Let the distance between the worker’s house and the factory be $d$ km.

Case 1: Speed = $5$ km/hr Time taken at $5$ km/hr = $d / 5$ hours. The worker is $3$ minutes late, so the actual time taken is $t + 3$ minutes (where $t$ is the ideal time). Case 2: Speed = $6$ km/hr Time taken at $6$ km/hr = $d / 6$ hours. The worker is $7$ minutes early, so the actual time taken is $t - 7$ minutes.

Set up the equation The difference in time between the two speeds is: $(d / 5) - (d / 6) = (3 + 7) / 60 = 10 / 60 = 1 / 6$ hours

Simplifying the left-hand side: $(d / 5) - (d / 6) = (6d - 5d) / 30 = d / 30$

Thus: $d / 30 = 1 / 6$ Multiplying both sides by $30$:

$d = 5$ km