What is the Rule of Three and How to Solve It
Updated: March 2026 · Category: Math & Education
The rule of three (also known as cross-multiplication) is one of the most useful mathematical tools for solving proportion problems. Whether you are scaling a recipe, converting currencies, calculating travel times, or figuring out unit prices, the rule of three gives you a simple, reliable method to find an unknown value when three related values are known. This guide covers both direct and inverse proportions with clear, step-by-step examples.
What Is the Rule of Three?
The rule of three is a method for solving proportions. A proportion is an equation stating that two ratios are equal. If you know three of the four values in a proportion, you can find the fourth.
The general setup looks like this: if A relates to B in the same way that C relates to an unknown X, you can write the proportion and solve for X.
Direct Proportion (Direct Rule of Three)
Two quantities are in direct proportion when they increase or decrease together at the same rate. If one doubles, the other doubles. If one is cut in half, the other is cut in half.
A → B
C → X
X = (B × C) ÷ A
Step-by-Step Method
- Identify the two related quantities and write down the known pair (A and B).
- Write down the second pair with the unknown (C and X).
- Set up the proportion: A/B = C/X.
- Cross-multiply: A × X = B × C.
- Solve for X: X = (B × C) ÷ A.
Example 1: Scaling a Recipe
A recipe for 4 servings calls for 6 cups of flour. How much flour do you need for 10 servings?
- 4 servings → 6 cups
- 10 servings → X cups
- X = (6 × 10) ÷ 4 = 60 ÷ 4 = 15 cups
Example 2: Unit Price at the Store
If 3 pounds of apples cost $5.25, how much do 8 pounds cost?
- 3 lb → $5.25
- 8 lb → X
- X = (5.25 × 8) ÷ 3 = 42 ÷ 3 = $14.00
Inverse Proportion (Inverse Rule of Three)
Two quantities are in inverse proportion when one increases as the other decreases at a proportional rate. For example, the more workers you assign to a job, the less time it takes to complete it.
A → B
C → X
X = (A × B) ÷ C
Notice the key difference: in the direct rule of three you multiply B × C, but in the inverse rule of three you multiply A × B.
Example 3: Workers and Time
If 5 workers can complete a project in 12 days, how many days would it take 8 workers (assuming equal productivity)?
- 5 workers → 12 days
- 8 workers → X days
- This is inversely proportional (more workers = fewer days)
- X = (5 × 12) ÷ 8 = 60 ÷ 8 = 7.5 days
Example 4: Speed and Travel Time
A car traveling at 60 mph takes 4 hours to reach a destination. How long would the trip take at 80 mph?
- 60 mph → 4 hours
- 80 mph → X hours
- Inversely proportional (faster speed = less time)
- X = (60 × 4) ÷ 80 = 240 ÷ 80 = 3 hours
Direct vs. Inverse: How to Tell the Difference
| Feature | Direct Proportion | Inverse Proportion |
|---|---|---|
| Relationship | Both increase or both decrease | One increases, the other decreases |
| Formula for X | (B × C) ÷ A | (A × B) ÷ C |
| Example | More items = higher total cost | More workers = less time |
| Key question | “If one goes up, does the other go up too?” | “If one goes up, does the other go down?” |
More Real-World Examples
| Scenario | Known Values | Type | Answer |
|---|---|---|---|
| Currency conversion | 100 USD = 92 EUR; 250 USD = ? | Direct | 230 EUR |
| Painting a fence | 3 painters in 8 hrs; 6 painters = ? | Inverse | 4 hrs |
| Gas mileage | 5 gal for 150 mi; 8 gal = ? | Direct | 240 mi |
| Filling a pool | 2 hoses in 6 hrs; 4 hoses = ? | Inverse | 3 hrs |
Common Mistakes to Avoid
- Confusing direct and inverse proportions. Always ask yourself: “If one quantity goes up, does the other go up (direct) or down (inverse)?”
- Mixing up units. Make sure both pairs of values use the same units. Do not mix kilograms with pounds, or miles with kilometers.
- Setting up the proportion incorrectly. Keep the same type of quantity on each side of the equation (e.g., servings on the left, cups on the right).
- Forgetting to simplify. After cross-multiplying, always double-check your arithmetic and simplify the result.
- Assuming proportionality when it does not exist. Not all relationships are proportional. For example, doubling the oven temperature does not halve the cooking time.
Solve Rule of Three Problems Instantly
Use our free Rule of Three Calculator to solve any direct or inverse proportion problem in seconds. Just enter your three known values, select the proportion type, and get the answer instantly with a full step-by-step explanation.