A boy goes from his house to school by bus at a speed of `20 km h^(-1)` and returns back through the same route at a speed of `30 km h^(-1)` . The average speed of his journey is :

Options

A. `24 km h^(-1)`
B. `25 km h^(-1)`
C. `30 km h^(-1)`
D. `20 km h^(-1)`

Answer

To find the average speed of the boy's journey from his house to school and back, we can follow these steps:

### Step 1: Define the distance
Let's assume the distance from the boy's house to the school is \( x \) kilometers.

### Step 2: Calculate the time taken to go to school
The speed of the bus when going to school is \( 20 \) km/h. The time taken to go to school can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So, the time taken to go to school is:
\[ t_1 = \frac{x}{20} \text{ hours} \]

### Step 3: Calculate the time taken to return home
The speed of the bus when returning home is \( 30 \) km/h. The time taken to return home is:
\[ t_2 = \frac{x}{30} \text{ hours} \]

### Step 4: Calculate the total time for the journey
The total time for the entire journey (to school and back) is the sum of the time taken to go and the time taken to return:
\[ \text{Total Time} = t_1 + t_2 = \frac{x}{20} + \frac{x}{30} \]

### Step 5: Find a common denominator and simplify
To add the fractions, we need a common denominator, which is \( 60 \):
\[ \frac{x}{20} = \frac{3x}{60} \quad \text{and} \quad \frac{x}{30} = \frac{2x}{60} \]
Thus,
\[ \text{Total Time} = \frac{3x}{60} + \frac{2x}{60} = \frac{5x}{60} = \frac{x}{12} \text{ hours} \]

### Step 6: Calculate the total distance
The total distance for the entire journey (to school and back) is:
\[ \text{Total Distance} = x + x = 2x \text{ kilometers} \]

### Step 7: Calculate the average speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we found:
\[ \text{Average Speed} = \frac{2x}{\frac{x}{12}} = 2x \times \frac{12}{x} = 24 \text{ km/h} \]

### Conclusion
The average speed of the boy's journey is \( 24 \) km/h.

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