MathsGCSE maths: ratio, probability and statistics30 min★★★ difficulty

Ratio, proportion and compound measures

Proportional reasoning runs through the whole specification: recipes, currency, speed, density, growth and scale drawings are all the same idea in different clothing.

Lesson overview

What you'll learn in this lesson

Use proportional reasoning, probability and statistics to interpret and solve problems.

Key learning points

• Ratio problems
• Direct and inverse proportion
• Compound measures and growth

This lesson at a glance

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

Words to know

proportionalreasoningprobabilitystatisticsinterpret

Scroll down — the lesson carries on below

Part 1 of 15 · Discover7%

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Part 1 of 15

Watch first

Ratio, proportion and compound measures

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

In a nutshell

Use proportional reasoning, probability and statistics to interpret and solve problems.

Learning cycle

Part 2 of 15

Learning cycle 1 of 2

Part 1 · Ratio problems

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

Explore the idea

Part 3 of 15

Learn

Ratio problems

For sharing, add the parts, divide the total by that number and multiply back. When a question gives the difference between shares rather than the total, that difference corresponds to the difference in parts — a distinction that decides whether the answer is right.

Explore the idea

Part 4 of 15

Learn

Direct and inverse proportion

In direct proportion y = kx, so doubling x doubles y and the graph is a straight line through the origin. In inverse proportion y = k ÷ x, so doubling x halves y and the graph is a curve. Find k from the given pair, then use it for the unknown.

Quick check

Part 5 of 15

Quick check

£120 shared in the ratio 2 : 3 gives the larger share as

Quick check

Part 6 of 15

Quick check

In inverse proportion, doubling x

Reset break

Part 7 of 15

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

Learning cycle

Part 8 of 15

Learning cycle 2 of 2

Part 2 · Compound measures and growth

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

Explore the idea

Part 9 of 15

Learn

Compound measures and growth

Speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area. For repeated percentage change, use a multiplier: a 4% annual rise over 3 years is × 1.04³. Reverse percentage questions divide by the multiplier, since the stated figure is the result, not the original.

Quick check

Part 10 of 15

Quick check

A 5% annual increase over 4 years uses the multiplier

Quick check

Part 11 of 15

Quick check

Density is

Explore the idea

Part 12 of 15

Worked example

Worked answer: after a 20% discount a coat costs £68. Find the original price (3 marks)

Paying 80% of the original gives a multiplier of 0.8 (1). The original price is therefore 68 ÷ 0.8 (1) = £85 (1). Adding 20% to £68 gives £81.60, which is wrong, because 20% of the larger original is more than 20% of the reduced price. Checking: 20% of £85 is £17, and £85 − £17 = £68.

Challenge round

Part 13 of 15

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

Mastery quiz

Part 14 of 15

Marked quiz

End of lesson quiz: Ratio, proportion and compound measures

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

1. £120 shared in the ratio 2 : 3 gives the larger share as
2. In inverse proportion, doubling x
3. A 5% annual increase over 4 years uses the multiplier
4. Density is

Lesson round-up

Part 15 of 15

Lesson round-up

Ready when you are

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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

Work out the total cost of the trip if 50 pupils go.
Show why the price per pupil falls when more pupils go, using numbers.
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

Write your answer, notes or working here

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Photos of paper work, diagrams or working outHow many minutes did the written work take?

Part 1 of 15 · Watch & discover

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