# Inverse Proportions

Two quantities are inversely proportional when one is multiplied or divided by any number, the other is divided or multiplied by the same number.

The relation is also commonly denoted as:

y ∝ x^{−1}

The graph of two variables that are inversely proportional is a hyperbola.

Speed and time are inversely proportional because as the speed increases, the time it takes to reach the destination decreases.

#### Examples

3 workers build a wall in 12 hours. How long would it have taken for 6 equally productive workers?

In this example, the number of workers and the time are inversely proportional, because when the quantity of people decreases, the total time increases and when the quantity of people increases, the total time decreases.

Follow these steps to complete an inverse proportionality word problem:

Write down the ratio using one type of term (number of workers or time).

Write down the ratio with the second type of term (number of workers or time).

Invert one of the ratios (flip it upside down).

Cross-multiply, divide and solve (the same method used for direct proportions).

It takes 14 hours for a faucet with a flow of 18 liters per minute to fill a reservoir with water. How long will it take if its flow is reduced to 7 liters per minute?

In this example, the flow and time are inversely proportional, because when the flow decreases, the total time increases and when the flow increases, the total time decreases.