Proportion Calculator
This proportion calculator solves a/b = c/d when one of the four values is missing, using cross multiplication, and shows the equation, the division step and a check line. Fill in all four boxes and it tells you whether the two ratios are proportional. Below the tool: the method, word problems, direct and inverse proportion, and common mistakes.
a/b = c/d — leave one box empty (or type x) to solve for it.
How to use the proportion calculator
- Enter three of the four values in the boxes laid out as
a/b = c/d. The defaults are a = 3, b = 4 and c = 9. - Leave the unknown box empty, or type
xin it. Exactly one box can be blank. - The answer appears straight away. With the defaults,
3/4 = 9/dgives d = 12. - Open the steps to see the cross-multiplication equation (
3 × d = 4 × 9), the division (d = 36 ÷ 3) and a check line showing both cross products are equal.
To test a proportion instead of solving one, fill in all four boxes. The calculator multiplies across both diagonals and tells you whether the products match. If they do, the ratios are proportional; if they do not, it shows both products so you can see how far apart they are.
Before you type, make sure the two fractions are set up the same way round. If the left side is cups over cookies, the right side must be cups over cookies too. The next sections show why.
How to solve a proportion with cross multiplication
A proportion is a statement that two ratios are equal. To solve one, multiply diagonally across the equals sign, then divide by the number beside the unknown.
Example: 3/4 = 9/d (the default)
Cross multiply, then isolate d.
The missing value is 12. You can see the same thing by scaling: the top went from 3 to 9, a factor of 3, so the bottom goes from 4 to 4 × 3 = 12.
Why cross multiplication works
Start from a/b = c/d and multiply both sides by b × d. On the left, the b cancels and leaves a × d. On the right, the d cancels and leaves c × b. Doing the same thing to both sides of an equation keeps it true, so a × d = b × c. Cross multiplication is just that two-step move written as one.
The same idea explains why equal ratios are interchangeable. 3/4 and 9/12 are the same number written two ways, exactly as the equivalent fractions guide shows, and the cross-product test is the quickest way to confirm it.
The unknown in any position
The method does not care where the blank sits. Multiply the diagonal that has two known numbers, then divide by the known number on the other diagonal.
| Proportion | Cross multiply | Divide | Answer |
|---|---|---|---|
| 3/4 = 9/d | 3d = 4 × 9 = 36 | 36 ÷ 3 | d = 12 |
| x/8 = 15/24 | 24x = 8 × 15 = 120 | 120 ÷ 24 | x = 5 |
| 7/x = 21/30 | 21x = 7 × 30 = 210 | 210 ÷ 21 | x = 10 |
| 2.5/3.5 = x/14 | 3.5x = 2.5 × 14 = 35 | 35 ÷ 3.5 | x = 10 |
| 7/9 = x/20 | 9x = 7 × 20 = 140 | 140 ÷ 9 | x ≈ 15.56 |
The last row does not come out even. 140 ÷ 9 = 15.555…, which rounds to 15.56 at two decimal places. Keep the full value until the end of any longer calculation, then use the rounding calculator to round once. Decimal entries such as the fourth row work the same way; dividing 35 by 3.5 is covered in how to divide decimals.
Proportion word problems, step by step
Most proportion questions come as a sentence, not an equation. The work is in the setup: write one ratio from the facts you know, write the matching ratio with the unknown, and keep the units in the same positions on both sides.
Example: scaling a recipe
A recipe uses 3 cups of flour for 24 cookies. How much flour for 40 cookies?
You need 5 cups of flour. Both fractions are cups over cookies, which is what makes the equation valid.
Example: price for a different quantity
Six apples cost $4.50. What do 10 apples cost at the same rate?
Ten apples cost $7.50. When the question is really about which pack is cheaper per item, the unit price calculator compares several sizes at once.
Example: fuel economy
A car goes 280 miles on 8 gallons. How many gallons for 455 miles?
The trip needs 13 gallons. The rate hidden inside is 280 ÷ 8 = 35 miles per gallon, and 455 ÷ 35 = 13 confirms it.
Percentages as proportions
Every “what percent” question is a proportion with 100 in one spot. “18 is what percent of 24?” becomes 18/24 = p/100, so 24p = 1,800 and p = 75. That is the same calculation the what percent of calculator runs, and the guide to calculating percentages explains the formula behind it.
Direct and inverse proportion
Everything above is direct proportion: when one quantity doubles, the other doubles too. More cookies need more flour; more apples cost more. The ratio between the two stays fixed.
The constant of proportionality
In a direct proportion, divide either value by its partner to get k. If 5 gallons of fuel cost $17.50, then k = 17.50 ÷ 5 = 3.50 dollars per gallon, and 12 gallons cost 12 × 3.50 = $42. Finding k first is often quicker than setting up a full proportion, especially when you have several values to work out at the same rate.
