📚 What's in This Guide
This proportion calculator solves for the missing number in a proportion, shows the cross-multiplication steps, and checks whether two ratios are equal. Enter three numbers, leave one box empty, and the missing value appears at once. If you searched for a proportions calculator, a proportional calculator, a proportion solver, or a proportion calc, this is the same tool. Two more tabs handle direct and inverse proportion, and the proportion of a group in statistics, including a confidence interval.
Below you will find the proportion formula, how to write and solve a proportion by hand, worked examples, and an explanation of the two different meanings of the word proportion.
If a / b = c / d, then a × d = b × c. To find the missing term, multiply the two numbers that are diagonal to each other and divide by the number that is left. For 4 / 25 = x / 100, x = 4 × 100 ÷ 25 = 16.
1. What Is a Proportion?
A proportion is an equation that says two ratios are equal. The ratio 3 : 4 and the ratio 15 : 20 describe the same relationship, so 3 : 4 = 15 : 20 is a proportion. It is usually written as fractions, a / b = c / d, and read as a is to b as c is to d. When one of the four numbers is unknown, solving the proportion tells you what it must be to keep the ratios equal.
The word is also used in statistics, where a proportion is a part divided by its whole. If 18 of 120 people said yes, the proportion who said yes is 18 / 120 = 0.15, or 15 percent. These are two different ideas that share a name, and this page covers both. In this guide, "proportion" means two equal ratios unless a section says it is about statistics.
2. How the Proportion Calculator Works
The calculator has three tabs.
Solve a proportion. The boxes are laid out as A / B = C / D. Fill in any three, leave one empty, and the calculator solves for it with cross-multiplication. If you fill in all four, it checks whether the two ratios are equal. Results include the completed proportion, the ratio form, the first ratio as a decimal and a percentage, the scale factor between the two sides, and the steps.
Direct and inverse. This tab is for quantities that change together or in opposite directions. Enter two cases of x and y, leave one value empty, and choose direct or inverse. It also shows the constant of proportionality.
Proportion in statistics. Enter the number of successes (x) and the sample size (n). You get the sample proportion as a decimal, a fraction, and a percent, the standard error, the margin of error, and two confidence intervals.
The solve and direct and inverse tabs use plain numbers, including decimals, with a size of 0 or between 0.000000000001 and 1,000,000,000,000, and the second tab needs values above zero. A number can be up to 100 characters long. When you check whether two ratios are equal, current browsers compare the exact decimal numbers you typed, with no rounding tolerance. The size limit applies to the numbers you enter. A result can be larger or smaller than that, and very large or very small results may be shown in scientific notation and are rounded for display. The calculator will not divide by zero, so a bottom number cannot be 0. The statistics tab assumes a simple random sample and gives an interval estimate, not a hypothesis test. It is for learning and quick checks, so verify results that matter.
3. How to Use the Proportion Calculator
- Write your problem as two fractions with an equal sign, A / B = C / D, with matching quantities in matching positions.
- Type the three numbers you know into the boxes and leave the unknown box empty.
- Read the missing value, the completed proportion, and the steps.
- To test whether two ratios are equal, fill in all four boxes and read the Yes or No answer.
- For direct or inverse problems, open the Direct and inverse tab, choose the type, enter two cases, and leave one value empty.
- For a statistics problem, open the Proportion in statistics tab, enter x and n, and choose a confidence level.
- Use Copy Result, Share Result, or Download PDF for the tab you are on, and Reset All to start again.
If a box cannot be read, a bottom number is zero, or more than one box is empty, the calculator tells you what to fix instead of showing a number.
4. Proportion Formula and Equation
a / b = c / d or a : b = c : d
Cross-multiplication: a x d = b x c
The outer terms a and d are called the extremes, and the inner terms b and c are called the means. In a true proportion, the product of the extremes equals the product of the means.
Rearranging the cross-product equation gives a formula for each unknown.
| Unknown | Formula | Cannot be solved if |
|---|---|---|
| a | a = (b × c) ÷ d | d = 0 |
| b | b = (a × d) ÷ c | c = 0 |
| c | c = (a × d) ÷ b | b = 0 |
| d | d = (b × c) ÷ a | a = 0 |
The constant of proportionality describes the single number that links two quantities. In a direct proportion, y = k × x, so k = y / x. In an inverse proportion, y = k / x, so k = x × y.
5. How to Solve Proportions and How to Write a Proportion
Writing a proportion. First name what each number measures. Put the two quantities of the first ratio in one fraction, then put the same two kinds of quantity in the same order in the second fraction. If grams are on top and muffins are on the bottom on one side, they must be on top and bottom on the other side too. Keeping the units consistent matters more than which way up you write it, because flipping both fractions gives the same answer, but flipping only one does not.
