Complete Guide to the Rule of Three
What is the rule of three?
The rule of three is a mathematical method for solving proportionality problems. It is used to find an unknown value when we know the proportional relationship between three other values.
Simple rule of three
The simple rule of three applies when there are two quantities that are proportionally related.
Direct rule of three
Used when quantities are directly proportional: if one increases, the other also increases in the same proportion.
X = (B × C) / A
Where:
- A = First known value
- B = Second known value (related to A)
- C = Third known value
- X = Value we want to find
If 3 workers build 6 tables, how many tables do 5 workers build?
X = (6 × 5) / 3 = 10 tables
Inverse rule of three
Used when quantities are inversely proportional: if one increases, the other decreases.
X = (A × B) / C
If 4 workers finish a job in 12 days, how many days will 6 workers take?
X = (4 × 12) / 6 = 8 days
Compound rule of three
The compound rule of three is used when there are more than two quantities involved. Each relationship can be direct or inverse.
How to solve a compound rule of three
- Identify all quantities involved
- Determine if each relationship is direct or inverse
- Multiply values according to proportion type:
- Direct: final value goes to numerator
- Inverse: initial value goes to numerator
- Calculate the result
If 4 machines working 8 hours produce 200 pieces, how many pieces will 6 machines working 10 hours produce?
- Machines: 4 → 6 (direct, more machines = more production)
- Hours: 8 → 10 (direct, more hours = more production)
X = 200 × (6/4) × (10/8) = 375 pieces
Common use cases
- Cooking recipes: Adjusting ingredient quantities
- Business: Calculating proportional costs
- Construction: Estimating required materials
- Speed and time: Calculating distances or travel times
- Productivity: Estimating team output
Frequently asked questions
Use the direct rule of three when both quantities move in the same direction (both increase or both decrease). Use the inverse when they move in opposite directions (one increases while the other decreases).
Ask yourself: if the first quantity increases, does the second increase or decrease? If it increases, it's direct. If it decreases, it's inverse. For example: more workers = less time (inverse), more hours = more production (direct).
Yes, the compound rule of three can combine any number of direct and inverse proportions. Each one is processed according to its type when calculating the final result.
The simple rule of three involves only two quantities (one relationship). The compound involves three or more quantities (multiple relationships) that are combined to find the result.
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