A proportion states that two ratios are equal, and cross multiplication is the standard way to solve one for its missing term. From a/b = c/x you get a × x = b × c, so x is b × c ÷ a. The method works because multiplying both sides of an equation by the same quantity leaves it true.
The constant of proportionality is the number that turns one column into the other. Once you have it, you can extend the table indefinitely without touching cross multiplication again. Recipes, map scales, currency conversion and unit pricing are all the same single relationship applied to different quantities.
The formula
a- First term of the known ratio
b- Second term of the known ratio
c- First term of the second ratio
x- The unknown you are solving for
How to use this calculator
- Enter the two terms of the ratio you already know, a and b.
- Enter c, the first term of the second ratio.
- The gauge shows x, the missing fourth term.
- Check the two cross products below — they must be equal.
Worked examples
Solving 3 : 8 = 12 : x
Cross multiplying gives 3x = 8 × 12 = 96, so x = 32. Both cross products come to 96, which is the check that the proportion holds.
Scaling a recipe
If 4 eggs serve 6 people, then for 15 people x = 6-to-15 scaled: x = 4 × 15 ÷ 6 = 10 eggs. The constant of proportionality is 0.6667 eggs per person.
How to read your score
Frequently asked questions
Does the position of the unknown matter?
Only for where you divide. Cross multiplication always gives the product of the two known diagonal terms divided by the third known term, wherever the unknown sits.
Why does cross multiplication work?
It is multiplying both sides of a/b = c/x by b and by x at once. Nothing is lost, so the resulting equation ax = bc has the same solutions.
What if a is zero?
The proportion has no solution, because a ratio with zero on top can only equal another ratio with zero on top. The calculator substitutes a tiny value rather than failing outright.
Is this the same as a direct proportion?
Yes. Direct proportion means one quantity is a constant multiple of the other, and that constant is shown below the result. Inverse proportion, where the product rather than the ratio is fixed, needs a different setup.
Cross multiplication worked across a table
| a | c = 6 | c = 12 | c = 20 | c = 30 | c = 45 |
|---|---|---|---|---|---|
| 2 | 24.000 | 48.000 | 80.000 | 120.000 | 180.000 |
| 3 | 16.000 | 32.000 | 53.333 | 80.000 | 120.000 |
| 4 | 12.000 | 24.000 | 40.000 | 60.000 | 90.000 |
| 5 | 9.600 | 19.200 | 32.000 | 48.000 | 72.000 |
| 6 | 8.000 | 16.000 | 26.667 | 40.000 | 60.000 |
| 10 | 4.800 | 9.600 | 16.000 | 24.000 | 36.000 |
Every cell is 8 multiplied by c and divided by a.