Direct Proportion Calculator

What “directly proportional” means

Two quantities are directly proportional when their ratio never changes. If a litre of fuel costs a fixed price, then 2 litres cost double, 3 litres triple, and half a litre costs half. Plotted on a graph, the points fall on a straight line that passes through the origin (0, 0): zero of one means zero of the other. That straight-line-through-zero shape is the visual signature of direct proportion, and it is what separates it from relationships that have a fixed starting cost.

The formula: y = kx

Every direct proportion can be written as:

Here k is the fixed ratio between the two quantities — the price per litre, the grams per serving, the dollars per euro. Once you know k, you can find y for any x, or x for any y. The rule of three is simply this idea applied without having to name k explicitly.

Finding the constant of proportionality

Divide a known y by its matching x. If 4 kg of apples cost £6, then k = 6 ÷ 4 = £1.50 per kg. Now any weight is easy: 10 kg costs 1.50 × 10 = £15. Finding k first is handy when you need many answers from the same rate; the rule of three is quicker when you just need one.

Solving with the rule of three

For a single unknown, set up two ratios with matching units in matching places and cross-multiply:

The calculator's Direct mode does exactly this. Type three of the four numbers, leave the unknown blank, and it cross-multiplies and shows the working.

Everyday examples of direct proportion

• Shopping — total cost is directly proportional to quantity at a fixed unit price.
• Recipes — every ingredient scales directly with the number of servings.
• Currency — the amount you receive is directly proportional to the amount you exchange.
• Distance at constant speed — distance is directly proportional to time travelled.
• Map and model scales — real distance is directly proportional to the distance on paper.

Direct versus inverse — a quick check

Before solving, confirm the relationship really is direct. Ask: if I double the first quantity, does the second double too? If yes, it is direct. If doubling one halves the other — more workers finishing a job sooner, faster speed cutting travel time — it is inverse, and you should switch to the inverse method instead. Choosing the wrong type is the single most common error in proportion problems, so it is worth a moment's thought.