Variation 
 Section 2.5  

Find the constant of proportionality. y is directly proportional to x. If x = 3, then y = 24. 
Solve 
Step 
Check 
The goal is to find k, the constant. 
y = 8x 

y = kx  Use the formula for direct variation 
If x = 3, then y = 24. 
If x = 3, then y = 24.  Given 
y = 8(3) 
24 = k(3)  Plug in x and y 
y = 24 
24 = 3k  Divide by 3  
k = 8  constant of proportionality  
y = 8x 
Plug 8 in formula for k  


Now we have a formula that can be used to find values for y for specific values of x.  
Find the constant of proportionality. y is directly proportional to x. If x = 12, then y = 4. 
[Solution] 

Find the constant of proportionality. L is inversely proportional to the square of d. If d = 2, then L = 3. 
Solve 
Step 
Check 
The goal is to find k, the constant.  Plug in the values 

Use the formula for inverse variation  
If d = 3, then L = 5.  Given 

Plug in d and L  
Square 3 and multiply both sides by 9  
k = 45  constant of proportionality  
Plug 45 in formula for k  
Find the constant of proportionality. 
[Solution] 

R varies jointly with p and q. If R = 20, when p = 5 and q = 8. Find R , when p = 3 and q = 16 
Solve 
Step 

The goal is to find k, then R. 
Plug in the values. Find the constant of proportionality.  
R = kpq 
Use
the formula for joint variation with p and q 

If R = 20, when p = 5 and q = 8 
Given  
20 = k(5)(8) = 40k 
Plug in p, q and R  

constant of proportionality  

The formula  

Plug in 3 and 16 to find R  
R = 24 
Done!!  

H varies jointly with the square g and the square root f. If
H = 160, when g = 4 and g = 25. Find H , when g = 1 and f = 81 
[Solution] 

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada