Given the following distances and speeds between points P, Q, R, and S:

| From | To | Distance (km) | Vehicle | Speed (km/h) |
|-------|----|---------------|---------|--------------|
| P | Q | 240 | Taxi | 80 |
| P | S | 520 | Taxi | 80 |
| Q | S | 400 | Bus | 50 |
| R | P | 280 | Car | 60 |
| R | Q | 480 | Car | 60 |
| S | R | 600 | Bike | 25 |

Note: There is no direct path between R and S.

Find the minimum possible sum of the time (in hours) taken to travel from P to S and from R to S.

Question

Given the following distances and speeds between points P, Q, R, and S:

| From | To | Distance (km) | Vehicle | Speed (km/h) |
|-------|----|---------------|---------|--------------|
| P     | Q  | 240           | Taxi    | 80           |
| P     | S  | 520           | Taxi    | 80           |
| Q     | S  | 400           | Bus     | 50           |
| R     | P  | 280           | Car     | 60           |
| R     | Q  | 480           | Car     | 60           |
| S     | R  | 600           | Bike    | 25           |

Note: There is no direct path between R and S.

Find the minimum possible sum of the time (in hours) taken to travel from P to S and from R to S.

Accepted Answer

**Solution:**

1. Calculate the time taken from P to S:
   - Distance from P to S is given as 520 km.
   - The vehicle used is a Taxi with speed 80 km/h.
   - Time = Distance / Speed = $\frac{520}{80} = 6.5$ hours.

2. Calculate the minimum time from R to S:
   - There is no direct route from R to S.
   - Possible routes:
     - Route 1: R -> P -> S
       - R to P distance = 280 km by Car at 60 km/h
       - P to S distance = 520 km by Taxi at 80 km/h
       - Time = $\frac{280}{60} + \frac{520}{80} = \frac{14}{3} + 6.5 = 4.6667 + 6.5 = 11.1667$ hours
     - Route 2: R -> Q -> S
       - R to Q distance = 480 km by Car at 60 km/h
       - Q to S distance = 400 km by Bus at 50 km/h
       - Time = $\frac{480}{60} + \frac{400}{50} = 8 + 8 = 16$ hours

- Minimum time from R to S is via Route 1: 11.1667 hours.

3. Sum of times from P to S and R to S:
   - $6.5 + 11.1667 = 17.6667$ hours
   - Expressed as a mixed fraction: $17\frac{2}{3}$ hours.

**Answer:** $17\frac{2}{3}$ hours

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