9.6 Proportion

In this section, we consider two important types of relationships that occur frequently in real life: direct and inverse proportion.

Direct Proportion

The first type of relationship is known as direct proportion. Two quantities are said to be directly proportional to each other if when one doubles, triples, and quadruples, the other also doubles, triples, and quadruples. For example, if you buy three times as many items as usual, you would expect to have to pay three times as much money (unless there were some special offer available) because the price is directly proportional to the number of items bought.

Activity Symbol Activity: Directly Proportional Relationships

Which of the following quantities are in direct proportion?

Select the two correct answers from the following:

Answer Symbol


(a) The number of kilometers and the equivalent number of miles.


(b) The number of euros that are exchanged for dollars.


(c) The monthly cost of a cell phone and the minutes used for talking in Section 9.3.3.

The correct answers are a and b.


The correct answers are (a) and (b).

Parts (a) and (b) are both examples of direct proportion. If you multiply the number of kilometers by any factor, the number of miles will also change by this factor.

For example, if you double the number of kilometers, the number of equivalent miles will also double, and similarly for exchanging currency.

If you triple the number of dollars you exchange, you would expect to get three times as many euros. However, part (c) is not a directly proportional relationship. If you use 30 minutes, the charge is $25. But if you double the time used to 60 minutes, the charge is $31, and the price has not doubled.

9.5.4 How Many Minutes for Chatting?

9.6.1 Other Proportion Solutions