How to Solve a Proportion and Simplify a Ratio (With Examples)
A proportion is solved by cross multiplication: rewrite A:B = C:D as A × D = B × C, then divide to isolate whichever value is missing. A ratio is simplified by dividing every term by their greatest common divisor until no whole number bigger than 1 divides them all evenly. Both take about a minute by hand once you know the pattern, and the calculator further down this page does either job instantly if you’d rather skip the arithmetic.
Reducing a ratio to its lowest terms
A ratio is in lowest terms when no whole number greater than 1 divides every part evenly. To get there, find the greatest common divisor (GCD) of the terms and divide each one by it. Say a batch of paint was mixed from 45 parts base to 75 parts tint. The GCD of 45 and 75 is 15: it divides into 45 exactly 3 times and into 75 exactly 5 times, and nothing larger divides both cleanly. Dividing each term by 15 reduces the ratio to 3:5. It’s still the same mix, described with the smallest pair of whole numbers that keeps the same proportion between base and tint.
Decimal ratios reduce the same way, just with an extra step first. Scale every term up until they’re all whole numbers (multiply by 10, 100, and so on until the decimals disappear), then divide by the GCD as usual. A ratio like 1.5:3 becomes 15:30 once scaled by 10, and 15:30 reduces to 1:2.
Solving a proportion for a missing value
A proportion states that two ratios are equal: A:B = C:D. Written as fractions, that’s A/B = C/D. Multiply both sides by B and by D and the denominators cancel out, leaving A × D = B × C. That’s the cross multiplication rule, and it holds no matter which of the four values happens to be the unknown one.
Once three of the four values are known, finding the fourth is just rearranging that same equation:
- Missing A: A = (B × C) / D
- Missing B: B = (A × D) / C
- Missing C: C = (A × D) / B
- Missing D: D = (B × C) / A
Exactly one of the four should be unknown at a time. A proportion with two missing values has infinitely many solutions, because scaling the whole ratio up or down by any factor would still satisfy A:B = C:D.
Worked examples
Scaling a recipe up
A recipe calls for 2 cups of flour for every 3 eggs. You want to make a bigger batch using 5 cups of flour and need to know how many eggs to crack. Set up the proportion with the unknown as x:
2 : 3 = 5 : xCross multiplying gives 2 × x = 3 × 5, so 2x = 15, and dividing both sides by 2 gives x = 7.5 eggs.
| Flour (cups) | Eggs |
|---|---|
| 2 | 3 |
| 5 | 7.5 (solved) |
Half an egg isn’t practical at the stove, so in practice you’d round to 7 or 8 depending on how forgiving the recipe is, or whisk one egg separately and use half of it. The math still says 7.5, and that’s the number the proportion actually returns.
Diluting a cleaning solution
Say the label on a cleaning concentrate calls for a 1:20 ratio of concentrate to water, and you need to mix up 6300 ml of finished solution for a big cleaning job. The two known parts of the ratio, 1 and 20, add up to 21 total parts in every batch. Divide the target volume by that total to find out how much one part is worth:
6300 ml / 21 parts = 300 ml per partFrom there, the concentrate is 1 part and the water is 20 parts:
| Component | Parts | Amount |
|---|---|---|
| Concentrate | 1 | 300 ml |
| Water | 20 | 6000 ml |
| Total | 21 | 6300 ml |
Checking the arithmetic: 300 ml + 6000 ml comes to 6300 ml, matching the target, and 300:6000 simplifies back down to 1:20 by dividing both sides by 300. That cross-check, does the simplified ratio match the one you started with, is a good habit any time you scale a mix up from a label instruction.
Solve your own ratio or proportion
Simplify a ratio
Use a colon, slash or space between terms, e.g. 1920:1080. Two or more terms.
Enter a ratio with at least two terms to reduce it.
Solve a proportion
Fill in three boxes of A : B = C : D and leave one empty. The missing value is solved for you.
Leave exactly one box empty to solve for it.
Common mistakes and edge cases
- Cross-multiplying the wrong pair. In A:B = C:D, the cross products are A×D and B×C, the two values on opposite corners of the equation. Multiplying A by C instead (the values sitting next to each other) gives a meaningless result.
- Reducing a decimal ratio before scaling it to whole numbers. Dividing 1.5 by a GCD found from the raw decimals produces a fraction of a fraction. Scale every term to a whole number first, then reduce.
- Losing track of which value is actually missing. It’s easy to write x in the wrong slot when a word problem is worded awkwardly. Reread the proportion once it’s set up and check that x sits where the unknown quantity from the question actually belongs.
- Dividing by zero. If the term diagonally opposite the unknown is 0, the proportion has no defined solution (or every value satisfies it, if the corresponding known term is also 0). A proportion built from real-world quantities should never have a zero in that position to begin with.
Frequently asked questions
How do you solve a proportion? Cross multiply. For A:B = C:D, that means A × D = B × C. Plug in the three values you know, then divide to isolate the one you don’t. For 2:3 = 5:x, cross multiplying gives 2x = 15, so x = 7.5.
How do you simplify a ratio to its lowest terms? Divide every term in the ratio by their greatest common divisor. A 45:75 ratio has a GCD of 15, so dividing both terms by 15 gives 3:5, the smallest whole-number pair with the same proportion.
What’s the difference between a ratio and a proportion? A ratio compares two or more quantities, like 3:5. A proportion is a statement that two ratios are equal, like 3:5 = 6:10. You simplify a ratio; you solve a proportion for a missing term.
Can a proportion have more than one missing value? Not a solvable one. With two unknowns in A:B = C:D there are infinitely many pairs of values that satisfy the equation, since the whole ratio can be scaled by any factor. A proportion needs exactly three known values to solve for the fourth.
Do ratios and proportions work with decimals or only whole numbers? Both work with decimals throughout the calculation. The lowest-terms result of a simplified ratio is conventionally shown in whole numbers, which is why decimal inputs get scaled up before dividing by the GCD, but the proportion solver itself handles decimal inputs and outputs without any extra steps.