# Proportion

Two numbers are said to be in **proportion** to each other if some numeric relationship exists between them. There are several types of proportions, each defined by a separate class of function.

## Contents

## Direct Proportion

Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers and can be expressed as:

where is some real number.

The graph of a direct proportion is always linear.

Often, this will be written as .

## Inverse Proportion

Inverse proportion is a proportion in which as one number's absolute value increases, the other's decreases in a directly proportional amount. It can be expressed as:

where is some real number that does not equal zero.

The graph of an inverse proportion is always a hyperbola, with asymptotes at the x and y axes.

## Exponential Proportion

A proportion in which one number is equal to a constant raised to the power of the other, or the logarithm of the other, is called an exponential proportion. It can be expressed as either:

- or

for some real number , where is not zero or one.

## Problems

### Introductory

- Suppose is either or in the following system:

Find the possible values of . (Source)

### Intermediate

- is directly proportional to the sum of the squares of and and inversely proportional to and the square of . If when and , find when and , what is ? (Source) (Thanks to Bicameral of the AoPS forum for this one)