Anju and Bobby simultaneously started climbing up an ascending escalator (a moving staircase). Since they were in a hurry, they also started climbing up the steps (taking one step at a time). Anju took k steps in the time that Bobby took 1 step, where k is an integer greater than 1. How many values can k assume, if it is known that the speed of the escalator was the same as that of Bobby and Anju took a total of 72 steps to climb up the escalator? (Assume that the number of steps taken by Bobby was also an integer)

This problem can be solved by understanding the relative speeds of Anju, Bobby, and the escalator.

Step 1: Define the variables Let's denote the number of steps on the escalator as 'n'. The speed of Bobby is 1 step at a time, and the speed of the escalator is also 1 step at a time (as given in the problem). The speed of Anju is 'k' steps at a time.

Step 2: Set up the equations Since Anju and Bobby started at the same time and ended at the same time, the time taken by both of them is the same. Therefore, we can write the equation for the time taken by Anju and Bobby as: Time taken by Anju = Time taken by Bobby n/k = n/1 This simplifies to n = k*n, which means that 'k' must be 1. But the problem states that 'k' is an integer greater than 1, so there seems to be a contradiction.

Step 3: Consider the escalator's movement However, we have not considered the movement of the escalator. The escalator is also moving at the same speed as Bobby, so for every step that Bobby takes, the escalator also takes one step. Therefore, the actual number of steps that Anju takes is not 'n', but 'n + the number of steps taken by the escalator'. Since the escalator takes one step for every step taken by Bobby, the number of steps taken by the escalator is the same as the number of steps taken by Bobby, which is 'n'. Therefore, the actual number of steps taken by Anju is 'n + n' = 2n.

Step 4: Set up the new equation Now we can set up the new equation for the time taken by Anju and Bobby: 2n/k = n/1 This simplifies to 2n = k*n, which means that 'k' must be 2.

Step 5: Check the possible values of 'k' The problem asks for the number of values that 'k' can assume. Since 'k' must be an integer greater than 1, and we have found that 'k' must be 2, the only possible value for 'k' is 2. Therefore, 'k' can assume 1 value.