Inverse proportion
In an inverse proportion, one quantity goes up as the other goes down, and their product stays the same. More workers finish a job in fewer days; a faster speed covers a fixed distance in less time. Cross multiplication of a/b = c/d gives the wrong answer here, because the ratio is not what stays constant.
Example: workers and days
Four workers take 12 days to finish a job. How long would 6 workers take, working at the same rate?
Setting it up as a direct proportion, 4/12 = 6/x, would give 18 days, which makes no sense: more workers cannot take longer.
A second check: a trip of 180 miles takes 3 hours at 60 mph. At 45 mph it takes 180 ÷ 45 = 4 hours. Speed went down by a quarter, time went up by a third, and the product of speed and time is still 180.
How to tell if two ratios are proportional
Fill in all four boxes and the calculator tests the proportion for you. The test is the cross-product rule: two ratios are proportional exactly when a × d = b × c.
Two other tests give the same verdict. You can reduce both fractions to lowest terms with the simplify fractions calculator: 10/15 reduces to 2/3, while 12/16 reduces to 3/4, which is not 4/5. Or you can divide each one out: 4 ÷ 5 = 0.8 and 12 ÷ 16 = 0.75. The cross-product test is usually fastest because it needs no division and no rounding.
In a table of values, a relationship is directly proportional when every y ÷ x gives the same k and the line through the points would pass through zero. A table where y = 2x + 1 is not proportional, even though y rises steadily, because 3 ÷ 1 and 5 ÷ 2 are not equal.
A proportion is an equation about ratios. For simplifying a single ratio or splitting a total in a ratio such as 2:3, use the ratio calculator; for similar shapes, maps and models, the scale factor calculator handles lengths, areas and volumes.
Common mistakes
- Mismatched setup. Writing
3/24 = 40/x(cups over cookies on the left, cookies over cups on the right) gives 320 instead of 5. Keep the same unit on top on both sides. - Using a direct proportion for an inverse situation. More workers, more speed or more pipes usually means less time. Multiply, do not cross multiply.
- Cross multiplying across a plus or minus sign. Cross multiplication only works when each side is a single fraction. For something like
x/3 + 1 = 5/6, isolate the fraction first. - Dividing by the wrong number. After
3d = 36, divide by 3 (the number attached to d), not by 36. - Rounding halfway through. Keep 15.555… as it is until the final answer, then round once.
- Forgetting units in the answer. A proportion solved for “x” still has units. Write 5 cups, not 5.
Frequently asked questions
How do you solve a proportion?
Cross multiply, then divide. For 3/4 = 9/d, multiply the diagonals to get 3 × d = 4 × 9, so 3d = 36. Divide both sides by 3 and d = 12. Check by multiplying the diagonals again with the answer filled in: 3 × 12 = 36 and 4 × 9 = 36, so the proportion holds.
What is x if x/8 = 15/24?
x = 5. Cross multiplying gives 24x = 8 × 15 = 120, and 120 ÷ 24 = 5. You can also simplify first: 15/24 reduces to 5/8, so x/8 = 5/8 and x must be 5. Both methods agree, and the check 5 × 24 = 8 × 15 = 120 confirms it.
How do you know if two ratios form a proportion?
Multiply across the diagonals. If a × d equals b × c, the ratios are proportional. For 2/3 and 10/15, both products are 30, so they are. For 4/5 and 12/16, the products are 64 and 60, so they are not. Reducing both fractions to lowest terms gives the same answer.
What is the difference between direct and inverse proportion?
In direct proportion both quantities rise together and their ratio stays constant, as with cost and number of items. In inverse proportion one rises as the other falls and their product stays constant, as with workers and days. Four workers taking 12 days means six workers take 48 ÷ 6 = 8 days, not 18.
What is the constant of proportionality?
It is the fixed number k that links two directly proportional quantities in y = kx. Find it by dividing any y by its matching x. If 5 gallons cost $17.50, then k = 17.50 ÷ 5 = 3.50, the price per gallon. Every pair in a proportional relationship gives the same k.
How do you use proportions to scale a recipe?
Write the ingredient over the yield on both sides and solve. For 3 cups of flour per 24 cookies, scaled to 40 cookies, write 3/24 = x/40. Cross multiplying gives 24x = 120, so x = 5 cups. Repeat with the same 40/24 scaling for every other ingredient in the recipe.
Why is cross multiplication allowed?
Because it is the same as multiplying both sides of a/b = c/d by b × d. The b cancels on the left and the d cancels on the right, leaving a × d = b × c. Doing the same operation to both sides of an equation keeps it true, so the shortcut is always valid when each side is a single fraction.
Related calculators
Guides that use this calculator
Longer explanations of the method behind it, with worked examples.
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