Choose which term is missing and fill in the other three; the fourth appears when you calculate, along with a check. Below is the method the calculator follows, with one case solved by hand. If your problem gives quantities and you must decide whether the relationship is direct or inverse, use the rule of three.
Worked example: 2/3 = x/12
In 2/3 = x/12, the numerator of the second ratio is missing. What is x?
- Cross-multiply. The extremes are 2 and 12, and the means are 3 and x, so 2 × 12 = 3 × x.
- Compute the known product. 2 × 12 = 24, so 24 = 3 × x.
- Solve. x = 24 ÷ 3 = 8.
- Check the denominators. 3 and 12 are not zero, so the solution is valid.
The result is x = 8.
Check. 8/12 simplifies by dividing by 4 to give 2/3, and the cross products match: 2 × 12 = 24 = 3 × 8.
Why cross products work
A proportion says two ratios are equal. If you multiply both sides of a/b = c/d by b × d, the denominators cancel and you get a × d = b × c. That step is valid only when b and d are not zero, which is why a known denominator equal to zero makes the proportion invalid. OpenStax puts the property this way: when a proportion is true, its cross products are equal (cross products and solving proportions, accessed October 17, 2026).
The unknown can be in any of the four positions, and it is always solved the same way: multiply the two known terms that sit diagonally from each other and divide by the term diagonal to the unknown. In 5/8 = 10/x, the unknown is in the denominator, so x = 8 × 10 ÷ 5 = 16. After solving, look again to make sure no denominator is zero; a result of x = 0 in a denominator is not a solution of the original proportion.
To test whether a given equation is a proportion, compare the cross products. In 3/4 = 6/9, 3 × 9 = 27 and 4 × 6 = 24. They do not match, so it is not a proportion: 3/4 is 0.75 and 6/9 is about 0.67.
Cases worth telling apart
Most proportions have a single answer, but in several situations the result changes character. The table puts them side by side, with the answer the calculator gives in each.
| Proportion | What happens | Result |
|---|---|---|
| x/6 = 5/4 | Unique, fractional solution | x = 15/2, which is 7.5 |
| 0/5 = x/7 | Zero is a valid solution | x = 0 |
| 2/x = 0/2 | Contradiction, because 2/x is never 0 | No solution |
| 0/x = 0/2 | Identity: both sides equal 0 | Any x other than 0 |
| x/0 = 1/2 | Known denominator equal to 0 | Rejected |
| 2/3 = 0/x | Solving would give x = 0 | Rejected |
In the last two rows, the cross products produce an equation even though the starting proportion makes no sense. The calculator checks the denominators at the end and does not invent a number to complete the equality.
Scales and recipes with units
A proportion works for any problem where two pairs of quantities share the same ratio, as long as each ratio compares the same quantities in the same order. In a recipe, if 250 g of flour takes 3 eggs, then 750 g takes x eggs. Set up 3/250 = x/750, and x = 3 × 750 ÷ 250 = 9 eggs. On a map at a scale of 1:50,000, 3 cm on the map equals 3 × 50,000 ÷ 100,000 = 1.5 km, because 1 km is 100,000 cm. To read a scale like 1:50,000 and turn it into a fraction, the guide on fractions and map scales gives more cases.
When to use another page
This page solves the equation once both ratios are already written. If your problem is something else, this table points you to the page that fits.
| Your problem | Page |
|---|---|
| Quantities where you must decide if they are directly or inversely proportional | Rule of three |
| Split a total into proportional parts | Proportional sharing |
| Reduce a ratio to its simplest form | Simplify ratios |
| Turn a ratio into a fraction | Ratio to fraction |
| Blank in an equivalent fraction | Missing fraction value |
| Compare two fractions without looking for a term | Compare fractions |
To see how the three ideas relate, read the guide on ratio, proportion, and fraction.
The calculator does not interpret word problems or units: you decide which quantity goes in each position and check that the units of both ratios match.
Frequently asked questions
Does it accept decimals and negative numbers?
Yes. The method is the same with signed integers, fractions, and decimals. With −3/7 = x/28 the result is x = −12, and with 2.5/4 = x/18 it is x = 45/4, which is 11.25 as a decimal. If the result is not a whole number, you can leave it as a fraction in lowest terms.
What is the fourth proportional?
It is the term d that completes a/b = c/d when a, b, and c are known. For 20, 30, and 45, set up 20/30 = 45/d, and d = 30 × 45 ÷ 20 = 135/2, which is 67.5 as a decimal. In the calculator, choose d as the unknown term and fill in the other three.