NDiwan wrote:If a car traveled from Townsend to Smallville at an average speed of 40 mph and then returned to Townsend along the same route 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 route between Townsend and Smallville is 165 miles long.
From Townsend to Smallville, the average speed = 40 mph.
To determine the average speed for the entire trip, we need to know the average speed on the return trip from Smallville to Townsend.
Let r = the average rate on the return trip to Townsend.
Question stem, rephrased:
What is the value of r?
Statement 1: The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend.
TIME and RATE are RECIPROCALS.
Since the TIME to Smallville was 3/2 of the TIME to Townsend, the RATE to Smallville -- 40mph -- was 2/3 of the RATE to Townsend.
Thus:
40 = (2/3)r
r = 60.
SUFFICIENT.
Statement 2: The route between Townsend and Smallville is 165 miles long.
No information about r.
INSUFFICIENT.
The correct answer is
A.
An alternate way to evaluate Statement 1 is to TEST TWO CASES.
If the average speed for the entire trip is THE SAME In each case, the statement is SUFFICIENT.
If the average speed is NOT the same in each case, the statement is INSUFFICIENT.
Statement 1: The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend.
Case 1: Distance in each direction = 120 miles.
At a rate of 40mph, the time to Smallville = d/t = 120/40 = 3 hours.
Since the time to Smallville is 50% longer than the time to Townsend, the time to Townsend = 2 hours.
Since the time for the entire 240-mile trip = 3+2 = 5 hours, the average rate for the entire trip = d/t = 240/5 = 48mph.
Case 2: Distance in each direction = 240 miles.
At a rate of 40mph, the time to Smallville = d/t = 240/40 = 6 hours.
Since the time to Smallville is 50% longer than the time to Townsend, the time to Townsend = 4 hours.
Since the time for the entire 480-mile trip = 6+4 = 10 hours, the average rate for the entire trip = d/t = 480/10 = 48mph.
Since the average speed in each case is the same, SUFFICIENT.
Algebra:
Let the distance in each direction = d miles.
At a rate of 40mph, the time to Smallville = d/40.
Since the time to Smallville is 3/2 the time to Townsend, the time to Townsend is equal to 2/3 the time to Smallville:
(2/3)(d/40) = (2d)/120.
Total time for the entire trip = d/40 + (2d)/120 = (3d)/120 + (2d)/120 = (5d)/120 = d/24.
Average rate for the entire trip of 2d miles = (total distance)/(total time) = (2d)/(d/24) = 48.
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