At what time between 4 and 5 o’clock are the hands of a clock at an angle of 48°?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4:13\(\frac{1}{11}\)
At 4:00 the minute hand is 120° behind the hour hand (hour hand at \(4\times30^\circ=120^\circ\), minute hand at 0°), and it gains 5.5° a minute. The angle is 48° once before the minute hand passes the hour hand and once after.
Before passing: gap \(120^\circ-5.5m=48^\circ\), so \(5.5m=72\) and \(\displaystyle m=\frac{72}{5.5}=\frac{144}{11}=13\frac{1}{11}\)
After passing: \(5.5m-120^\circ=48^\circ\), so \(5.5m=168\) and \(\displaystyle m=\frac{336}{11}=30\frac{6}{11}\), a time that is not among the options
So the time in the options is \(13\frac{1}{11}\) minutes past 4
Why the other options are incorrect
Option (2) 4:32\(\frac{8}{11}\) — Incorrect. The minute hand has gained \(5.5\times\frac{360}{11}=180^\circ\), so it is 60° past the hour hand.
Option (3) 4:30\(\frac{8}{11}\) — Incorrect. The minute hand has gained \(5.5\times\frac{338}{11}=169^\circ\), so it is 49° past the hour hand, not 48°.
Option (4) 4:10\(\frac{10}{11}\) — Incorrect. The minute hand has gained \(5.5\times\frac{120}{11}=60^\circ\), so it is still 60° behind the hour hand.