Proportion Calculator

A proportion calculator is only worth using if it answers the question you actually have. Most of them solve one equation, A/B = C/D, and hand back a number with a tick beside it. That is the right equation for a recipe, a map or a medicine dose. It is the wrong equation for workers and days, speed and journey time, or taps filling a tank, and it gives a confidently wrong answer for all three. This calculator solves for the missing value either way, shows every line of the cross multiplication, and prints the answer the other method would have given so the difference is impossible to miss.

What the calculator solves

A proportion says that two ratios match, written A/B = C/D. Fill any three of the four boxes, leave the unknown term blank, and the calculator rearranges the equation for that position. There are only four possible rearrangements, and they are worth knowing by heart because every proportion problem you meet is one of them:

The same move is sometimes taught as the rule of three: multiply the two known numbers that sit diagonally across from each other, then divide by the third. Leave all four boxes filled and the calculator stops solving and starts testing, checking the statement as a direct proportion and as an inverse one and telling you which, if either, is true.

Cross multiplication, and why it is allowed

Cross multiplication turns A/B = C/D into A × D = B × C. It looks like a trick, but it is nothing more than ordinary multiplication done twice. Multiply both sides of the equation by B and the B on the left cancels. Multiply both sides by D and the D on the right cancels. What is left is A × D = B × C, and every step was legal because both sides were multiplied by the same number, which is fine as long as that number is not zero.

The older vocabulary says the same thing. In A : B = C : D the outer terms, A and D, are the extremes and the inner terms, B and C, are the means, and the product of the means equals the product of the extremes. You can check 6/5 = 36/30 this way without dividing anything: 6 × 30 = 180 and 5 × 36 = 180, so the proportion is true. The calculator prints that check as the last line of every answer, because a result you can verify in your head is a result you can trust.

Set it up right and the answer takes care of itself

Almost every wrong answer on a proportion problem comes from the set up, not the arithmetic. The rule is simple: the same kind of quantity has to sit in the same position on both sides. If cups of flour are on top on the left, cups of flour go on top on the right.

Take a recipe that uses 3 cups of flour for 24 cookies. For 40 cookies the proportion is 3/24 = C/40. Cross multiply to get 24 × C = 3 × 40 = 120, then divide by 24, and C = 5 cups. Put the 40 on top instead and you get 3/24 = 40/D, which asks a different question entirely. The two situations sit side by side in the calculator, labelled Situation 1 and Situation 2, so each fraction describes one real case with its two quantities in a fixed order.

The other classic slip is cross addition. Adding across the equals sign has no meaning at all, and it tends to happen when a learner half remembers the diagonal arrows. Multiplication is the only operation that crosses.

Direct or inverse proportion: the question nobody asks

In a direct proportion both quantities rise together. Double the cookies and you double the flour. The ratio between them stays fixed, which is why A/B = C/D works. In an inverse proportion one quantity rises as the other falls. Double the workers and the job takes half as long. Here the ratio changes, and what stays fixed is the product: A × B = C × D.

Apply the direct formula to an inverse problem and the arithmetic still runs, which is exactly what makes the mistake dangerous. Suppose 3 workers finish a task in 8 days and you want to know how long 5 workers take. The inverse method gives 3 × 8 = 5 × D, so D = 24 ÷ 5 = 4.8 days. The direct method gives 3/8 = 5/D, so D is about 13.3 days, a plan in which hiring more people makes the job slower. Of the fifteen pages that rank highest for this keyword, only two can solve an inverse problem in the same tool, and five describe inverse proportion in their text while their own tool can only do the direct case.

The fastest test is to picture the first quantity doubling. If the second should double too, the problem is direct. If it should halve, it is inverse. The calculator makes you choose, then shows the other method's answer in an amber row beneath the check. If that other number looks more sensible than yours, you picked the wrong method.

The constant of proportionality

Every proportional relationship has one number that never changes, the constant of proportionality, written k. In a direct proportion it is the ratio, k = x ÷ y, and the relationship can be written y = kx, a straight line through the origin when you graph it. In an inverse proportion it is the product, k = x × y, written y = k/x or y ∝ 1/x, and its graph is a curve called a hyperbola that bends toward both axes without ever touching them, because dividing by zero is undefined.

