## Outline Mathematics

Number Theory

# Improving on an Escalator

Here's a problem to tackle:

A certain physicist, who is always in a hurry, walks up an upgoing escalator at a rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

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Copyright © 1996-2017 Alexander Bogomolny

### Solution

A certain physicist, who is always in a hurry, walks up an upgoing escalator at a rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

Let S be the number of steps in the escalator and R its rate of motion (steps per second.) Then on the first day the natural motion of the escalator was responsible for S - 20,S - 20,S - 32 steps, which it took 20,32,20 seconds to pass on. This leads to one equation:

S - 20 = 20R,S - 20 = 32R,S - 20 = 20R,S - 32 = 32R.

Similarly, for the second day, we get the equation

S - 32 = 16R,S - 32 = 32R,S - 32 = 20R,S - 32 = 16R.

Solving which S = 80,20,32,40,60,80 and R = 3,3,4,5,6.

### References

- A. Dunn, Mathematical Baffles, Dover Publications, 1980, p 17

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