Enter the pair you already know (a and b) and the new amount (c), then choose “Direct” or “Inverse”. The result comes out in the unit of b. If what you have is an equality between two ratios with the unknown in any position, that is a job for the proportion calculator.
Decide first whether the relationship is direct or inverse
Double one quantity in your head and see what happens to the other. If it also doubles, the relationship is direct. If it is cut in half, it is inverse. And if it does neither, there is no rule of three.
| Situation | Type | What does not change |
|---|---|---|
| Pints of paint and area covered | Direct | Square yards per pint |
| Servings and amount of each ingredient | Direct | Amount per serving |
| Workers and days for the same job | Inverse | Total work, in worker-days |
| Speed and time on the same trip | Inverse | Distance traveled |
| Miles and fare for a taxi with a flat starting fee | Neither | Nothing: there is a fixed charge |
The table works as a quick test before you type any number. The last row is the one that fools people most, and it is explained below.
A direct case with paint
Paint spreads over the wall at a fixed rate per pint, so the problem is direct.
Worked example: 3 pints of paint cover 7.5 square yards
3 pints of paint cover 7.5 square yards of wall. How many square yards do 5 pints cover?
- Check the type. More paint covers more wall and the coverage per pint does not change, so it is direct.
- Place the data. a = 3 pints, b = 7.5 square yards, c = 5 pints.
- Apply the formula. x = b × c / a = 7.5 × 5 / 3 = 37.5 / 3.
- Divide. 37.5 / 3 = 12.5.
5 pints cover 12.5 square yards of wall.
Check. Each pint covers 7.5 / 3 = 2.5 square yards. With 5 pints that is 2.5 × 5 = 12.5 square yards, the same as before.
How each type is calculated and why it works
In the direct case, the ratio between the two quantities stays the same, that is, b / a = x / c and therefore x = b × c / a. Multiplying b by c / a scales the known amount by however much the other quantity has grown or shrunk.
In the inverse case, the product stays the same, so a × b = c × x and therefore x = a × b / c. If 5 workers finish a wall in 9 days, the job is worth 45 worker-days, and with 15 workers it takes 45 / 15 = 3 days.
The calculator works with fractions and reduces the result, so an answer that is not a whole number appears as a fraction in lowest terms, like 25/2 in the paint example, which equals 12.5.
When there is no rule of three
The rule of three requires the rate or the product to stay constant, and many real situations do not meet that. A taxi with a $3 starting fee and $2 per mile charges $7 for 2 miles and $11 for 4 miles, not $14, because the fixed charge does not double. The rule of three would give a false result even though the arithmetic is right. Volume discounts, flat shipping fees, and capacity limits break the proportion in the same way.
Inverse proportion also fails when the task cannot be split up, and nine months of pregnancy do not get shorter by adding people. This page solves the simple rule of three with three positive values. The compound rule of three, with more than two quantities, is solved by chaining simple rules, and the calculator does not set it up for you.
To continue with other proportions
When the problem asks you to split a total among several parts with different weights, proportional sharing solves it in one go. If you need to scale a whole recipe and not just one ingredient, scale recipes multiplies every amount by the same factor. For drawings and maps, the guide to fractions and map scales shows how to read a scale as a ratio. And to find a part of a quantity, such as 3/5 of a total, use fraction of a number.
Sources
The rule of three as a shorthand for cross-multiplication, which finds the unknown from three known values as x = bc/a, was checked against Wikipedia, article Rule of three (mathematics) (accessed October 17, 2026). The formulas on this page come from solving for the unknown in each proportion. The numerical examples were checked with the calculator’s engine.
Frequently asked questions
What do I do if one of the amounts is zero or negative?
The tool does not accept it. In direct mode, a zero in a means dividing by zero; in inverse mode, the same happens with a zero in c. Any other zero would give a meaningless result of zero. A negative value almost always means a sign is wrong, because an amount of paint or a number of people cannot drop below zero. Check the problem and enter all three values as positive numbers.
How do I check an inverse rule of three?
Multiply each pair and see that the products match. If 14 workers take 15 days and 21 workers take 10 days, then 14 × 15 = 210 and 21 × 10 = 210, so it checks out. If the products differ, check whether you chose the opposite mode or copied a value wrong.
Is speed direct or inverse?
It depends on which quantity you hold fixed. For the same time, distance and speed are direct: if you travel 135 miles at 45 mph, at 75 mph you travel 135 × 75 / 45 = 225 miles. For the same distance, they are inverse: if the trip takes 4 hours at 45 mph, at 90 mph it takes 45 × 4 / 90 = 2 hours.
How do I use the rule of three for a percent?
A percent is a direct rule of three with 100 as the reference. To find what percent of 80 is 28, take a = 80, b = 100%, and c = 28, which gives 100 × 28 / 80 = 35%. Going the other way, 35% of 80 is 80 × 35 / 100 = 28. The percentage calculator solves both cases directly.
How do I enter a time in hours and minutes?
Convert it to a single unit before calculating. If you read 15 pages in 1 h 15 min, that is 75 minutes, and 21 pages take 75 × 21 / 15 = 105 minutes, which is 1 h 45 min. Fraction of an hour to minutes explains why 1.5 h is 1 h 30 min and not 1 h 50 min.