A bo walks to his school at a distance of 6 km with constant speed of `2.5 kmh^(-1)` and walks back with a constant speed of `4 kmh^(-1)`. His average speed for round trip expressed in `kmh^(-1)`, is

Options

A. 24/13
B. 40/13
C. 3
D. `1//2`

Explanation

To find the average speed of the boy for the round trip to school and back, we can follow these steps:

### Step 1: Understand the Problem
The boy walks to school and back, covering a total distance of 12 km (6 km each way). He walks to school at a speed of 2.5 km/h and returns at a speed of 4 km/h.

### Step 2: Calculate the Time Taken for Each Leg of the Trip
1. **Time to walk to school (T1)**:
\[ T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{6 \text{ km}}{2.5 \text{ km/h}} = 2.4 \text{ hours} \]

2. **Time to walk back home (T2)**:
\[ T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{6 \text{ km}}{4 \text{ km/h}} = 1.5 \text{ hours} \]

### Step 3: Calculate the Total Time for the Round Trip
\[ T_{\text{total}} = T_1 + T_2 = 2.4 \text{ hours} + 1.5 \text{ hours} = 3.9 \text{ hours} \]

### Step 4: Calculate the Total Distance for the Round Trip
\[ D_{\text{total}} = 6 \text{ km} + 6 \text{ km} = 12 \text{ km} \]

### Step 5: Calculate the Average Speed
Average speed is defined as total distance divided by total time:
\[ \text{Average Speed} = \frac{D_{\text{total}}}{T_{\text{total}}} = \frac{12 \text{ km}}{3.9 \text{ hours}} \approx 3.08 \text{ km/h} \]

### Step 6: Use the Formula for Average Speed
We can also use the formula for average speed when two distances are covered at different speeds:
\[ \text{Average Speed} = \frac{2 v_1 v_2}{v_1 + v_2} \]
Where \(v_1 = 2.5 \text{ km/h}\) and \(v_2 = 4 \text{ km/h}\):
\[ \text{Average Speed} = \frac{2 \cdot 2.5 \cdot 4}{2.5 + 4} = \frac{20}{6.5} = \frac{40}{13} \text{ km/h} \approx 3.08 \text{ km/h} \]

### Final Answer
The average speed for the round trip is approximately \( \frac{40}{13} \text{ km/h} \).

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