Let $x$ denote the angle between the hour hand and the vertical, and $y$ the angle between the minute hand and the vertical.

$\frac{360}{12}=30$ so we calculate that $x = \left( 4+\frac{2}{3} \right) \times 30 =140^\circ$ and $y = 8 \times 30 = 240^\circ$.

The angle between the hands is $y - x = 240^\circ - 140^\circ = 100^\circ$.

*This problem is taken from the UKMT Mathematical Challenges.*