Direct and Indirect Variation

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!

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/