Master the Rule of Three with our comprehensive step-by-step calculator that shows every calculation detail. This educational resource is perfect for mathematical learning, helping students understand proportional relationships through detailed problem-solving demonstrations. Whether you're learning direct proportions or inverse proportions, our calculator breaks down each mathematical concept into clear, understandable steps.
Interactive Step-by-Step Calculator
Enter your values below to see detailed mathematical calculations with step-by-step explanations:
Direct Proportion - Step by Step
Learn how direct proportions work with detailed mathematical reasoning.
Inverse Proportion - Step by Step
Understand inverse proportions with comprehensive mathematical explanations.
Worked Examples with Detailed Steps
Learn from these comprehensive examples that demonstrate proportional calculations in real-world scenarios:
Example 1: Direct Proportion - Recipe Scaling
Problem: If 2 cups of flour make 12 cookies, how much flour is needed for 30 cookies?
Step-by-Step Solution:
- Identify the relationship: This is a direct proportion - more flour means more cookies
- Set up the proportion: 2 cups flour → 12 cookies, x cups flour → 30 cookies
- Apply the formula: x = (b × c) ÷ a = (12 × 30) ÷ 2
- Calculate: x = 360 ÷ 2 = 180
- Answer: You need 5 cups of flour for 30 cookies
Example 2: Inverse Proportion - Workforce Planning
Problem: If 3 workers can complete a project in 8 days, how long will it take 6 workers?
Step-by-Step Solution:
- Identify the relationship: This is an inverse proportion - more workers means less time
- Set up the proportion: 3 workers → 8 days, 6 workers → x days
- Apply the formula: x = (a × b) ÷ c = (3 × 8) ÷ 6
- Calculate: x = 24 ÷ 6 = 4
- Answer: 6 workers will complete the project in 4 days
Example 3: Business Mathematics - Cost Analysis
Problem: If 5 units cost $125, what is the cost for 18 units?
Step-by-Step Solution:
- Identify the relationship: This is a direct proportion - more units means higher cost
- Set up the proportion: 5 units → $125, 18 units → $x
- Apply the formula: x = (b × c) ÷ a = (125 × 18) ÷ 5
- Calculate: x = 2250 ÷ 5 = 450
- Answer: 18 units will cost $450
Understanding the Mathematical Concepts
Master the mathematical principles behind proportional relationships and enhance your problem-solving skills:
Direct Proportion Explained
Direct proportions occur when two quantities increase or decrease together at the same rate. In mathematical education, this concept is fundamental to understanding ratio calculations and proportional reasoning.
- Mathematical relationship: y = kx (where k is constant)
- Real-world examples: Distance and time (constant speed), price and quantity
- Problem-solving approach: Set up equivalent ratios and solve for unknown
Inverse Proportion Explained
Inverse proportions occur when one quantity increases while the other decreases. This mathematical concept is essential for engineering calculations and business mathematics.
- Mathematical relationship: y = k/x (where k is constant)
- Real-world examples: Workers and time, speed and time
- Problem-solving approach: Use the formula x = (a × b) ÷ c
Related Educational Resources
Expand your understanding of mathematical concepts and problem-solving techniques:
Key Takeaways
- Step-by-step learning enhances understanding of proportional relationships
- Mathematical education benefits from detailed problem-solving demonstrations
- Direct proportions and inverse proportions require different calculation approaches
- Real-world applications make mathematical concepts more accessible
- Practice with worked examples improves mathematical problem-solving skills