CAT 2022 Slot 1 · DILR · Question 20

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 in this city?

Answer & solution

Answer: 48

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 the table in Step 4: A–B 8, C–D 8, M–N 16, P–Q 16. Total = 48.

Hence, the answer is 48.

CAT 2022 Slot 1 · DILR · Question 20

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