The hour hand and the minute hand of a correct clock are together at a certain moment. After how many minutes will they next be together?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 65\(\frac{5}{11}\) minutes
The minute hand turns 6° a minute and the hour hand 0.5° a minute, so the minute hand gains 5.5° a minute on the hour hand. After the hands are together, they meet again when the minute hand has gained one full round, 360°, on the hour hand.
Gain needed \(=360^\circ\)
Gain in one minute \(=6^\circ-0.5^\circ=5.5^\circ\)
Time \(=\displaystyle\frac{360}{5.5}=\frac{720}{11}=65\frac{5}{11}\) minutes
For example, together at 12:00, they are next together at 1:05\(\frac{5}{11}\)
Why the other options are incorrect
Option (2) 60 minutes — Incorrect. In 60 minutes the minute hand is back at its place but the hour hand has moved on 30°, so they are not together.
Option (3) 64 minutes — Incorrect. In 64 minutes the gain is only \(64\times5.5^\circ=352^\circ\), 8° short of a full round.
Option (4) 66 minutes — Incorrect. In 66 minutes the gain is \(363^\circ\), so the minute hand is already 3° past the hour hand.