A man travels from Point P to Q at $90 ext{ km/hr}$ and from Q to R at $60 ext{ km/hr}$. The total distance between P and R is $200 ext{ km}$. If his average speed for the entire journey is $75 ext{ km/hr}$, find the distance between P and Q.

Question

A man travels from Point P to Q at $90 	ext{ km/hr}$ and from Q to R at $60 	ext{ km/hr}$. The total distance between P and R is $200 	ext{ km}$. If his average speed for the entire journey is $75 	ext{ km/hr}$, find the distance between P and Q.

Accepted Answer

**Solution:**

1. Let the distance from P to Q be $x$ km and from Q to R be $y$ km.

2. Given total distance:
   $$x + y = 200$$

3. Speeds:
   - From P to Q: $90 	ext{ km/hr}$
   - From Q to R: $60 	ext{ km/hr}$

4. Average speed for the whole journey is $75 	ext{ km/hr}$.

5. Total time taken for the journey:
   $$	ext{Time} = \frac{x}{90} + \frac{y}{60}$$

6. Average speed formula:
   $$	ext{Average Speed} = \frac{	ext{Total Distance}}{	ext{Total Time}}$$

Substitute values:
   $$75 = \frac{200}{\frac{x}{90} + \frac{y}{60}}$$

7. Rearranging:
   $$\frac{x}{90} + \frac{y}{60} = \frac{200}{75} = \frac{8}{3}$$

8. Multiply both sides by 180 (LCM of 90 and 60):
   $$2x + 3y = 180 	imes \frac{8}{3} = 480$$

9. Now solve the system of equations:
   $$\begin{cases}
x + y = 200 \
2x + 3y = 480
\end{cases}$$

10. From the first equation:
    $$y = 200 - x$$

11. Substitute into second equation:
    $$2x + 3(200 - x) = 480$$
    $$2x + 600 - 3x = 480$$
    $$-x + 600 = 480$$
    $$-x = -120$$
    $$x = 120$$

12. Therefore, distance from P to Q is $120 	ext{ km}$.

**Answer:** $120 	ext{ km}$

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