Rule of Three — Definition, Formula & Examples
The Rule of Three is a method for finding an unknown value when three values in a proportion are known. You set up a ratio equation and cross-multiply to solve for the missing number.
Given that two quantities are proportional and three of the four values in the proportion are known, the Rule of Three determines the fourth value by computing .
Key Formula
Where:
- = Known value paired with b
- = Known value paired with a
- = Known value paired with the unknown x
- = The unknown fourth value
How It Works
Start by identifying which quantities are related proportionally. Write the known pair as a fraction on one side and the pair with the unknown on the other side, keeping corresponding units in the same position (numerator with numerator, denominator with denominator). Cross-multiply and divide to isolate the unknown. The key is making sure both ratios compare the same types of quantities in the same order.
Worked Example
Problem: If 4 notebooks cost $10, how much do 6 notebooks cost?
Set up the proportion: Place the known pair on one side and the pair with the unknown on the other, keeping notebooks in the numerators and cost in the denominators.
Cross-multiply: Multiply diagonally: 4 times x equals 10 times 6.
Solve for x: Divide both sides by 4.
Answer: 6 notebooks cost $15.
Why It Matters
The Rule of Three appears constantly in everyday calculations — scaling recipes, converting currencies, and figuring out travel times. It is one of the most practical tools taught in middle school math and forms the basis for more advanced work with rates and variation in algebra and science courses.
Common Mistakes
Mistake: Mixing up the positions of quantities so the ratios compare different things (e.g., notebooks/cost on one side but cost/notebooks on the other).
Correction: Always place the same type of quantity in the same position on both sides of the proportion. If notebooks are in the numerator on the left, they must be in the numerator on the right.