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SavannahYear 8
← Maths
MathsRatio, proportion and rates30 min★★★ difficulty

Direct and inverse proportion

In direct proportion, doubling one thing doubles the other. In inverse proportion, doubling one halves the other. Deciding which you are looking at is the whole skill.

Lesson overview

What you'll learn in this lesson

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

Key learning points

  • • The unitary method
  • • Direct proportion on a graph
  • • Inverse proportion
  • • Deciding which is which

This lesson at a glance

  • 30 minutes
  • 19 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 19 · Discover5%
1

Watch & discover

Part 1 of 19

Watch first

Direct and inverse proportion

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

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

2

What you already know

Part 2 of 19

Before we start

What you already know

You can find one part in a ratio and read a straight-line graph.

Recap: Sharing in a ratioRecap: Straight-line graphs in four quadrants
3

Key words

Part 3 of 19

Key words

Words you'll need today

direct proportion
Two quantities that increase in the same ratio; y = kx.
inverse proportion
As one increases the other decreases in the same ratio; y = k/x.
unitary method
Finding the value of one, then scaling.
4

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Part 1 — Direct proportion

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

5

Explore the idea

Part 5 of 19

Learn

The unitary method

If 6 identical pens cost £4.50, one pen costs 75p, so 11 pens cost £8.25. Find one, then scale — it works for recipes, currency and fuel alike. Direct proportion always passes through the origin: zero pens cost nothing.

6

Explore the idea

Part 6 of 19

Learn

Direct proportion on a graph

Direct proportion graphs are straight lines through (0, 0), and the gradient is the constant k in y = kx. If 4 litres of paint cover 30 m², then k = 7.5 m² per litre and y = 7.5x. Any straight line that does not pass through the origin is not direct proportion, even though it is linear.

7

Quick check

Part 7 of 19

Quick check

5 books cost £12.50. What do 8 cost?

8

Quick check

Part 8 of 19

Quick check

Which is inverse proportion?

9

Reset break

Part 9 of 19

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
10

Learning cycle

Part 10 of 19

Learning cycle 2 of 2

Part 2 — Inverse proportion

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

11

Explore the idea

Part 11 of 19

Learn

Inverse proportion

If 4 people build a wall in 6 hours, 8 people take 3 hours: twice the workers, half the time. Here x × y stays constant at 24 worker-hours, so y = 24/x. Its graph is a curve, never touching either axis, because no number of workers finishes in zero time.

12

Explore the idea

Part 12 of 19

Learn

Deciding which is which

Ask what happens when the first quantity doubles. Cost of petrol for more litres — doubles, so direct. Time to finish a job with more people — halves, so inverse. Speed and journey time are inverse; distance and fuel used are direct. Real problems are usually decided by common sense before any algebra.

13

Quick check

Part 13 of 19

Quick check

3 taps fill a tank in 40 minutes. How long do 5 taps take?

14

Quick check

Part 14 of 19

Quick check

A direct proportion graph passes through...

15

Common mix-ups

Part 15 of 19

Common mix-ups

Things people often get wrong

  • Treating 'more people, less time' as direct proportion.
  • Calling any straight-line graph direct proportion even when it misses the origin.
  • Scaling by adding a difference instead of multiplying.
16

Challenge round

Part 16 of 19

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

17

Challenge round

Part 17 of 19

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

____ proportion always passes through the origin: zero pens cost nothing.

Direct ____ graphs are straight lines through (0, 0), and the gradient is the constant k in y = kx.

Time to finish a job with more people — halves, so ____.

18

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Direct and inverse proportion

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

  1. 1. 7 kg of apples cost £10.50. Cost of 4 kg?

  2. 2. In y = k/x, the quantities are...

  3. 3. 6 machines take 12 days. How long for 9?

  4. 4. Speed and journey time for a fixed distance are...

  5. 5. If y is directly proportional to x and y = 20 when x = 4, find k.

  6. 6. Inverse proportion graphs are...

19

Lesson round-up

Part 19 of 19

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 19 · Watch & discover

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