Ratio and Proportion

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 $3 : 5$: add the parts ($3 + 5 = 8$), size one share (120 \div 8 = \15$), then **multiply** ($3 \times 15 = $45$and$5 \times 15 = $75$).

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.

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 $6, then the cost-per-pen is 6 \div 4 = \1.50$— and that constant holds for any number of pens. So 10 pens cost$10 \times 1.50 = $15$.

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.

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 $\times$ hours $= 24$ every time — that 24 is the total amount of work.

So to find how long 12 workers would take, use the constant: $12 \times t = 24$, giving $t = 2$ hours. Unlike direct proportion, doubling one quantity halves the other.

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.

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.

• This is direct — more machines, more boxes.
• One machine: $240 \div 6 = 40$ boxes per hour.
• Scale up: $10 \times 40 = 400$ boxes.

For an inverse problem, work to the constant product instead. If 5 hoses fill a pool in 12 hours, the constant is $5 \times 12 = 60$, so 4 hoses take $60 \div 4 = 15$ hours. Decide direct or inverse, find your constant, then scale.

Ratio and proportion are among the most practical skills in maths:

• Scale drawings and maps use ratio to shrink real distances onto a page.
• Similar shapes in geometry have sides in equal ratios — direct proportion in a picture.
• Science is full of both kinds: speed and time for a fixed distance are inversely proportional, while distance and time at a fixed speed are directly proportional.
• Everyday decisions lean on the unitary method, from best-value shopping to splitting a bill fairly.

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.