## Proportion, Direct Variation, Inverse Variation, Joint Variation

This section defines what proportion, direct variation, inverse variation, and joint variation are and explains how to solve such equations.

#### Proportion

A proportion is an equation stating that two rational expressions are equal. Simple proportions can be solved by applying the cross products rule.

If , then ab = bc.

More involved proportions are solved as rational equations.

##### Example 1

Solve . Apply the cross products rule. The check is left to you.

##### Example 2

Solve . Apply the cross products rule. The check is left to you.

##### Example 3

Solve . However, x = 4 is an extraneous solution, because it makes the denominators of the original equation become zero. Checking to see if is a solution is left to you.

#### Direct variation

The phrase “ y varies directly as x” or “ y is directly proportional to x” means that as x gets bigger, so does y, and as x gets smaller, so does y. That concept can be translated in two ways.

• for some constant k.

The k is called the constant of proportionality. This translation is used when the constant is the desired result.

• This translation is used when the desired result is either an original or new value of x or y.

• ##### Example 4

If y varies directly as x, and y = 10 when x = 7, find the constant of proportionality. The constant of proportionality is .

##### Example 5

If y varies directly as x, and y = 10 when x = 7, find y when x = 12. Apply the cross products rule. #### Inverse variation

The phrase “ y varies inversely as x” or “ y is inversely proportional to x” means that as x gets bigger, y gets smaller, or vice versa. This concept is translated in two ways.

• yx = k for some constant k, called the constant of proportionality. Use this translation if the constant is desired.

• y 1 x 1 = y 2 x 2.

Use this translation if a value of x or y is desired.

##### Example 6

If y varies inversely as x, and y = 4 when x = 3, find the constant of proportionality. The constant is 12.

##### Example 7

If y varies inversely as x, and y = 9 when x = 2, find y when x = 3. #### Joint variation

If one variable varies as the product of other variables, it is called joint variation. The phrase “ y varies jointly as x and z” is translated in two ways.

• if the constant is desired.

• if one of the variables is desired.

##### Example 8

If y varies jointly as x and z, and y = 10 when x = 4 and z = 5, find the constant of proportionality. ##### Example 9

If y varies jointly as x and z, and y = 12 when x = 2 and z = 3, find y when x = 7 and z = 4. Occasionally, a problem involves both direct and inverse variations. Suppose that y varies directly as x and inversely as z. This involves three variables and can be translated in two ways:

• if the constant is desired.

• ##### Example 10

If y varies directly as x and inversely as z, and y = 5 when x = 2 and z = 4, find y when x = 3 and z = 6. 