Direct and Inverse Variation

Date: 04/25/2001 at 15:40:39
From: Krysten Heath
Subject: Direct and Inverse Variation

I don't understand how to solve inverse and direct variations. I have 
an example of each type of problem:

Inverse
If P is 12 when Q is 6, and P varies inversely as Q, what is the 
value for Q when P is 8?

Direct
If P is 12 when Q is 18, and P varies directly as Q, what is Q when P 
is 30?

Hope you can help... thanks.
Krysten

Date: 04/25/2001 at 19:34:44
From: Doctor Schwa
Subject: Re: Direct and Inverse Variation

Inverse Variation
-----------------
What I think of as a standard example of inverse variation is speed 
and time. If it takes me a certain amount of time to get to school at 
a certain speed, then if I want to get there in half the time, I'll 
have to go twice the speed.

In other words, since (rate) * (time) is a constant, namely the
distance to school, when one thing increases the other one decreases.
That's why it's called an inverse variation.

So, I would reword your problem as

   If at 12 miles per hour it takes me 6 hours to get there, what
   is the value for the time when my speed is 8 miles per hour?

Instead of the abstract letters, when I see inverse variation I
think of one letter (we could choose P) as speed, and the other as 
time. Can you solve the reworded problem?

You can also try to do it with the abstraction:

   initial P * initial Q = final P * final Q

will be the pattern for an inverse variation.

Direct Variation
----------------

When I think of direct variation, I need an example where when one 
thing gets bigger, the other one gets bigger. For instance, buying 
chocolate bars at the store: twice as many chocolate bars will cost 
twice as much money.

In this case, it's (cost) / (number of bars) that is a constant, and 
when one thing increases the other one increases as well.

So, again, let's try a reworded problem:

   If 12 chocolate bars cost $18, what is the cost for 30 chocolate 
   bars?

Here I let P stand for the number of chocolate bars, and Q for the 
cost. Can you solve the reworded problem?

If not, you can still try the abstraction:

    initial P     final P
   ----------- = ---------
    initial Q     final Q

and you should get the same answer.

I hope that helps clear things up for you. The vocabulary of this 
topic can be confusing indeed. Feel free to write back if you have
any more questions about direct and inverse variation.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/