Direct and Inverse Proportion

In Grade 6 you met direct proportion — two quantities that grow and shrink together at the same rate. Double one and the other doubles; halve one and the other halves.

If 3 notebooks cost $6, then 6 notebooks cost $12 and 1 notebook costs $2 — the cost keeps perfect step with the number of notebooks.

This year you look at proportion more closely. The deep idea is this: in direct proportion, dividing one quantity by the other always gives the same number. For the notebooks, cost $\div$ number is $2 \div 1 = 2$, $6 \div 3 = 2$, $10 \div 5 = 2$ — always 2.

That fixed number is the constant of proportionality. It is the rate that links the two quantities, and once you know it, you can find any matching value.

How can you tell two quantities really are in direct proportion? There are two clear tests — one for a table, one for a graph.

From a table: divide one quantity by the other in every row. If you get the same number each time, the quantities are in direct proportion.

From a graph: plot the points. Direct proportion gives a straight line that passes through the origin $(0, 0)$.

Why through the origin? Because if you have none of one quantity, you have none of the other — 0 notebooks cost $0.

Watch out for graphs that are straight lines but miss the origin. A taxi fare graph might be a straight line, but if it starts at a fixed charge it does not pass through $(0, 0)$ — so the fare is not in direct proportion to the distance.

The most reliable way to solve a direct proportion problem is the unitary method — find the value of one unit, then scale.

Suppose 4 identical bricks have a mass of 10 kg. What is the mass of 7 bricks?

1. Find one unit. One brick has mass $10 \div 4 = 2.5$ kg.
2. Scale up. Seven bricks have mass $7 \times 2.5 = 17.5$ kg.

The unitary method handles recipes, prices, currency, fuel use and best-buy comparisons — anything where two quantities keep step.

A useful check: the constant of proportionality here is 2.5 kg per brick. Whatever number of bricks you choose, mass $\div$ number should always come back to 2.5.

Inverse proportion is the opposite relationship: as one quantity grows, the other shrinks — but in a balanced way, so their product stays fixed.

Think of a job that takes a fixed total amount of work. If 2 people take 6 hours, then 4 people take only 3 hours, and 1 person takes 12 hours. More people, less time.

The deep idea: in inverse proportion, multiplying the two quantities always gives the same number. For the job above, workers $\times$ hours is $1 \times 12 = 12$, $2 \times 6 = 12$, $3 \times 4 = 12$ — always 12.

That fixed number is the constant product. It represents the total work — in this case, 12 worker-hours. Knowing the constant product lets you answer any question about the job.

Spotting inverse proportion uses two tests, mirroring the direct-proportion ones.

From a table: multiply the two quantities in every row. A constant product means inverse proportion.

From a graph: inverse proportion gives a smooth curve that falls steeply then levels off, never touching either axis.

To solve an inverse proportion problem, find the constant product, then use it. If 5 machines fill an order in 8 hours, how long would 4 machines take?

1. Find the constant product: $5 \times 8 = 40$ machine-hours.
2. Use it for 4 machines: time $= 40 \div 4 = 10$ hours.

So 4 machines take 10 hours. Notice how the answers behave correctly — fewer machines means more time, just as inverse proportion should.

Direct and inverse proportion are some of the most practical ideas in maths:

• Science is full of proportion — speed and time for a fixed distance are in inverse proportion, while distance and time at a fixed speed are in direct proportion.
• Recipes and mixtures scale by direct proportion; sharing a fixed job among more people is inverse proportion.
• Graphs of proportion connect straight to your work on linear graphs and gradients.
• Best-value decisions at the shop are direct proportion questions in disguise.

The key skill is deciding which kind of proportion a situation shows. Ask yourself: when one quantity goes up, does the other go up too (direct) or down (inverse)? Then reach for the matching tool — constant ratio or constant product.

If proportion work feels hard later, it is usually telling the two types apart that needs a refresh. Come back to this guide whenever you need it — there is no prize for rushing.