Two boys enter a running escalator at the ground floor in a shopping mall and they do some fun on it. The first boy repeatedly foolows `p_(1)=1` step up and then `q_(1)=2` steps down whereas the second body repeatedly follws `p_(2)=2` steps up and then `q_(2)=1` step down. Both of them move rlative to escalator with speed `v_(r) =50 cm s^(-1)`. If the first boy takes `t_(1)=250 s` and the second boy takes the first boy takes `t_(1)=50 s ` to reach the first floor, how fast is escalator running ?.

Explanation

To solve the problem, we need to analyze the movements of both boys on the escalator and set up equations based on their movements relative to the escalator.

### Step 1: Define Variables
Let:
- \( v \) = speed of the escalator (in cm/s)
- \( v_r = 50 \) cm/s (relative speed of each boy with respect to the escalator)
- \( t_1 = 250 \) s (time taken by the first boy)
- \( t_2 = 50 \) s (time taken by the second boy)
...