Henry Ernest Dudeney/Puzzles and Curious Problems/61 - Right and Left/Solution

by : $61$

 * Right and Left

Solution

 * $41 \tfrac 7 {13}$ past $3$.


 * Dudeney-Puzzles-and-Curious-Problems-61-Solution.png

Proof
Let $T$ be the time in question.

Let $m$ be the number of minutes past $3:00$ that $T$ is.

Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock at time $T$.

Let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock at time $T$.

Between $3:00$ and $4:00$ we have that $90 < \phi < 120$.

Hence from Condition for Valid Time Indication, we have that:
 * $12 \paren {\phi - 90} = \theta$

Then we are told that:
 * $\theta = 360 - \phi$

Hence:

Hence the result.