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You can already simplify ratios, share amounts and use direct proportion. Stage 9 adds the other side of the story — inverse proportion, where more of one thing means less of another — and sharper proportional reasoning.
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
Ratio work should now be quick — share by counting parts and sizing one share.
By Stage 9 ratio work should be fast and sure. A ratio compares amounts with a colon; the order matters and every part must be in the same units before you simplify.
The three-step sharing method should run automatically. To split $120 in the ratio : add the parts (), size one share (120 \div 8 = \153 \times 15 = $455 \times 15 = $75$).
Two amounts in direct proportion grow together — their ratio stays constant.
Two quantities are in direct proportion when they change together at the same rate. Double one and the other doubles; halve one and the other halves. When one is zero, so is the other.
The signature of direct proportion is a constant ratio: divide one quantity by the other and you always get the same number.
If 4 pens cost $6, then the cost-per-pen is 6 \div 4 = \1.5010 \times 1.50 = $15$.
In inverse proportion, more of one amount means less of the other.
Stage 9 introduces the opposite pattern: inverse proportion. Here, as one quantity grows, the other shrinks — and they balance so that the product stays the same.
The classic example is workers and time. If 4 people paint a fence in 6 hours, then 8 people — twice as many — share the work and finish in 3 hours — half the time.
The signature of inverse proportion is a constant product: multiply the two quantities and you always get the same number. For the fence, workers hours every time — that 24 is the total amount of work.
Before calculating, ask whether the second amount rises or falls with the first.
The single most important Stage 9 skill here is deciding which kind of proportion a situation shows — before you reach for a method.
Ask one question: as the first quantity increases, does the second one increase too, or does it decrease?
Try a few. More petrol means a longer journey — both rise together, so direct. More taps filling a tank means less time — one rises, one falls, so inverse. More guests means each gets a smaller slice of cake — inverse again.
Get this decision right and the rest is easy: direct proportion uses a constant ratio, inverse proportion uses a constant product. Decide first, calculate second.
Harder problems chain proportional steps — work to one unit, then scale.
Stage 9 proportion problems often need a chain of proportional reasoning, not a single step. The reliable route is still the unitary method: work back to one of something, then scale to whatever the question wants.
Say 6 identical machines fill 240 boxes in an hour, and you want to know how many boxes 10 machines fill.
Ratio and proportion reach into geometry, science and everyday decisions.
Ratio and proportion are among the most practical skills in maths:
If a later topic about scaling or comparing feels tricky, it is often a ratio or proportion step underneath. The habit of asking "direct or inverse?" first is worth practising until it feels natural.
Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 9). Last reviewed 2026-05-19.
Step-by-step solutions for ratio and proportion, written exactly the way a tutor would explain them at the board.
Question
If 3 notebooks cost $7.50, how much do 8 notebooks cost?
Step-by-step solution
Step 1
More notebooks means more cost, so this is direct proportion. Find the cost of one notebook.
Question
It takes 6 workers 10 days to build a wall. How long would 4 workers take, working at the same rate?
Step-by-step solution
Step 1
Fewer workers means more time, so this is inverse proportion. Find the constant product.
Question
A tank is filled by 5 identical pumps in 8 hours. One pump breaks down. How long will the remaining pumps take to fill the same tank?
Step-by-step solution
Step 1
Fewer pumps means more time, so this is inverse proportion — work with a constant product.
Step 2
Find the constant product of pumps and hours.
The important words for ratio and proportion and what they mean — learn these so you can explain your thinking clearly.
A comparison of two or more amounts, written with a colon. The order of the parts matters.
A relationship in which two quantities change together according to a fixed rule.
A relationship where two quantities grow and shrink together at the same rate.
Example
Double the number of items, double the cost.
A relationship where one quantity grows as the other shrinks, so their product stays constant.
Example
Twice as many workers take half the time.
The fixed value found by dividing one quantity by the other in direct proportion.
The fixed value found by multiplying the two quantities in inverse proportion.
Finding the value of one item first, then multiplying to scale up to any amount.
The amount each part of a ratio receives when a total is divided up.
A ratio where the only number dividing every part exactly is 1.
A ratio that describes the same comparison, found by multiplying or dividing every part by the same number.
A comparison of two quantities measured in different units, such as price per kilogram.
A ratio that links a length on a drawing or map to the matching real-life length.
The slip-ups students most often make with ratio and proportion — and simple ways to avoid them.
Why it happens
The ratio numbers are the most visible, so one of them gets used directly.
How to avoid it
Add the parts first. For the total is 9, so divide the amount by 9.
Why it happens
Direct proportion is met first and becomes the default for any 'proportion' question.
How to avoid it
Ask whether the second amount rises or falls. 'More workers, less time' is inverse.
Why it happens
The fixed value is found by addition in some other contexts.
How to avoid it
Inverse proportion keeps the product constant — multiply the two quantities.
Why it happens
The numbers are simplified before checking that the units agree.
Why it happens
The colon is treated as a join rather than an ordered comparison.
How to avoid it
Match the order of the ratio to the order named in the question — flour to water is flour first.
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The reliable check still holds: the shares add back to the original total. And when a question gives you one share or a difference instead of the total, the golden rule is unchanged — find the value of one part first, then everything else is a quick multiplication.
Graphed, direct proportion is always a straight line through the origin: zero items, zero cost, then a steady climb. Spotting direct proportion tells you straight away that the unitary method will solve the problem.
So to find how long 12 workers would take, use the constant: , giving hours. Unlike direct proportion, doubling one quantity halves the other.
For an inverse problem, work to the constant product instead. If 5 hoses fill a pool in 12 hours, the constant is , so 4 hoses take hours. Decide direct or inverse, find your constant, then scale.
Step 2
Scale up to 8 notebooks.
Answer
\20$
Step 2
The product of workers and days is always 60. So 4 workers take 60 ÷ 4 days.
Answer
days
Step 2
The smaller share is 2 parts.
Step 3
Add both shares to find the total.
Answer
sweets
Step 3
With one pump broken, 4 pumps remain.
Step 4
Divide the constant product by the number of pumps left.
Answer
hours
How to avoid it
Convert to the same unit first: .