M46-57

Question

A courier traveled from Hillford to Lakeshore in three consecutive legs, with no stops between the legs. On the first leg, the courier’s average (arithmetic mean) speed was 40 kilometers per hour; on the second leg, the courier’s average (arithmetic mean) speed was 60 kilometers per hour; and on the third leg, the courier’s average (arithmetic mean) speed was 80 kilometers per hour. What was the courier’s average speed for the entire trip?

(1) The ratio of the times the courier spent on the first, second, and third legs, respectively, was \(3:2:1\).

(2) The ratio of the lengths of the first, second, and third legs, respectively, was \(3:3:2\).

Show Answer

Official Answer

D

(1) The ratio of the times the courier spent on the first, second, and third legs, respectively, was  3:2:1 .
 
(2) The ratio of the lengths of the first, second, and third legs, respectively, was  3:3:2 .

Bunuel · Answer

Official Solution:

A courier traveled from Hillford to Lakeshore in three consecutive legs, with no stops between the legs. On the first leg, the courier’s average (arithmetic mean) speed was 40 kilometers per hour; on the second leg, the courier’s average (arithmetic mean) speed was 60 kilometers per hour; and on the third leg, the courier’s average (arithmetic mean) speed was 80 kilometers per hour. What was the courier’s average speed for the entire trip? 
 
The average speed for the entire trip is:
 
 \frac{	ext{total distance}}{	ext{total time}} 
 
(1) The ratio of the times the courier spent on the first, second, and third legs, respectively, was  3:2:1 .
 
Let the times spent on the first, second, and third legs be  3t ,  2t , and  t , respectively.
 
Then the distances were:
 
• First leg:  40 * 3t = 120t 
 
• Second leg:  60 * 2t = 120t 
 
• Third leg:  80 * t = 80t 
 
So the total distance was:
 
 120t + 120t + 80t = 320t 
 
The total time was:
 
 3t + 2t + t = 6t 
 
Therefore, the average speed for the entire trip was:
 
 \frac{320t}{6t} = \frac{160}{3}  kilometers per hour
 
Sufficient.
 
(2) The ratio of the lengths of the first, second, and third legs was  3:3:2 .
 
Let the lengths of the first, second, and third legs be  3d ,  3d , and  2d , respectively.
 
Then the times were:
 
• First leg:  \frac{3d}{40} 
 
• Second leg:  \frac{3d}{60} 
 
• Third leg:  \frac{2d}{80} 
 
So the total distance was:
 
 3d + 3d + 2d = 8d 
 
The total time was:
 
 \frac{3d}{40} + \frac{3d}{60} + \frac{2d}{80} = \frac{3d}{20} 
 
Therefore, the average speed for the entire trip was:
 
 \frac{8d}{\frac{3d}{20}} = \frac{160}{3}  kilometers per hour
 
Sufficient.
 
Answer: D

M46-57

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