Given above is the schematic map of the metro lines in a city with rectangles denoting terminal stations (e.g. A), diamonds denoting junction stations (e.g. R) and small filled-up circles denoting other stations. Each train runs either in east-west or north-south direction, but not both. All trains stop for 2 minutes at each of the junction stations on the way and for 1 minute at each of the other stations. It takes 2 minutes to reach the next station for trains going in east-west direction and 3 minutes to reach the next station for trains going in north-south direction. From each terminal station, the first train starts at 6 am; the last trains leave the terminal stations at midnight. Otherwise, during the service hours, there are metro service every 15 minutes in the north-south lines and every 10 minutes in the east-west lines. A train must rest for at least 15 minutes after completing a trip at the terminal station, before it can undertake the next trip in the reverse direction. (All questions are related to this metro service only. Assume that if someone reaches a station exactly at the time a train is supposed to leave, (s)he can catch that train.)

Q. 1  If Hari is ready to board a train at 8:05 am from station M, then when is the earliest that he can reach station N?

1)  9:13 am        2)  9:01 am        3)  9:11 am        4)  9:06 am

Q. 2  If Priya is ready to board a train at 10:25 am from station T, then when is the earliest that she can reach station S?

1)  11:28 am      2)  11:12 am     3)  11:22 am      4)  11:07 am

Q. 3  Haripriya is expected to reach station S late. What is the latest time by which she must be ready to board at station S if she must reach station B before 1 am via station R?

1)  11:49 am      2)  11:43 pm      3)  11:35 pm      4)  11:39 pm

Q. 4   What is the minimum number of trains that are required to provide the service on the AB line (considering both north and south directions)?

Q. 5   What is the minimum number of trains that are required to provide the service in this city?

Solutions

1. Given that Hari is ready to board a train at 8:05 am from station M. Trains leave after every 10 minutes. So Hari can catch the 8:10 train from M. There are 17 stations, 2 junctions and 20 intervals, so time taken to travel = 20 × 2 + 17 × 1 + 2 × 2 = 61 minutes. Hence, time at which Hari reaches station N = 8:10 + 1:01 = 9:11 am.

2. There can be two routes that Priya can take.

Case 1: First travel by East-West line and then North-South line.

Trains leave after every 10 minutes on the East- West line.

The 10:20 am train from P reaches and then leaves T at = 10:20 + 00:11 + 00:02 = 10:33 am In the first case Priya will have to wait for 8 minutes at the station before boarding the train.

So let us see if the waiting time is reduced in case 2.

Case 2: First travel by North-South line and then East-West line.

Trains leave after every 15 minutes on the North- South line. The 10:15 am train from B will reach T at = 10:15 + 00:11 = 10:26 am and leave T at 10:28 am. In this case Priya’s train leaves within 3 minutes of her reaching the station. This train reaches R at = 10:28 + 00:15 = 10:43 am.

From R Priya has to catch the East-West train for station S.

The 10:30 am train from M reaches R at = 10:30 + 00:11 = 10:41 am and leaves at 10:43 am.

Priya can catch this train and reach station S at = 10:43 + 00:29 = 11:12 am.

3. The last (midnight) train from A reaches R at 12:11 am and leaves at 12:13 am. It will reach B at = 12:13 + 00:17 + 00:11 = 12:41 am.

From S to R the travel time is 29 minutes. To catch the 12:13 am train Haripriya will have to start from S at = 12:13 – 00:29 = 11:44 pm or before that.

Trains start from N after every 10 minutes and the time taken to reach S is 17 minutes.

There is a stoppage of 2 minutes at S.

The 11:30 pm train from N reaches S at 11:47 pm, which does not serve the purpose.

The 11:20 pm train from N reaches S at 11:37 pm and leaves at 11:39 pm.

This train will reach R by 11:39 + 00:29 = 12:08 am.

Then she can catch the last train from A going to B at 12:13 am.

Haripriya can catch the train at S if she reaches the station latest by 11:39 pm.

4. Total time for a train to travel from A to B and to start back = 11 + 2 + 15 + 2 + 11 + 15 = 56 minutes.

Trains leave after every 15 minutes.

The minimum number of trains that are required to provide the service on the AB line (in the North-South direction) = [56/15] = [3.733] = 4 trains

Hence, for both directions = 4 + 4 = 8 trains.

5. The minimum number of trains that are required to provide the service in this city
= Trains in AB and CD lines + Trains in MN and PQ lines.
Time taken for a train to travel from M to N and start back
= 11 + 2 + 29 + 2 + 17 + 15 = 76 minutes.
Trains leave after every 10 minutes on this line.
The minimum number of trains required on the MN line in the East-West direction
= [76/10] = [7.6] = 8 trains
Minimum number of trains required in both directions on the MN line = 16 trains Hence, total number of trains required = (AB + CD) + (MN + PQ) = (8 + 8) + ( 16 + 16) = 48.