Proportion Calculator

Where cross multiplication comes from

Cross multiplication is usually taught as a trick: multiply diagonally. It is worth seeing that it is not a trick at all.

Start with 3/4 = 9/d. Multiply both sides by 4: 3 = 36/d. Multiply both sides by d: 3d = 36. That is the cross product, reached by the ordinary rules of equations.

In general, multiplying a/b = c/d by bd clears both denominators and leaves ad = bc. Every proportion problem is that one line followed by a division.

Knowing the derivation matters because it tells you when the shortcut applies: only when both sides really are single fractions. a/b + 1 = c/d is not a proportion, and cross multiplying it gives nonsense.

Setting it up the same way round

The arithmetic is easy. Setting the proportion up correctly is where the errors are.

If 3 apples cost £1.50, what do 7 cost? Write apples over pounds on both sides: 3/1.50 = 7/x. Or pounds over apples on both: 1.50/3 = x/7. Both give x = 3.50.

What does not work is mixing them — 3/1.50 = x/7 — which gives 14, four times too much. Nothing about the equation looks wrong; only the units do.

So the habit worth building: label each position with its unit before solving. If the units read the same way on both sides, the setup is right.

Direct and inverse proportion

Direct. One goes up, so does the other, by the same factor. Twice the flour needs twice the sugar. This is what a : b = c : d describes, and it is what this page solves.

Inverse. One goes up, the other goes DOWN. Twice as many workers, half the time. Here the PRODUCT is constant, not the ratio: a × b = c × d.

Using the direct form on an inverse problem gives an answer that is wrong by the square of the factor, and it always looks plausible. The tell is in the sentence: “more of this means less of that” means inverse, and this page is not the tool.

Speed and time for a fixed distance are inversely proportional, which is also why their correct average is a harmonic mean rather than an arithmetic one.

Scaling recipes and drawings

Recipes. A recipe for four scaled to six is one proportion per ingredient: 4 : 200g = 6 : x, giving 300g. Every ingredient uses the same 1.5 factor, so do it once and multiply through.

Maps and drawings. A 1 : 25,000 map means 1 cm is 25,000 cm, which is 250 m. A measured 3.4 cm is 850 m.

Similar shapes. Corresponding sides are in proportion, which is what makes similar triangles solvable from three known lengths.

Currency and unit conversion. Any fixed-rate conversion is a proportion, which is why the arithmetic feels identical whatever the subject.

When a proportion is the wrong model

A proportion assumes strict linearity through the origin: double one side, the other doubles; zero on one side means zero on the other. Plenty of real relationships fail that.

Anything with a fixed cost. A taxi at £3 plus £2 a mile is not proportional. Ten miles costs £23, not twice the £13 of five miles.

Areas and volumes. Doubling the side of a square quadruples the area. A length proportion does not carry across to area, and reaching for it is a factor-of-two error.

People and time on a task. Two painters may not paint twice as fast, and nine women cannot make a baby in one month. Inverse proportion is the better model, and often even that is generous.

Anything that saturates. Twice the fertiliser is not twice the yield. Real systems have limits, and a proportion has none.

The cases with no answer

Some proportions cannot be solved, and the page says which rather than returning something.

Dividing by zero. 0 : 4 = 9 : d has no solution for d: the cross product gives 0 × d = 36, and nothing satisfies that.

Zero in a denominator position. a : 0 is not a ratio at all, for the same reason a fraction cannot have a zero denominator.

Four terms that simply do not match. With all four given, this page checks rather than solves, and reports honestly when the cross products differ. 2 : 5 = 6 : 14 is not a proportion, and saying so is the answer.

Sources and methodology

The definition and the linearity assumption behind it are curriculum standards; these are the references.

• National Council of Teachers of Mathematics — where proportional reasoning and its limits are set out
• NIST Office of Weights and Measures — on scaling measured quantities
• Ratio Calculator — simplifying a ratio and sharing a total by it
• How we calculate — methodology

Method. Every term is carried as an exact rational, so 3/4 : 1/2 = 9 : x is solved on fractions rather than on 0.75 and 0.5. Whichever position is left blank is solved from the same identity rather than from four separate formulas, and the answer is substituted back into ad = bc and the result displayed, so the verification is computed rather than claimed. The suite solves three thousand generated proportions and requires every one to satisfy ad = bc. That engine is verified on every change against 81 hand-written assertions, including that every solved proportion satisfies ad = bc across three thousand generated cases, in all four positions. The count and the per-case breakdown are published on the formula verification page.

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Read the guide

To simplify a ratio or split a total by one rather than solve an equation, the Ratio Calculator covers that and shows each term’s share of the whole.