Hi M7MBA,
Though I am not a "GMAT Expert," I can help you with this problem.
First, let's write down what we know:
X% of the distance was traveled at 60mph
The remaining distance was traveled at 50 mph.
We also know that the total trip must add up to 100%, so we can say
X% @ 60 mph
(100-X)% @ 50mph
Since the question asks for average speed, we need to use the average speed equation:
$$Average\ Speed\ =\ \frac{Total\ Dis\tan ce}{Total\ Time}$$
In this case, we do not have a total distance provided, but we do know that it does need to add up to 100%. So let's assume the distance is 100 miles (if you don't like assuming distances, you could alternatively express the total distance as 100% *D and the individual distances as X%*D and (100-X)%*D, but assuming 100 miles is better and more efficient):
So Total distance = 100 miles
Distance travelled at 60 mph = X miles
Distance travelled at 50 mph = 100-X miles
We are missing the time to calculate average speed, so we can go ahead and find that using the standard rate/speed equation:
$$Speed\ =\ \frac{dis\tan ce}{time}$$
Therefore, total time can be calculated using:
$$Total\ time\ =\frac{Dist_1}{Speed_1}+\frac{Dist_2}{Speed_2}\ =\ \frac{X\ miles}{60\ mph}+\frac{100-X\ miles}{50\ mph}$$
To simplify the fraction, use 300 as a common denominator and fire away:
$$Total\ time\ =\ \ \frac{5\left(X\ \right)}{300}+\frac{6\left(100-X\ \right)}{300}\ =\ \frac{5X\ -\ 6X\ +\ 600}{300}\ =\ \frac{600-X}{300}$$
Plug this into the average speed equation:
$$Average\ Speed=\ \frac{100}{\left(\frac{600-X}{300}\right)}\ =\ \frac{300\times100}{600-X}\ =\ \frac{30,\ 000}{600-X}$$
Which gives you answer choice E. In reality, you could have stopped once you came up with the common denominator and arrived at the same answer; however, because the question asks for the numerator in its reduced form, it is important to simplify the final equation as a whole in case the equation can further be reduced once you combine like terms, etc.
