Proportion Calculator

Solve a proportion for the missing value, or check whether two ratios really are proportional. Enter three of the four numbers and leave the fourth empty — the calculator finds it — or enter two ratios and compare their cross products. It runs entirely in your browser: free, instant, no sign-in.

How the proportion calculator works

A proportion is an equation between two ratios: a / b = c / d. The two ratios are equal, so their cross products are equal too: a × d = b × c. That one fact does both jobs on this page. To solve for a missing term, cross-multiply and divide: if 3 / 4 = x / 12, then 3 × 12 = 4 × x, so x = 36 ÷ 4 = 9. To check whether two ratios form a proportion, compare the cross products: 2 / 3 and 4 / 6 give 2 × 6 = 12 and 3 × 4 = 12 — equal, so the ratios are proportional. For 1 / 2 and 3 / 7 the cross products are 1 × 7 = 7 and 2 × 3 = 6 — different, so they are not proportional.

The calculator shows the working for every answer, so you can follow the cross-multiplication yourself. It accepts whole numbers and decimals, and it never divides by zero: a denominator of 0 is not a valid ratio, and the calculator says so instead of giving a wrong answer.

A proportion is an equation between two ratios: a / b = c / d. It shows up everywhere — scaling a recipe, converting a map distance, working out a sale price, or solving a homework problem. The Proportion Calculator does the two jobs proportions are used for: it solves a proportion for the missing value when you know the other three, and it checks whether two ratios really form a proportion. Enter the numbers, and it shows the answer plus the working, so you can follow the cross-multiplication yourself. It runs entirely in your browser, so nothing you enter is uploaded anywhere, and there is no sign-in and no limit on how often you use it.

The whole tool rests on one fact: in a true proportion, the cross products are equal. If a / b = c / d, then a × d = b × c. That single equality is what lets the calculator find a missing term, and it is also the test for whether two ratios are proportional.

How to use the calculator, step by step

There are two modes, on two tabs. The first solves for a missing value; the second checks two ratios.

• Step 1 — Solve for x: fill in three boxes. On the “Solve for x” tab, type three of the four numbers into the boxes a, b, c and d. Leave exactly one box empty — that is the value the calculator finds. For example, to solve 3 / 4 = x / 12, type 3, 4 and 12, and leave the c box empty.
• Step 2 — Read the answer and the working. The result updates as you type. The big number is the missing value, and underneath it the calculator shows the full proportion, the cross-multiplication step, and a check that the two cross products are equal.
• Step 3 — Check a proportion: fill in all four boxes. On the “Check a proportion” tab, type both ratios in full — for example 2, 3, 4 and 6 to check 2 / 3 against 4 / 6. The calculator multiplies across and tells you straight away whether the ratios form a proportion, with both cross products shown.
• Step 4 — Start again with Clear. The Clear button empties all four boxes in the current mode, ready for the next problem.

How the maths works

A ratio compares two numbers: 3 / 4 means 3 parts out of 4. A proportion says two ratios are equal: 3 / 4 = x / 12 means the second ratio has the same value as the first, and x is whatever makes that true.

Cross-multiplication turns a proportion into a plain equation. Multiply each numerator by the other ratio’s denominator:

• a / b = c / d means a × d = b × c. The two cross products are equal in any true proportion.
• Missing a: a = b × c ÷ d. For x / 8 = 5 / 10, the missing term is a, so a = b × c ÷ d = 8 × 5 ÷ 10 = 4.
• Missing b: b = a × d ÷ c. For 2 / x = 6 / 15, the missing term is b, so b = a × d ÷ c = 2 × 15 ÷ 6 = 5.
• Missing c: c = a × d ÷ b. For 3 / 4 = x / 12, the missing term is c, so c = a × d ÷ b = 3 × 12 ÷ 4 = 9.
• Missing d: d = b × c ÷ a. For 2 / 5 = 6 / x, the missing term is d, so d = b × c ÷ a = 5 × 6 ÷ 2 = 15.

To check whether two ratios form a proportion, you do not need to solve anything — just compare the cross products. If a × d equals b × c, the ratios are proportional; if they differ, they are not. For 2 / 3 and 4 / 6, the cross products are 2 × 6 = 12 and 3 × 4 = 12, so they are proportional. For 1 / 2 and 3 / 7, the cross products are 1 × 7 = 7 and 2 × 3 = 6, so they are not.

One rule has no exceptions: a denominator cannot be 0. A ratio with a denominator of 0 has no value, so the calculator refuses it and explains why instead of giving a wrong answer.

Worked examples

Example 1 — 3 / 4 = x / 12. The missing term is c. Cross-multiply: 3 × 12 = 4 × x, so 36 = 4x and x = 36 ÷ 4 = 9. Check: 3 × 12 = 36 and 4 × 9 = 36 — the cross products match.

