Problem Solving

A car traveled 75% of the way from town A to town B by traveling at T hours at an average speed of V mph. The car travels at an average speed of S mph for the remaining part of the trip. Which of the following expressions represents the average speed for the entire trip?

$$A.\ 0.75V+0.25S$$
$$B.\ 0.75T+0.25S$$
$$C.\ \frac{VT}{3S}$$
$$D.\ \frac{4VT}{\frac{\left(T+S\right)}{3}}$$
$$E.\ \frac{4VS}{3S+V}$$

The OA is E.

Can I say that total distance is 100 miles, then

75% of the total distance will be, 75 miles at an average speed of V mph, that's mean that the time for this 75% will be 75/V.

25% of the total distance will be, 25 miles at an average speed of S mph, that's mean that the time for this 25% will be 25/S.

Hence, the total average speed will be,
$$A_s=\frac{100}{\frac{75}{V}+\frac{25}{S}}=\frac{100}{\frac{75S+25V}{VS}}=\frac{100VS}{25\left(3S+V\right)}=\frac{4VS}{3S+V}$$

Experts, any suggestion about this PS question? Thanks in advance.

LUANDATO wrote:A car traveled 75% of the way from town A to town B by traveling at T hours at an average speed of V mph. The car travels at an average speed of S mph for the remaining part of the trip. Which of the following expressions represents the average speed for the entire trip?

$$A.\ 0.75V+0.25S$$
$$B.\ 0.75T+0.25S$$
$$C.\ \frac{VT}{3S}$$
$$D.\ \frac{4VT}{\frac{\left(T+S\right)}{3}}$$
$$E.\ \frac{4VS}{3S+V}$$

Since 75% = 3/4, let the total distance = 4 miles.

A car traveled 75% of the way from town A to town B by traveling at T hours at an average speed of V mph.
3/4 of 4 miles = 3 miles.
Let V = 3 mph and T = 1 hour, with the result that rt = 3*1 = 3 miles.

The car travels at an average speed of S mph for the remaining part of the trip.
Remaining distance = 4-3 = 1 mile.
Let S = 1 mph, with the result that the car travels for 1 more hour, bringing the total time for the 4-mile trip to 2 hours.

Average speed:
(total distance)/(total time) = (4 miles)/(2 hours) = 2 mph. This is our target

Now plug V=3, T=1 and S=1 into the answers to see which yields the target value of 2 mph.
Only E works:
(4VS)/(3S + V) = (4*3*1)/(3*1 + 3) = 12/6 = 2.

The correct answer is E.