Proportional Relationships

Two quantities have a proportional relationship when their ratio stays the same no matter how big or small they get. Whatever pair of values x and y you pick, dividing one by the other gives the very same number every time:

\frac{y}{x} = k \quad\text{(always the same } k\text{)}

That fixed number k is the constant ratio. It is exactly the unit rate — the amount of y for one unit of x. Once the ratio is locked, the two quantities move together: triple x and y triples too, because the ratio has to stay put.

A food stall charges the same for every taco. Check three different orders and work out the ratio £\div tacos each time:

food food £6 · 4 = £12 · 6 = £18

6 \div 2 = 3, 12 \div 4 = 3, 18 \div 6 = 3. Every order gives the same ratio k = 3 pounds per taco, so the price and the number of tacos are proportional. That single number 3 prices any order.

Two ways to spot it

A proportional relationship shows itself in two matching ways — one in a table, one on a graph:

Table test: divide y \div x in every row. If you get the same answer each time, it is proportional.
Graph test: plot the points. If they form a straight line that passes through the origin (0, 0), it is proportional.

Worked example. A table pairs hours worked with pay: 2 \to 16, 3 \to 24, 5 \to 40. Testing: 16 \div 2 = 8, 24 \div 3 = 8, 40 \div 5 = 8. The ratio is always 8, so pay is proportional to hours, at £8 per hour.

Worked example. Another table: 1 \to 5, 2 \to 8, 3 \to 11. Testing: 5 \div 1 = 5 but 8 \div 2 = 4 — the ratios disagree, so this is not proportional. (Look closely: it climbs by 3 each step but starts at 2, so it is y = 3x + 2.)

A hot-air balloon rises a steady 3 metres every second — but it launches from a 2-metre platform. So "more seconds means more height", yet it is not proportional: at 0 seconds the height is 2, not 0.

balloon height = 3t + 2

The ratio height \div time keeps changing (5 \div 1 = 5, then 8 \div 2 = 4, then 11 \div 3 \approx 3.7), and its graph is a straight line that misses the origin. Growing together is not enough — the ratio must be constant.

Quantities x and y are proportional when:

the ratio y \div x = k is the same for every pair — this constant k is the unit rate;
they are linked by y = kx;
the graph is a straight line through the origin (0, 0).

See it: straight, and through the origin

Below is a random proportional line y = kx. Press Play to reveal it step by step: first the line, then a couple of table points marked on it. Every marked point has the same ratio y \div x = k, and the line drives straight through the origin. Press Refresh for a new k.

Drive the ratio yourself

Pull the slider to change the constant ratio k. However you set it, the line stays straight and stays pinned to the origin — a bigger k just tilts it steeper. That is what "proportional" looks like — the same idea as direct proportion. Later, that multiplier k gets its own name: the constant of proportionality.

The classic trap

The most common mistake is thinking any "more means more" pairing is proportional. A phone plan that charges £5 to join plus £2 per month costs more the longer you stay — but it is not proportional, because month 0 already costs £5. Always run the ratio test and check the origin.

Two traps to avoid:

"More of one means more of the other" is NOT enough. A straight line with a starting amount (like y = x + 2) also grows together, yet its ratio y \div x keeps changing — so it is not proportional.
Proportional means the graph is straight AND passes through (0, 0). If there is a joining fee, flag-fall, or any fixed starting amount, the line misses the origin and the relationship is not proportional.