To say that y is proportional to x is the same as saying that y varies with x or y varies directly as x. Any of these can be replace with the following notation: y∝x

If this is the case, our job is often to find out the constant of proportionality, which shows the relative increase in y as x increases. This is represented by the letter k in the equation y = kx.

Typically we will know the value of y at one value of x, and this will let us distinguish k, then allowing us to find the value of y at other values of x.

Examples

Exercise

Let’s complete exercise 10 on pages 185 and 186 of the extended textbook:

The answers are below:

Saying that y is inversely proportional to x, or that y varies inversely as x, means that as x increases y decreases and vice versa, which is written in mathematics using the following notation: y∝1/x

As above, our job will often be to find out the constant of proportionality, which shows the relative decrease in y as x increases. This is represented by the letter k in the equation y = k/x.

Examples

Exercise

Let’s complete exercise 11 on pages 187 to 189 of the textbook:

The answers are below: