Rule of Three Calculator
Enter three values and find the fourth. The tool solves direct or inverse rule of three problems and shows how the proportion was set up.
Educational use: results are estimates made for study and reference. Checking the answer and deciding what to do with it is the sole responsibility of whoever uses the tool. Read the legal notice.
On this page
- Step by step: how to use it
- The formula, in both cases
- Direct or inverse? The graph tells you
- Solved examples
- Where the rule of three tends to trip you up
- Accuracy, limits and responsibility
- Direct or inverse? Some common pairs
- Common questions about the rule of three
- Where you will use this
- A much older rule than it looks
- Proportion trivia
- Proportion vocabulary
- See also
Step by step: how to use it
The rule of three works whenever you know one pair of related values and want to find a new value with the same relationship. In the calculator:
- Decide the type of relationship. If both quantities grow together (more product, higher price), choose directly proportional. If one grows while the other shrinks (more people working, fewer days), choose inversely proportional.
- Fill in A and B with the pair you already know. For example: 3 notebooks (A) cost 12 (B).
- Fill in C with the new value of the first quantity. For example: what about 5 notebooks (C)?
- Click Calculate. The value of x appears along with the full calculation. For the notebooks, x = 20.
A way to avoid getting lost: say the sentence out loud. "If 3 notebooks cost 12, 5 notebooks cost x." First quantity with first quantity (notebooks), second with second (price). If the sentence makes sense, the form will too.
The formula, in both cases
The idea is to keep one relationship constant. What changes is which relationship stays the same.
Directly proportional quantities
The ratio between the two quantities does not change. What holds for the first pair holds for the second:
Inversely proportional quantities
Here, what stays the same is the product of the quantities, because when one doubles, the other drops by half:
Notice the difference: in the direct case, x comes out as B × C ÷ A. In the inverse case, as A × B ÷ C. Swapping one for the other is by far the most common mistake, which is why the calculator asks you to choose the type before anything else.
Compound rule of three: when more than two quantities are involved (for example, machines, days and parts), solve it in stages or multiply the proportion factors, as the last example below shows.
Direct or inverse? The graph tells you
If you plot the two quantities on a graph, a direct relationship forms a rising straight line. An inverse one forms a falling curve that never touches the axes.
Solved examples
1. Direct: the notebooks
3 notebooks cost 12. How much do 5 cost?
Five notebooks cost 20. Checking: each one costs 4 (12 ÷ 3), and 5 × 4 = 20.
2. Direct: the recipe that grew
A recipe uses 250 g of flour for 4 people. How much flour for 10 people?
625 g. (If the result does not match your intuition, remember that 10 people is 2.5 times more than 4, and 250 × 2.5 = 625.)
3. Inverse: the construction job
4 bricklayers finish a wall in 6 days. How many days would 8 bricklayers need to build the same wall, working at the same pace?
Three days. Twice the people, half the time.
4. Inverse: the trip
At 60 km/h, a trip takes 2 h 30 min (2.5 h). How long would it take at 90 km/h?
About 1 h 40 min. The distance (150 km) is the same, only the pace changes.
5. Compound: machines, days and parts
6 machines produce 1,200 parts in 5 days. How many parts do 10 machines produce in 8 days?
3,200 parts. In compound problems, it is worth asking about each quantity separately: is it direct or inverse in relation to the one I am looking for?
Where the rule of three tends to trip you up
Using direct when it is inverse (and vice versa)
This is mistake number one. Before calculating, do the doubling test: if one of the quantities doubles, what happens to the other? If it doubles, it is direct. If it drops by half, it is inverse.
Mixing units
If A is in hours and C is in minutes, convert first. Dividing 2 hours by 30 minutes without converting is asking to be off by a factor of 60.
Swapping the order of the pairs
The first value of each pair must be the same quantity. If A is "notebooks," C must also be "notebooks." Putting notebooks and price on the same side gives a meaningless number.
Assuming everything is proportional
Not every relationship between quantities is proportional. A person who is 1.70 m tall does not weigh twice as much as one who is 0.85 m. A price with a fixed delivery fee does not follow the rule of three either. Use the tool when the proportionality is real.
Forgetting that the pace must be constant
In the bricklayer example, the calculation assumes everyone works equally and that they do not get in each other's way. In real life, more people do not always speed things up in the same proportion.
Accuracy, limits and responsibility
The tool does the exact calculation for a proportion, but it does not know whether your situation is truly proportional. That judgment is yours: if the problem has fixed fees, limits, price tiers or changing paces, the rule of three will only give an approximation (or not even that).
