Year 9 - Direct proportion, rates and scale - Free practice

Year 9 Direct Proportion, Rates and Scale Worksheets & Practice

Model real situations with ratios, rates and scale. Worked solutions every time, free.

Quick answer

How do you solve problems involving direct proportion, rates and scale?

Direct proportion means two quantities increase together at a constant ratio, written y = kx, where k is the constant of proportionality. Find k by dividing a known y by its matching x, then use it to solve for unknowns. For example, if 4 litres of paint covers 24 m², then k = 6 m²/L, so 7 litres covers 42 m².

Generate a free maths worksheet

Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

Want more than a worksheet?

Practice built from mistakes.

A worksheet on Year 9 direct proportion, rates and scale is a good start. Tuterly goes further: it turns any question students get wrong into fresh practice, at the right level, on what’s likely to be in the next test or exam.

~90% of Bayside Academics students using it weekly improved their test marks

Of the Bayside Academics students who used Tuterly at least once a week, about 90% improved their school test marks.

Based on the actual school test marks of 100+ Bayside Academics students (in person and online) in Years 7 to 10, before and after using Tuterly at least once a week. How we measured this

Free worksheet Tuterly

The questionsFree worksheetThe same questions for every student on this topic.TuterlyWritten from the questions they got wrong in homework and school tests, at their level.

How manyFree worksheetOne sheet at a time.TuterlyAs many fresh questions as they need, each one different, never just the numbers swapped.*

Before a testFree worksheetPick a topic and hope it’s the right one.TuterlyA practice test covering every part of the topic. Upload the term planner and it focuses on what’s coming up.

MarkingFree worksheetCheck the answer key yourself.TuterlyMarked as they go, with the exact step that went wrong explained.

ProgressFree worksheetNothing is saved.TuterlyEvery question tracked until it’s mastered, and a weekly review for parents.

For Years 7 to 10: $8.99 a week or $29.99 a month at the launch price (from 1 April 2027, $17.99 a week or $59.99 a month), with a 7-day free trial on the monthly plan only. Better marks in 6 months, or your money back.

Or try it free now on one of their questions →

Coming soon on iPhone. Android soon after.

Free to download · 7-day free trial on monthly · Years 7 to 10 maths · On iPhone. Android coming soon

* With a subscription. Fair use applies: up to 80 snapped questions and 120 new practice questions a day per student.

Worked examples

Year 9 direct proportion, rates and scale: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Finding the constant of proportionality
A tap fills a tank at a constant rate. After 3 hours the tank holds 480 litres. Write an equation in the form $y = kx$ relating the volume of water $y$ (in litres) to the time $x$ (in hours), and find the volume after 7 hours.
Show worked solution
Key idea: In a direct proportion relationship $y = kx$ , the constant $k$ is found by dividing a known $y$ -value by the matching $x$ -value.
1. Find $k$ : $k = \frac{y}{x} = \frac{480}{3} = 160$
2. Write the equation: $y = 160x$
3. Substitute $x = 7$ : $y = 160 \times 7 = 1120$
Answer: The equation is $y = 160x$ and the volume after 7 hours is 1120 litres.
Example 2 · Converting and comparing rates
Runner A completes 10 km in 48 minutes. Runner B runs at 13.5 km/h. Which runner is faster, and by how much (in km/h)?
Show worked solution
Key idea: Convert both speeds to the same units before comparing.
1. Convert Runner A's time to hours: $48 \text{ min} = \frac{48}{60} \text{ h} = 0.8 \text{ h}$
2. Find Runner A's speed: $\text{Speed} = \frac{10}{0.8} = 12.5 \text{ km/h}$
3. Compare: Runner B runs at 13.5 km/h and Runner A at 12.5 km/h, so Runner B is faster.
4. Find the difference: $13.5 - 12.5 = 1 \text{ km/h}$
Answer: Runner B is faster by 1 km/h.
Example 3 · Scale diagrams in a real-life context
A landscape architect draws a plan of a rectangular garden using a scale of 1 : 250. On the plan, the garden measures 8 cm long and 5 cm wide. Paving costs $34 per square metre. Find the total cost to pave the entire garden.
Show worked solution
Key idea: Use the scale factor to convert plan measurements to real measurements, then calculate area and cost.
1. Convert plan dimensions to real dimensions (1 cm on the plan = 250 cm = 2.5 m in real life): $\text{Length} = 8 \times 2.5 = 20 \text{ m}, \text{Width} = 5 \times 2.5 = 12.5 \text{ m}$
2. Find the real area: $\text{Area} = 20 \times 12.5 = 250 \text{ m}^2$
3. Calculate the cost: $\text{Cost} = 250 \times 34 = 8500$
Answer: The total cost to pave the garden is $8500.

Common mistakes

Where students usually lose marks

Dividing in the wrong order to find k

Given $y = 480$ when $x = 3$ , a student writes $k = 3 \div 480 = 0.00625$ .

