Correct Option

Let the distance from the house to the factory be $x$ km and the scheduled time to reach the factory be $t$ hours.

According to the problem statement, two scenarios are given:

When the worker walks at 5 km/hr, he is 3 minutes late. The time taken is $\frac{x}{5}$ hours. Converting 3 minutes to hours: $\frac{3}{60}$ hours.

So, the equation is: $\frac{x}{5} = t + \frac{3}{60}$ …(1)

When the worker walks at 6 km/hr, he is 7 minutes early. The time taken is $\frac{x}{6}$ hours. Converting 7 minutes to hours: $\frac{7}{60}$ hours.

So, the equation is: $\frac{x}{6} = t - \frac{7}{60}$ …(2)

To find $x$ , subtract equation (2) from equation (1):

$\frac{x}{5} - \frac{x}{6} = (t + \frac{3}{60}) - (t - \frac{7}{60})$

Simplify the left side:

$\frac{6 x - 5 x}{30} = \frac{x}{30}$

Simplify the right side:

$t + \frac{3}{60} - t + \frac{7}{60} = \frac{3}{60} + \frac{7}{60} = \frac{10}{60}$

Equating both sides:

$\frac{x}{30} = \frac{10}{60}$

$\frac{x}{30} = \frac{1}{6}$

Solve for $x$ :

$x = \frac{30}{6}$

$x = 5 km$

Thus, the distance of the factory from his house is 5 km.

Incorrect Options:

Options 1 (3 km), 2 (4 km), and 4 (6 km) are incorrect because they do not satisfy the conditions derived from the given speeds and time differences. The calculated distance based on the established equations is uniquely 5 km.