The Hands of a Clock

The clock can be an interesting source of mathematical applications. These can be applications in mathematics and not in other disciplines as we usually find mathematics being applied.

Begin by asking someone to determine the exact time that the hands of a clock will overlap after 4:00 o’clock. Their first reaction to the solution to this problem will most likely be that the answer is simply 4:20.

When you remind them that the hour hand moves uniformly while the minute hand moves faster, they will begin to estimate the answer to be between 4:21 and 4:22. They should realize that the hour hand moves through an interval between minute markers every 12 minutes. Therefore, it will leave the interval 4:21–4:22 at 4:24. This, however, doesn’t answer the original question about the exact time of this overlap.

You could show them a technique, once they realize that this is not the correct answer, since the hour hand does not remain stationary and moves when the minute hand moves.

The trick:    Simply multiply the 20 (the wrong answer) by ¹²/₁₁ to get 21(⁹/₁₁), which yields the correct answer: 4: 21(⁹/₁₁).