Raghu and Rajan are stationed at points A and B respectively. Raghu starts driving towards Rajan at 12 noon at a uniform speed. On the other hand, Rajan starts driving towards Raghu at 2 PM at a uniform speed. They cross each other at point P at 5 PM. After crossing each other, Raghu continues to drive towards point B and Rajan continues to drive towards point A at their respective uniform speeds. After reaching their respective destinations, they immediately reverse their directions and continue driving at their respective uniform speeds. If they meet again at point P in their return journey, what is the ratio of the speeds of Raghu and Rajan?

Let's analyze the problem step by step to find the ratio of the speeds of Raghu and Rajan.

### Step 1: Define the variables
- Let the distance between points A and B be \( D \).
- Let the speed of Raghu be \( v_R \).
- Let the speed of Rajan be \( v_J \).
- The time Raghu drives from 12 noon to 5 PM (when they meet at point P) is 5 hours.
- The time Rajan drives from 2 PM to 5 PM (when they meet at point P) is 3 hours.

### Step 2: Determine the distances covered by Raghu and Rajan until they meet at point P
- Distance covered by Raghu until they meet at P:
\[
\text{Distance covered by Raghu} = v_R \times 5
\]
- Distance covered by Rajan until they meet at P:
\[
\text{Distance covered by Rajan} = v_J \times 3
\]

Since they meet at point P, the sum of these distances is equal to the total distance \( D \):
\[
v_R \times 5 + v_J \times 3 = D \quad \text{(Equation 1)}
\]

### Step 3: Determine the time to meet again at point P after reversing
After reaching their respective destinations (A and B), they reverse their directions and meet again at point P. Since they meet at point P again, the time taken for both to reach P from their respective destinations must be the same.

The time taken by Raghu to travel back to P from B can be determined by the distance \( D - v_R \times 5 \) (since the remaining distance to B after meeting at P is \( D - v_R \times 5 \)):

\[
\text{Time taken by Raghu to travel back to P} = \frac{D - v_R \times 5}{v_R}
\]

Similarly, for Rajan:

\[
\text{Time taken by Rajan to travel back to P} = \frac{v_R \times 5}{v_J}
\]

Since they meet again at the same point, these times are equal:

\[
\frac{D - v_R \times 5}{v_R} = \frac{v_R \times 5}{v_J} \quad \text{(Equation 2)}
\]

### Step 4: Solve the equations

**From Equation 1:**
\[
D = v_R \times 5 + v_J \times 3 \quad \text{(Equation 1)}
\]

**Substitute \( D \) from Equation 1 into Equation 2:**

\[
\frac{v_R \times 3}{v_R} = \frac{v_R \times 5}{v_J}
\]

This simplifies to:
\[
3 = \frac{5 \times v_R}{v_J}
\]

So, the ratio of speeds is:
\[
\frac{v_R}{v_J} = \frac{3}{5}
\]

Thus, the ratio of the speeds of Raghu and Rajan is \( \frac{3}{5} \).

Would you like more details on any step or have any questions?

Here are 5 related questions for further practice:
1. If Raghu and Rajan had met at 4 PM instead of 5 PM, what would be the ratio of their speeds?
2. How far is point P from point A?
3. If Raghu's speed was increased by 20%, at what time would they have met at point P?
4. What is the distance between A and B if Raghu's speed is 60 km/h?
5. How long would it take Rajan to reach point A if Raghu’s speed was double Rajan's speed?

**Tip:** In relative motion problems, always compare the distances and times traveled in a consistent frame of reference to simplify calculations.

Learn how to solve a relative motion problem involving Raghu and Rajan's speeds. Find out the ratio of their speeds when they cross each other and meet again after reversing directions. Detailed step-by-step solution provided.

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