Results are shown with up to six decimal places. If the result must be a whole number (people, parts, boxes), round it according to the context: there is no such thing as 2.5 bricklayers.
Bottom line: this tool is educational. Results do not replace your own checking, and how you use them is entirely your responsibility. More details in the legal notice.
Direct or inverse? Some common pairs
| Pair of quantities | Type of relationship | Why |
|---|---|---|
| Number of products × total price | Direct | more items, more cost |
| Distance × fuel used | Direct | more road, more liters (in the same car) |
| Servings × amount of ingredients | Direct | the recipe grows together |
| Speed × travel time | Inverse | for the same distance, going faster takes less time |
| Workers × days to finish | Inverse | more people share the work |
| Open taps × time to fill | Inverse | more flow, less time |
| A person's height × age | Neither | not proportional, do not use the rule of three |
Common questions about the rule of three
What is the simple rule of three?
It is a method for finding a fourth value from three known ones, when two quantities are proportional. You set up a proportion between the pairs and solve for the unknown x.
How do I know if it is a direct or inverse rule of three?
Use the doubling test: if doubling one quantity doubles the other, it is direct. If doubling one makes the other drop by half, it is inverse.
What is the formula for the direct rule of three?
If A matches B and C matches x, then x = (B × C) ÷ A. This is cross multiplication.
What is the formula for the inverse rule of three?
x = (A × B) ÷ C. Here the product of the two quantities stays the same, so you multiply A by B and divide by C.
What is the compound rule of three?
It is when three or more quantities are involved, such as machines, days and parts. Compare each quantity with the one you are looking for, decide whether it is direct or inverse, and multiply the factors, as in example 5.
Can I use the rule of three with percentages?
Yes. Saying that 30 is 20% of a total is a proportion: 30 is to 20 as x is to 100. Solving it: x = 150. The percentage calculator does this for you.
Can the result be a decimal?
Yes. The calculation is exact, and the result may have decimals. Round according to the context: people and parts are whole numbers, liters and kilos accept decimals.
Do I need to convert the units first?
Yes, always. The two quantities of the same type must be in the same unit (hours with hours, meters with meters). If needed, use the site's unit converter.
Why does the rule of three not work for some problems?
Because it only applies to quantities that are truly proportional. Age and height, for example, are not, and fixed fees also break the proportion.
Can zero be used in the calculation?
As the value of B or C, yes. As a divisor (A in the direct case, C in the inverse case), no: dividing by zero has no result, and the calculator will warn you.
Where you will use this
- Cooking: scaling recipes up or down without upsetting the balance of ingredients.
- Shopping: comparing the price of different package sizes by price per unit or per kilo.
- Maps and floor plans: using the scale to find real distances from a ruler measurement. On a 1:50,000 map, 3 cm represents 1.5 km.
- Travel: estimating travel time when the speed changes, or fuel use for a longer route.
- Work and production: dividing tasks, planning shifts and estimating deadlines.
- Chemistry and the lab: dilutions and proportions between reagents and solutions.
- Exams and civil service tests: it is one of the most frequently tested topics in basic math.
A much older rule than it looks
Indian mathematicians were already describing the "rule of three terms" (trairāśika) more than a thousand years ago, and it reached Europe through Arabic translations. During the Middle Ages and the Renaissance, merchants used it so much that it earned the nickname "golden rule": anyone who mastered the rule of three knew how to convert currencies, split profits and work out the price of goods.
The name comes from the fact that you always know three values and look for the fourth. Cross multiplication, which is taught today as a shortcut, is the result of a deeper idea: two equal ratios form a proportion, and in a proportion the product of the means equals the product of the extremes.
Proportion trivia
- Map scales are pure rule of three: 1:100,000 means that 1 cm on paper stands for 100,000 cm (1 km) on the ground.
- The golden ratio. The number 1.618... comes from a proportion in which the whole is to the larger part as the larger part is to the smaller. Many people see it in art and nature, although you should be careful with exaggerated claims.
- Screens and photos. When a 1920 × 1080 image is reduced to a width of 1280, the height comes from a rule of three: 1080 × 1280 ÷ 1920 = 720.
Proportion vocabulary
- Quantity
- Anything that can be measured or counted: time, price, distance, people, parts.
- Ratio
- The division of one value by another. The ratio of 12 to 3 is 4.
- Proportion
- An equality between two ratios, such as 3/12 = 5/20.
- Directly proportional
- Quantities that rise or fall together, at the same ratio.
- Inversely proportional
- When one quantity increases, the other decreases in the opposite proportion. Their product is constant.
- Unknown (x)
- The value you want to find.
- Cross multiplication
- A shortcut for solving a proportion: the product of the means equals the product of the extremes.