Discussion
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?
Answer: A
The minimum time is 6 hours 30 minutes.
The arrows leave little choice. From R the only route is to Q; from Q the only route is to P; from P the person can go straight to S; and from S there is a direct route back to R. So the journey is R, Q, P, S, R, and the time is set by the waits for connecting trains:
- R: first train at 7 AM, leave 7:00, reach Q at 8:00.
- Q: trains at 5, 7, 9 AM (every 120 min); wait for 9:00, reach P at 10:00.
- P: trains at 8:00, 9:30, 11:00 (every 90 min); wait for 11:00, reach S at 12:00.
- S: trains from 8:00 every 45 min, so 11:45 then 12:30; leave 12:30, reach R at 1:30 PM.
From 7:00 AM to 1:30 PM is 6 hours 30 minutes. The shorter options count only the four hours of travel plus some of the waits; every connection here forces a wait, and none can be skipped because each city has only one useful outgoing route.
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