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### 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/
```
Associated Topics:
High School Basic Algebra
Middle School Algebra

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