A and B are travelling towards each other from the points P and Q respectively. After crossing each other, A and B take `6(1)/(8)` hours and 8 hours respectively to reach their destinations Q and P respectively. If the speed of B is 16.8 km/h, then the speed (in km/h) of A is

Options

A. `20.8`
B. `19.8`
C. `19.2`
D. `20.4`

Answer

To find the speed of A, we can use the concept of relative speed and the time taken by A and B to reach their respective destinations after crossing each other.

### Step-by-Step Solution:

1. **Convert the time taken by A into hours**:
A takes `6(1)/(8)` hours to reach Q after crossing B.
This can be converted to an improper fraction:
\[ 6 \frac{1}{8} = \frac{6 \times 8 + 1}{8} = \frac{48 + 1}{8} = \frac{49}{8} \text{ hours} \]

2. **Use the speed of B**:
The speed of B is given as 16.8 km/h.

3. **Calculate the distance covered by B after crossing A**:
Since B takes 8 hours to reach P after crossing A, we can calculate the distance B covers in that time:
\[ \text{Distance covered by B} = \text{Speed of B} \times \text{Time taken by B} = 16.8 \text{ km/h} \times 8 \text{ hours} = 134.4 \text{ km} \]

4. **Set up the relationship between the distances covered by A and B**:
After crossing each other, the distances covered by A and B are proportional to the times they take to reach their destinations. Therefore, we can set up the following ratio:
\[ \frac{\text{Distance covered by A}}{\text{Distance covered by B}} = \frac{\text{Time taken by B}}{\text{Time taken by A}} \]
Let the distance covered by A be \(d_A\) and the distance covered by B be \(d_B = 134.4 \text{ km}\).

5. **Substituting the times**:
The time taken by A is \(\frac{49}{8}\) hours and the time taken by B is 8 hours. Thus, we have:
\[ \frac{d_A}{134.4} = \frac{8}{\frac{49}{8}} \]
Simplifying the right side:
\[ \frac{8}{\frac{49}{8}} = \frac{8 \times 8}{49} = \frac{64}{49} \]

6. **Cross-multiplying to find \(d_A\)**:
\[ d_A = 134.4 \times \frac{64}{49} \]

7. **Calculating \(d_A\)**:
\[ d_A = \frac{134.4 \times 64}{49} = \frac{8601.6}{49} \approx 175.2 \text{ km} \]

8. **Finding the speed of A**:
Now, we can find the speed of A using the distance covered by A and the time taken by A:
\[ \text{Speed of A} = \frac{d_A}{\text{Time taken by A}} = \frac{175.2 \text{ km}}{\frac{49}{8} \text{ hours}} = 175.2 \times \frac{8}{49} \]
\[ \text{Speed of A} = \frac{1401.6}{49} \approx 28.6 \text{ km/h} \]

### Final Answer:
The speed of A is approximately **28.6 km/h**.