Learn4024Paper 2 · CalculatorNumberRatio, proportion and rates

Number · Subtopic 4 of 8

Ratio, Proportion and Rates

A ratio compares parts: add the parts, find one part, then multiply. Direct proportion keeps the ratio fixed; inverse proportion keeps the product fixed. A rate is one quantity per unit of another: speed is distance per hour, density is mass per cm³, unit price is cost per item.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Simplify ratios and share in a given ratio.
Solve direct and inverse proportion problems.
Use speed, density and other rates, with correct units.
Compare best buys and convert currencies.

Pause and think

$6$ workers build a wall in $10$ days. How long would $4$ workers take? Does "fewer workers" mean more or fewer days?

Section 2 of 14

Core idea

Start hereMixing paint

A shade of green is 2 tins of yellow to 3 tins of blue. Need 60 litres? Each "set" of 5 tins makes the colour, so there are $60\div5=12$ litres per tin-part. Ratios are all about finding the size of one part.

The big idea

Share in a ratio: total parts → one part → each share. Rates compare different units: speed, density, unit price.

1. Sharing

Sharing £60 in the ratio 2 : 3

2+3=5

parts. One part

=60\div5=£12

. Ali gets

2\times12

, Bea gets

3\times12

Worked exampleShare 60 in the ratio 2 : 3

Split 60.

Goal: Two shares.
Strategy: One part.

$2+3=5\text{ parts}$
Total.
$60\div5=12$
One part.
$2\times12=24,\quad3\times12=36$
Shares.

Answer

24

and

36

✓ Check

24+36=60

;

24:36=2:3

✓.

Worked exampleDifference known

Ages in the ratio 5 : 3 differ by 12 years.

Goal: Ages.
Strategy: The difference is $5-3=2$ parts.

$2\text{ parts}=12\Rightarrow1\text{ part}=6$
One part.
$5\times6=30,\quad3\times6=18$
Ages.

Answer

30

and

18

✓ Check

30-18=12

✓.

2. Scales and rates

\text{speed}=\frac{\text{distance}}{\text{time}}\qquad\rho=\frac mV

$\rho$: density.
$m$ , $V$: mass, volume.

Worked exampleFour rate questions

(a) 4 cm on a 1 : 50 000 map. (b) 150 km in $2\frac12$ h. (c) 540 g in 200 cm³. (d) 500 g for 2.40 or 750 g for 3.45?

Goal: Real distance, speed, density, best buy.
Strategy: Divide to "per one unit".

$4\times50000=200000\text{ cm}=2\text{ km}$
Scale; 100 000 cm per km.
$150\div2.5=60\text{ km/h}$
Speed.
$540\div200=2.7\text{ g/cm}^3$
Density.
$0.48\text{ vs }0.46\text{ per 100 g}$
$2.40\div5$ , $3.45\div7.5$ .

Answer2 km; 60 km/h; 2.7 g/cm³; the 750 g pack.

✓ CheckLarger pack cheaper per gram ✓.

Common misconceptionMixed units Simplify 50 cm : 2 m as 50 : 200 = 1 : 4, not 25 : 1.

Pause and think

Simplify $1.5 : 0.25$ .

Summary

Total parts → one part → shares.
Rates: divide to get "per one".

Ratio: Comparison of quantities.
Rate: Comparison of different units.

HistoryThe golden ratio. The Greeks studied the "golden ratio"

\varphi\approx1.618:1

, where the larger part is to the smaller as the whole is to the larger. Euclid described it around

300

BC; it appears in sunflower spirals and was used by Renaissance artists.

Fun factCooking for 100. Scaling a recipe for

4

people up to

100

is direct proportion: multiply every ingredient by

25

. Except cooking time, which is not proportional to quantity. Proportion only works when it really applies!

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