Inverse proportion

In an inverse proportion, when one quantity increases by a certain factor, the other quantity decreases by the same factor. Keep reading to see a real-life example of this situation.

Suppose x and y are variables and k is a constant. Then, y = 8x and y = -2x are examples of direct proportions.

However, y = 3/x and y = -4/x are examples of inverse proportions.

A real life example of inverse proportion

A good real-life example of inverse proportion is the speed you drive and the time it takes to travel a certain distance. Suppose you need to drive to a city that is located 200 miles away.

The following table shows the time it will take you to get to your destination based on the speed. 

Speed (miles per hour) Time
20 m/h 10 hours
40 m/h 5 hours
80 m/h 2.5 hours

Did you make the following observations?

  • 20 × 2 = 40  and  40  × 2 = 80
  • 10 ÷ 2 = 5    and   5 ÷ 2 = 2.5

This means that every time you multiply the speed by 2, the time it takes to get to your destination is divided by 2 or multiplied by 1/2.

In general, with inverse proportion, when one quantity is multiplied by x, the other quantity is divided by x or multiplied by 1/x

Furthermore, notice the following:

20 × 10 = 200

40 × 5 = 200

80 × 2.5 = 200

200 is a constant, so let k = 200.

20, 40, or 80 is the independent variable, so let x = 20, 40, or 80 

10, 5, or 2.5 is the dependent variable, so let y = 10, 5, or 2.5 

We get  x × y = 200  or  y = 200/x

In general, with inverse proportion, y = k/x and we say that y is inversely proportional to x.

Other real-life examples of inverse proportion

  • The number of people doing something and the time it takes to do it. As the number of people increases, the time it takes to finish decreases.
  • Sharing a specific amount of money with a certain amount of people. As the number of people increases, the amount decreases.
  • Suppose your income is constant. The money you save is inversely proportional to your expenses. 

The figure below nicely summarizes the difference between direct proportion and inverse proportion

Inverse proportion

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