Inverse proportion

Two quantities are in inverse proportion when multiplying one of them by a number divides the other by the same number. For a fixed journey, doubling the speed halves the time taken; for a fixed job, doubling the number of identical workers halves the time.

A \(90\) km journey at different steady speeds

Why it works: the distance stays \(90\) km, so speed \(\times\) time is always \(90\). For inverse proportion the product stays fixed: \(x_1y_1=x_2y_2\).

Method

1. Check that the total amount of work, distance or supply stays fixed and that every worker, machine or pipe works at the same steady rate.
2. Unitary method: find the time for one unit first. One machine takes longer than several, so multiply. Then divide by the new number of units.
3. Product method: write \(x_1y_1=x_2y_2\) and solve.

Example

Six identical machines can harvest a field in \(10\) hours. How long will \(4\) such machines take?

Unitary method: \(1\) machine takes \(6\times10=60\) hours, so \(4\) machines take \(\dfrac{60}{4}=15\) hours.

Product method: \(4t=6\times10\), so \(t=15\).

Check: fewer machines must take longer, and \(15\) h is more than \(10\) h. A common error is to use the direct proportion method, \(\dfrac{10}{6}\times4\), which gives a shorter time. State the assumption: every machine works at the same steady rate.