Solution to the Escalator Problem

The problem was: Shyama and Vyom walk up an escalator (moving stairway). The escalator moves at a constant speed. Shyama takes three steps for every two of Vyom’s steps. Shyama gets to the top of the escalator after having taken 25 steps, while Vyom (because his slower pace lets the escalator do a little more of the work) takes only 20 steps to reach the top. If the escalator were turned off, how many steps would they have to take to walk up?
1) 40 2) 50 3) 60 4) 80

Since the ratio of speeds of Shyam and Vyom is 3 : 2 and the distance that each travels is 25 : 20, the ratio of the time in which they reach the top is (25/3) : (20/2) i.e. 5 : 6.

Thus, if the escalator has a total of n steps, the escalator covers the balance (n – 25) and (n – 20) steps in time intervals in the ratio 5 : 6.

Since speed of escalator is same in both the cases, the ratio of distances covered is same as ratio of time i.e. (n – 25) : (n – 20) is in the ratio 5 : 6.

The difference of 5 parts and 6 parts i.e. 1 part corresponds to an actual difference between (n – 25) and (n – 20) i.e. 5. Thus (n – 25), corresponding to 5 parts, is 25 (or n – 20, corresponding to 6 parts is 30), giving n = 50.

Excellent problem to learn putting ratios to their optimum use.