A person goes from point A to B and comes back. His average speed for the whole journey is 65 km/hr. If his speed while coming back from B to A is 40 km/hr, then what will be the speed of the person (in km/hr) while going from A to B?

Options

A. 90
B. `173.33`
C. 104
D. 112

Answer

To find the speed of the person while going from point A to point B, we can use the concept of average speed and the relationship between distance, speed, and time.

### Step-by-Step Solution:

1. **Define Variables**:
Let the distance from A to B be \( x \) km. The speed from A to B is \( S \) km/hr, and the speed from B to A is given as 40 km/hr.

2. **Calculate Total Distance**:
The total distance for the round trip (A to B and back to A) is:
\[ \text{Total Distance} = x + x = 2x \text{ km} \]

3. **Calculate Time Taken for Each Leg of the Journey**:
- Time taken to go from A to B:
\[ \text{Time}_{AB} = \frac{x}{S} \text{ hours} \]
- Time taken to return from B to A:
\[ \text{Time}_{BA} = \frac{x}{40} \text{ hours} \]

4. **Calculate Total Time**:
The total time for the journey is:
\[ \text{Total Time} = \text{Time}_{AB} + \text{Time}_{BA} = \frac{x}{S} + \frac{x}{40} \]

5. **Use Average Speed Formula**:
The average speed for the whole journey is given as 65 km/hr. The formula for average speed is:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the known values:
\[ 65 = \frac{2x}{\frac{x}{S} + \frac{x}{40}} \]

6. **Simplify the Equation**:
To simplify, multiply both sides by the total time:
\[ 65 \left( \frac{x}{S} + \frac{x}{40} \right) = 2x \]
Dividing through by \( x \) (assuming \( x \neq 0 \)):
\[ 65 \left( \frac{1}{S} + \frac{1}{40} \right) = 2 \]

7. **Rearranging the Equation**:
\[ \frac{1}{S} + \frac{1}{40} = \frac{2}{65} \]
Now, isolate \( \frac{1}{S} \):
\[ \frac{1}{S} = \frac{2}{65} - \frac{1}{40} \]

8. **Finding a Common Denominator**:
The least common multiple of 65 and 40 is 520. Convert each fraction:
\[ \frac{2}{65} = \frac{16}{520}, \quad \frac{1}{40} = \frac{13}{520} \]
Thus,
\[ \frac{1}{S} = \frac{16}{520} - \frac{13}{520} = \frac{3}{520} \]

9. **Reciprocate to Find S**:
\[ S = \frac{520}{3} \approx 173.33 \text{ km/hr} \]

### Final Answer:
The speed of the person while going from A to B is approximately **173.33 km/hr**.