Inverse Variation
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Inverse Variation
(The Opposite of Direct Variation)
 

In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases.

For instance, a biker traveling at 8 mph can cover 8 miles in 1 hour.  If the biker's speed decreases to 4 mph, it will take the biker 2 hours (an increase of one hour), to cover the same distance. 

Inverse variation:  when one variable increases,
the other variable decreases.   


As speed decreases, the time increases.

Notice the shape of the graph of inverse variation.
If the value of x is increased, then y decreases.    
If x decreases, the y value increases.  We say that y varies inversely as the value of x

An inverse variation between 2 variables, y and x, is a relationship that is expressed as:


where the variable k is called the constant of proportionality.

As with the direct variation problems, the k value needs to be found using the first set of data.


Find the Constant, k

 

 

The number of hours, h, it takes for a block of ice to melt varies inversely as the temperature, t.  If it
takes 2 hours for a square inch of ice to melt at 65,
find the constant of proportionality.  
Start with the formula:        

Substitute the values  :           

then solve for k:              


 


Typical Inverse Variation Problem: 

In a formula, Z varies inversely as p
If Z is 200 when p = 4, find Z when p = 10.

Use the same three process steps that were used in direct variation problems:

1.  Set up the formula.

2.  Find the missing constant, k, by     
     using the first set of data given.

3.  Using the formula and constant, k,
   find the missing value in the problem.




 


Inverse
Variation Example:

 

 

In kick boxing, it is found that the force, f, needed to break a board, varies inversely with the length, l, of the board.  If it takes 5 lbs of pressure to break a board 2 feet long, how many pounds of pressure will it take to break a board that is 6 feet long?
 

1.  Set up the formula.
2.  Find the missing constant, k,      
     using the first set of data given.
3.  Using the formula and constant, k,
   find the missing value in the problem.



 


Combination Variation Example:

Variable M varies directly as variable t and inversely as variable s.
If M = 24 when t = 3 and s = 2,
find M when t = 5 and s = 8.
( In combination problems, there is only one constant value, k, used with the direct and inverse variables.)

1.  Set up the formula

2.  Find the missing constant
    of proportionality, k.

3.  Using the formula and the
    constant, k, find the new
    value in the problem