Question

An escalator moves up at a constant rate. John walks up the escalator at the rate of one step per second and reaches the top in twenty seconds. The next day John’s rate was two steps per second, and he reached the top in sixteen seconds. The number of steps in the escalator is

Answer 1

Hint: Using the given conditions we can make two equations with respect to the time. By solving the equation we will get the number steps in the escalator.

Complete step-by-step answer:
It is given that an escalator moves up at a constant rate. John walks up the escalator at the rate of one step per second and reaches the top in twenty seconds. The next day John’s rate was two steps per second, and he reached the top in sixteen seconds.
We have to find the number of steps.
Let us consider the number of steps of the escalator \[n\].
The rate of the escalator is \[k\] steps per seconds.
On first day,
John’s rate is \[1\] steps per seconds.
In twenty seconds, he reached bottom to top. So, the escalator has gone up \[20k\] steps.
Also, John has gone up \[20\] steps.
So, we have,
\[20k + 20 = n\]… (1)
On second day,
In \[16\] seconds it takes John from bottom to top at \[2k\] steps per seconds, the escalator has gone up \[16n\] steps. Also, John has gone up \[2(16)\] steps.
So, we have
\[16k + 32 = n\]… (2)
From (1) and (2) we can equate the equation since the right hand sides are equal,
\[20k + 20 = 16k + 32\]
By simplifying the equation we get,
\[4k = 12\]
Which in turn implies that, \[k = 3\]
Let us now substitute k in (1) we get,
\[n = 20 \times 3 + 20 = 80\]
Hence, the number of steps escalator is \[80\].

Note: The answer can also be found by substituting the value of k in (2), even though we substitute in the other equation we get the same answer.
That is \(n = 16 \times 3 + 32 = 48 + 32 = 80\) hence the total number of steps is 80.