Solving a proportion.
- Write the proportion with a letter, such as x, for the unknown.
- Cross-multiply: multiply the top of each fraction by the bottom of the other, and set the two products equal.
- Divide both sides by the number that multiplies x.
- Check by substituting the answer back and confirming that both cross-products match.
There is also a shortcut. Because the two fractions are equal, one is a scaled copy of the other. If you can see the scale factor between two numbers that sit side by side, multiply the remaining pair by the same factor. In 3 / 8 = x / 40, the bottoms go from 8 to 40, which is a factor of 5, so x = 3 × 5 = 15.
6. Worked Examples
Finding a percentage: 4 out of 25
To turn 4 out of 25 into a percentage, set it equal to x out of 100: 4 / 25 = x / 100. Cross-multiplying gives 4 × 100 = 25 × x, so 400 = 25x and x = 16. The proportion of 4 and 25 is 0.16, which is 16 percent. This is the calculator's default example.
Scaling a recipe
A recipe uses 250 g of flour for 12 muffins. How much flour is needed for 30 muffins? Write 250 / 12 = x / 30. Cross-multiplying gives 12x = 250 × 30 = 7,500, so x = 625 g.
Reading a map
On a map, 3 cm represents 12 km. A distance of 7.5 cm on the map is how many kilometers? Write 3 / 12 = 7.5 / x, which gives 3x = 12 × 7.5 = 90, so x = 30 km. In the calculator, enter 3, 12, and 7.5, and leave the D box empty.
What x-value makes the ratios equivalent?
Find x so that 3 / 8 = x / 40. Cross-multiplying gives 8x = 3 × 40 = 120, so x = 15. The scale factor from 8 to 40 is 5, and 3 × 5 = 15 agrees. The ratios 3 : 8 and 15 : 40 are equivalent.
Checking whether two ratios are in proportion
Are 2 / 4 and 3 / 6 in proportion? The cross-products are 2 × 6 = 12 and 4 × 3 = 12, so yes. Are 2 / 4 and 3 / 7 in proportion? The cross-products are 2 × 7 = 14 and 4 × 3 = 12, so no. Fill in all four boxes to run this check.
7. Direct and Inverse Proportion
Two quantities are in direct proportion when doubling one doubles the other, so y / x is constant. If 5 pens cost $15, then each pen costs $3, and 12 pens cost 12 × 3 = $36. As a proportion, 15 / 5 = y / 12, which gives y = 36.
Two quantities are in inverse proportion when doubling one halves the other, so x × y is constant. If 6 workers finish a job in 8 days, the job is 6 × 8 = 48 worker-days. With 4 workers it takes 48 ÷ 4 = 12 days. The equation is 6 × 8 = 4 × y, so y = 12.
| Type | What stays constant | Equation | Typical example |
|---|---|---|---|
| Direct | y / x = k | y = kx | Cost and number of items at a fixed price |
| Inverse | x × y = k | y = k / x | Workers and days, or speed and travel time for a fixed distance |
A quick test is to ask what happens when one quantity increases. If the other increases by the same factor, the relationship is direct. If it decreases by that factor, the relationship is inverse. The calculator treats the quantities as values above zero.
8. Proportion in Statistics
In statistics, a proportion is a count divided by a total. If x items in a sample of n have a property, the sample proportion is p = x / n. It is a number between 0 and 1, and multiplying by 100 gives a percent.
A sample proportion is an estimate, so it comes with a measure of its uncertainty. As Penn State's introductory statistics lesson on confidence intervals explains, the standard error of a sample proportion is the square root of p × (1 − p) ÷ n, and for large random samples a confidence interval is the sample proportion plus or minus a z value times the standard error. The z value is about 1.645 for 90 percent, 1.96 for 95 percent, and 2.576 for 99 percent confidence.
p = x / n
SE = sqrt( p x (1 - p) / n )
Wald interval = p +/- z x SE
Suppose 18 of 120 survey respondents said yes. The sample proportion is 18 / 120 = 0.15, or 15 percent. The standard error is the square root of 0.15 × 0.85 ÷ 120, which is about 0.0326. At 95 percent confidence, the margin of error is 1.96 × 0.0326 = 0.0639, so the Wald interval runs from 8.61 percent to 21.39 percent.
