A boy goes to a school from his home at a speed of x km/hr and comes back at a speed of y km/hr, then the average speed is given by

Options

A. AM
B. GM
C. HM
D. none of these

Explanation

To find the average speed of a boy who travels to school at a speed of \( x \) km/hr and returns at a speed of \( y \) km/hr, we can follow these steps:

### Step 1: Define the Distance
Let the distance from the boy's home to the school be \( s \) kilometers.

### Step 2: Calculate the Time Taken for Each Journey
- The time taken to go to school is given by:
\[ t_1 = \frac{s}{x} \]
- The time taken to return home is given by:
\[ t_2 = \frac{s}{y} \]

### Step 3: Calculate the Total Time Taken
The total time \( T \) for the round trip is the sum of the time taken to go and the time taken to return:
\[ T = t_1 + t_2 = \frac{s}{x} + \frac{s}{y} \]
We can factor out \( s \):
\[ T = s \left(\frac{1}{x} + \frac{1}{y}\right) \]

### Step 4: Calculate the Total Distance
The total distance \( D \) for the round trip is:
\[ D = s + s = 2s \]

### Step 5: Calculate the Average Speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} = \frac{2s}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{2s}{s \left(\frac{1}{x} + \frac{1}{y}\right)} \]
The \( s \) cancels out:
\[ V_{avg} = \frac{2}{\left(\frac{1}{x} + \frac{1}{y}\right)} \]

### Step 6: Simplify the Expression
To simplify \( \frac{2}{\left(\frac{1}{x} + \frac{1}{y}\right)} \):
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} \]
Thus,
\[ V_{avg} = \frac{2}{\frac{y + x}{xy}} = \frac{2xy}{x + y} \]

### Final Result
The average speed of the boy for the round trip is:
\[ \boxed{\frac{2xy}{x + y}} \]