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Up an Escalator

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1. Michael takes the underground train to work and uses an escalator at the railway station. If Michael runs up 6 steps of the escalator, then it takes him 45.0 seconds to reach the top of the escalator. If he runs up 14 steps of the escalator, then it takes him only 25.0 seconds to reach the top.

How many seconds would it take Michael to reach the top if he did not run up any steps of the escalator at all?
Answer: 60.0

Solution:

If he runs up 6 steps, then he needs 45.0 seconds to reach the top.
If he runs up 14 steps, then he needs 25.0 seconds to reach the top.

The 8 additional steps take 20.0 seconds.
Therefore, each step takes 2.5 seconds.

Total steps in escalator = 6 + 45.0 / 2.5 = 24 or Total steps in escalator = 14 + 25.0 / 2.5 = 24.

If Michael did not run up any steps at all,
he would reach the top of the escalator in 60.0 seconds (i.e., 24 steps × 2.5 seconds/step).

Alternative Solution through Equations:

Let the total number of steps in the escalator be x.
The escalator moves at a constant speed given by

Speed of escalator = (x − 6)/45.0 = (x − 14)/25.0

The above equation may be solved as follows.

25.0 (x − 6) = 45.0 (x − 14); or
x = (14 × 45.0 − 6 × 25.0) / (45.0 − 25.0) = 24.

Now, Speed of escalator = (24 − 6)/45.0 = (24 − 14)/25.0 = 1/2.5 steps/second.
Time to reach top = Total Steps / Speed = 60.0 seconds.

Food for thought:

Does it matter whether the escalator is moving up or down?

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puzzle : image for escalator

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