Mohan drives to a marketplace which is 30 km away from his home at an average speed of 75 km per hour. After he completes his shopping, he decides to visit his friend. If he drives at the same speed as on his way to the market and it takes him 30 minutes to reach his friend's house, find the distance between the market and the friend's house.

Answer

To solve the problem step by step, we will use the relationship between speed, distance, and time.

### Step 1: Calculate the time taken to reach the marketplace
We know that:
- Distance to the marketplace = 30 km
- Average speed = 75 km/h

Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
we can calculate the time taken to reach the marketplace.

\[ \text{Time to marketplace} = \frac{30 \text{ km}}{75 \text{ km/h}} = \frac{30}{75} \text{ hours} \]

### Step 2: Simplify the time calculation
Now, simplify \(\frac{30}{75}\):
\[ \frac{30}{75} = \frac{2}{5} \text{ hours} \]

### Step 3: Convert time from hours to minutes
To convert hours to minutes, we multiply by 60:
\[ \text{Time in minutes} = \frac{2}{5} \times 60 = 24 \text{ minutes} \]

### Step 4: Determine the time remaining to reach the friend's house
Mohan takes a total of 30 minutes to reach his friend's house. So, the time remaining after reaching the marketplace is:
\[ \text{Time remaining} = 30 \text{ minutes} - 24 \text{ minutes} = 6 \text{ minutes} \]

### Step 5: Convert the remaining time to hours
Convert 6 minutes into hours:
\[ \text{Time in hours} = \frac{6}{60} = \frac{1}{10} \text{ hours} \]

### Step 6: Calculate the distance to the friend's house
Now, using the speed of 75 km/h, we can find the distance to the friend's house using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Substituting the values:
\[ \text{Distance to friend's house} = 75 \text{ km/h} \times \frac{1}{10} \text{ hours} = 7.5 \text{ km} \]

### Final Answer
The distance between the marketplace and the friend's house is **7.5 km**.

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