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.
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:
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.
