Rule of Three Calculator
Direct: when one quantity goes up, the other goes up too.
Result (X):
Calculation:
The rule of three finds a missing value when you know three values from two proportional quantities. In US classrooms the same idea is usually taught as setting up a proportion and cross-multiplying. This calculator solves both direct and inverse cases and shows the cross-multiplication with your numbers.
Below you will find how to tell whether a problem is direct or inverse, with examples about recipes, prices, speed and percentages.
How to use the rule of three calculator?
Step 1: Choose Direct or Inverse. If you are not sure which one, see the section Direct or inverse? below.
Step 2: Enter in A and B the two values you already know that go together. For example, 4 servings (A) need 3 cups of flour (B).
Step 3: Enter in C the new value of the same quantity as A. For example, 6 servings.
Step 4: Click Calculate or press Enter. You get the value of X and, in the Calculation box, the full operation.
Rule of three formula
The data is always arranged the same way: A is to B as C is to X. A and C are the same kind of quantity (for example, servings), and B and X are the other kind (for example, cups).
| A | → | B |
| C | → | X |
Direct rule of three. Multiply B and C crosswise, then divide by A:
| X = (B × C) ÷ A |
Inverse rule of three. Multiply A and B straight across, then divide by C:
| X = (A × B) ÷ C |
In the inverse case the product of the two quantities stays constant: A × B = C × X. That is why you solve for X by dividing by C.
Direct or inverse?
Ask what happens to one quantity when the other goes up. If both go up together (or both go down together), the relationship is direct. If one goes up while the other goes down, it is inverse.
| Situation | Type |
|---|---|
| More pounds of an item, more money to pay | Direct |
| More servings in a recipe, more of every ingredient | Direct |
| More hours worked at the same hourly wage, more pay | Direct |
| More gallons of gas, more miles driven at the same mpg | Direct |
| Map scale: more inches on the map, more real miles | Direct |
| Percentages: a part compared with the whole | Direct |
| More workers, fewer days to finish the same job | Inverse |
| Higher speed, less time to cover the same distance | Inverse |
| More faucets open, less time to fill a tank | Inverse |
| More people sharing the same pizza, a smaller slice each | Inverse |
| More animals on the same feed, fewer days it lasts | Inverse |
Rule of three examples
Example 1 (direct): A recipe
A recipe for 4 servings uses 3 cups of flour. How much flour is needed for 6 servings?
More servings need more flour, so it is direct. A = 4, B = 3, C = 6:
X = (B × C) ÷ A
X = (3 × 6) ÷ 4 = 18 ÷ 4 = 4.5
You need 4.5 cups of flour.
Example 2 (direct): A price
If 3 pounds of tomatoes cost $4.50, how much do 7 pounds cost?
More pounds cost more, so it is direct. A = 3, B = 4.5, C = 7:
X = (B × C) ÷ A
X = (4.5 × 7) ÷ 3 = 31.5 ÷ 3 = 10.5
The 7 pounds cost $10.5.
Example 3 (inverse): Speed and time
A trip takes 5 hours at 60 mph. How long does the same trip take at 75 mph?
A higher speed means less time, so it is inverse. A = 60, B = 5, C = 75:
X = (A × B) ÷ C
X = (60 × 5) ÷ 75 = 300 ÷ 75 = 4
The trip takes 4 hours.
Example 4 (inverse): Workers and days
4 workers paint a house in 12 days. How many days do 6 workers need?
More workers finish in fewer days, so it is inverse. A = 4, B = 12, C = 6:
X = (A × B) ÷ C
X = (4 × 12) ÷ 6 = 48 ÷ 6 = 8
The 6 workers need 8 days.
Example 5 (percentage): What is 15% of 240?
The whole, 240, is 100%. Set it up as: 100 is to 15 as 240 is to X. It is direct. A = 100, B = 15, C = 240:
X = (B × C) ÷ A
X = (15 × 240) ÷ 100 = 3600 ÷ 100 = 36
15% of 240 is 36.
Example 6 (percentage): 18 is what percent of 72?
Now we are looking for the percentage. 72 is 100%, so: 72 is to 100 as 18 is to X. A = 72, B = 100, C = 18:
X = (B × C) ÷ A
X = (100 × 18) ÷ 72 = 1800 ÷ 72 = 25
18 is 25% of 72.
Frequently asked questions
How do I know if a rule of three is direct or inverse?
Imagine doubling one of the quantities. If the other one also doubles, it is direct. If the other one is cut in half, it is inverse. For example, twice the pounds costs twice as much (direct), but twice the workers take half the days (inverse).
How do I find a percentage with the rule of three?
Treat the whole as 100%. To find how much a percentage is, set up 100 is to that percentage as the whole is to X. To find what percent a part is, set up the whole is to 100 as the part is to X. Both cases are solved in examples 5 and 6. For plain percentage problems there is also a percentage calculator.
Is the rule of three the same as cross multiplication?
It is the same idea. The direct rule of three is exactly what you do when you solve the proportion A/B = C/X by cross-multiplying. The name rule of three is more common in Latin America and Europe; US textbooks usually call it solving a proportion.
What is the compound rule of three?
It relates three or more quantities at once, for example workers, hours per day and days. You can solve it by applying the simple rule of three one quantity at a time, checking in each step whether the relationship is direct or inverse.
Can I use the rule of three for medication doses?
A proportion is the usual way to go from a concentration to a volume, for example how many milliliters hold a given number of milligrams. It is a direct relationship. Even so, a dose should always be set and double-checked by a health professional.
Learn more
- Direct and inverse proportionality
- Examples of direct proportion
- Examples of inverse proportion
- Examples of compound proportion
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