Time, Speed & Distance
Aptitude

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Q.

Twenty six men - 1,2,3,....25 and 26 participate in 10km running race on a circular track of length 100m. All of them start at the same time, from the same point and run in the same direction. Their speeds, taken in the order, are in increasing $AP$. The time taken by 26 to meet 1, for the first time after they start is 20 sec and the time taken by 13 to complete the race is 52 minutes and 5 seconds. Find the time taken (in seconds), for all the twenty six men to meet for the first time at the starting point.

 A.

1000

 B.

500

 C.

625

 D.

400

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Solution:
Option(B) is correct

Let 1,2,3,4...are in AP $2=x+d$, $3=x+2d$... and so on...

Let assume 1(variable) = $x$

Given time taken by 26 to meet 1 for the first time is 20 sec

\(\Rightarrow \dfrac{100}{x+25d-x}=20\)

\(⇒ d=0.5\) m/sec

Time taken by $13=x+12d$ is 52 minutes and 5 seconds

\(100\times \left(\dfrac{100}{x+12d}\right)=3125\)

\(⇒ x=0.8\) m/sec

Time taken by all of them to meet for the first time at the starting point is:

\(\text{L.C.M}\left[\dfrac{100}{x},\dfrac{100}{x+d},....\dfrac{100}{x+25d}\right]\)

\(\Rightarrow \left(\dfrac{\text{L.C.M}(100,10,100,100,100)}{\text{H.C.F}(0.8,1,1.2,1.4.....5.8)}\right)\)

\(=\dfrac{100}{0.2}\)

\(=500\)


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