Lesson 11.2 · 25 min

Direct Proportion

Quantities that grow at the same rate: the unitary method, multiplier tables, and straight-line graphs through the origin.

In this lesson you will learn to…

Recognise when two quantities are directly proportional
Use the unitary method or a multiplier to find missing values
Know that a direct proportion graph is a straight line through the origin
Write and use the equation $y = m x$

Why this matters

Exchange rates, recipes, fuel costs and unit prices are all direct proportion. If you know what one costs — or how much one litre covers — you know everything.

Is it direct proportion?

Situation	Direct?
Cost of apples and number of apples	Yes — twice as many cost twice as much
Paint used and area covered	Yes
Euros and US dollars (fixed rate)	Yes
Taxi fare with a fixed starting charge	No — the graph does not start at 0
Age and height	No — you do not keep growing at the same rate

The gradient is the multiplier

For apples at 30 cents each, the graph goes through $(4, 120)$ and $(10, 300)$ . The gradient is

$\frac{300 - 120}{10 - 4} = \frac{180}{6} = 30$

— exactly the cost of one apple — so the equation is $y = 30x$ .

Key rules and formulas

Direct proportion
Two quantities are directly proportional if, when one is doubled (or tripled, or halved), the other is too. The ratio between them never changes.
The unitary method
Find the value of one, then multiply. 3 oranges cost 81 cents, so 1 costs 27 cents and 7 cost $7 \times 27 = 189$ cents.
The multiplier
$y = m x$
The number you multiply $x$ by to get $y$ is always the same. It is also the gradient of the graph.
The graph
A graph of two directly proportional quantities is a straight line through the origin $(0, 0)$ .

Try it

Direct proportion is y = mx

Gradient, m1

y-intercept, c0

Equation: y = 1x + 0
Crosses the y-axis at: (0, 0)
For every 1 across: 1 up

Keep $c = 0$ and change $m$ : every line passes through the origin. That is what makes it direct proportion. Set $c$ to anything else and the proportion is broken — doubling $x$ no longer doubles $y$ .

Worked examples

Worked example 1

A multiplier table

x

and

y

are directly proportional. When

x = 4

y = 10

. Find

y

when

x = 7

, and

x

when

y = 35

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Mixing paint

Green paint is yellow and blue in the ratio

5 : 2

. How much blue is needed with 28 litres of yellow?

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

An equation

I change 80 British pounds into 6584 rupees. How many rupees do I get for 75 pounds?

Try it yourself first, then reveal the solution one step at a time.

Practice: direct proportion

9 questions

Nine questions. Find the multiplier, then use it.

Question 1 of 9Type the answerFoundation
3 oranges cost 81 cents. How much do 7 oranges cost?
Your answercents
Question 2 of 9Complete the table
$x$ and $y$ are directly proportional. Complete the table.
x y
5 8
Row 2, x 12
7 Row 3, y
Question 3 of 9Explore and match
$x$ and $y$ are directly proportional, with $y = 6$ when $x = 2$ . Set the graph of $y = m x + c$ to show this relationship.
Move the controls until you match the question, then press Check.
Gradient, m1
y-intercept, c0
Equation
y = 1x + 0
Crosses the y-axis at
(0, 0)
For every 1 across
1 up
Question 4 of 9Type the answer
Green paint is yellow and blue in the ratio $5 : 2$ . How many litres of blue go with 28 litres of yellow?
Your answerlitres
Question 5 of 9Step by step
Work with an equation.
A 5-litre tin of paint covers 60 square metres. Area covered, $A$ , is directly proportional to volume, $V$ .
1. Write the equation for $A$ in terms of $V$ .
  Step 1 answer
Question 6 of 9Type the answer
I change 80 British pounds into 6584 rupees. How many pounds would I get for 16 460 rupees?
Your answerpounds
Question 7 of 9Choose all that apply
Which of these are directly proportional?
Select every correct answer — there may be more than one.
The number of workers and the time to build a wallA taxi fare with a 3 dollar starting charge and the distance travelledThe weight of flour and the number of cakes bakedThe cost of petrol and the number of litres bought
Question 8 of 9True or false
The graph of two directly proportional quantities is a straight line through the origin.
Question 9 of 9Type the answer
$x$ and $y$ are directly proportional: $x = 3$ gives $y = 7.5$ . Find $x$ when $y = 23$ .
Your answer

x	y
5	8
Row 2, x	12
7	Row 3, y

In this lesson you will learn to…

Why this matters

Is it direct proportion?

The gradient is the multiplier

Key rules and formulas