CAT 2022 Slot 1 — DILR Question 5
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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 in this city?
Answer & solution
Answer: 48
Consider the solution to previous question.
We need at least 8 trains on AB route as well as 8 trains on CD route.
∴ 16 trains on north-south routes.
Total time taken by a east-west train = 61 mins (calculated in first question of this set)
A train going from M to N at 6 am will reach N at 7:01 and will be ready for journey back at 7:16 am. A train starts from N every 10 mins. Hence, this train can start its journey at 7:20 am.
∴ There have to be trains starting at 6 am, 6:10 am, 6:20 am, 6:30 am, 6:40 am, 6:50 am, 7 am and 7:10 am before trains coming from M can start their journey back.
Hence, we need at least 8 trains from N to M and similarly from M to N.
∴ We need at least 16 trains on the route MN and 16 trains on PQ route
∴ 32 trains on east-west routes.
⇒ we need at least 16 + 32 = 48 trains.
Hence, 48.