Example 2 — 2 / 5 = 6 / x. The missing term is d. Cross-multiply: 2 × x = 5 × 6, so 2x = 30 and x = 30 ÷ 2 = 15. Check: 2 × 15 = 30 and 5 × 6 = 30 — the cross products match.

Example 3 — x / 8 = 5 / 10. The missing term is a. Cross-multiply: x × 10 = 8 × 5, so 10x = 40 and x = 40 ÷ 10 = 4. Check: 4 × 10 = 40 and 8 × 5 = 40 — the cross products match.

Example 4 — Are 2 / 3 and 4 / 6 proportional? Cross-multiply: 2 × 6 = 12 and 3 × 4 = 12. The cross products are equal, so yes — the ratios form a proportion. Notice that 4 / 6 is just 2 / 3 with the top and bottom both doubled, which is why they match.

Example 5 — Are 1 / 2 and 3 / 7 proportional? Cross-multiply: 1 × 7 = 7 and 2 × 3 = 6. The cross products are different (7 is not 6), so no — the ratios do not form a proportion, even though 3 / 7 looks close to 1 / 2.

Example 6 — 1 / 3 = x / 7 (a decimal answer). The missing term is c. Cross-multiply: 1 × 7 = 3 × x, so 7 = 3x and x = 7 ÷ 3 = 2.3333…, shown as 2.3333. Check: 1 × 7 = 7 and 3 × 2.3333 = 6.9999, which is 7 once the rounding is undone.

Common mistakes

• Leaving two boxes empty. A proportion with two unknowns cannot be solved from the other two numbers alone — there are endless answers. Fill in three boxes and leave only one empty.
• Cross-multiplying the wrong pair. The cross products are a × d and b × c — each numerator times the other ratio’s denominator. Multiplying a × b or c × d is not cross-multiplication and gives the wrong answer.
• Putting the answer in the wrong place. In 3 / 4 = x / 12, x is a numerator, so it goes in the top box of the second ratio. Putting 9 in the denominator box instead would solve a different problem — 3 / 4 = 12 / 9 is false.
• Dividing by zero. A denominator of 0 makes the ratio meaningless, and a zero in the dividing step makes the answer undefined. If the calculator reports a zero denominator, check which box holds the 0.
• Rounding too early. For decimal answers, keep the full value through the working and round only at the end. Rounding 7 ÷ 3 to 2.3 before checking gives cross products of 6.9 and 7, which look wrong even though the maths is right.
• Assuming close ratios are proportional. 1 / 2 and 3 / 7 look almost the same, but their cross products are 7 and 6 — they are not equal, so the ratios are not proportional. “Close” is not “equal”.

Related tools

The proportion calculator sits in the Math group, and these companion tools cover the nearby maths — percentages, discounts and growth.

• Percentage Calculator — work out a percentage of a number, a percentage increase or decrease, or what percentage one number is of another.
• Discount Calculator — find the price after a discount, or the discount percentage from the old and new prices.
• Compound Interest Calculator — see how savings or an investment grow when the interest compounds over time.

The Ratio Calculator and the LCM & GCF Calculator are companion maths tools that are coming to ToolZana — the ratio tool simplifies ratios to their lowest terms, and the LCM & GCF tool finds lowest common multiples and greatest common factors. For all of ToolZana’s free tools, start at the ToolZana home page.

Tips

• Always check with the cross products. After solving, multiply across both ways. If the two products match, the answer is right; if they do not, something went wrong in the arithmetic.
• Simplify first when you can. In 5 / 10 = x / 8, notice that 5 / 10 is 1 / 2, so the problem becomes 1 / 2 = x / 8 and x = 4 by inspection. Simplifying makes the numbers smaller and the mistakes fewer.
• Use proportions for scaling. A recipe for 4 people that needs 3 cups of flour needs 3 / 4 × 12 = 9 cups for 12 people — the same 3 / 4 = x / 12 calculation as Example 1.
• Watch the units. Both ratios must compare the same kind of thing in the same order. If the first ratio is cups per person, the second must be cups per person too — not people per cup.
• Decimals are fine. Proportions work exactly the same with decimals: 0.5 / 2 = x / 8 gives x = 2. Enter decimals with a point, not a comma.

The short version

The Proportion Calculator solves a / b = c / d for the one missing value by cross-multiplication, or checks whether two ratios form a proportion by comparing their cross products. 3 / 4 = x / 12 gives x = 9; 2 / 5 = 6 / x gives x = 15; x / 8 = 5 / 10 gives x = 4. The ratios 2 / 3 and 4 / 6 are proportional because both cross products are 12, while 1 / 2 and 3 / 7 are not, because their cross products are 7 and 6. A denominator of 0 is never allowed. The tool is free, runs entirely in your browser, and needs no sign-in.

