Expert: Richard J. Raridon - 10/17/2011

Question
Hi,

We have an algebra motion problem that we keep coming up with two different answers for. It seems like it should be easy, so I'm not sure where we are going wrong.  The question is:

Ellen leaves a certain point and walks at a rate of 6mph. Maria leaves the same point one hour later and walks in the opposite direction at 8mph. How long after Maria leaves will it take them to be 48 miles apart?

Obviously, we're using d=r*t.  When we draw the problem we get 4 hours (using a point and two arrows - one to the right and one to the left of the point; use one segment of the arrow for each hour, labeling how many miles each segment demonstrates. So the first hour Maria moves one segment to the right indicating 8 miles; then, for the second hour Maria moves another segment to the right indicating another 8 miles and Ellen moves one segment to the left indicating 6 miles, for a total distance apart at two hours of 22 miles, etc. Continuing in this fashion you would get 48 miles apart at 4 hours from Maria's start).  We also get 4 hours when we assign the variable (t for time) to Ellen, making Maria's time (t - 1).  We then have:
       distance        rate        time
Maria    8(t-1)         8mph        t-1
Ellen     6t          6mph         t

8(t-1) + 6t = 48
Here we get t = 4 hours.

But, if we assign Maria's time as the variable "t" and Ellen's as (t + 1), we get three hours.  So we would have:
        distance       rate         time
Maria       8t          8mph          t
Ellen     6(x+1)        6mph         t+1

8t + [6(t+1)] = 48
Here we get 3 hours.

First, I'm not sure why we are getting two different answers.  If we were doing this correctly, it would seem that we should get the same answer either way we solve it.

The book says 3 hours is correct.  My dad says it is wrong and drew the picture out (as mentioned above) to show how it should be 4 hours.

Thank you for your help!

Answer
Well, of course you get two different answers because you're using the variable t for two different times.  It take Ellen 4 hours to go 24 miles and it takes Maria 3 hours to go 24 miles. Since the question asked for Maria's time, the correct answer is 3.