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Proportion Calculator

Enter three known values and choose how the quantities relate — the fourth follows, with the working shown.

More kilometres, more fuel. Three oranges cost £2, so nine cost £6.

Known pair
The pair with the unknown

Direct and inverse proportion, and how to tell them apart

A proportion states that two ratios are equal: a is to b as c is to x. Cross multiplying gives a·x = b·c, so x = b·c ÷ a. If three oranges cost £2 and you want nine, then 3 is to 2 as 9 is to x, and x = 2 × 9 ÷ 3 = £6. The same shape answers most everyday scaling questions — 350 g of flour serves four, so ten need 350 × 10 ÷ 4 = 875 g.

That formula only holds when the quantities move together. Plenty of them move apart, and then the arithmetic is different: x = a·b ÷ c. Four workers finish a job in six days, so eight workers finish it in 4 × 6 ÷ 8 = 3 days. Feeding those numbers into the direct formula would give twelve days and claim that hiring people slows the work down. This is the single most common mistake with proportions, and roughly half of the exercises in a school textbook are the inverse kind — which is why the direction is a choice here rather than an assumption.

The test is quick: double one quantity and see what happens to the other. Twice the distance needs twice the fuel, so distance and fuel are directly proportional. Twice the workers need half the time, so workers and time are inversely proportional. Another way to check is the product: in an inverse relationship a·b stays constant — four workers times six days is 24 worker-days, and so is eight times three. When the constant is a product rather than a ratio, the proportion is inverse.

Proportions turn up far outside the classroom. Scaling a recipe up or down, reading a map (at 1:50,000, seven centimetres on paper is 3.5 km on the ground), converting currency at a known rate, working out how long a journey takes at a different speed, and figuring out unit prices to compare two package sizes are all the same three-term sum. Speed is worth a second look, because it is the inverse case in disguise: covering the same distance at 90 km/h for two hours takes 90 × 2 ÷ 60 = 3 hours at 60 km/h.

Frequently asked questions

How do I solve a proportion?

Cross multiply. If a is to b as c is to x, then a·x = b·c, so x = b·c ÷ a. For 3 is to 2 as 9 is to x: x = 2 × 9 ÷ 3 = 6. The value you divide by is always the one diagonally opposite the unknown.

What is the difference between direct and inverse proportion?

In direct proportion the two quantities rise and fall together and x = b·c ÷ a — more kilometres, more fuel. In inverse proportion one rises as the other falls and x = a·b ÷ c — more workers, less time. Using the direct formula on an inverse problem is the classic error, and the answer it gives is not slightly off but backwards.

How do I know which one my problem is?

Double one of the quantities and ask what should happen to the other. If it doubles too, the relationship is direct. If it halves, it is inverse. You can also check whether the product stays constant: four workers over six days and eight workers over three days both come to 24 worker-days, which marks it as inverse.

Is a proportion the same as cross multiplication?

Cross multiplication is the method, the proportion is the statement. Writing a/b = c/x and multiplying each numerator by the opposite denominator clears both fractions and leaves a·x = b·c, which is a single step away from the answer.

Can I use it with decimals?

Yes. The three terms can be any numbers — prices, distances, weights. When all three are whole numbers the answer is also shown as an exact fraction, since a proportion is an equality between two fractions and the result often is not a round number.

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