Direct and Inverse Variation — Mathematics Knowledge System

Overview

Two quantities vary directly if their ratio is constant (one grows proportionally as the other grows), and vary inversely if their product is constant (one shrinks proportionally as the other grows). These two relationships describe an enormous number of real-world quantities — from speed and travel time to pressure and volume — using just a single constant of proportionality.

Interactive Graph

Formal Definition

Definition

Direct and inverse variation are defined by:

y = kx

y varies directly with x; k is the constant of variation

Direct variation

y = \frac{k}{x}

y varies inversely with x

Inverse variation

Properties

Direct variation constant

k = \frac{y}{x} \text{ is constant for all corresponding pairs}

Inverse variation constant

k = xy \text{ is constant for all corresponding pairs}

Direct variation graph

\text{A line through the origin with slope } k

Worked Examples

Find k using the given pair.
$k=\frac{y}{x}=\frac{15}{3}=5$
Use y = kx with the new x-value.
$y=5(8)=40$

Answer: y = 40

Practice Problems

Difficulty 3/10

The time to complete a trip varies inversely with speed. At 60 mph, a trip takes 3 hours. How long does it take at 45 mph?

Common Mistakes

Common Mistake

Using y = kx for a relationship that is actually inverse variation, or vice versa.

Check whether the RATIO y/x is constant (direct variation) or the PRODUCT xy is constant (inverse variation) before choosing which model to use — they behave oppositely as x increases.

Summary

Direct variation: y = kx — y and x increase/decrease together, with constant ratio k.
Inverse variation: y = k/x — as x increases, y decreases, with constant product k.
Find k from one known pair of values, then use it to find any other unknown value.

References

WebsiteWikipedia — Proportionality (mathematics)