Problem Solving

Escalator is moving from first floor to ground floor in a shopping complex. Jack took 30 seconds to reach the ground floor if he takes 25 steps but if he triples his speed it takes him only 18 seconds .
a) Total steps in the stationary escalator.
b) Time taken if only the escalator was in motion

The OA are
a) 75
b) 45 seconds

smanstar wrote:Escalator is moving from first floor to ground floor in a shopping complex. Jack took 30 seconds to reach the ground floor if he takes 25 steps but if he triples his speed it takes him only 18 seconds .
a) Total steps in the stationary escalator.
b) Time taken if only the escalator was in motion

Let e = the escalator's rate.

Case 1: time = 30 seconds.
Since Jack takes 25 steps in 30 seconds, his rate = 25/30 = 5/6 steps per second.
Combined rate for Jack and the escalator = 5/6 + e.
Number of steps traveled = r*t = (5/6 + e)(30) = 25 + 30e.

Case 2: time = 18 seconds.
Since Jack triples his speed, his rate = 3(5/6) = 5/2 steps per second.
Combined rate for Jack and the escalator = 5/2 + e.
Number of steps traveled = r*t = (5/2 + e)(18) = 45 + 18e.

Since the number of steps between the first floor and the ground floor is the same in each case, we get:
25 + 30e = 45 + 18e
12e = 20
e = 20/12 = 5/3 steps per second.

Referring to case 1:
Number of steps traveled = (5/6 + 5/3)(30) = 25 + 50 = 75 steps.

Since the escalator's rate = 5/3 steps per second, the time for the escalator alone = 75/(5/3) = 45 seconds.

Thank You Mitch for this wonderful explanation.

GMATGuruNY wrote:

smanstar wrote:Escalator is moving from first floor to ground floor in a shopping complex. Jack took 30 seconds to reach the ground floor if he takes 25 steps but if he triples his speed it takes him only 18 seconds .
a) Total steps in the stationary escalator.
b) Time taken if only the escalator was in motion

Let e = the escalator's rate.

Case 1: time = 30 seconds.
Since Jack takes 25 steps in 30 seconds, his rate = 25/30 = 5/6 steps per second.
Combined rate for Jack and the escalator = 5/6 + e.
Number of steps traveled = r*t = (5/6 + e)(30) = 25 + 30e.

Case 2: time = 18 seconds.
Since Jack triples his speed, his rate = 3(5/6) = 5/2 steps per second.
Combined rate for Jack and the escalator = 5/2 + e.
Number of steps traveled = r*t = (5/2 + e)(18) = 45 + 18e.

Since the number of steps between the first floor and the ground floor is the same in each case, we get:
25 + 30e = 45 + 18e
12e = 20
e = 20/12 = 5/3 steps per second.

Referring to case 1:
Number of steps traveled = (5/6 + 5/3)(30) = 25 + 50 = 75 steps.

Since the escalator's rate = 5/3 steps per second, the time for the escalator alone = 75/(5/3) = 45 seconds.

Hello Mitch!

I have a query regarding the combined rates. If the escalator and Jack are moving in the same direction then why are we adding their respective rate? shouldn't they be subtracted?

sakshis wrote:

GMATGuruNY wrote:

smanstar wrote:Escalator is moving from first floor to ground floor in a shopping complex. Jack took 30 seconds to reach the ground floor if he takes 25 steps but if he triples his speed it takes him only 18 seconds .
a) Total steps in the stationary escalator.
b) Time taken if only the escalator was in motion

Let e = the escalator's rate.

Case 1: time = 30 seconds.
Since Jack takes 25 steps in 30 seconds, his rate = 25/30 = 5/6 steps per second.
Combined rate for Jack and the escalator = 5/6 + e.
Number of steps traveled = r*t = (5/6 + e)(30) = 25 + 30e.

Case 2: time = 18 seconds.
Since Jack triples his speed, his rate = 3(5/6) = 5/2 steps per second.
Combined rate for Jack and the escalator = 5/2 + e.
Number of steps traveled = r*t = (5/2 + e)(18) = 45 + 18e.

Since the number of steps between the first floor and the ground floor is the same in each case, we get:
25 + 30e = 45 + 18e
12e = 20
e = 20/12 = 5/3 steps per second.

Referring to case 1:
Number of steps traveled = (5/6 + 5/3)(30) = 25 + 50 = 75 steps.

Since the escalator's rate = 5/3 steps per second, the time for the escalator alone = 75/(5/3) = 45 seconds.

Hello Mitch!

I have a query regarding the combined rates. If the escalator and Jack are moving in the same direction then why are we adding their respective rate? shouldn't they be subtracted?

If the escalator and Jack were COMPETING, we would subtract their rates.
Here, the escalator and Jack are WORKING TOGETHER to travel the distance from the first floor to the ground floor.
When elements WORK TOGETHER, we ADD their rates.
In case 1:
Jack travels 5/6 of a step every second.
As Jack walks down the escalator, it moves him downward 5/3 steps every second.
Thus, the total number of steps traveled every second by Jack and the escalator = 5/6 + 5/3 = 5/2 -- a COMBINED rate of 5/2 steps per second.