At what time after 4:00 PM will the hands of the clock be together?

4:15
4:40
4:30
None of the above

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Hour handtakes a complete rotation of $360^\circ$ in 12 hours i.e 720 minutes. so hour hand traces $\frac {1}{2}^\circ$ in a minute.Minute handtraces $6^\circ$ in a minute. So, Angular speed ofminute handrelative tohour handis $6-\frac{1}{2}=\frac{11}{2}^\circ$ per minute.At

4:00, the angle between the hour and minute hand will be $120^\circ$ . so to overlap, minute hand will have to cover $120^\circ$ relative to hour hand. so time taken to overlap = $\frac{120}{\frac{11}{2}}=\frac{240}{11}=$21.81minutes(approx.).Hence, they will overlap 21.81 minutes past 4.