At what time between 4 o'clock and 5 o'clock will the hour hand and the minute hand coincide?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(4:21\frac{9}{11}\)
The minute hand turns 6° a minute and the hour hand 0.5° a minute, so the minute hand gains 5.5° on the hour hand every minute. At 4:00 the hour hand is on the 4 and the minute hand on the 12, so the minute hand starts 120° behind; the hands coincide when it has made up this gap.
Gap at 4:00: \(4\times30=120\)°
Time to make it up: \(\displaystyle\frac{120}{5.5}=\frac{240}{11}=21\frac{9}{11}\) minutes, so the hands meet at \(4:21\frac{9}{11}\)
Tip: Between H and (H + 1) o'clock the hands coincide \(\displaystyle\frac{60H}{11}\) minutes past H. For H = 4 this gives \(\frac{240}{11}\) minutes at once.