Henry Ernest Dudeney/Modern Puzzles/25 - A Dreamland Clock

by : $25$

 * A Dreamland Clock
 * In a dream, I was travelling in a country where they had strange ways of doing things.
 * One little incident was fresh in my memory when I awakened.
 * I saw a clock and announced the time as it appeared to be indicated.
 *  but my guide corrected me.
 * He said, "You are apparently not aware that the minute hand always moves in the opposite direction to the hour hand.
 * Except for this improvement, our clocks are precisely the same as those you have been accustomed to."


 * Now, as the hands were exactly together between the hours of $4$ and $5$ o'clock,
 * and they started together at noon,
 * what was the real time?

Solution

 * $36 \tfrac {12} {13}$ minutes after $4$.

Proof
We have that:
 * the minute hand takes $1$ hour to go $360 \degrees$ around the dial anticlockwise
 * the hour hand takes $1$ hour to go $30 \degrees$ around the dial clockwise.

Hence in one minute:
 * the minute hand travels $6 \degrees$ anticlockwise
 * the hour hand travels $\dfrac 1 2 \degrees$ clockwise.

At $4$ o'clock, the minute hand was on the $12$ and the hour hand was on the $4$.

Let the minute hand and hour hand be coincident at some point in time $t$ minutes after $4$, between $4$ and $5$.

We need to find $t$.

Let $\theta$ degrees be the angle the hour hand has moved (clockwise), in time $t$.

Then we have:
 * $\theta = \dfrac t 2 \degrees$

During $t$, the minute hand, moving backwards as it does, has moved anticlockwise from $12$ o'clock to $5$ o'clock plus $30 - \theta \degrees$.

That is, the minute hand has moved $7 \times 30 + \paren {30 - \theta} \degrees$, that is, $\paren {240 - \theta} \degrees$.

Hence:
 * $t = \dfrac {240 - \theta} 6$

as the minute hand moves $1 \degrees$ every $\dfrac 1 6$ of a minute.

Hence we have:

That is, the time is $36 \tfrac {12} {13}$ minutes after $4$.