At what time between 7 and 8 o’clock are the hands of a clock at an angle of 12°?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 7:36
At 7:00 the minute hand is 210° behind the hour hand (hour hand at \(7\times30^\circ=210^\circ\), minute hand at 0°), and it gains 5.5° a minute. The angle is 12° once before the minute hand passes the hour hand and once after.
Before passing: gap \(210^\circ-5.5m=12^\circ\), so \(5.5m=198\) and \(\displaystyle m=\frac{198}{5.5}=36\)
After passing: \(5.5m-210^\circ=12^\circ\), so \(5.5m=222\) and \(\displaystyle m=\frac{444}{11}=40\frac{4}{11}\), a time that is not among the options
So the time in the options is 36 minutes past 7
Why the other options are incorrect
Option (1) 7:33 — Incorrect. The minute hand has gained \(5.5\times33=181.5^\circ\), so it is still 28.5° behind the hour hand.
Option (3) 7:37 — Incorrect. The minute hand has gained \(5.5\times37=203.5^\circ\), so it is only 6.5° behind the hour hand, not 12°.
Option (4) 7:30\(\frac{6}{11}\) — Incorrect. The minute hand has gained \(5.5\times\frac{336}{11}=168^\circ\), so it is still 42° behind the hour hand.