Proportion Calculator

A proportion calculator finds the missing value in an equation of two equal ratios, a / b = c / x, by cross-multiplication. Example: in 3 / 4 = 9 / x, multiply 4 × 9 = 36 and divide by 3 to get x = 12. This free proportion solver shows each step, tests whether two ratios are proportional, and handles percent problems as proportions.

Two sister tools cover neighboring questions: the Ratio Calculator simplifies and scales a single ratio when you are not solving an equation, and the Percentage Calculator answers "what is x% of y" directly when you do not need the proportion written out. Use this page when three values are known and a fourth is missing, or when you want to verify that two ratios really are equal.

What a proportion is

A ratio compares two quantities, such as 3 cups of flour to 4 cups of sugar, written 3 : 4 or 3/4. A proportion is a statement that two ratios are equal: 3/4 = 9/12. Reading it aloud, "3 is to 4 as 9 is to 12." Proportions describe every situation where two quantities grow together at a constant rate - recipes, map scales, unit prices, fuel economy, drawings, shadows, currency exchange, and percentages. When one of the four numbers is unknown, solving the proportion tells you what it must be to keep the relationship intact.

The formula: cross-multiplication

The single rule behind this calculator is that in a true proportion the products of the diagonals are equal:

Why it works: multiplying both sides of a/b = c/d by b × d cancels both denominators and leaves a × d = b × c. From that single equation you can isolate whichever term is unknown with one division. The only restriction is that the number you divide by cannot be zero, which is why the calculator refuses a zero in that position instead of printing "Infinity."

Worked example 1: solving for the missing value

A cookie recipe uses 2 cups of flour for 36 cookies. How much flour for 60 cookies? Set flour over cookies on both sides: 2 / 36 = x / 60.

1. Cross-multiply: 2 × 60 = 36 × x, so 120 = 36x.
2. Divide by 36: x = 120 ÷ 36 = 3.3333.
3. Check: 2 × 60 = 120 and 36 × 3.3333 ≈ 120. The cross products match.

You need about 3 1/3 cups of flour. The scale factor is 60 ÷ 36 = 1.6667, so every other ingredient in the recipe is multiplied by the same 1.6667 - which is exactly what the result card's "scale factor" tile is for.

Worked example 2: the unknown in the denominator

A person 6 feet tall casts a 4-foot shadow. At the same moment a tree casts a 30-foot shadow. How tall is the tree? Height over shadow on both sides: 6 / 4 = x / 30, so x = (6 × 30) ÷ 4 = 180 ÷ 4 = 45 feet. Now flip the question: if the tree is 45 feet and you want the shadow length, the unknown moves to the denominator: 6 / 4 = 45 / x, so 6x = 4 × 45 = 180 and x = 180 ÷ 6 = 30 feet. Cross-multiplication does not care which position is empty - the calculator lets you tap any of the four letters to mark it as x.

Solutions for a / b = c / x at a glance

Each row applies the same two operations: multiply b by c, then divide by a. The last two rows show that decimals and large numbers behave identically - a 16 : 9 screen 1,920 pixels wide is 1,080 pixels tall, and 3.5 inches on a 1 inch = 25 miles map is 87.5 miles.

Checking whether two ratios are proportional

Sometimes all four numbers are known and the question is simply "are these equal?" Cross-multiply both diagonals; if the products match, the ratios are proportional. As a second check, both ratios should reduce to the same simplest form and give the same decimal. The table shows both tests side by side - notice that 4/6 and 6/8 look close (0.6667 vs 0.75) but fail, while 7/4 and 21/12 pass even though the numbers are larger on the right.

When a proportion fails, the calculator also tells you what would fix it. For 9/12 and 8/10, keeping 9, 12 and 8 means the fourth value would have to be (12 × 8) ÷ 9 = 10.6667, not 10.

Percent problems as proportions

Every percentage question is a proportion in disguise, because "percent" means "per 100." Write part / whole = percent / 100, fill in the two values you know, and cross-multiply for the third. The three classic question types all land on the same rule:

The first three rows are deliberately chosen so the answer is 75 each time: same arithmetic engine, three different unknowns. Once you see percent problems this way you never have to memorize three separate formulas again.

