What the rule of three is
The rule of three is a shortcut for solving proportion problems: you know three values and need the fourth, the unknown X. You state it as “A is to B, as C is to X” and solve it with one multiplication and one division — the classic cross-multiplication in a single step. It shows up everywhere: scaling recipes, pricing by unit rate, reading map distances.
There are two kinds of proportion, and telling them apart is the whole game:
- Direct proportion: both quantities move in the same direction. More cookies require more flour; more miles burn more gas. If A goes up and X goes up too, it is direct.
- Inverse proportion: the quantities move in opposite directions. More painters means fewer hours; higher speed means less travel time. If A goes up while X goes down, it is inverse.
How to use the calculator
- Pick direct or inverse, based on how your quantities behave.
- Enter value A and its partner B — the complete pair you already know.
- Enter value C, the new amount you are asking about.
- Read the highlighted result X; the card below shows the exact operation performed so you can double-check it.
The formulas
- Direct:
X = (B × C) ÷ A - Inverse:
X = (A × B) ÷ C
Worked example: direct
A recipe uses 2 cups of flour for 12 cookies. How much for 30 cookies? More cookies means more flour, so it is direct: X = (2 × 30) ÷ 12 = 60 ÷ 12 = 5 cups.
Worked example: inverse
3 painters can paint a fence in 8 hours. How long would 6 painters take? More painters means less time, so it is inverse: X = (3 × 8) ÷ 6 = 24 ÷ 6 = 4 hours. The total effort never changes: 3 painters × 8 hours equals 24 painter-hours — exactly what 6 painters deliver in 4 hours.
Quick reference: classic problems
| Problem | Type | Setup | X |
|---|---|---|---|
| 2 cups of flour make 12 cookies; 30 cookies? | Direct | 2 × 30 ÷ 12 | 5 cups |
| Shrimp costs 18 for 3 lb; 5 lb? | Direct | 18 × 5 ÷ 3 | 30 |
| 1 inch on the map equals 5 miles; 7 inches? | Direct | 5 × 7 ÷ 1 | 35 miles |
| At 60 mph the trip takes 4 hours; at 80 mph? | Inverse | 60 × 4 ÷ 80 | 3 hours |
| 3 painters need 8 hours; 6 painters? | Inverse | 3 × 8 ÷ 6 | 4 hours |
Everyday uses
Recipe scaling. Any “serves 4, but I need 10” situation is a direct rule of three on every ingredient.
Map scales. If 1 inch stands for 5 miles, a 7-inch route on paper is 35 real miles.
Kitchen conversions. Since 1 cup equals 16 tablespoons, 2.5 cups equal (16 × 2.5) ÷ 1 = 40 tablespoons — every unit conversion is quietly a direct rule of three.
Frequently asked questions
When do I need the inverse rule of three?
Whenever the quantities pull in opposite directions: workers and days, speed and time. A sanity check catches mistakes: the direct formula on the painters example would give (8 × 6) ÷ 3 = 16 hours — more painters taking longer, which is absurd. If the answer defies common sense, switch modes.
What is the compound rule of three?
It is the version for problems with three or more quantities at once — say, workers, days, and hours per day. You solve it by chaining simple rules of three: decide, one quantity at a time, whether it is direct or inverse with respect to the unknown, and apply each factor in turn. This calculator handles the simple case; run it twice for a compound problem.
How do I set up the proportion without mixing things up?
Write the data in two labeled columns with units (pounds on the left, dollars on the right). Make sure A and C measure the same thing, and B and X the other. Then ask the direction question — does X grow when A grows? — to pick the mode. The most common errors: mixing units within a column (ounces with pounds) and swapping the pairs.
Can I use it for percentages?
Yes — a percentage is just a direct proportion with a base of 100. What is 25% of 80? State it as “100 is to 25, as 80 is to X”: X = (25 × 80) ÷ 100 = 20.