Inverse Proportion

Two quantities are in inverse proportion when one increases exactly as fast as the other decreases – double one, and the other halves. Unlike direct proportion, where the ratio y/x stays constant, in inverse proportion it's the product x × y that stays constant: xy = k. The classic example is workers and time: more workers on a job means less time needed to finish it (assuming everyone works at the same steady rate).

The Rule for Inverse Proportion

If y is inversely proportional to x, then xy = k for a constant k. This means x₁y₁ = x₂y₂ for any two pairs of matching values.

4 painters can finish a mural in 9 days. How many days would 6 painters take?

Workers × Days is constant: 4 × 9 = 36. With 6 painters: 36 ÷ 6 = 6 days.

Spotting the Difference from Direct Proportion

Ask: as one quantity gets bigger, does the other get bigger too (direct), or smaller (inverse)? More speed means less travel time for a fixed distance – that's inverse. More hours worked at a fixed hourly wage means more pay – that's direct.

Real-Life Application

Construction: more workers, less time to finish a project.
Speed and time: for a fixed distance, faster speed means less travel time.
Sharing: more people sharing a fixed prize means a smaller share each.

Key Takeaways

Inverse proportion means xy = k – as one quantity increases, the other decreases to match.
x₁y₁ = x₂y₂ lets you solve for an unknown value directly.
Always check whether a real-world problem is direct or inverse before solving it.

Practice: Inverse Proportion

Inverse Proportion: Workers and Time