1. **Problem statement:** A train starts its journey from station P at 7:00 am, reaches station Q at 7:45 am, and station R at 8:25 am. The mean speed of the train is 100 km/h. The train reaches its destination at 9:30 am. We need to find the total distance traveled by the train and the distance between stations Q and R. 2. **Given data:** - Start time at P: 7:00 am - Arrival at Q: 7:45 am - Arrival at R: 8:25 am - Arrival at destination: 9:30 am - Mean speed: 100 km/h 3. **Step 1: Calculate total travel time.** - From 7:00 am to 9:30 am is $2$ hours and $30$ minutes. - Convert to hours: $2 + \frac{30}{60} = 2.5$ hours. 4. **Step 2: Calculate total distance using mean speed formula:** $$\text{Distance} = \text{Speed} \times \text{Time}$$ $$= 100 \times 2.5 = 250 \text{ km}$$ 5. **Step 3: Calculate time intervals between stations:** - Time from P to Q: 7:45 am - 7:00 am = 45 minutes = $\frac{45}{60} = 0.75$ hours - Time from Q to R: 8:25 am - 7:45 am = 40 minutes = $\frac{40}{60} = \frac{2}{3}$ hours 6. **Step 4: Calculate distance from P to Q:** $$d_{PQ} = 100 \times 0.75 = 75 \text{ km}$$ 7. **Step 5: Calculate distance from Q to R:** $$d_{QR} = 100 \times \frac{2}{3} = \frac{200}{3} \approx 66.67 \text{ km}$$ 8. **Step 6: Calculate remaining distance from R to destination:** - Time from R to destination: 9:30 am - 8:25 am = 1 hour 5 minutes = $1 + \frac{5}{60} = \frac{13}{12}$ hours - Distance from R to destination: $$d_{R\text{dest}} = 100 \times \frac{13}{12} = \frac{1300}{12} \approx 108.33 \text{ km}$$ 9. **Step 7: Verify total distance:** $$d_{PQ} + d_{QR} + d_{R\text{dest}} = 75 + 66.67 + 108.33 = 250 \text{ km}$$ **Final answers:** - Total distance traveled by the train: $250$ km - Distance between stations Q and R: approximately $66.67$ km