How to use this proportion calculator

1. Pick a mode. "Solve for x" finds a missing value, "Check" tests four known numbers, and "Percent" sets up part / whole = percent / 100.
2. Mark the unknown. In Solve mode, tap the letter A, B, C or D above or below the box you want to find. That box turns into a dashed x.
3. Enter the three known values. Keep the same kind of quantity on top on both sides and the same kind on the bottom - flour over cookies, inches over miles, part over whole.
4. Read the result. The headline number is x; the tiles show the simplified ratio, the shared decimal value, the scale factor and both cross products.
5. Follow the steps. The numbered working below the result is the same sequence you would write on paper, ready to copy into homework or a report.

Everything recalculates as you type. If a required box is empty or a division by zero would occur, a short message appears instead of a wrong number.

Who this calculator is for

• Students in pre-algebra, algebra and geometry who need to solve proportions and show the cross-multiplication steps.
• Cooks and bakers scaling a recipe from one yield to another without redoing every ingredient by hand.
• Builders and DIYers converting plan or model dimensions to real-world sizes, or mixing concrete, fertilizer and cleaning solutions to a stated ratio.
• Drivers and travelers estimating fuel needed for a trip from a known miles-per-gallon figure, or distance from a map scale.
• Shoppers comparing unit prices: if 3 lb costs $4.47, then 5 lb at the same rate is 4.47 / 3 = x / 5, so x = $7.45.
• Anyone doing percentages who prefers one method for "what percent," "how much is x%," and "x% of what."

Key terms

• Ratio: a comparison of two quantities by division, written a : b or a/b.
• Proportion: an equation stating that two ratios are equal, a/b = c/d.
• Means and extremes: in a : b = c : d, the outer terms a and d are the extremes and the inner terms b and c are the means. "The product of the means equals the product of the extremes" is cross-multiplication in older textbook language.
• Cross product: either diagonal product, a × d or b × c.
• Scale factor: the multiplier that turns the left ratio into the right one, c ÷ a (equal to d ÷ b).
• Direct variation: a relationship y = kx in which two quantities keep a constant ratio - the situation a proportion models.
• Percent proportion: part / whole = percent / 100, the bridge between ratios and percentages.

What changes the result

• Which position is unknown: the same three numbers give different answers depending on where x sits. 3/4 = 9/x gives 12, but 3/4 = x/9 gives 6.75.
• Order within each ratio: swapping numerator and denominator on one side only breaks the proportion. 2/36 = 60/x returns 1,080, not 3.3333.
• Rounding of inputs: a rounded rate such as 33 mpg instead of 32 shifts every result by the same percentage. Enter the most precise figure you have.
• Units: the calculator does not convert units, so 12 inches over 1 foot must be entered as 12/1, not 12/12.
• Zero and negative values: a zero in the divisor position makes the proportion unsolvable; negatives are allowed and simply carry their sign through.

Tips for setting up a proportion correctly

• Label before you type. Write the units next to each position (inches / miles = inches / miles). If the labels do not line up, the numbers will not either.
• Estimate first. If 2 cups make 36 cookies, 60 cookies clearly need "a bit under double" - about 3 to 4 cups. An answer of 1,080 is an instant red flag.
• Use the decimal tile as a sanity check. Both sides of a solved proportion share one decimal value; if a manual calculation gives a different decimal on each side, a step went wrong.
• Copy the scale factor for multi-ingredient jobs. Solve one proportion, read the factor, and multiply everything else by it rather than setting up ten proportions.
• For percent questions, put the whole on the bottom. The whole is the base you are taking a percent of - the original price, the total students, the full tank.

Limitations and assumptions

• The calculator assumes direct proportion: doubling one quantity doubles the other. Inverse situations (more workers, fewer hours; higher speed, less time) need a × b = c × d instead and are not handled here.
• Only one unknown is solved. A proportion with two missing values has infinitely many solutions.
• Results display to four decimal places; simplified ratios treat inputs as exact to four decimals, so 0.33333 will not simplify to 1 : 3.
• Real-world relationships are not always perfectly proportional - fuel economy changes with speed, and recipes do not scale baking times linearly. Treat the result as the arithmetic answer, not a guarantee about the physical situation.

How it compares to related calculators

This page answers "what is the fourth number?" and "are these two ratios equal?" For a different shape of question, a sister tool fits better: the Ratio Calculator when you have one ratio to simplify or scale rather than an equation to solve, the Percentage Calculator when you want a percent answer without the proportion written out, the Percentage Change Calculator when the question is how much a value rose or fell, the Fraction Calculator for adding, subtracting, multiplying and dividing fractions, and the Aspect Ratio Calculator when the proportion you are solving is specifically an image or screen size.