Direct proportion | Year 9

01 / LEARN

The idea, step by step

A constant multiplier

Two quantities are directly proportional when one is always the same multiple of the other. If the input doubles, the output doubles; if it halves, the output halves. For positive inputs, the quotient output ÷ input stays constant.

Use $y = k x$ . The constant $k$ is the output for one unit of input, so its units matter. If $x$ is kilograms and $y$ is dollars, $k$ has units dollars per kilogram.

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

Find the unit rate from more than one row of a table. A constant difference alone does not prove direct proportion: the differences must correspond to equal input steps, and a proportional model has no starting charge.

The graph of $y = k x$ is a straight line through the origin. A taxi fare with a booking fee may be linear, but it is not directly proportional to distance. Always read the situation before extending a rule.

A common trap

Equal increases in output always mean direct proportion.

Instead: A rule such as $y = 4 x + 1$ increases at a constant rate but includes a starting amount. Direct proportion must have the form $y = k x$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Price from a unit rate

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

The price per notebook is $14 \div 4 = 3.50$ .
Use the same price for seven: $7 \times 3.50 = 24.50$ .

24.50 dollars

Example 2

Test a table

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

The output/input quotients are $9 / 2 = 4.5$ , $17 / 4 = 4.25$ and $25 / 6 \approx 4.167$ . They differ.
The rule is $y = 4 x + 1$ , which includes an initial amount. It is linear, but not direct proportion.

No; the unit rate is not constant.

Check your understanding

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

Show answer

25 dollars

$15 \div 6 = 2.50$ dollars per metre, then $10 \times 2.50 = 25$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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
$5 / 1 = 15 / 3 = 40 / 8 = 5$ .
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 $C / d$ 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 ↗

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