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 |
 |
|
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