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, she can catch that train).
Solution
Understanding the set :-
- This is a routes and networks set, in this type of set you basically get some junction points and multiple routes to reach that point
- In this type of set, after getting a broad idea of the details and clues (if given), will mainly be based on questions of the set.
- In this particular question, we are given metro lines (routes) with their stations (junctions).
- Also, Each train runs either in east-west or north-south direction, but not both
Let’s try to simplify the set using clues given
Clues given to us :-
- 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.
- 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.
First we need to find the time taken from all terminals to all junctions through stations.
M → R = 2 + 2 + 2 + 2(3 stations gap is 4) + 3(every station stop ) = 11 min
P → T = 2 + 2 + 2 + 2 + 3 (same as above) = 11 min
R → S = 2 + 2 + 2+ 2 + 2 + 2 + 2 + 2 + 2 + 2(9 stations so gap is 10) + 9(9 stations ) = 20 min + 9 min = 29 min
T → V = symmetrical as R → S = 20 + 9 = 29 min
S → N = 2 + 2 + 2 + 2 + 2 + 2 + 5 = 12 + 5 = 17 min
V → Q = 12 + 5 = 17 min
A → R = 3 + 3 + 3 + 2 = 11 min
C → S = 3 + 3 + 3 + 2 = 11 min
R → T = 3 + 3 + 3 + 3 + 3 = 15 min
S → V = 3 + 3 + 3 + 3 + 3 = 15 min
T → B = 3 + 3 + 3(for north south direction train takes 3 minutes) + 2 = 11 min
V → D = 3 + 3 + 3 + 2 = 11 min
For minimum number of trains
we need to check that by the time first train reach at the other terminal, we keep on sending the trains.
For AB line total time taken = ARTB = 11 + 2 + 15 + 2 + 11 + 15 = 56 min
For BA line total time taken = BTRA = 15 + 11 + 2 + 15 + 2 + 11 = 56 min
Total time for AB + BA = 56 + 56 = 112 min
For north south direction we can send trains after every 15 mins Total number of trains = = 7.4 approx,
which implies there are more than 7 gaps, so 8 trains can be sent. Therefore 8 trains is the answer