Algebraic Tutorial
How to Solve Proportions
A proportion is a mathematical statement demonstrating that two ratios are equal. Learn how to solve for any unknown variable.
The Cross-Multiplication Rule
In any proportion where A / B = C / D (with B ≠ 0 and D ≠ 0), the cross products of the terms are strictly equal:
A × D = B × C
"The product of the extremes equals the product of the means."
Solving for the Missing Variable (X)
Depending on which position contains the unknown variable, isolate it using basic algebraic division:
When X is in Numerator:
Example: X / 8 = 3 / 4
- Cross-multiply: 4 × X = 8 × 3 = 24.
- Divide by 4: X = 24 ÷ 4 = 6.
When X is in Denominator:
Example: 5 / X = 15 / 27
- Cross-multiply: 15 × X = 5 × 27 = 135.
- Divide by 15: X = 135 ÷ 15 = 9.
Direct vs. Inverse Proportions
Not all proportional relationships behave identically. Understanding the difference prevents costly calculation errors:
- Direct Proportion: As one value increases, the other increases at a constant rate (e.g. buying 2 items costs $10 → buying 4 items costs $20). Formula:
Y = k × X. - Inverse Proportion: As one value increases, the other decreases proportionally (e.g. 2 workers take 6 hours to paint a fence → 4 workers take 3 hours). Formula:
X × Y = k.