Ladies' Diary/Hour and Minute Hand in Straight Line
Puzzle
- "Being at so large a distance from the dial-plate of a great clock, that I could not distinguish the figures;
- but as the hour and minute hands were very bright and glaring,"
- the correspondent noted that they were in a straight line and pointing upwards to the right.
- It was evening.
- What was the time?
Solution
Proof
Let it be assumed from the description of the general direction of the hands, and the fact that it is evening, that it is some short time after $7$ p.m.
Thus the hour hand is pointing a little way past the $7$.
Hence the minute hand is pointing a little way past the $1$.
Let $x$ be the number of minutes after $7$ that it is.
Let $\theta$ be the angle made by the hands of the clock from the vertical at that time.
The minute hand rotates at $1$ revolution every $60$ minutes.
That is, $\dfrac {360 \degrees} {60} = 6 \degrees$ per minute.
The hour hand rotates at $1$ revolution every $12$ hours.
That is, $\dfrac {360 \degrees} {12 \times 60} = \dfrac 1 2 \degrees$ per minute.
But at $7$, the hour hand is already at $\dfrac {360 \degrees} {12} = 30 \degrees$ from the vertical.
Thus:
| \(\ds \theta\) | \(=\) | \(\ds \dfrac x 2 + 30\) | angle of hour hand | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 x\) | angle of minute hand | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\paren {12 - 1} x} 2\) | \(=\) | \(\ds 30\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \dfrac {60} {11}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \tfrac 5 {11}\) |
So it is $5 \frac 5 {11}$ minutes past $7$.
$\dfrac 5 {11}$ minutes is $\dfrac {5 \times 60} {11}$ seconds, which evaluates to $27.\dot 2 \dot 7$ seconds.
$\blacksquare$
Sources
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Ladies' Diary or Woman's Almanac, $\text {1704}$ – $\text {1841}$: $142$