Logical Reasoning

Routes & Networks

Easy

Figure for CAT 2022 DILR question 20 (Logical Reasoning)

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).

What is the minimum number of trains that are required to provide the service in this city?

Entered answer:

✅ Correct Answer: 48

Slide 1/4

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

Solution figure for CAT 2022 DILR question 20 (Logical Reasoning)

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

Total number of trains that required to provide in the city can be calculated through total number of trains across all the roots

For east-west root trains leave after every 10 minutes and rest of 15 minutes is also given after reaching the last terminal

M $\rightarrow$ R $\rightarrow$ S $\rightarrow$ N = 11 + 2 + 29 + 2 + 17 = 61 min

P $\rightarrow$ T $\rightarrow$ V $\rightarrow$ Q = 11 + 2 + 29 + 2 + 17 = 61 min

For east west trains total time = (61 + 61 + 15) = 76 minutes each for MN and PQ

Total time = 76 $\times$ 2 = 152

Every train comes after 10 minutes, number of trains = $\frac{152}{10}$ = 15.2 approx,

which implies there are more than 15 so number of trains in east-west direction = MN + PQ = 16 + 16 = 32 trains

Similarly, for north-south direction trains leave after every 15 minute and rest of 15 minutes is same

A $\rightarrow$ R $\rightarrow$ T $\rightarrow$ B =

C $\rightarrow$ S $\rightarrow$ V $\rightarrow$ D = 11 + 2 + 15 + 2 + 11 + 15 = 56 min + 56 min = 112 min

Number of trains = $\frac{112}{15}$ = 7.4 approx

implies more than 7 gaps are there so number of trains each side = 8

Total trains north-south direction = 8 + 8 = 16

Therefore, total number of trains = trains in north-south direction + trians in east-west direction = 16 + 32 = 48

Question 16

If Hari is ready to board a train at 8:05 am from station M, then when is the earliest that he can reach station N?

Question 17

If Priya is ready to board a train at 10:25 am from station T, then when is the earliest that she can reach station S?

Question 18

Haripriya is expected to reach station $S$ late. What is the latest time by which she must be ready to board at station $S$ if she must reach station $B$ before $1$ am via station $R$ ?

Question 19

What is the minimum number of trains that are required to provide the service on the AB line (considering both north and south directions)?

What is the minimum number of trains that are required to provide the service in this city?

Solution

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