A person walks along a straight road from his house to a market 2.5kms away with a speed of 5 km/hr and instantly turns back and reaches his house with a speed of 7.5 kms/hr. The average speed of the person during the time interval 0 to 50 minutes is (in m/sec)

Options

A. `4 2/3`
B. `5/3`
C. `5/6`
D. `1/3`

Answer

To solve the problem, we need to find the average speed of a person who walks from his house to a market and back. Here's a step-by-step breakdown of the solution:

### Step 1: Calculate the time taken to walk to the market
- **Distance to the market** = 2.5 km
- **Speed while going to the market** = 5 km/hr

Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
\[ t_1 = \frac{2.5 \text{ km}}{5 \text{ km/hr}} = 0.5 \text{ hours} = 30 \text{ minutes} \]

### Step 2: Calculate the time taken to return home
- **Speed while returning home** = 7.5 km/hr

Using the same formula:
\[ t_2 = \frac{2.5 \text{ km}}{7.5 \text{ km/hr}} = \frac{1}{3} \text{ hours} = 20 \text{ minutes} \]

### Step 3: Calculate the total time taken for the round trip
\[ \text{Total time} = t_1 + t_2 = 0.5 \text{ hours} + \frac{1}{3} \text{ hours} \]
To add these, we need a common denominator:
\[ 0.5 = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
So,
\[ \text{Total time} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \text{ hours} \]

### Step 4: Calculate the total distance traveled
\[ \text{Total distance} = \text{Distance to market} + \text{Distance back home} = 2.5 \text{ km} + 2.5 \text{ km} = 5 \text{ km} \]

### Step 5: Calculate the average speed
Average speed is given by the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \]
Substituting the values:
\[ \text{Average speed} = \frac{5 \text{ km}}{\frac{5}{6} \text{ hours}} = 5 \text{ km} \times \frac{6}{5} = 6 \text{ km/hr} \]

### Step 6: Convert the average speed to meters per second
To convert km/hr to m/sec, we use the conversion factor:
\[ 1 \text{ km/hr} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/sec} \]
Thus,
\[ \text{Average speed in m/sec} = 6 \text{ km/hr} \times \frac{5}{18} = \frac{30}{18} = \frac{5}{3} \text{ m/sec} \]

### Final Answer
The average speed of the person during the time interval from 0 to 50 minutes is:
\[ \frac{5}{3} \text{ m/sec} \]