Inverse Proportion

Inverse proportion animation

The heights of these two bars vary in inverse proportion to one another.

When one gets large, the other gets small; when one gets small, the other gets large - but in a special way.

When the blue bar grows so that it doubles (times 2) its size, the green bar shrinks to half (times 1/2) its size.

If the green bar shrinks so that its height is cut to one third (times 1/3), then the blue bar grows to triple (times 3) its height.

The height of one of the bars will change by the inverse of the factor by which the other bar's height changes.

Further explanations follow.

It is probably best to understand direct proportions first.

If you are not familiar with direct proportions, perhaps it would help to see that material before going further here. You can find an explanation of direct proportions here.


Inverse proportions have reciprocal factor changes.

Probably better stated as a reciprocal proportion, the inverse proportions relates two quantities through factors that are multiplicative inverses. That is, through factors that are reciprocals, such as 3 and 1/3.

Drive the same distance in twice (2) the time at half (1/2) the speed.

For example, let us say that you are driving a car and you are going to travel 60 miles. Consider this to be a constant distance throughout the following discussion.

Case #1:

Suppose that you spent 1 hour driving. Your average speed would be 60 mph.

Case #2:

Suppose that you spent 2 hours driving. Your average speed would be 30 mph.

So, changing the number of hours that you drive will change the average speed that you will travel.

Notice that the number of hours, the time, that is, changed by a factor of 2, since 1 hour times 2 is 2 hours.

Also, notice that the speed at which you were traveling changed by a factor of 1/2, since 60 mph times 1/2 is 30 mph.

The two quantities, time and speed, changed by reciprocal factors. Time changed by a factor of 2; speed changed by a factor of 1/2.


Definition of Inverse Proportion

When quantities are related this way we say that they are in inverse proportion. That is, when two quantities change by reciprocal factors, they are inversely proportional.

In the above example the time is in inverse proportion to the average speed. One could also say that the average speed was in inverse proportion to the time.

The formal definition of inverse proportion:

Two quantities, A and B, are in inverse proportion if by whatever factor A changes, B changes by the multiplicative inverse, or reciprocal, of that factor.


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