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A train overtakes two people walking along a railway track.

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A train overtakes two people walking along a railway track.  [#permalink]

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New post Updated on: 19 Sep 2019, 01:53
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Question Stats:

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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

Originally posted by Hovkial on 11 Sep 2019, 10:30.
Last edited by Hovkial on 19 Sep 2019, 01:53, edited 1 time in total.
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Re: A train overtakes two people walking along a railway track.  [#permalink]

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New post 11 Sep 2019, 14:33
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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

let the length of train be L
and speed of train be S km/hr
as all are moving in same direction then relative speed of train with first person = S-4.5
and relative speed of train with second person= S-5.4

thus as Distance /speed=time
As time given for each personrespt.

we can write
L/(S-4.5) = 8.4-----(i)
L/(S-5.4) = 8.5----(ii)

solving both we get
S=81kmph

Ans D
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Re: A train overtakes two people walking along a railway track.  [#permalink]

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New post 12 Sep 2019, 19:38
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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Re: A train overtakes two people walking along a railway track.  [#permalink]

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New post 01 Oct 2019, 18:02
length of train = x
and speed of train =y
we are given
x/y-4.5=8.4 and x/y-5.4= 8.5
solve for y= 81 km/hr
IMO D

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
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A train overtakes two people walking along a railway track.  [#permalink]

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New post 02 Oct 2019, 12:55
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


Set the train's speed as x km/h. The train is faster than the first person by \((x-4.5)\) km/h, and the train needs 8.4s to overtake, therefore the train's length is \((x - 4.5)*8.4\). Similarly, for the second person, the speed of the train is faster by \((x - 5.4)\) km/h so it takes longer to overtake him, and the train's length is \(8.5(x - 5.4)\). We have the train's length in two different expressions so we can set them equal to each other and solve for x.

\(8.5(x - 5.4) = 8.4(x - 4.5)\)

\(0.1x = 8.5*5.4 - 8.4*4.5 = (8.4*5.4 + 0.1*5.4) - 8.4*4.5 = 8.4*(5.4 - 4.5) + 0.1*5.4 = 8.4*0.9 + 0.1*5.4\)

\(x = 8.4*9 + 5.4 = 72 + 3.6 + 5.4 = 81.\)

Ans: D
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A train overtakes two people walking along a railway track.   [#permalink] 02 Oct 2019, 12:55
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