CAT 2022 Slot 1 · DILR · Question 19

CAT 2022 Slot 1 previous year question on Caselet DI, with the answer and a step-by-step solution.

Caselet DIData InterpretationType in the answer (TITA)Hard

Set 4: Metro Lines in a City — this question belongs to a set. Read the full set with all 5 questions →

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

Solving the set

Step 1 – Count the stations on each line from the map. M–N: M, 3 stations, R, 9 stations, S, 5 stations, N (20 links). P–Q: same pattern through T and V (20 links). A–B: A, 2, R, 3, T, 2, B (10 links). C–D: same through S and V (10 links).

Step 2 – One-way trip times.

Line	Direction	Links	Time per link	Intermediate stops	One-way trip
M–N and P–Q	East–west	20	2 min	17 ordinary (1 min) + 2 junctions (2 min) = 21 min	40 + 21 = 61 min
A–B and C–D	North–south	10	3 min	7 ordinary + 2 junctions = 11 min	30 + 11 = 41 min

Step 3 – Times at the junctions (t = departure from the starting terminal). Trains leave each terminal every 10 min (east–west) or 15 min (north–south) from 6:00 am to midnight.

Train	Leaves terminal at	Reaches 1st junction	Leaves 1st junction	Reaches 2nd junction	Leaves 2nd junction	Reaches end
M → N	t	R: t+11	t+13	S: t+42	t+44	t+61
N → M	t	S: t+17	t+19	R: t+48	t+50	t+61
P → Q	t	T: t+11	t+13	V: t+42	t+44	t+61
A → B	t	R: t+11	t+13	T: t+28	t+30	t+41
B → A	t	T: t+11	t+13	R: t+28	t+30	t+41
D → C	t	V: t+11	t+13	S: t+28	t+30	t+41

Step 4 – Number of trains on a line. After a trip a train rests 15 minutes, then waits for the next departure slot at that terminal; it repeats this at the other end.

Line	Trip	Ready after rest	Next departure slot	Round-trip cycle	Headway	Trains
A–B	41	56	60	120 min	15 min	120/15 = 8
C–D	41	56	60	120 min	15 min	8
M–N	61	76	80	160 min	10 min	160/10 = 16
P–Q	61	76	80	160 min	10 min	16
Total						48

This question

From Step 4, each A–B train completes a full cycle (A → B → A with rests) every 120 minutes, and a train must leave each terminal every 15 minutes, so 120/15 = 8 trains are needed. (At 6:00 am, 4 trains start from A and 4 from B; from 7:00 the arriving trains cover the departures.)

Hence, the answer is 8.

CAT 2022 Slot 1 · DILR · Question 19

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