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-by-Step Solution:

1. **Define the Distance**:
Let the distance between the boy's home and the school be \(s\) kilometers.

2. **Calculate Time Taken to Reach School**:
The time taken to go from home to school at a speed of \(x\) km/hr is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{s}{x} \text{ hours} \]

3. **Calculate Time Taken to Return Home**:
The time taken to return from school to home at a speed of \(y\) km/hr is:
\[ \text{Time} = \frac{s}{y} \text{ hours} \]

4. **Total Distance**:
The total distance traveled (to school and back) is:
\[ \text{Total Distance} = s + s = 2s \text{ kilometers} \]

5. **Total Time**:
The total time taken for the round trip is:
\[ \text{Total Time} = \frac{s}{x} + \frac{s}{y} \]

6. **Finding the Average Speed**:
The average speed is calculated using the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we have:
\[ \text{Average Speed} = \frac{2s}{\frac{s}{x} + \frac{s}{y}} \]

7. **Simplifying the Expression**:
We can factor out \(s\) from the denominator:
\[ \text{Average Speed} = \frac{2s}{s\left(\frac{1}{x} + \frac{1}{y}\right)} = \frac{2}{\frac{1}{x} + \frac{1}{y}} \]
This simplifies to:
\[ \text{Average Speed} = \frac{2xy}{x + y} \text{ km/hr} \]

8. **Identifying the Type of Mean**:
The expression \(\frac{2xy}{x + y}\) is recognized as the harmonic mean of \(x\) and \(y\).

### Final Answer:
The average speed of the boy is given by:
\[ \text{Average Speed} = \frac{2xy}{x + y} \text{ km/hr} \]
This corresponds to the harmonic mean of the speeds \(x\) and \(y\).

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