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A person leaves his house between 4.00 and 5.00 pm. He carefully notes the position of the minute hand and hour hand when he leaves the house. He returns back between 7.00 and 8.00 pm.He notices that the hour hand and the minute hand have exactly interchanged their positions. What time did he leave his house?

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    Hi, welcome to Math.StackExchange. Since you're new here, I would like to mention that it's considered good form, when posting homework questions, to not just give us the task, but also tells us what you have tried so far, and where you are stuck.2012-05-15

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$\def\deg{{^\circ}}$ The key thing with these clock puzzles is that the hands are not independent; when the minute hand goes forward by some angle $x$, the hour hand also goes forward by the amount $x\over 12$. The hour hand turns $360\deg$ in 12 hours, so $30\deg$ in one hour, and that is $\frac12\deg$ per minute. The minute hand turns $6\deg$ in one minute.

If the time is h:mm, where $0\le mm \lt 60$ and $0\le h \lt 12$, then the minute hand is at position $6\deg\cdot mm$ degrees and the hour hand is at position $h\cdot30\deg+mm\cdot{1\over 2}\deg$ degrees—the $h\cdot30\deg$ term is how far it has turned to get the the beginning of hour $h$, and then the $mm\cdot{1\over 2}\deg$ is how many degrees it has turned in the $mm$ minutes since the beginning of the hour.

Now we need two times, 4:mm and 7:nn, where the hand positions are reversed. The first has the minute hand at $6\deg mm$ and the hour hand at $120\deg + mm\cdot{1\over 2}\deg$. The second has the minute hand at $6\deg nn$ and the hour hand at $210\deg + nn\cdot{1\over 2}\deg$. So we need:

$ \begin{eqnarray} 6\deg mm & = & 210\deg + nn\cdot{1\over 2}\deg \\ 6\deg nn & = & 120\deg + mm\cdot{1\over 2}\deg \end{eqnarray} $

Dividing through by $6\deg$ gives:

$ \begin{eqnarray} mm & = & 35 + {nn\over 12} \\ nn & = & 20 + {mm\over 12} \end{eqnarray} $

By substitution we get $mm = 36+\frac{132}{143}$ minutes, and $nn = 23+\frac{11}{143}$ minutes. The two times are therefore approximately 4:36:55.359 and 7:23:04.615.

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    There is [another exposition of similar techniques here](http://math.stackexchange.com/questions/103117/deriving-two-properties-of-clocks).2012-07-15