Solving Problems Using Proportions
Proportions are powerful tools that help us solve real-world problems involving relationships between quantities. Let's master the art of setting up and solving proportion problems!
What is a Proportion?
A proportion is an equation stating that two ratios are equal:
a/b = c/d or a : b = c : d
Fundamental Property: In any proportion, the product of the means equals the product of the extremes:
a × d = b × c
(Cross-multiplication)Two Types of Proportionality
Direct Proportion
When one quantity increases, the other increases too.
Setup: x₁ : y₁ = x₂ : y₂
Example: More items → More cost
Inverse Proportion
When one quantity increases, the other decreases.
Setup: x₁ : x₂ = y₂ : y₁
Example: More workers → Less time
5 Steps to Solve Any Proportion Problem
Interactive visualization — available in the full lesson
Quick Decision Guide
| Ask Yourself | Type |
|---|---|
| More items → More cost? | Direct |
| More workers → Less time? | Inverse |
| Faster speed → Less time? | Inverse |
| Bigger scale → Bigger size? | Direct |
| More machines → More output? | Direct |