While coming down on an escalator of a station, when a man walks down twenty-six steps of the staircase, he requires thirty seconds to reach the bottom. However, if he steps down thirty-four steps, it takes eighteen seconds to reach the bottom. How many steps are there in the escalator ?
Let V_m be the speed of the man relative to the escalator (steps/second).
Let V_e be the speed of the escalator (steps/second).
The total distance covered by the man relative to the ground is E steps. This distance is the sum of the steps the man walks relative to the escalator and the steps the escalator moves relative to the ground during the time taken.
E = (Man’s speed relative to ground) * Time
Man’s speed relative to ground = V_m + V_e.
E = (V_m + V_e) * T.
Alternatively, consider the total steps E. In time T, the man takes N steps relative to the escalator (so V_m = N/T). The escalator moves V_e * T steps.
E = N + V_e * T.
Case 1: Man takes N1 = 26 steps, T1 = 30 seconds.
E = 26 + V_e * 30 (Equation 1)
Case 2: Man takes N2 = 34 steps, T2 = 18 seconds.
E = 34 + V_e * 18 (Equation 2)
We have a system of two linear equations with two variables (E and V_e).
Equating the expressions for E:
26 + 30 * V_e = 34 + 18 * V_e
30 * V_e – 18 * V_e = 34 – 26
12 * V_e = 8
V_e = 8 / 12 = 2/3 steps/second.
Substitute the value of V_e into either equation. Using Equation 1:
E = 26 + (2/3) * 30
E = 26 + 2 * 10
E = 26 + 20
E = 46.
Using Equation 2:
E = 34 + (2/3) * 18
E = 34 + 2 * 6
E = 34 + 12
E = 46.
Both equations give the same result.