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Four cities $\text{P, Q, R}$ and $\text{S}$ 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 $\text{P}$, the first trains of the day start at $8\;\text{AM}$ with a frequency of $90$ minutes to each of $\text{R}$ and $\text{S}$. 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 $\text{R}$ at $7\;\text{AM}$ and is required to visit $\text{S}$ and return to $\text{R}$, what is the minimum time required?

  1. $6$ hours $30$ minutes
  2. $3$ hours $45$ minutes
  3. $4$ hours $30$ minutes
  4. $5$ hours $15$ minutes
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If the person starts from $\text{R}$ at $\text{7 AM}$ and is required to visit $\text{S}$ and return to $\text{R},$ then he must visit $\text{Q}$ and $\text{P}$ in between.

$\underbrace{\underbrace{\text{R}}_{{\color{Green}{\text{Start at: 7 AM}}}} \overset{\text{1 hour}}{\Longrightarrow} {\underbrace{\text{Q}}_{{\color{Red}{\text{Next train at: 9 AM}}}}} \overset{\text{1 hour}}{\Longrightarrow} \underbrace{\text{P}}_{{\color{Blue}{\text{Next train at: 11 AM}}}} \overset{\text{1 hour}}{\Longrightarrow} \underbrace{\text{S}}_{{\color{Purple}{\text{Next train at: 12:30 PM}}}} \overset{\text{1 hour}}{\Longrightarrow} \underbrace{\text{R}}_{{\color{Lime}{\text{Reach at: 1:30 PM}}}}}_{\text{Total time taken by person = 1 + 1 (waiting) + 1 + 1 (waiting)  + 1 +  ½ (waiting) + 1 = 6:30 hours}}$

$\therefore$ If the person starts from $\text{R}$ at $\text{7 AM}$ and is required to visit $\text{S}$ and return to $\text{R},$ then the minimum time required is $\text{6 hours 30 minutes}.$

Correct Answer $:\text{A}$

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