Finding k is often the quickest way into a problem. If 16 workers finish a job in 3 days, k = 16 × 3 = 48 worker days, and any crew and duration that multiply to 48 will do the same job: 8 workers take 6 days. If a car travels at 60 miles per hour for 3 hours, k = 180 miles, so covering the same distance in 4 hours needs 180 ÷ 4 = 45 miles per hour. The calculator prints k on every answer so you can reuse it for the next question without solving again.

Applied problems: maps, doses and models

Map scales. A ratio scale such as 1 : 50 000 means one unit on the map is 50 000 of the same unit on the ground. So 3 cm on the map gives 1/50 000 = 3/D, and D = 150 000 cm. The answer comes out in centimetres because the scale uses the same unit on both sides. Divide by 100 000 to turn centimetres into kilometres and the real distance is 1.5 km. At 1 : 25 000, 10 cm on the map is 2.5 km on the ground. A scale written in mixed units works the same way: if 2 inches represents 50 miles, two cities 7 inches apart are 2/50 = 7/D, so D = 175 miles.

Liquid doses. A stock solution of 100 mg per 4 mL and an order for 250 mg gives 100/4 = 250/D, so D = 10 mL. This is the setting where the position rule matters most. Always double check a dose calculation against the label and with a pharmacist or prescriber; the calculator checks arithmetic, not prescriptions.

Scale models and materials. A 1 : 100 aircraft model with a 3.5 inch part gives 1/100 = 3.5/D, so the full size part is 350 inches. A deck that needs 14 boards for every 84 square feet needs 35 boards for 210 square feet, and a printer that makes 240 pages in 8 minutes makes 450 pages in 15.

Inverse cases. If 2 workers build a fence in 9 hours, k = 18 worker hours and 6 workers take 3 hours. If 5 taps fill a pool in 360 minutes, k = 1800 tap minutes and 3 taps take 600 minutes. If a journey at 8 km/h takes 40 minutes, the same journey at 16 km/h takes 20. Every one of these quietly assumes each worker, tap or vehicle runs at the same steady rate and nobody gets in anyone's way. Past a certain crew size that stops being true, and the answer becomes a best case rather than a forecast.

Zero, negatives, fractions and decimal answers

Solving a proportion always ends in a division, and you cannot divide by zero. If the number you would divide by is 0, the calculator says so rather than printing infinity. Either no value makes the two sides equal, or every value does because both sides are already 0, and in both cases the numbers you entered do not pin the unknown down. In a direct proportion B and D sit below a fraction line, so neither can be 0 at all.

Negative numbers are algebraically fine and the calculator accepts them, but in a real proportion problem the quantities are lengths, prices, people or hours, so a minus sign is almost always a typing error. The working table flags it. Decimals, simple fractions such as 3/4 and mixed numbers such as 2 3/4 can all be typed straight into the boxes, and when the answer is a fraction the table shows it exactly as well as in decimals.

One last trap sits in decimal time. An answer of 3.1 hours is 3 hours and 6 minutes, not 3 hours and 10, because an hour has 60 minutes rather than 100. Multiply the decimal part by 60 before you write the answer down: 4.8 days of work is 4 days and 0.8 of a day, not 4 days and 8 hours.

What belongs on a different page

This page solves proportions for an unknown term. Simplifying a ratio to lowest terms, sharing an amount in a given ratio and working with aspect ratios are handled by the ratio calculator. Adding, subtracting and simplifying fractions belongs to the fraction calculator, and questions such as what percentage one number is of another are quicker on the percentage calculator, even though they can be written as proportions. Two rarer forms are worth knowing about. A continued proportion, a/b = b/c, has a middle term shared by both ratios, so b² = a × c. A compound proportion links three or more quantities at once, such as workers, hours and job size, and is best solved one pair at a time: direct in the size of the job, inverse in the size of the crew.