Proportion - Revision Notes

In these revision notes for Proportion, we cover the following key points:

• What is proportion?
• What is the difference between proportion and ratio?
• What is the difference between proportion and rate?
• What are the names of the outer and inner terms of a proportion?
• How many types of proportion are there? What is the difference between them?
• How do you express a direct proportion graphically?
• How do you express an inverse (indirect) proportion graphically?
• What is the graph shape for direct/indirect proportion? Why?

Proportion Revision Notes

If two quantities involved in a situation (we call them variables) are related in such a way that if one quantity changes, the other quantity changes at the same (or opposite) degree, we say these quantities are in proportion with (or proportional to) each other. In simpler words, a proportion indicates us how one thing changes in relation to another.

Any proportion includes two equal fractions, i.e. if

then a and c are proportional (as well as b and d).

In other words, if a/b and c/d are two equal ratios or rates, they are in proportion.

We can also write the above proportion in the form

where a is the first term, b is the second term, c is the third term and d is the fourth term of the proportion. The terms a and d are called outer terms or extremes, while b and c are called inner terms or means of the proportion.

There are two main types of proportion: direct and inverse proportion. In direct proportion, the quantities a and c change in the same way as well as b and d.

By definition, direct proportion is the relation between quantities whose ratio (or rate) is constant.

The graph representing a direct proportion is a straight (sloped) line that starts from the origin (otherwise it is not a direct proportion), where the slope is determined by the simplest form of the ratio or of the unit rate.

We can write a direct proportion as:

This method of expressing a proportion as equality of products is known as cross product. The cross product method of writing a proportion allows checking the veracity of proportion in an easier way, without involving rational numbers, simplifications or GCF calculations.

By definition, inverse (or indirect) proportion occurs when a decrease in one quantity or variable causes an increase by the same factor in another quantity or variable.

We can express an inverse proportion in symbols as

or

Two quantities involved in an inverse proportion have a relation of type a × b = constant, which we can write as

or

The graph of such a relation is called hyperbola, it is a curved line that approaches the axes without touching them with the increase in the values of a and b. To increase the accuracy of an inverse proportion graph, we must use as many points as possible.