9.6.3 Inverse Proportion

The second type of relationship is known as inverse proportion.

Suppose you have decided to hire a taxi to take a group of colleagues from work to the train station. If the taxi charges a set fee for the journey, then the more people who go in the taxi, the less each person has to pay: If two people go, each pays half the cost; if three people go, each pays a third of the cost; and if four people go, each person pays a quarter of the cost. This is an example of inverse proportion. As one quantity (the number of people) doubles or triples, the other quantity (the cost) halves or is reduced to a third of the original value. In other words, if you multiply one quantity by a factor, the other quantity is divided by that factor.

The cost per person can be calculated directly or by using the following formula:

cost per person in dollar equals taxi fare in dollar division number of people

For example, if the fare is $23.00 and four people go in the taxi, each person will have to pay dollar equation left hand side 23.00 division four equals right hand side dollar 5.75.

Two quantities are said to be inversely proportional if their product is a constant. For example, suppose you drive 30 miles and it takes you an hour to get there. Your average speed for the trip is 30 miles per hour. Now if you doubled your speed to 60 miles per hour, it would take you half as long (that is, half an hour) to get there; if you halved your speed, it would take twice as long, and so on. In each case, the following relationship holds:

left parenthesis speed in miles per hour right parenthesis multiplication left parenthesis time for trip in hours right parenthesis equals left parenthesis distance traveled in miles right parenthesis

This shows that for a given distance, speed and time are inversely proportional to each other.

This formula can also be written as: speed in miles per hour left parenthesis mph right parenthesis equals distance traveled in miles divided by time for trip in hours.

Activity Symbol Activity: How Fast?

For driving the 30 miles above, what is the average speed if the trip took hours?

Answer Symbol


Substituting the distance as 30 and the time as the decimal 1.5 in the formula above gives:

average speed equation sequence equals 30 divided by 1.5 equals 20 mph

So, the average speed for the trip is 20 miles per hour.

9.6.2 The More, the Merrier

9.7 Writing Inequalities