5  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.

5.1 What this chapter helps you do

Symbols to keep handy

These are the bits of notation you'll see a lot. If a line of symbols feels like a fence, read it out loud once, then keep going.

  • a:b: the ratio of a to b

  • ∝: is proportional to

  • k: the constant of proportionality

Here is the main move this chapter is making, in plain terms. You do not need to be fast. You just need to keep the thread.

  • Coming in: You can express any quantity as a number with a unit. You can divide one number by another.

  • Leaving with: A ratio is a comparison by division. Two quantities are proportional when their ratio stays constant — when one changes, the other changes by the same factor. This single structure underlies scale, dilution, gear ratios, map reading, and most of the relationships in science.

5.2 What the notation is saying

Start with a simple picture. If a drink is mixed in the ratio 2:3, that means 2 equal parts of one ingredient for every 3 equal parts of the other. Ratio compares quantities by how many equal parts they have.

The ratio a:b means “for every a of the first thing, there are b of the second.” It is a comparison by division. We can also write that comparison as the fraction \frac{a}{b}.

This means a ratio tells us how one quantity matches another, not just their total.

Now think about scaling. If 4 people need 300 g of flour, then 8 people need twice as much. If 2 people need 300 mL of paint, then 6 people need three times as much. In both cases, the quantities change together.

When two quantities always change by the same scale factor, they are called proportional.

y = kx

Here, x and y are the two quantities, and k is a fixed number. That fixed number tells us how much of y goes with 1 unit of x.

This means k is the constant multiplier linking the two quantities.

We also write this as y \propto x, read as “y is proportional to x.”

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.

This means two comparisons are describing the same relationship.

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

5.3 The method

Finding a missing value in a proportion

Suppose you know

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

and you want to find d.

Multiply both sides by bd. This clears the denominators and gives

ad = bc

Now divide by a:

d = \frac{bc}{a}

Here, a, b, and c are known values, and d is the missing one. This means the missing value is chosen so the two ratios stay equal.

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.
  2. Find one part: T \div (a + b).
  3. Multiply by each share: first amount = a \times one part, second amount = b \times one part.

Here, T is the total being shared. This means we first split the whole into equal pieces, then count how many pieces each side gets.

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.

This means one scale change happens after another.

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.

This means a larger driven gear turns more slowly, because each turn of the small gear moves only part of the larger one.

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.

Here, (x_0, y_0) is one matching pair of values. This means once you know one pair, you can build the whole proportional relationship.

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.

5.4 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 number of people changes from 4 to 10, so we find the scale factor:

\frac{10}{4} = 2.5

Now multiply each ingredient by 2.5:

\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}

This means every ingredient grows by the same factor, so the recipe keeps the same taste.

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.

So 7.4 cm on the map means:

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

This means every map centimetre stands for a much larger real distance.

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?

First count the total parts:

2 + 3 + 1 = 6

Now find 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.

This means the cost is shared by use, not equally.

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?

First count the total parts:

3 + 5 = 8

Then find one part:

\$24,000 \div 8 = \$3,000

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

Check: $$9,000 + $$15,000 = \$24,000. Yes.

This means the profit is divided in the same ratio as the investment.

5.5 Where this goes

This chapter gives us a way to compare quantities by scale instead of by difference. That idea comes back immediately in the next chapter.

Percent and rates (Chapter 3) is a special case of proportion, where the comparison is made out of 100.

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.

5.6 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?