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The Bird and the Train

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The distance between Station Atena and Station Barcena is 78 miles. A train starts from Atena towards Barcena. A bird starts at the same time from Barcena straight towards the moving train. On reaching the train, it instantaneously turns back and returns to Barcena. The bird makes these journeys from Barcena to the train and back to Barcena continuously till the train reaches Barcena. The bird finally returns to Barcena and rests. Calculate the total distance in miles the bird travels in the following two cases:

1. the bird flies at 70 miles per hour and the speed of the train is 60 miles per hour

Answer: 91

2. the bird flies at 60 miles per hour and the speed of the train is 70 miles per hour

Answer: 72

Solution:

Case 1: Bird flies at a speed greater than that of the train

The train (at a speed of 60 miles per hour) travels 60 miles in 60 minutes.
Therefore, the train travels from Atena to Barcena (78 miles) in 78 minutes.

Importantly, the bird makes the journeys continuously back and forth for this same amount of time (namely, 78 minutes).
Thus, the total distance traveled by the bird
= 70 miles per hour × 78 minutes = 70 × 78 / 60 miles = 91 miles.

Case 2: Bird flies at a speed less than that of the train

In 36 minutes, the bird travels 36 miles, the train travels 42 miles, and the two meet.
Now, the train (which is traveling at a speed greater than that of the bird) will reach Barcena before the bird.
So, the bird simply returns to Barcena (a return journey of 36 miles).
Thus, the total distance traveled by the bird is 72 miles.

Food for thought:

How many journeys back and forth does the bird make in Case 1?
Would the distances in these back and forth journeys form an infinite series with a finite sum?

Wish to try a simple puzzle similar in concept to Case 2? Then click here.


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