
Distance problem
On a drive from your ranch to Austin, you
wish to average 48 mph. The distance from
your ranch to Austin is 78 miles. However,
at 39 miles (half way), you find you have
averaged only 36 mph.
What average speed must you maintain in
the remaining distance in order to have an
overall average speed of 48 mph? Answer in
units of mph.
I thought the answer would be 60MPH but apparently that's wrong. Do I have to factor time into this equation?

The answer is 72 MPH.
For an overall average speed of 48MPH, the trip would take her 1.625 hours. After driving 39 miles, however, her trip has already taken her 1.083333 hours. Therefore, she needs to travel the remaining 39 miles in .54166667 hours. :)

Ok lets check for the expected time how long it will take us to arrive.
We find that by $\displaystyle t=\frac{d}{r}$ plug in values and get $\displaystyle t = \frac{78}{48}$ which we can reduce to $\displaystyle t = \frac{13}{8}$
Now to solve for how much time has elapsed $\displaystyle t = \frac{39}{36}$ which can be reduced to $\displaystyle t = \frac{13}{12} $
Now we want to do $\displaystyle t_{1}  t_{2} = \frac{\frac{1}{2}d}{r} $
So we get $\displaystyle \frac{39}{24}  \frac{26}{24} = \frac{39}{r}$
$\displaystyle \frac{13}{24} = \frac{39}{r}$
Now solve for r and we find the $\displaystyle r = 72mph $

Your answer is much less messy and doesn't have those infinite decimals, I like it :)