Hovkial wrote:
A train overtakes two people walking along a railway track. The two people are walking in the same direction as the train at the rates of 4.5 km/hr and 5.4 km/hr respectvely. The train takes 8. 4 seconds and 8.5 seconds respectively to overtake them. What is the speed of the train?
(A) 66 km/hr
(B) 72 km/hr
(C) 78 km/hr
(D) 81 km/hr
(E) 96 km/hr
We should notice that the time it takes a train to overtake a walker is between the moment when the nose of the train catches the walker and the moment when the tail of the train passes the walker.
Let d be the length of the train and r be the rate of the train.
From the moment the nose of the train catches the first walker, the distance between the walker and the nose increases at a rate of r - 4.5 kilometers per hour. By the time the tail of the train passes the walker, the nose has traveled a distance of d; therefore the time it takes for the train to overtake the first walker is given by d/(r - 4.5). We are told that this time is equal to 8.4 seconds, or 8.4/3600 hours; thus we can write the equation:
d/(r - 4.5) = 8.4/3600
Applying the same argument to the second walker, we get:
d/(r - 5.4) = 8.5/3600
Since we are trying to find r, let’s isolate d in each equation and set them equal to each other:
d = [(r - 4.5)(8.4)]/3600
d = [(r - 5.4)(8.5)]/3600
[(r - 4.5)(8.4)]/3600 = [(r - 5.4)(8.5)]/3600
Canceling 3600 from each side of the equation and opening up the parentheses, we obtain:
8.4r - (8.4)(4.5) = 8.5r - (8.5)(5.4)
(8.5)(5.4) - (8.4)(4.5) = 0.1r
Let’s multiply each side by 10:
r = (8.5)(54) - (8.4)(45) = 9[(8.5)6 - (8.4)5] = 9[(85/10)*6 - (84/10)*5] = 9[17*3 - 42] = 9*9 = 81
Answer: D
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