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

From ProofWiki
Jump to navigation Jump to search

Modern Puzzles by Henry Ernest Dudeney: $25$

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:

\(\ds \theta\) \(=\) \(\ds \dfrac t 2\) angle moved by hour hand
\(\ds t\) \(=\) \(\ds \dfrac {240 - \theta} 6\) angle moved by minute hand
\(\ds t\) \(=\) \(\ds \dfrac {240 - \paren {t / 2} } 6\) substituting for $\theta$
\(\ds \leadsto \ \ \) \(\ds t\) \(=\) \(\ds \dfrac {480} {13}\)
\(\ds \) \(=\) \(\ds 36 \tfrac {12} {13}\)

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

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Modern_Puzzles/25_-_A_Dreamland_Clock/Solution&oldid=667002"