The calculator also shows the Wilson interval, which behaves better when the sample is small or the proportion is close to 0 or 1. For the same survey, the Wilson interval is 9.70 percent to 22.47 percent. With a smaller sample the two methods can differ much more. For 4 out of 25, the proportion is 0.16, the Wald interval is 1.63 percent to 30.37 percent, and the Wilson interval is 6.40 percent to 34.65 percent. A common rule of thumb is to rely on the simple Wald interval only when there are at least 10 successes and 10 failures, but that is a rough guideline and not a guarantee, so the Wilson interval is the safer choice whenever the sample is small or the proportion is close to 0 or 1.
Note that this is an estimate of a population proportion, and not a hypothesis test. If you need to test whether an observed proportion differs from a stated value, you need a test of one proportion, which uses the same standard-error idea but compares against a null value.
9. Ratios, Fractions, and Percentages
A ratio, a fraction, a decimal, and a percent can all describe the same comparison. The ratio 4 : 25, the fraction 4 / 25, the decimal 0.16, and 16 percent are different ways to write the same part-to-whole comparison of 4 out of 25. The calculator shows the first ratio as a decimal and a percent so you can move between them.
There is one trap. "4 to 25" can mean 4 parts compared with 25 parts, which is a part-to-part ratio, or it can mean 4 out of 25, which is part-to-whole. As part-to-whole, 4 out of 25 is 16 percent. As part-to-part, 4 parts to 25 parts make a total of 29 parts, so the first part is 4 / 29, or about 13.79 percent. Always check which meaning a problem intends. To work with ratios directly, use our Ratio Calculator.
10. Common Mistakes
- Mixing the order. If grams are on top on one side, they must be on top on the other side.
- Cross-multiplying the wrong pairs. Multiply diagonally, not straight across.
- Dividing by the wrong number. Divide by the number that sits next to the unknown.
- Treating an inverse relationship as direct. More workers means fewer days, not more.
- Mixing units. Convert minutes and hours, or centimeters and kilometers, before setting up the proportion.
- Using a zero as a bottom number. A fraction cannot have 0 in the denominator.
- Confusing part-to-part and part-to-whole. 4 : 25 and 4 out of 25 are different comparisons.
- Reading a confidence interval as a certainty. It describes a method, not a guarantee about one sample.
11. Related Calculations and Use Cases
Proportions appear in recipes, maps and scale drawings, currency and unit conversion, mixing ratios, and similar triangles, where matching sides are in proportion. For comparisons that do not need a missing value, such as simplifying or comparing two quantities, the Ratio Calculator is the better starting point. Use this calculator when one of the four numbers is unknown.
12. Frequently Asked Questions
In math, a proportion is a statement that two ratios are equal, such as 3/4 = 15/20 or 3 : 4 = 15 : 20. In statistics, a proportion is the part of a group that has some property, found by dividing the count by the total, such as 18 out of 120 = 0.15. The calculator handles both meanings in separate tabs.
Write the proportion as a/b = c/d, then cross-multiply to get a x d = b x c. Divide both sides by the number that sits next to the unknown. For 4/25 = x/100, cross-multiplying gives 4 x 100 = 25 x x, so 400 = 25x and x = 16. Substitute the answer back in to check that both cross-products match.
Divide the number of items with the property (x) by the total number of items (n). A sample with 18 successes out of 120 has a sample proportion of 18 / 120 = 0.15, or 15 percent. The standard error is the square root of p x (1 - p) / n, and a confidence interval is the proportion plus or minus a z value times the standard error.
A ratio compares two quantities, such as 3 : 4. A proportion says that two ratios are equal, such as 3 : 4 = 15 : 20. In statistics, the word proportion also means a part divided by its whole, which is a fraction between 0 and 1.
Name each quantity, then put matching quantities in matching positions. If 250 g of flour makes 12 muffins, the proportion for 30 muffins is 250 / 12 = x / 30, with grams on top and muffins on the bottom on both sides. You can also write it with colons as 250 : 12 = x : 30.
In a direct proportion, two quantities rise and fall together, and their ratio y / x stays constant. In an inverse proportion, one rises when the other falls, and their product x times y stays constant. Pens at a fixed price per pen are direct, and the number of workers against the days to finish a job is inverse.
Divide the first number by the second to get a decimal, then multiply by 100. The proportion 4 / 25 is 0.16, which is 16 percent. If you are solving 4 / 25 = x / 100, the missing value x is already the percentage, because 100 is the whole.
Conclusion
This proportion calculator finds the missing value in a proportion, checks whether two ratios are equal, solves direct and inverse problems, and estimates a proportion with a confidence interval. Set up your fractions with matching quantities in matching positions, cross-multiply, and check the result by substituting it back. Use the steps shown in each tab to learn the method, and double-check anything important, especially when you are working with real measurements or survey data.
This calculator is for learning and general calculation. Check important results independently, especially for health, financial, or research decisions.