Rule of Three Calculator
Solve proportional relationships with ease. Calculate direct and inverse proportions using the classic rule of three method with step-by-step solutions.
Direct Proportion Setup
If A gives B, then C gives X
Direct Proportion Example
Problem: If 3 books cost $15, how much do 7 books cost?
Solution: X = (7 × $15) ÷ 3 = $35
Inverse Proportion Example
Problem: If 4 workers finish in 12 days, how long for 6 workers?
Solution: X = (4 × 12) ÷ 6 = 8 days
What is the Rule of Three?
The Rule of Three is a fundamental mathematical method used to solve proportion problems. It's based on the principle that if three quantities are known in a proportional relationship, the fourth quantity can be calculated.
This method is widely used in everyday situations, from cooking recipes and currency conversion to business calculations and scientific applications. The rule of three provides a simple yet powerful way to find unknown values when dealing with proportional relationships.
Basic Formula:
If A is to B as C is to X, then X = (B × C) ÷ A
Types of Proportions
Direct Proportion
In direct proportion, as one quantity increases, the other increases proportionally. Both quantities change in the same direction.
Example:
If 3 apples cost $6, how much do 5 apples cost?
Solution: (5 × $6) ÷ 3 = $10
Inverse Proportion
In inverse proportion, as one quantity increases, the other decreases proportionally. The quantities change in opposite directions.
Example:
If 4 workers complete a job in 12 days, how long for 6 workers?
Solution: (4 × 12) ÷ 6 = 8 days
Step-by-Step Method
Identify the Known Values
Write down the three known quantities and identify which value you need to find.
Set Up the Proportion
Arrange the values in the format: A is to B as C is to X, where X is the unknown.
Apply the Formula
For direct proportion: X = (B × C) ÷ A. For inverse proportion: X = (A × B) ÷ C.
Calculate and Verify
Perform the calculation and check if the result makes logical sense in the context of the problem.
Real-World Applications
🍳 Cooking & Recipes
Scale ingredients up or down when adjusting recipe serving sizes.
💱 Currency Exchange
Convert between different currencies using exchange rates.
🏗️ Construction
Calculate material quantities based on project scale and dimensions.
⚡ Speed & Distance
Calculate travel time, fuel consumption, and distance relationships.
💼 Business Finance
Calculate costs, profits, and resource allocation in business scenarios.
🔬 Science & Medicine
Dosage calculations, concentration ratios, and experimental scaling.
Common Examples & Solutions
Example 1: Shopping Problem
Problem: If 8 pencils cost $12, how much do 15 pencils cost?
Solution:
8 pencils : $12 = 15 pencils : X
X = (15 × $12) ÷ 8 = $22.50
Example 2: Work Rate Problem
Problem: If 6 workers can build a wall in 10 days, how many days will 4 workers take?
Solution (Inverse Proportion):
6 workers × 10 days = 4 workers × X days
X = (6 × 10) ÷ 4 = 15 days
Example 3: Recipe Scaling
Problem: A recipe for 4 people uses 2 cups of flour. How much flour for 7 people?
Solution:
4 people : 2 cups = 7 people : X cups
X = (7 × 2) ÷ 4 = 3.5 cups
Quick Reference
Direct Proportion
X = (B × C) ÷ A
Both quantities change in same direction
Inverse Proportion
X = (A × B) ÷ C
Quantities change in opposite directions
Tips & Tricks
Always check if your answer makes logical sense
Keep units consistent throughout calculations
Practice with real-world examples to build intuition
Double-check by working backwards from your answer