Mikrislac wrote: ↑Mon Aug 10, 2020 11:07 pm
Raymond and Ronald are moving upwards on an escalator that is travelling up. The speeds at which Raymond and Ronald are moving are in the ratio 2 : 1. Raymond has to climb 40 steps to reach the top, while Ronald has to climb 30 steps to reach the top. The escalator had to be turned off due to an emergency. How many steps would either of Raymond or Ronald need to climb to reach the top?
(A) 10
(B) 30
(C) 40
(D) 60
(E) 120
Solution:
Let T and t be the time of Raymond and Ronald needs to travel up the escalator when it’s running. Let R and r be their respective rates. Let e = the rate of the escalator. Since the distance they travel up the escalator is the same, we can create the equation:
T(R + e) = t(r + e)
Since the speeds at which Raymond and Ronald are moving are in the ratio 2 : 1, Raymond’s rate is twice Ronald’s. That is, R = 2r. Replacing this into the equation, we have:
T(2r + e) = t(r + e)
T/t = (r + e) / (2r + e)
Furthermore, since Raymond takes 40 steps to go through the escalator versus Ronald’s 30 steps, we have R = 40/T and r = 30/t. Since R = 2r, we have R = 2(30/t) = 60/t. However, since R = 40/t, we have:
60/t = 40/T
T/t = 40/60
T/t = 2/3
However, since T/t = (r + e)/(2r + e), we have:
(r + e)/(2r + e) = 2/3
3(r + e) = 2(2r + e)
3r + 3e = 4r + 2e
e = r
From this, we can see that e = r. Since Ronald takes 30 steps to travel up the escalator and his rate is the same as the escalator’s rate, the escalator must help him with another 30 steps. That is, the escalator has 30 + 30 = 60 steps when it’s not moving.
(Note: This is from Raymond’s perspective: We see that R = 2r = 2e. Since Raymond takes 40 steps to travel up the escalator and his rate is twice the escalator’s rate, the escalator must help him with half as many steps, i.e., 20 steps. That is, the escalator has 40 + 20 = 60 steps when it’s not moving.)
Answer: D
