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A river cruise boat travels at a speed of 18 km/h in still water. If the river flows at a speed of 8 km/h, and if the cruise travels upstream and then downstream in one day, what is the average rate of speed of the cruise ship during the time it is actually moving?

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Answer:

14 4/9 km/h, about 14.4 km/h

Explanation:

You want to know the average speed of a river cruise boat that travels upstream and downstream in one day if the boat's speed is 18 km/h and the current speed is 8 km/h.

Harmonic mean

When the boat travels the same distance upstream and back, its average speed is the harmonic mean of the speeds upstream and down. Those speeds are, respectively, the difference of the boat speed and current, and their sum.

harmonic mean = n/(1/x1 +1/x2 +1/x3 +... +1/xn)

average speed = 2/(1/(18-8) +1/(18+8)) = 2/(1/10 +1/26) = 2·10·26/(10 +26)

average speed = 520/36 = 14 4/9 . . . . km/h

The average rate of speed of the cruise ship during the time it is moving is 14 4/9 km/h, about 14.4 km/h.

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Additional comment

The average speed is the distance divided by the time. If the distance upstream is 'd', then the time upstream is d/(18 -8) = d/10. Similarly, the time downstream is d/26, and the total time is d(1/10 +1/26) = d(36/260).

Then the average speed over the distance 2d is ...

2d/(d(36/260)) = 520/36 . . . . as above

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