SAT Algebra Question

Date: 06/20/97 at 13:05:29
From: Kristen
Subject: SAT Algebra Question

A woman drove to work at an average speed of 40 miles per hour and 
returned along the same route at 30 miles per hour. If her total 
traveling time was 1 hour, what was the total number of miles in the 
round trip?

I know how to find the answer to this problem by using logic, but 
I don't know what the formula is. The answer is 34 2/7. I added 30 and 
40 together to come up with 70, and then I divided 30 by 70 times 40: 
30/(70x40)
     
The answer I got when I did this was 17 1/7. Then I multiplied this by 
two to come up with the correct answer that was written in the back of 
my study book.
     
Could you please send me the correct equation for finding the answer 
to this problem? 

Thank you for your time.

Date: 06/20/97 at 13:19:14
From: Doctor Daniel
Subject: Re: SAT Algebra Question

Hi Kristen,

Your reasoning was pretty good, but you want a formula or perhaps some 
other way of knowing you were on the right track.

Here's my idea for how to address this problem:

Suppose that her commuting distance is x miles. Then we know that it 
takes her x/40 hrs to get to work (distance divided by rate equals 
time). For example, if she's driving 10 miles, it'll take her 10/40 of 
an hour. Similarly, it takes her x/30 hrs to get back.

We know from the problem that the total of these two times equals one 
hour. So we have:

x/40 + x/30 = 1

We are told that we want to compute 2x.

When we compute the common denominator and solve, we get that: 

3x/120 + 4x/120 = 1
             7x = 120
              x = 120/7 
              x = 17.143...

So the right answer is 2X, or 34 2/7, as you got.

Here's another "guess" sort of method for this problem, which takes
advantage of the fact that the SAT is multiple choice.  We know that 
the right answer is between 30 and 40, since it's however far she can 
go in an hour. It's probably pretty close to the average of these two 
numbers, but a little below, since it'll take her longer to get home 
than to get to work. So it's probably somewhere around 33 or 34 miles 
an hour.  

Also, you can plug in the numbers you're given and see if they'll 
work; if one of the choices were 35 miles, you could just compute 
17.5/30 + 17.5/40 and realize that it's more than 1 and thus not the 
right answer.

I hope this helps.  Good luck on the test!

-Doctor Daniel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/