Proportion Calculator
Solve a proportion a/b = c/x for the missing value by cross multiplication, and check whether two ratios are equal — the maths of scaling and conversion.
How to use this calculator
- 1Enter the three known values a, b, and c in the proportion a/b = c/x.
- 2Leave the x field to be solved — its entered value is ignored.
- 3Read the value of x that makes the two ratios equal.
- 4Check the cross products: a × x should equal b × c.
How it works
Proportion
a / b = c / x cross multiply: a · x = b · c solve: x = b · c ÷ a two ratios are equal ⇔ their cross products are equal
A proportion is a statement that two ratios are equal, written a/b = c/x. Because the two fractions are equal, their cross products must match: multiplying the numerator of one by the denominator of the other, in both directions, gives the same value. This is the rule of cross multiplication, and it turns a proportion into a simple linear equation, a·x = b·c, that can be solved for whichever value is unknown. Solving for the fourth term when three are known is one of the most-used calculations in everyday mathematics, because so many real problems — scaling a recipe up, converting between units, working out a distance from a map, adjusting a mixture — reduce to 'these two ratios are equal, find the missing part.'
Worked example
If 3 is to 4 as 9 is to x, cross multiplication gives 3·x = 4·9 = 36, so x = 36 ÷ 3 = 12. You can check it: 3/4 and 9/12 both equal 0.75, confirming the ratios are genuinely equal and the proportion holds.
Proportion Calculator: the complete guide
Why cross multiplication works
Cross multiplication can feel like a magic trick — draw an X across the equals sign, multiply along each diagonal, and set the products equal — but it is just ordinary algebra in disguise. Starting from a/b = c/x, you can multiply both sides of the equation by b and by x to clear the denominators. Doing so leaves a·x = b·c, which is exactly the cross-product relationship. The diagonal shortcut is simply a fast way to reach the equation you would get by clearing fractions the long way.
Seeing it as clearing denominators rather than a memorised rule helps avoid errors. It also makes clear why the method requires the unknown to appear in a denominator or numerator of one ratio, and why it fails when a term is zero in the wrong place — dividing by zero has no meaning. Once the fractions are cleared, what remains is a linear equation, the simplest kind to solve, which is why proportions are usually the first place students learn to 'solve for x' in a practical context.
Proportions everywhere in daily life
Proportional reasoning is one of the most transferable pieces of mathematics, because an enormous range of practical problems share the same structure. Doubling a recipe, converting miles to kilometres, working out how much paint covers a wall, reading a scale on a map or model, mixing a fuel-to-oil ratio, calculating a tip, figuring a discount, and adjusting medication to body weight are all proportion problems. In each, two quantities vary together in a fixed ratio, and you know three of the four numbers involved.
Recognising this common shape is more valuable than memorising a separate method for each situation. Once you see 'this many of these corresponds to that many of those, so how much for a different amount?', you can set up a proportion and cross multiply, whatever the context. The units on each side act as a built-in check: if the ratios are set up consistently — the same kind of quantity in each numerator and each denominator — the answer comes out in the right units, and a mismatch signals a setup error before you even compute.
Direct and inverse proportion
The proportions solved here are direct: as one quantity grows, the other grows in step, keeping their ratio constant. More servings need proportionally more flour; a longer distance takes proportionally more time at a fixed speed. This is the most common case and the one cross multiplication handles directly. But not every relationship is direct, and applying the direct method to an inverse one is a classic mistake.
In an inverse proportion, one quantity rises as the other falls, so their product rather than their ratio stays constant. If a job takes 6 workers 4 days, it does not take 12 workers 8 days — it takes them 2, because more workers means less time. Here the relationship is workers × days = constant, so you multiply rather than form equal ratios. Before setting up any proportion, it pays to ask whether the quantities move together or in opposition; that single check determines whether you keep the ratio constant or the product constant, and it is where proportional reasoning most often goes wrong.
Frequently asked questions
How do I solve a proportion?
Use cross multiplication. For a/b = c/x, multiply diagonally to get a·x = b·c, then divide to find x = b·c ÷ a. For 3/4 = 9/x, that gives 3x = 36, so x = 12. It works because equal ratios always have equal cross products.
What is cross multiplication?
It is a shortcut for solving a proportion: multiply the numerator of each ratio by the denominator of the other and set the products equal. It is really just clearing the fractions by multiplying both sides by the denominators, which turns the proportion into a simple linear equation.
How do I know if two ratios are equal?
Compare their cross products. Two ratios a/b and c/d are equal exactly when a·d = b·c. If the cross products match, the ratios are equivalent; if not, they aren't. You can also simplify each ratio to lowest terms and check whether they're identical.
What's the difference between direct and inverse proportion?
In direct proportion, quantities rise and fall together, keeping their ratio constant — solve with cross multiplication. In inverse proportion, one rises as the other falls, keeping their product constant, so you multiply instead. More workers taking less time is inverse; more servings needing more flour is direct.