Given:
- Total distance = $6$ km
- Total time = $45$ min = $\frac{45}{60}$ hours = $\frac{3}{4}$ hours
- Distance covered in $\frac{1}{2}$ of the distance = $\frac{1}{2} \times 6 = 3$ km
- Time taken for this distance = $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ hours
Remaining distance = $6 - 3 = 3$ km
Remaining time = $\frac{3}{4} - \frac{1}{2} = \frac{6}{12} - \frac{3}{6} = \frac{1}{4}$ hours
Speed required for the remaining distance:
$ Speed = \frac{Remaining Distance}{Remaining Time} = \frac{3}{1/4} = 3 \times 4 = 12 $ kmph
Thus, the required speed is 12 kmph