubling x

7

Learning cycle

Part 7 of 17

Learning cycle 2 of 2

Part 2 · Recognising which is which

A short piece of teaching, then a check to make sure it has landed.

8

Reset break

Part 8 of 17

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
9

Explore the idea

Part 9 of 17

Learn

Recognising which is which

Ask what happens when one quantity doubles. If the other doubles, the relationship is direct; if it halves, it is inverse. Recipes, currency conversion and fuel use are direct; the number of workers against time, and speed against journey time, are inverse.

10

Explore the idea

Part 10 of 17

Learn

Modelling assumptions

Inverse proportion for workers assumes everyone works at the same constant rate and does not get in each other's way. Stating that assumption is part of a full answer, because in reality doubling the workforce rarely halves the time exactly.

11

Quick check

Part 11 of 17

Quick check

A direct proportion graph is

12

Quick check

Part 12 of 17

Quick check

5 taps fill a tank in 12 minutes. 3 taps take

13

Explore the idea

Part 13 of 17

Worked example

Worked answer: 4 workers build a wall in 9 hours. How long would 6 identical workers take? (3 marks)

The total work is 4 × 9 = 36 worker-hours, which stays constant (1). With 6 workers, time = 36 ÷ 6 = 6 hours (1). This assumes all workers work at the same steady rate and can work at once without obstruction (1). Answering 13.5 hours treats the relationship as direct, which is the classic error.

14

Challenge round

Part 14 of 17

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

15

Challenge round

Part 15 of 17

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

If the other doubles, the relationship is ____; if it halves, it is inverse.

Recipes, currency conversion and fuel use are direct; the number of workers against time, and speed against journey time, are ____.

Inverse ____ for workers assumes everyone works at the same constant rate and does not get in each other's way.

16

Mastery quiz

Part 16 of 17

Marked quiz

End of lesson quiz: Direct and inverse proportion

4 questions, marked with the reasoning shown. No timer.

  1. 1. If 6 books cost £27, 10 books cost

  2. 2. y is inversely proportional to x. Doubling x

  3. 3. A direct proportion graph is

  4. 4. 5 taps fill a tank in 12 minutes. 3 taps take

17

Lesson round-up

Part 17 of 17

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 17 · Watch & discover

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    Solve problems involving direct and inverse proportion, including graphical and algebraic representations

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