Question 167572
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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