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
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Raymond and Ronald are moving upwards on an escalator that is travelling up. The speeds at which Raymond and Ronald
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Solution:Mikrislac wrote: ↑Mon Aug 10, 2020 11:07 pmRaymond 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
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
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It is interesting to note that even though Ronald is travelling at a slower speed than Raymond, he takes fewer number of steps to reach the top. How could this happen?
We can realize that since Ronald travels slower, much of his walking effort is taken up by the escalator. Consider an extreme case. Assume Ronald only takes two steps to reach the top. He will definitely reach the top as almost of all his effort in reaching the top will be done by the escalator.
Two methods to solve:
METHOD 1:
Let us assume that the escalator has 'x' total steps. Now, Raymond reaches the top after having taken 40 steps. This means that in the same time, the escalator would have moved through (x-40) steps.
Since Ronald travels at half the speed of Raymond, we can say that in the same time in which Raymond travels 40 steps, Ronald would have travelled only 20 steps.
Therefore, when Ronald has taken 20 steps, the escalator would have moved through [(x-40)/20]*30 = 1.5*(x-40) steps.
But, by now, Ronald has reached the top. So, the total number of steps taken by Ronald and the escalator together is 30 + 1.5*(x-40) = x. Solving, we get x = 60.
Therefore, the number of steps in the escalator is 60. This is also the number of steps that would be taken by Ronald and the escalator together to reach the top.
METHOD 2:
Let us assume that Raymond reaches the top in one minute. Then, the speed of Raymond is 40 steps/min and the speed of Ronald is 20 steps/min.
Let the speed of the escalator be 'u'.
Then, the downstream speed of Raymond = (40 + u) steps/min.
From this statement, we can also say that the total number of steps in the escalator is (40 + u).
The downstream speed of Ronald is (20 + u) steps/min. Ronald walks 20 steps in one minute. Therefore, he would walk 30 steps in 1.5 minutes.
In 1.5 minutes, Ronald, together with the escalator, would have moved through 1.5*(20 + u) steps. This is also the total number of steps in the escalator.
We have,
1.5*(20 + u) = (40 + u)
or, u = 20 steps/min.
Therefore, the total number of steps in the escalator = 40 + 20 = 60 steps.
[spoiler]Answer: D[/spoiler]
We can realize that since Ronald travels slower, much of his walking effort is taken up by the escalator. Consider an extreme case. Assume Ronald only takes two steps to reach the top. He will definitely reach the top as almost of all his effort in reaching the top will be done by the escalator.
Two methods to solve:
METHOD 1:
Let us assume that the escalator has 'x' total steps. Now, Raymond reaches the top after having taken 40 steps. This means that in the same time, the escalator would have moved through (x-40) steps.
Since Ronald travels at half the speed of Raymond, we can say that in the same time in which Raymond travels 40 steps, Ronald would have travelled only 20 steps.
Therefore, when Ronald has taken 20 steps, the escalator would have moved through [(x-40)/20]*30 = 1.5*(x-40) steps.
But, by now, Ronald has reached the top. So, the total number of steps taken by Ronald and the escalator together is 30 + 1.5*(x-40) = x. Solving, we get x = 60.
Therefore, the number of steps in the escalator is 60. This is also the number of steps that would be taken by Ronald and the escalator together to reach the top.
METHOD 2:
Let us assume that Raymond reaches the top in one minute. Then, the speed of Raymond is 40 steps/min and the speed of Ronald is 20 steps/min.
Let the speed of the escalator be 'u'.
Then, the downstream speed of Raymond = (40 + u) steps/min.
From this statement, we can also say that the total number of steps in the escalator is (40 + u).
The downstream speed of Ronald is (20 + u) steps/min. Ronald walks 20 steps in one minute. Therefore, he would walk 30 steps in 1.5 minutes.
In 1.5 minutes, Ronald, together with the escalator, would have moved through 1.5*(20 + u) steps. This is also the total number of steps in the escalator.
We have,
1.5*(20 + u) = (40 + u)
or, u = 20 steps/min.
Therefore, the total number of steps in the escalator = 40 + 20 = 60 steps.
[spoiler]Answer: D[/spoiler]
Last edited by Mikrislac on Tue Aug 18, 2020 10:49 pm, edited 1 time in total.
