Proportional Relationships
A proportional relationship is one in which two quantities vary directly with each other. We say the variable varies directly as if:
for some constant , called the constant of proportionality .
(Some textbooks describe a proportional relationship by saying that " varies proportionally with " or that " is directly proportional to .")
This means that as increases, increases and as decreases, decreases-and that the ratio between them always stays the same.
The graph of the proportional relationship equation is a straight line through the origin.
Example 1:
Given that varies proportionally with , with a constant of proportionality , find when .
Write the equation of the proportional relationship.
The variable varies proportionally with with a constant of proportionality equal to .
So,
Substitute the given value.
Example 2:
Given that varies proportionally with , find the constant of proportionality if and .
Write the equation of the proportional relationship.
Substitute the given and values, and solve for .
Example 3:
Suppose varies proportionally with , and when . What is the value of when ?
Write the equation of the proportional relationship.
Substitute the given and values, and solve for .
The equation is . Now substitute and find .