A person went from A to B at an average speed of x km/hr and returned from B to A at an average speed of y km/hr. What was his average speed during the total journey?

Options

A. `(x+y)/(2xy)`
B. `(2xy)/(x+y)`
C. `(2)/(x+y)`
D. `(1)/(x) + (1)/(y)`

Answer

To find the average speed during the total journey from A to B and back from B to A, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the distance from A to B be \( d \) km.
- The average speed from A to B is \( x \) km/hr.
- The average speed from B to A is \( y \) km/hr.

2. **Calculate the Time Taken for Each Leg of the Journey:**
- Time taken to go from A to B (denote as \( t_1 \)):
\[ t_1 = \frac{d}{x} \]
- Time taken to return from B to A (denote as \( t_2 \)):
\[ t_2 = \frac{d}{y} \]

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

4. **Calculate the Total Time:**
- The total time for the round trip is:
\[ \text{Total Time} = t_1 + t_2 = \frac{d}{x} + \frac{d}{y} \]
- To combine these fractions, find a common denominator:
\[ \text{Total Time} = \frac{dy + dx}{xy} = \frac{d(x + y)}{xy} \]

5. **Calculate the Average Speed:**
- The average speed for the entire journey is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- Substituting the values we found:
\[ \text{Average Speed} = \frac{2d}{\frac{d(x + y)}{xy}} = \frac{2d \cdot xy}{d(x + y)} = \frac{2xy}{x + y} \]

### Final Answer:
The average speed during the total journey is:
\[ \text{Average Speed} = \frac{2xy}{x + y} \text{ km/hr} \]