Question

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.​

Answer 1

Given: Twenty six men - 1,2,3,...25 and 26 participate in 10 km running race on a circular track of length 100 m. All of  them start at the same time, from the same point and run in the same direction.

To find:  The time taken (in seconds), for all the  twenty six men to meet for the first time at the starting point.​

Solution:

• Now there are twenty six men, so:

                 1, 2, 3, 4, ...are in AP.

• Assume 1 = x, then 2 = x + d, 3 = x + 2d, ............

• According to the question, the time taken by 26 to meet 1, for the first time after they start is 20 sec, so:

                 100 / x+25d-x = 20

                 100 = 25x20d

                 100 = 500d

                 d = 1/5 m/sec

• Again, the  time taken by 13 to complete the race is 52 minutes and 5 seconds.

                 100 (100/x+2d) = 3125

                 10000 = 3125(x+2d)

                 x = 4/5 m/sec

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

             LCM of { 100/x , 100/x+d , ............, 100/x+25d }

             LCM of (100, 100 ,100 , 100,...... 100 ) / HCF of ( 0.8, 1, 1.2, 1.4 ....5.8)

             100 / 0.2

             500

Answer:

             So the time taken is 500 sec.