In the hour after the time shown, for how long is the angle between the hands less than 30°?

At 4:00 the minute hand is $120^{\circ }$ behind and gains $5.5^{\circ }$/min. The angle is under $30^{\circ }$ while it is between $30^{\circ }$ behind and $30^{\circ }$ ahead, i.e. while its gain runs from $90^{\circ }$ to $150^{\circ }:\frac{60}{5.5}=10\frac{10}{11}$ min. (10 min uses $6^{\circ }$/min.)