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### Direct and Indirect Variation

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Date: 07/09/98 at 17:42:08
From: Eric Sparrow
Subject: Word problems

I am having trouble with direct and indirect variation. I don't really
understand the difference. I have tried to work the problems over and
over, and I still can't get it. I would appreciate it if you could
help me with this problem.

Thank you!
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Date: 07/09/98 at 19:54:50
From: Doctor Pat
Subject: Re: Word problems

Eric,

Many simple problems can be examples of both direct and indirect
variation, depending on which variable you make a constant. For
example, the traditional formula distance = rate x time (or D = R x T)
is an example of direct variation if we make the rate or the time a
constant.

Let's say you walk 5 mph. The distance you walk is then directly
proportional to the time you walk. If you walk twice as long, you go
twice as far, and the ratio of the distance and time is always the
same. In this case the rate is the "constant of proportionality."

Now let's change it to an indirect variation. To do so, we merely keep
the distance constant. Suppose you live five miles from school. If
you walk slower, it takes longer. As your speed goes up, the time goes
down. That's why it is called indirect variation (it is often called
inverse variation also).

Two variables, call them x and y, are directly related if they can be
expressed so that their ratio is always a constant. This means either
y/x = k or y = kx (where k is a constant). The two equations are
really the same. Just multiply both sides of the left one by x and
you get the one on the right.

The same two variables are indirectly related if their product is
always a constant. This means either xy = k or x = k/y or y = k/x.
If you have indirect variation, problems usually tell you the values
of x and y at two different times (except one of them is left for you
to find). They often look something like this:

If x and y are indirectly related, and y = 6 when x = 4, find y
when x = 8.

The key is to recognize that we have two x,y pairs. For indirect
variation we just set up an equation that says:

first x * first y = last x * last y

Using the numbers in the example problem, this becomes:

6 * 4 = 8 * last y

which you can easily solve to get y = 3.

I hope this helps. If I didn't make it clear, please write back and
give examples of some of the problems, and what you do and don't
understand.

Good luck,

- Doctor Pat, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
High School Equations, Graphs, Translations
Middle School Graphing Equations
Middle School Word Problems

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