If a car traveled from Townsend to Smallville

[bluescale]

Hello,

I understand how to solve this DS problem, but when (for the sake of curioisity) I solve the problem mathematically, I get a different answer than what is given in the answer explanation. Please help straighten me out:

If a car traveled from Townsend to Smallville at an average speed of 40 mph and then returned to Townsend later that evening, what was the average speed for the entire trip?

(1) The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend.
(2) The distance from Townsend to Smallville is 165 miles.

The answer on the GMAT would be A, and I'm fine with that. However, here's my problem.

Based on statement 1, the trip from Townsend to Smallville takes 50% LONGER than the return trip. The CAT explanation calculates the total average speed to be 32 mph. How is this possible? Mustn't the total average speed be greater than 40 mph since the average speed on the return trip was faster?

By my calculation, the average speed for the entire trip is 50 mph.

Thanks!

[atanup]

I think you are right unless somebody has a different explanantion.

The actual answer is 53.33. The average speed would be total distance upom total time and used both the options and even then see the same 53.33.

[editor: this number is incorrect. you are right that you use total distance divided by total time, but doing so will give an average of 48 miles per hour, not 53.33. see my post below. --ron]

[keanuxie]

Assume the distance between Townsend and Smallville is X,
the time consumed from Townsend to Smallville is X/40,
and the time consumed from Smallville to Townsend is (X/40)*3/2,
so clearly the total time spent equals 2X/[X/40+(X/40)*3/2]=32.

Hope it explains clearly to you.

[mccollud]

I think the original poster is right. The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend. That means that the car would have been traveling faster on the from Smallville to Townsend. I agree that A is the correct answer, but the math showed in the explanation is incorrect. The average mph couldn't possibly be below 40, since the car traveled faster than 40 to get from Smallville to Townsend.

I agree with the math done by the second poster

[jayram.m.v]

Can you please post the original explanation that was given?

32 is the average time of the trip, not the average speed, as per the explanation offered by Keanuxie below. I just wanted to check how the original explanation is given.

The average speed is certainly greater than 40 mph.

Lets assume the distance is 40 miles and traveling from T to S it takes 1 hr.
So as per the given data in statement 1 the return journey for the same 40 miles would have taken 40 mins.
80/5/3 = 48 mph.

[rohit.shah7]

The answer is not 32 mph. It is tempting to do that.
the mistake you made in arriving there is that the

time from town T to smallsville = 1.5 ( distance / 40)
and not the other way around.
the actual answer using only 1) is 48 mph.
not 32. here s how in detail

let distance = x
time from town T to smallsville = (40 /x)
now time from town T to smallsville = 3/2 * (return trip time)
therefore return trip time = [2/3 * (40/x)] [not 3/2*40/x) that s your mistake] [ by cross multiplication]

so continuing on you will get 48 mph.

[RonPurewal]

the previous posters (the ones who calculated a speed of 48 mph) pretty much have it.

but a comment is definitely due on the following:

By my calculation, the average speed for the entire trip is 50 mph.

oh no. absolutely not. it is important to note that this will always, ALWAYS, give you the wrong answer.

the average speed for a round trip is NEVER the average of the two speeds going each way.

the average speed for the whole round trip will always be LESS than the average of the two speeds, since you spend more time traveling at the slower speed.

this is actually impossible, unless the two speeds are the same.

there's no shortcut to taking the average speed; you have to work out the total distance and the total time, and then calculate the average speed in longhand fashion.

for instance, let's say that the distance of the trip is 120 miles. (this is a good number to pick, because both 40 and 60 go into it evenly.)
then:
the first leg of the trip takes 3 hours.
the second leg of the trip takes 2 hours.
so we have a total distance of 240 miles, covered in a total time of 5 hours. that's an average speed of (240 miles)/(5 hours), or 48 miles per hour.

--

note:
if you travel at two different speeds for the same TIME each, then your average speed will be the average of the two speeds.
...but that's not what happens on a round trip. on a round trip, you travel the same distance (not the same time) at each speed.