Solution

Suppose we let T be the time in minutes past 3 o'clock. In other words, T minutes past 3 o'clock, the minute hand points to T minutes on the clock, and the hour hand points to (1/12) T + 15 minutes. (The hour hand starts 15 minutes ahead of the minute hand, but it goes only 1/12th as fast.)

Number of minutes the minute hand is after 3 = T
Number of minutes the hour hand is after 3 = (1/12)T + 15

Hence, the two hands point in the same direction when these values are the same, or when

T = (1/12) T + 15

Solving this equation, gives

T = (12/11)(15) = 16 (4/11) = 16.363636 ...

In other words, 16.363636... minutes past 3. Converting to seconds, this is approximately 16 minutes and 22 seconds past 3 o'clock.

Note: You can also find the time between 3 o'clock and 4 o'clock when the two hands point in the opposite direction by solving

T - 30 = (1/12) T + 15

which gives T = 49.090909... , or approximately 49 minutes and 5 seconds past 3 o'clock.

Now that you know what happens between 3 o'clock and 4 o'clock, you can find the other times during the day the hands of the clock point in the same direction. In fact, what is the total number of times the two hands point in the same direction in a day ? Most people give the wrong answer to this question.

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Last modified on Tuesday, January 12, 1999