Look at which term is missing, multiply the two known terms that sit diagonally, and divide by the third. If the terms are not integers, use the proportion calculator; and to generate or check equivalent fractions without solving for any term, there is equivalent fractions. That recipe works in all four possible positions of the unknown; the only thing that changes is which product goes on top and which term you divide by.
Worked example: 3/4 = x/20
A shape is cut into 4 parts and 3 are shaded. How many shaded parts do you need if it is cut into 20 equal parts to cover the same amount?
- Look at the denominator. It goes from 4 to 20, so it is multiplied by 20 ÷ 4 = 5.
- Apply the same factor to the numerator. 3 × 5 = 15.
- Confirm with cross products. 3 × 20 = 60 and 4 × 15 = 60.
Result: x = 15, and 3/4 = 15/20.
Check: 15/20 simplifies by dividing by 5 and goes back to 3/4.
The factor works when one denominator is a multiple of the other. When it is not, cross products work just the same and do not require you to see the factor.
The four positions of the unknown
With a/b = c/d, the missing term is found by multiplying the two terms that sit diagonally from each other and dividing by the term that sits diagonally from the unknown. The table shows one equation of each type and the value it gives; in every case, the result is checked by cross-multiplying.
| Missing | Equation | Solving | Example |
|---|---|---|---|
| Numerator of the first | x/b = c/d | x = b × c ÷ d | x/12 = 5/6 gives x = 10 |
| Denominator of the first | a/x = c/d | x = a × d ÷ c | 2/x = 6/15 gives x = 5 |
| Numerator of the second | a/b = x/d | x = a × d ÷ b | 5/8 = x/24 gives x = 15 |
| Denominator of the second | a/b = c/x | x = b × c ÷ a | 2/5 = 8/x gives x = 20 |
If the unknown is on the bottom, solving changes which product is divided by which term, and it is easy to swap them. Always substitute the value back into the starting equation to catch that.
No integer solution, no solution, or infinitely many
Sometimes the calculation does not land on an integer. With 5/6 = x/8, the cross products give 5 × 8 = 6 × x, or x = 40/6 = 20/3. That number makes the equation true, but it is not an integer term, so the calculator answers that there is no integer solution. To allow a rational number, use the proportion calculator.
There are two more cases, and both come from zero. In 0/x = 0/5, the left side equals 0 for any nonzero x, and the calculator says there are infinitely many solutions. In 2/x = 0/5 no value works, because 2/x is never 0: you get the contradiction 10 = 0.
When to choose another tool
The calculator covers a single integer unknown. If the terms are decimals or fractions, the proportion calculator accepts them, and if x appears in several places, fraction equations solves them. To generate a list of fractions with the same value, use equivalent fractions, and to check whether a fraction is already in lowest terms, use simplify fractions. OpenStax covers cross products in proportions, including solving when the variable is in a denominator (accessed October 17, 2026).
Frequently asked questions
Do the fractions have to be simplified before I enter them?
No. Enter them as given, because cross products do not depend on the fractions being in lowest terms. In 15/25 = x/35, x = 21, and you would have reached the same value with 3/5 = x/35.
How do I enter and check a negative term?
Type the minus sign right next to the number. In −11/x = 22/18, x = −9, and to check it you substitute: −11 × 18 = −198 and 22 × (−9) = −198, so the cross products match.
Does it work for a problem with “out of every”?
Yes, as long as the ratio can be written as a fraction. If 4 out of every 9 people in a group of 63 prefer tea, you set up x/63 = 4/9 and get x = 28. When there are units or inverse quantities, the rule of three sets them up more directly.