Direct and Inverse Variation in One Problem

Date: 03/23/2006 at 15:44:58
From: Nicole
Subject: Direct and Inverse Variation in one problem

Dr. Math,

I understand the difference between direct and inverse variation, but 
I'm having a hard time figuring out this problem that combines the 
two together: 

If s varies directly as r and inversely as t, and s = 10 when r = 5
and t = 3, for what value of t will s = 3 when r = 4?

The answer in the back of the book is 8, but I can't figure out how to
get that number!  I'm not sure if you do it out as 2 separate problems
and then combine them or if it should be one problem together.

For Inverse: 
  k = sr
  k = (10)(5)
  k = 50

For Direct:
  k = s/t
  k = 10/3

Then to combine them:
  3 = (10/3)t

Which is completely wrong!

Thanks!

Date: 03/23/2006 at 16:15:42
From: Doctor Douglas
Subject: Re: Direct and Inverse Variation in one problem

Hi Nicole,

Note that the problem stated that S varies directly as R and inversely
as T (not the other way round as you seem to have worked out above). 
We can write two equations:

   s = b*r    b is a constant that is independent of r, but not
              necessarily independent of t.

   s = c/t    c is a constant that is independent of t, but not
              necessarily independent of r.

Combining these two equations, we have

   s = k*r/t  we can now see that everything works out (because s
              varies directly with r (with b = k/t), and inversely
              with t (with c = k*r).

Now you can finish from here, right?

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