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GATE 2022 ME (ME2) – Question 9

General Aptitude · Analytical Aptitude: Logical Reasoning and Deduction · 2 marks · Multiple choice

Four cities P, Q, R and S are connected through one-way routes as shown in the figure. The travel time between any two connected cities is one hour. The boxes beside each city name describe the starting time of first train of the day and their frequency of operation. For example, from city P, the first trains of the day start at 8 AM with a frequency of 90 minutes to each of R and S. A person does not spend additional time at any city other than the waiting time for the next connecting train.

If the person starts from R at 7 AM and is required to visit S and return to R, what is the minimum time required?

routes R to Q, Q to P, P to R and S, and S to Q and R; first departures/frequencies P 8 AM/90 min, Q 5 AM/120 min, R 7 AM/60 min, S 8 AM/45 min.
  1. 6 hours 30 minutes
  2. 3 hours 45 minutes
  3. 4 hours 30 minutes
  4. 5 hours 15 minutes

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Correct answer: (A) 6 hours 30 minutes

Explanation

**Setup.** All trips take 1 hour. The first train of each route leaves at the given time and then repeats at the given interval. Routes (one-way): R→Q, Q→P, P→R and P→S, S→Q and S→R.

To visit S and come back to R the cheapest path is R→Q→P→S→R (R has no direct link to S).

**Step 1: R to Q.** Start at R at 7 AM. R's trains start at 7 AM and run every 60 minutes, so the 7 AM train leaves at once. Arrive at Q at **8 AM**.

**Step 2: Q to P.** Q's first train is at 5 AM with a frequency of 120 minutes: 5, 7, 9, 11 and so on. The 7 AM train has gone, so wait until **9 AM**. Arrive at P at **10 AM**.

**Step 3: P to S.** P's first train is at 8 AM with a frequency of 90 minutes: 8:00, 9:30, 11:00. The 9:30 train has gone, so wait until **11:00** and arrive at S at **noon**.

**Step 4: S to R.** S's first train is at 8 AM with a frequency of 45 minutes: 8:00, 8:45, 9:30, 10:15, 11:00, 11:45, **12:30**. Wait for 12:30 and arrive at R at **1:30 PM**.

**Total time:** 7:00 AM to 1:30 PM = **6 hours 30 minutes**.