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Proportional Relationships

A proportional relationship is one in which two quantities vary directly with each other. We say the variable y varies directly as x if:  

y = k x

for some constant k , called the constant of proportionality .

(Some textbooks describe a proportional relationship by saying that " y varies proportionally with x " or that " y is directly proportional to x .")

This means that as x increases, y increases and as x decreases, y 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 y varies proportionally with  x , with a constant of proportionality k = 1 3 , find y when x = 12 .

Write the equation of the proportional relationship.

The variable x varies proportionally with y with a constant of proportionality equal to 1 3 .

So,

Substitute the given x value.

Example 2:

Given that y varies proportionally with x , find the constant of proportionality if y = 24 and x = 3 .

Write the equation of the proportional relationship.

y = k x

Substitute the given x and y values, and solve for k .

24 = k 3 k = 8

Example 3:

Suppose y varies proportionally with x , and y = 30 when x = 6 . What is the value of y when x = 100 ?

Write the equation of the proportional relationship.

y = k x

Substitute the given x and y values, and solve for k .

30 = k 6

k = 5

The equation is y = 5 x . Now substitute x = 100 and find y .

y = 5 100 y = 500