WaywardHouse

6  Ratio and proportion

Comparing two quantities of the same kind

A recipe for 4 people needs 300 g of flour. You’re cooking for 10. How much flour do you need?

A map is drawn at 1:25,000. A road measures 8 cm on the map. How long is it on the ground?

You and two friends split a streaming subscription. The monthly cost is £14.99 for three people. One friend joins. What does each person pay now?

These three questions share the same calculation underneath.

6.1 What the notation is saying

The ratio \(a:b\) means “for every \(a\) of the first thing, there are \(b\) of the second.” It is a comparison by division. Writing \(a:b\) is the same as writing the fraction \(\frac{a}{b}\).

Two quantities \(y\) and \(x\) are proportional when:

\[y = kx\]

where \(k\) is a constant. Double \(x\), you double \(y\). Halve \(x\), you halve \(y\). The ratio \(\frac{y}{x} = k\) stays fixed.

Write this as \(y \propto x\) (read: “\(y\) is proportional to \(x\)”). The constant \(k\) is called the constant of proportionality — it is the fixed multiplier that links the two quantities.

A proportion equation sets two ratios equal:

\[\frac{a}{b} = \frac{c}{d}\]

Read: “\(a\) is to \(b\) as \(c\) is to \(d\).” If you know three of the four values, you can find the fourth. The next section shows how.

Adjust the two parts of a ratio below. The bars show how A and B relate to each other and to the total.

6.2 The method

Finding a missing value in a proportion

Given \(\frac{a}{b} = \frac{c}{d}\), find \(d\).

Multiply both sides by \(bd\) — this clears both denominators at once, leaving \(ad = bc\). Rearranging for \(d\):

\[d = \frac{bc}{a}\]

Dividing a quantity in a given ratio

To divide total \(T\) in the ratio \(a:b\):

  1. Count the total number of parts: \(a + b\) — this tells you how many equal slices the whole is cut into.
  2. Value of one part: \(T \div (a + b)\) — the size of each slice.
  3. First share: \(a \times\) (one part); second share: \(b \times\) (one part) — multiply each person’s slices by the slice size.

Cascaded ratios

Two ratios applied in series multiply. A 3:1 ratio followed by a 4:1 ratio gives a combined ratio of \(3 \times 4 = 12:1\).

Gear ratio from tooth counts

For meshing gears, the ratio equals the number of teeth on the driven gear divided by the teeth on the driving gear.

Finding a constant of proportionality

Given one known pair \((x_0, y_0)\) where \(y \propto x\):

\[k = \frac{y_0}{x_0}\]

Then for any other \(x\): \(y = kx\).

Why this works

A ratio is a relative relationship — it does not depend on the size of either quantity, only on their comparison. When two quantities are proportional, their ratio \(k\) is constant by definition. Cross-multiplication works because multiplying both sides of \(\frac{a}{b} = \frac{c}{d}\) by \(bd\) gives \(ad = bc\) — a balanced equation.

Enter three known values in a proportion. The fourth is calculated for you. Use this to check your worked-example answers.

6.3 Worked examples

Example 1 — Recipe scaling. A recipe for 4 people needs 300 g flour, 150 g butter, and 80 g sugar. How much of each do you need for 10 people?

The quantities scale in proportion with the number of people. Find the ratio of new to old: \(\frac{10}{4} = 2.5\).

Multiply each ingredient by 2.5 — this keeps everything in the same ratio:

\[\text{flour: } 300 \times 2.5 = 750 \text{ g}\] \[\text{butter: } 150 \times 2.5 = 375 \text{ g}\] \[\text{sugar: } 80 \times 2.5 = 200 \text{ g}\]

Example 2 — Geography: map scale. A road appears as 7.4 cm on a 1:50,000 map. What is the actual length of the road?

Map scale \(1:50,000\) means 1 unit on the map = 50,000 units on the ground. Multiply to convert:

\[\text{actual length} = 7.4 \text{ cm} \times 50,000 = 370,000 \text{ cm} = 3.7 \text{ km}\]

Example 3 — Splitting costs. Three friends split a monthly phone plan costing £36 in the ratio 2:3:1 (based on how much data each uses). How much does each pay?

Total parts: \(2 + 3 + 1 = 6\). One part = \(£36 \div 6 = £6\).

Friend A (2 parts): \(2 \times £6 = £12\) Friend B (3 parts): \(3 \times £6 = £18\) Friend C (1 part): \(1 \times £6 = £6\)

Check: \(£12 + £18 + £6 = £36\). Yes.

Example 4 — Finance: dividing profit. Two partners invest in a business in the ratio 3:5. The year’s profit is £24,000. How much does each receive?

Total parts: \(3 + 5 = 8\). One part = \(£24,000 \div 8 = £3,000\).

Partner A (3 parts): \(3 \times £3,000 = £9,000\) Partner B (5 parts): \(5 \times £3,000 = £15,000\)

Check: \(£9,000 + £15,000 = £24,000\). Yes.

6.4 Where this goes

Proportion is the first multiplicative relationship you encounter. Percent and rates (Chapter 3) is proportion restricted to a denominator of 100 — a special case of this chapter.

Later, proportion reappears inside trigonometry (the sine ratio is a ratio of lengths) and in logarithms (where proportional growth becomes additive). In engineering mathematics, dimensional analysis — checking that equations are consistent in their units — is a systematic application of ratio.

The notation \(y \propto x\) is also the simplest case of a functional relationship. Volume 3’s treatment of functions generalises this to any rule that maps inputs to outputs.

Where this shows up

  • Scaling a recipe up or down is proportion — every ingredient changes by the same factor.
  • Reading a map involves ratio: every measurement on paper is a proportional reduction of the real distance.
  • Splitting a bill by how much each person ordered is dividing a quantity in a ratio.
  • A gear ratio on a bike tells you how many times the back wheel turns for each turn of the pedals.

The arithmetic is identical in every case.

6.5 Exercises

  1. A recipe for 4 people needs 300 g flour, 150 g butter, and 80 g sugar. Scale the recipe to serve 7 people. (Exact grams to one decimal place.)

  2. A map has scale 1:25,000. Two towns are 14.6 cm apart on the map. What is the actual distance in kilometres?

  3. A car engine produces 250 Nm of torque. It drives through a gearbox with a 3.8:1 ratio, then a differential with a 3.2:1 ratio. What is the torque at the rear wheels?

  4. Silver and copper are mixed in the ratio 7:3 by mass to make an alloy. You need 2.5 kg of alloy. How many grams of each metal do you need?

  5. A 500 mL bottle of concentrated cleaning fluid requires diluting 1:15 before use. How many litres of diluted product can you make from one bottle?

  6. Three investors share profits in the ratio 2:3:7. Total profit this year is £48,000. How much does each investor receive?

  7. A gear system has a driver gear with 24 teeth and a driven gear with 60 teeth. What is the gear ratio? If the driver rotates at 900 rpm, what is the speed of the driven gear?