Frequently Asked Questions

What is a proportion calculator?

A proportion calculator solves proportions — equations between two ratios, written a / b = c / d. It has two jobs: find the missing value when you know the other three numbers, and check whether two ratios really form a proportion. For example, it solves 3 / 4 = x / 12 to get x = 9, and it checks 2 / 3 against 4 / 6 and confirms they are proportional because their cross products are both 12.

How do I solve a proportion for x?

Use the “Solve for x” tab. Fill in three of the four boxes (a, b, c and d) and leave the box for x empty. The calculator cross-multiplies and divides to find the missing value, and shows the working. For 3 / 4 = x / 12, type 3, 4 and 12 and leave the c box empty — the answer is x = 9, because 3 × 12 ÷ 4 = 9.

How do I check if two ratios form a proportion?

Use the “Check a proportion” tab and fill in all four boxes — both ratios in full. The calculator multiplies across: a × d against b × c. If the two cross products are equal, the ratios form a proportion; if they differ, they do not. For 2 / 3 and 4 / 6 the cross products are both 12, so they are proportional. For 1 / 2 and 3 / 7 the cross products are 7 and 6, so they are not.

What is the cross-multiplication method?

Cross-multiplication is the standard way to solve or check a proportion. In a / b = c / d, multiply each numerator by the other ratio’s denominator to get the cross products a × d and b × c. In a true proportion these two products are equal, so you can solve for a missing term by dividing: for example, in 3 / 4 = x / 12, cross-multiplying gives 3 × 12 = 4 × x, so x = 36 ÷ 4 = 9.

Can a proportion have a zero in it?

A zero is allowed in a numerator but never in a denominator. A ratio with a denominator of 0 has no value, so the calculator refuses it and explains why instead of giving a wrong answer. A zero numerator is fine: 0 / 4 = x / 12 gives x = 0, because 0 × 12 ÷ 4 = 0. A zero in the dividing step is also an error — for example, solving for d in 0 / 5 = 6 / x would mean dividing by 0, which has no answer.

Can I use decimals or fractions in a proportion?

Yes — decimals work exactly the same as whole numbers. Enter them with a point, not a comma: 0.5 / 2 = x / 8 gives x = 2. Decimal answers are shown rounded to four decimal places at most, so 1 / 3 = x / 7 gives x = 2.3333. If you have a fraction as an input, enter its decimal value — for example, enter 0.75 instead of 3 / 4.

What if two values are missing?

Then the proportion cannot be solved. With two unknowns and only two known numbers, there are endless pairs of values that satisfy the equation — for example, in x / 4 = y / 12, any pair where y is three times x works. Fill in three of the four boxes and leave only one empty, and the calculator will find it.

What is the difference between a ratio and a proportion?

A ratio is a single comparison between two numbers, like 3 / 4 or 2 : 3. A proportion is an equation saying that two ratios are equal, like 3 / 4 = 9 / 12. Every proportion contains two ratios, but a single ratio on its own is not a proportion. The check mode on this page tests exactly that: whether the two ratios you enter are equal to each other.

How do I solve 3 / 4 = x / 12?

Cross-multiply: 3 × 12 = 4 × x, so 36 = 4x and x = 36 ÷ 4 = 9. Check it by multiplying across the other way: 3 × 12 = 36 and 4 × 9 = 36 — the cross products match, so the answer is right. On the calculator, type 3, 4 and 12 into the a, b and d boxes, leave the c box empty, and it shows x = 9 with this working.

Why are my two ratios not proportional?

Because their cross products are different. Two ratios form a proportion only when a × d equals b × c exactly — “almost equal” is not enough. For example, 1 / 2 and 3 / 7 look close, but 1 × 7 = 7 while 2 × 3 = 6, so they are not proportional. Check that you entered each number in the right box: a swapped numerator and denominator is the most common reason for a surprising “not proportional” result.

Can proportions have negative numbers?

Yes — the cross-multiplication rule works the same with negative numbers. For example, -3 / 4 = x / 12 gives x = -9, because -3 × 12 ÷ 4 = -9, and the check is -3 × 12 = -36 against 4 × -9 = -36. Negative proportions are rare in everyday problems like recipes and scaling, but they come up in algebra, where a negative ratio is just as valid as a positive one.

Is the proportion calculator free?

Yes — the Proportion Calculator is completely free, with no sign-in and no limits. It runs entirely in your browser, so nothing you enter is uploaded anywhere, and you can use it as often as you like, for homework, recipes, scaling, map distances and any other proportion problem.