Inverse Variation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inverse Variation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A relationship where y = \frac{k}{x}: as one quantity doubles, the other halves—their product stays constant.

More workers means less time: if 4 workers take 6 hours, 8 workers take 3 hours.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Inverse variation means xy = k, so the product is constant.

Common stuck point: Inverse variation is NOT y = -kx (that's direct with negative k).

Sense of Study hint: Multiply x and y for each data pair -- if the product is always the same, you have inverse variation.

Worked Examples

Example 1

medium
\(y\) varies inversely with \(x\), and \(y = 8\) when \(x = 3\). Find \(k\) and the equation. Then find \(y\) when \(x = 6\).

Solution

  1. 1
    Inverse variation: \(y = k/x\), so \(k = xy\).
  2. 2
    \(k = 3 \times 8 = 24\).
  3. 3
    Equation: \(y = 24/x\).
  4. 4
    When \(x=6\): \(y = 24/6 = 4\).

Answer

\(k = 24\); \(y = 4\) when \(x = 6\)
In \(y = k/x\), as \(x\) doubles from 3 to 6, \(y\) halves from 8 to 4. The product \(xy = 24\) stays constant.

Example 2

hard
Speed and travel time are inversely proportional for a fixed distance. At 60 km/h the trip takes 4 hours. How long at 80 km/h? Identify \(k\).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
If \(y = k/x\) and \(y = 12\) when \(x = 5\), find \(y\) when \(x = 15\).

Example 2

hard
The pressure \(P\) and volume \(V\) of a gas at constant temperature satisfy \(PV = k\). If \(P = 4\) atm when \(V = 6\) L, find \(P\) when \(V = 8\) L.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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