Two men A and B start walking simultaneously from P to Q, a distance of 21 kms, at the speeds of 3 km and 4 km an hour respectively. B reaches Q, returns immediately and meets A at R. Then the distance from P to R is:

We are given that two men, A and B, start walking simultaneously from point P to point Q. The total distance from P to Q is 21 km. A walks at a speed of 3 km/h, and B walks at a speed of 4 km/h.

To solve this problem, we will:

1. Calculate the time it takes for B to reach Q.
2. Determine the distance B covers after reaching Q and meeting A at point R.
3. Finally, compute the distance from P to R.

### Step 1: Time for B to reach Q
The distance between P and Q is 21 km, and B walks at a speed of 4 km/h. Thus, the time taken by B to reach Q is:

\[
t_B = \frac{\text{Distance}}{\text{Speed}} = \frac{21}{4} = 5.25 \text{ hours}.
\]

### Step 2: Distance A covers during this time
A walks at a speed of 3 km/h. In the same 5.25 hours (time for B to reach Q), A covers:

\[
\text{Distance covered by A} = \text{Speed of A} \times \text{Time} = 3 \times 5.25 = 15.75 \text{ km}.
\]

At this point, B has reached Q and A has covered 15.75 km towards Q.

### Step 3: B walks back towards A after reaching Q
After reaching Q, B immediately turns back and starts walking towards A at 4 km/h. Meanwhile, A continues walking towards Q at 3 km/h. 

Now, we need to find the point R where they meet. The relative speed between A and B is:

\[
\text{Relative speed} = 4 + 3 = 7 \text{ km/h}.
\]

The remaining distance between them is:

\[
\text{Remaining distance} = 21 - 15.75 = 5.25 \text{ km}.
\]

The time taken for them to meet is:

\[
t = \frac{\text{Remaining distance}}{\text{Relative speed}} = \frac{5.25}{7} = 0.75 \text{ hours}.
\]

### Step 4: Distance from P to R
In 0.75 hours, A walks a further distance of:

\[
\text{Distance covered by A in 0.75 hours} = 3 \times 0.75 = 2.25 \text{ km}.
\]

Thus, the total distance from P to R is:

\[
\text{Total distance from P to R} = 15.75 + 2.25 = 18 \text{ km}.
\]

### Final Answer:
The distance from P to R is 18 km.

---

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

Here are 5 related questions to expand your understanding:

1. How would the solution change if A and B had different speeds?
2. What if the distance between P and Q were doubled? 
3. How can we generalize this solution for different speeds and distances?
4. What happens if A and B start at different times instead of simultaneously?
5. How long will it take both men to meet if they keep walking in opposite directions after meeting?

**Tip**: When solving problems involving relative speeds, focus on how the speeds of the individuals or objects contribute to the rate at which they approach or move away from each other.

Learn how to solve a relative speed problem involving two men walking at different speeds from P to Q, a distance of 21 km. Using time, speed, and distance relationships, find where they meet and determine the distance from P to R. Suitable for Grades 8-10.

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