Always divide the $y$ -value by the $x$ -value: $k = y \div x$ . Check that $k$ makes sense in context; here, $k$ should be the number of litres per hour, which must be greater than 1.

Forgetting to convert units before calculating a rate

A runner completes 10 km in 48 minutes, so speed $= 10 \div 48 \approx 0.208$ km/h.

Convert time to hours before dividing: 48 minutes $= 0.8$ hours, giving $10 \div 0.8 = 12.5$ km/h. Always check that your units match the rate units required.

Squaring the scale factor incorrectly for area

Scale is 1 : 250, plan area is $8 \times 5 = 40$ cm², so real area $= 40 \times 250 = 10\,000$ cm².

When scaling an area, the scale factor must be applied to each dimension separately before multiplying, or the scale factor must be squared: real area $= 40 \times 250^2$ cm². It is safer to convert each length first, then find the area.

Assuming a relationship is proportional when it is not

A taxi charges a $3.50 flag fall plus $2.10 per km, so a student writes the direct proportion equation $y = 2.10x$ .

A direct proportion relationship must pass through the origin (when $x = 0$ , $y = 0$ ). A fixed starting fee means the relationship is linear but not directly proportional. The correct equation here is $y = 2.10x + 3.50$ .

Try one yourself

Your turn

A car travels at a constant speed and covers 270 km in 3 hours. A map of the route uses a scale of 1 : 4 000 000. How long, in centimetres correct to 1 decimal place, is the route on the map?

Got a different direct proportion, rates and scale question wrong? Upload it to Tuterly and get the worked solution and a fresh one to practise, free.

About this worksheet

Why we built it

Direct proportion, rates and scale sit at the heart of Year 9 mathematical modelling, connecting classroom maths to everyday situations like speed, unit pricing, map reading and mixing solutions. These worksheets build your student's skill from recognising proportional relationships all the way to setting up and solving multi-step problems, with fully worked solutions so they can follow the reasoning at every stage.

What's covered

Sub-skills your student will practise

✓Identifying direct proportion relationships and writing the equation y = kx
✓Finding the constant of proportionality k from tables, graphs and word problems
✓Solving practical rate problems including speed, density and unit cost
✓Converting and comparing rates expressed in different units
✓Interpreting and applying map and scale diagram ratios
✓Solving multi-step problems involving ratio, rate and proportion together

Other free Year 9 worksheet generators

Year 9

Quadratics

Year 9

Algebra

Year 9

Linear equations and graphs

Year 9

Trigonometry

Year 9

Pythagoras (applied)

Year 9

Surface area and volume

Year 9

Coordinate geometry

Year 9

Exponent laws

Year 9

Scientific notation

Year 9

Simple interest

Year 9

Statistics

Year 9

Probability

Year 9

Measurement error

Year 9

Enlargements and scale factors

Year 9

Comparing data sets

Year 9

Venn diagrams and two-way tables

Or generate any topic from the full worksheet builder.

From our families

Parents on the practice questions.

“Tuterly has been exactly what we needed since Tom started secondary school. Once the topic tests began, he was struggling to keep up. Now he has unlimited practice, all matched to what his class is actually covering, and his Maths results have improved noticeably.”

Rebecca M.

Year 8 · Maths

“My daughter used to leave revision until the night before. Having a platform she can dip into for fifteen minutes after dinner has completely changed that. She's calmer going into assessments, and honestly, so am I.”

Priya S.

Year 9 · Maths

“I like that I can just keep doing questions until it clicks. If I get something wrong, it shows me why, and then gives me another one like it.”

Ella, student

Year 10 · Maths

Verified parent

Frequently asked

Questions parents ask about direct proportion, rates and scale

What is direct proportion and how is it taught in Year 9?

Two quantities are in direct proportion when they increase or decrease at the same rate, written as y = kx where k is the constant of proportionality. In Year 9, students move beyond simple ratio to set up and solve these equations in practical contexts like speed, fuel consumption and ingredient scaling.

How are rates different from ratios at Year 9 level?

A ratio compares two quantities of the same kind, such as 3 : 1. A rate compares quantities of different kinds, such as 60 km per hour or $4.50 per kilogram. Year 9 students work with both, including converting between units and choosing the most useful form for a given problem.

What kinds of scale problems appear in Year 9 maths?

Students read and apply scale ratios on maps and technical drawings, calculate real distances or dimensions from scaled measurements, and work backwards to find what scale was used. Problems often involve unit conversion as an extra step.

Better maths marks start with the next wrong answer.

Tuterly turns every missed maths question into practice tailored towards the student, not just a generic worksheet.

Coming soon on iPhone. Android soon after.

Free to download · 7-day free trial on monthly · Years 7 to 10 maths · On iPhone. Android coming soon
Or try it free now with your own question →