CAT 2022 Slot 1 — DILR Question 4
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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.)
What is the minimum number of trains that are required to provide the service on the AB line (considering both north and south directions)?
Answer & solution
Answer: 8
A train leaves A for B and B to A every 15 minutes starting from 6 am.
Total time taken by a north-south train:
There are total 10 stations from M till N, hence total travelling time = 10 * 3 = 30 minutes.
There are 2 junctions, hence stoppage time = 2 * 2 = 4 mins
There are 7 other stations, hence stoppage time = 7 * 1 = 7 mins
∴ Total time taken = 30 + 4 + 7 = 41 mins.
The train which starts from A to B at 6 am reaches B at 6:41 am. It needs to rest for 15 minutes before it can start the journey back to A i.e., it can start it’s journey at 6:56 or after that.
As per schedule north-south trains leave every 15 minutes starting from 6 am. Hence, the train that starts from A to B at 6 am, can start its backward journey at 7 am.
∴ There have to be trains starting at 6 am, 6:15 am, 6:30 am and 6:45 am before trains coming from A can start their journey back.
Hence, we need at least 4 trains from B to A and similarly from A to B.
∴ We need at least 8 trains on the route AB.
Hence, 8.