Direct proportion | Year 9

01 / LEARN

Understand the idea.

A constant multiplier

In direct proportion, each unit of input contributes the same amount of output. The rule is $y = k x$ : doubling $x$ doubles $y$ , and zero input gives zero output. There is no extra starting amount.

For nonzero input, calculate $k = \frac{y}{x}$ . This is a unit rate, so keep its units: a cost divided by a mass gives dollars per kilogram. Multiply that rate by a new mass to predict its cost.

y = k x

Fruit sold at a constant price
Mass (kg)	Cost (dollars)
1	3.20
2	6.40
5	16.00

Testing the rule

Test the quotients from several rows, rather than only comparing increases. Equal output increases for equal input steps show a constant rate of change, but a fixed starting charge could still be present.

A direct-proportion graph lies on a straight line through the origin. A taxi booking fee shifts the line away from the origin. Data consistent with one multiplier support a proportional model only while the same pricing conditions apply.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Price from a unit rate

Four identical notebooks cost 14 dollars. What do seven cost?

Worked solution

1
Find the cost of one notebook.
$\begin{aligned} k & = \frac{14}{4} \\ = 3.50 dollars/notebook \end{aligned}$
Identical notebooks have the same unit price.
2
Multiply by seven notebooks.
$C = 7 \times 3.50 = 24.50 dollars$
The total cost is the number of notebooks times the unit price.
3
Check the original purchase.
$4 \times 3.50 = 14 dollars$
The calculated unit price reproduces the given cost.

Answer

24.50 dollars

EXAMPLE 02

Test a table

A table has input-output pairs $(2, 9), (4, 17), (6, 25)$ . Is output directly proportional to input?

Worked solution

1
Calculate the first two output-to-input quotients.
$\begin{aligned} \frac{9}{2} & = 4.5 \\ \frac{17}{4} & = 4.25 \end{aligned}$
Direct proportion requires the same quotient at every nonzero input.
2
Compare with the final pair.
$\frac{25}{6} = 4.1 \bar{6}$
The third quotient also differs, so no single multiplier fits the table.
3
Explain the constant increases.
$\begin{aligned} y & = 4 x + 1 \\ 4 (2) + 1 & = 9 \\ 4 (4) + 1 & = 17 \\ 4 (6) + 1 & = 25 \end{aligned}$
These pairs fit a linear rule with a starting amount of one, rather than direct proportion.

Answer

No; the unit rate is not constant.

03 / YOUR TURN

Try it yourself.

Check your understanding

Six metres of ribbon cost 15 dollars at a constant price per metre. Find the cost of ten metres.

Show answer & reasoning

25 dollars

The unit price is $\frac{15}{6} = 2.50$ dollars per metre. At this constant rate, ten metres cost $10 \times 2.50 = 25$ dollars.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Three kilograms of rice cost 8.40 dollars. Find the price per kilogram.
2 marks
$y$ is directly proportional to $x$ . If $y = 18$ when $x = 6$ , write the rule.
2 marks
Use $y = 2.4 x$ to find $y$ when $x = 7$ .
2 marks
A printer produces 96 pages in four minutes at a constant rate. How long for 168 pages?
2 marks
Decide whether the pairs $(1, 5), (3, 15), (8, 40)$ show direct proportion. Explain.
2 marks
A delivery costs 6 dollars plus 2 dollars per kilometre. Is the total cost directly proportional to kilometres? Explain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Three kilograms of rice cost 8.40 dollars. Find the price per kilogram.
2 marks
$8.40 \div 3 = 2.80$ .
$2.80$ dollars per kg
$y$ is directly proportional to $x$ . If $y = 18$ when $x = 6$ , write the rule.
2 marks
$k = 18 \div 6 = 3$ .
Substitute into $y = k x$ .
$y = 3 x$
Use $y = 2.4 x$ to find $y$ when $x = 7$ .
2 marks
$y = 2.4 \times 7 = 16.8$ .
$16.8$
A printer produces 96 pages in four minutes at a constant rate. How long for 168 pages?
2 marks
$96 \div 4 = 24$ pages/min.
$168 \div 24 = 7$ min.
$7$ minutes
Decide whether the pairs $(1, 5), (3, 15), (8, 40)$ show direct proportion. Explain.
2 marks
$\frac{5}{1} = \frac{15}{3} = \frac{40}{8} = 5$ .
The same multiplier applies to every pair.
Yes; $y = 5 x$ .
A delivery costs 6 dollars plus 2 dollars per kilometre. Is the total cost directly proportional to kilometres? Explain.
3 marks
The rule is $C = 6 + 2 d$ .
The fixed 6 dollars charge means $\frac{C}{d}$ changes; the graph does not pass through the origin.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 proportional modelling, rates, scale, similarity and enlargement. Assumptions, interpretation and model limitations are taught alongside calculation.

Official curriculum source ↗

In this lesson

Understand the idea.

A constant multiplier

Testing the rule

See the method in action.

Price from a unit rate

Find the cost of one notebook.

Multiply by seven notebooks.

Check the original purchase.

Test a table

Calculate the first two output-to-input quotients.

Compare with the final pair.

Explain the constant increases.

Try it yourself.

Check your understanding

Keep practising