Directions: Read the table carefully and answer the following question. The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one person wants to travel from P to S, then he/she can only use a vehicle available at the point P. (ii) ** means there is no direct connection between given points. | | P | Q | R | S | Transport Vehicle | Speed (km/h) | |---|---|---|---|---|---|---| | P | 0 | 360 | ** | 780 | Train | 60 | | Q | 540 | 0 | ** | 600 | Taxi | 50 | | R | 420 | 720 | 0 | ** | Bus | 40 | | S | 630 | ** | 900 | 0 | Bike | 45 | Find the minimum possible sum of the time (in hours) taken to cover distance from P to S and from R to S.

### Concept & Shortest Path Time We need to independently calculate the minimum time to travel from P to S and from R to S, then sum these two minimum values. $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$ ### Step-by-Step Solution 1. **Find minimum time for $P \to S$:** - Direct route: $P \to S = 780$ km at 60 km/h (Train) = $780 / 60 = 13$ hours. - Route via Q: $P \to Q \to S$. - $P \to Q$: 360 km at 60 km/h = 6 hours. - $Q \to S$: 600 km at 50 km/h = 12 hours. - Total = $6 + 12 = 18$ hours. - Minimum time for $P \to S = 13$ hours. 2. **Find minimum time for $R \to S$:** - Direct route: $R \to S = $ "**" (No direct connection). - Route via P: $R \to P \to S$. - $R \to P$: 420 km at 40 km/h (Bus) = $420 / 40 = 10.5$ hours. - $P \to S$: 780 km at 60 km/h (Train) = $780 / 60 = 13$ hours. - Total = $10.5 + 13 = 23.5$ hours. - Route via Q: $R \to Q \to S$. - $R \to Q$: 720 km at 40 km/h = $720 / 40 = 18$ hours. - $Q \to S$: 600 km at 50 km/h = $600 / 50 = 12$ hours. - Total = $18 + 12 = 30$ hours. - Minimum time for $R \to S = 23.5$ hours. 3. **Calculate the sum:** - Sum $= 13 \text{ hours} + 23.5 \text{ hours} = 36.5 \text{ hours}$. ### Exam Strategy & Shortcut Whenever checking paths, start with the direct path. If indirect paths have distances that are vastly larger, you can often visually skip the full calculation. For instance, $R \to Q \to S$ covers $720 + 600 = 1320$ km at relatively slow speeds compared to $420 + 780 = 1200$ km. ### Common Pitfall A frequent error is misreading the direction of travel (e.g., using $S \to P$ distances instead of $P \to S$). Always strictly read rows as origins and columns as destinations. ### Final Answer Therefore, the correct answer is **36.5**.