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 65º,
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 


