SOLUTION: Leon, who is always in a hurry, walked up an escalator, while it was moving, at the rate of one step per second and reached the top in 28 steps. The next day he climbed two steps p

Question 167572: Leon, who is always in a hurry, walked up an escalator, while it was moving, at the rate of one step per second and reached the top in 28 steps. The next day he climbed two steps per second (skipping none), also while it was moving, and reached the top in 40 steps.
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Found 2 solutions by Mathtut, gonzo:
Answer by Mathtut(3670) (Show Source): You can put this solution on YOUR website!
lets call S the number of steps on the escalator:the escalator is moving
with Leon, so we must add the number of steps taken by Leon to the # of steps when it is still. d=rt so lets call d in this instance S+28 and S+40. time is 28 sec in 1st scenario and 20 sec in the 2nd instance(40/2).
S+28=r(28)
S+40=r(20)
solve the system:
subtract the 2nd equation from the 1st to eliminate the S terms
-12=8r---->r=-3/2--->now plug r's value into either equation I choose the 2nd equation
S+40=(-3/2)20
S+40=-30
feet on the escalator when still

Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
Rate * Time = Distance.
let R = rate of the escalator in steps per second.
let T = time required to travel D.
let D = distance from the bottom of the escalator to the top of the escalator.
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D is measured in steps based oan the question:
If the escalator had been stoppped, how many steps did the escalator have from the bottom to the top?
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if the escalator is moving at rate R, then D will be covered in time T.
we have the equation:
R*T = D
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if Leon takes a step a second, and Leon reaches the top of the escalator in 28 steps, then the time for Leon to travel the distance D from the bottom of the escalator to the top of the escalator is 28 seconds.
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Leon's Rate of Travel is R + 1 (the rate of the escalator steps per second plus the additional 1 step per second that Leon takes).
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the equation for leon traveling the distance is:
(R+1)*28 = D (equation 1).
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if leon takes 2 steps a second, and leon reaches the top of the escalator in 40 steps, then the time for leon to travel the distance D from the bottom of the escalator to the top of the escalator is 20 seconds (40 steps at 2 steps per second = 20 second).
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leon's rate of travel is R + 2 (the rate of the escalator steps per second plus the additional 2 steps per second that leon takes).
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the equation for leon traveling the distance is:
(R+2)*20 = D (equation 2).
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since equation 1 and 2 both equal D, then these equations are equal to each other.
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(R+1)*28 = (R+2)*20
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remove parenthese:
28*R + 28 = 20*R + 40
subtract 20*R from both sides of the equation:
8*R + 28 = 40
subtract 28 from both sides of the equation:
8*R = 40 - 28 = 12
divide both sides of the equation by 8:
R = 12/8 = 3/2
this is the rate that the escalator is traveling.
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since R = 3/2,
R+1 = 5/2
R+2 = 7/2
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substituting R+1 = 5/2 in equation 1:
(R+1)*28 = D (equation 1) becomes:
(5/2)*28 = D
this becomes:
(5*28)/2 = D
which becomes:
(5*14) = D
which becomes:
70 = D
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substituting R+21 = 7/2 in equation 2:
(R+2)*20 = D (equation 2) becomes:
(7/2)*20 = D
this becomes:
(7*20)/2 = D
which becomes:
7*10 = D
which becomes:
70 = D
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the escalator distance is 70 steps.
that is the number of steps required to get from the bottom to the top if the escalator is not moving at all.
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