Bunuel wrote:
A train traveled the first d miles of its journey it an average speed of 60 miles per hour, the next d miles of its journey at an average speed of y miles per hour, and the final d miles of its journey at an average speed of 160 miles per hour. If the train’s average speed over the total distance was 96 miles per hour, what is the value of y?
(A) 68
(B) 84
(C) 90
(D) 120
(E) 135
We are given that a train traveled the first d miles of its journey at an average speed of 60 miles per hour, the next d miles of its journey at an average speed of y miles per hour, and the final d miles of its journey at an average speed of 160 miles per hour. Since time = distance/rate:
The time of the first d miles = d/60
The time of the next d miles = d/y
The time of the final d miles = d/160
We are also given that the overall average rate was 96 mph. We can use the average rate formula to determine y.
average rate = (total distance)/(total time)
96 = (d + d + d)/(d/60 + d/y + d/160)
96 = 3d/(d/60 + d/y + d/160)
We can divide the numerator and denominator by d:
96 = 3/(1/60 + 1/y + 1/160)
Now we get the common denominator of 480y:
96 = 3/(8y/480y + 480/480y + 3y/480y)
96 = 3/[(11y + 480)/480y]
96 = 3(480y)/(11y + 480)
96(11y + 480) = 1440y
11y + 480 = 15y
480 = 4y
y = 120
Answer: D
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