P, Q, and R start their journeys at 8 am, 9 am, and 10 am respectively. The speeds of P, Q, and R are in the ratio $x : 1.25x : 0.5x$ respectively. After 4 hours from Q's start, Q meets P, and both immediately begin returning to their starting points. At what time will Q meet R?

Question

Accepted Answer

**Solution:**

1. Let the speed of P be $x$ km/h.
   - Then speed of Q = $1.25x = \frac{5x}{4}$ km/h
   - Speed of R = $0.5x = \frac{x}{2}$ km/h

2. Q starts at 9 am and meets P after 4 hours, i.e., at 1 pm.
   - Distance covered by Q in 4 hours = $\frac{5x}{4} 	imes 4 = 5x$ km
   - P starts at 8 am, so by 1 pm (5 hours), distance covered by P = $x 	imes 5 = 5x$ km

3. Distance covered by R from 10 am to 1 pm (3 hours) = $\frac{x}{2} 	imes 3 = \frac{3x}{2}$ km

4. After meeting at 1 pm, Q and P start returning to their starting points.
   - Now, Q and R move towards each other (opposite directions).
   - Distance between Q and R at 1 pm = $5x - \frac{3x}{2} = \frac{7x}{2}$ km

5. Relative speed of Q and R when moving towards each other = $\frac{5x}{4} + \frac{x}{2} = \frac{5x}{4} + \frac{2x}{4} = \frac{7x}{4}$ km/h

6. Time taken for Q and R to meet after 1 pm:
   $$
   	ext{Time} = \frac{	ext{Distance}}{	ext{Relative Speed}} = \frac{\frac{7x}{2}}{\frac{7x}{4}} = 2 	ext{ hours}
   $$

7. Therefore, Q meets R at $1 	ext{ pm} + 2 	ext{ hours} = 3 	ext{ pm}$.

**Answer:** 3 pm

Answer

Answer

Answer