Henry Ernest Dudeney/Puzzles and Curious Problems/59 - Mistaking the Hands/Solution

by : $59$

 * Mistaking the Hands

Solution
The time was $02:05 \tfrac 5 {11}$, but it appeared to be $01 : 10 \tfrac 5 {11}$.


 * Dudeney-Puzzles-and-Curious-Problems-59-Solution.png

Proof
Let $T$ be the time in question.

Let $m$ be the number of minutes after $2:00$ at $T$.

The time mistaken is $T - 55$ minutes, that is $m + 5$ minutes after $1:00$.

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 $T$.

At $T - 55$ minutes, the angle made by the minute hand with respect to twelve o'clock is $\theta + 30 \degrees$.

But this is the angle of the hour hand at $T$.

Hence from Condition for Valid Time Indication, we have that:
 * $12 \paren {\phi - 60} = \theta$

as $T$ is between $2:00$ and $3:00$.

Hence:

Hence the time was $02:05 \tfrac 5 {11}$.