An accurate clock is started at noon. By 10 minutes past 3, the hour hand has turned through:
Answer: B
The hour hand turns \( 30^{\circ} \) an hour, or \( 0.5^{\circ} \) a minute.
From noon to 3:10 is \( 3\tfrac{1}{6} \) hours:
\[ 3\tfrac{1}{6} \times 30 = 95^{\circ} \]
The extra ten minutes contribute \( 10 \times 0.5 = 5^{\circ} \), which is what lifts the answer above the whole-hour figure.
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At what angle are the hands of a clock inclined at 15 minutes past 4?
Answer: D
At 4 o’ clock, the hour hand is at 120° from the vertical.
In 15 minutes,
Hour hand = 120 + 15 x (0.5) = 120 + 7.5 = 127.5° (The hour hand moves at 0.5 dpm)
Minute hand = 15 x (6) = 90° (The minute hand moves at 6 dpm)
Difference or Angle between the hands = 127.5 - 90 = 37.5°
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At what time, in minutes, between 3 o'clock and 3.25 pm, both the needles of the clock will coincide each other?
Answer: D
At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand.
To be coincident, it must gain 15 min. spaces.
55 min. are gained in 60 min
15 min. spaces are gained in 60/55 x 15 = 16 4/11 min
The hands are co-incident at 16 (4/11) past 3.
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Rohit has a watch that gains 5 seconds in 3 minutes. It was set right at 9 a.m. in the morning.
In the afternoon of the same day, when the watch indicated 4 o'clock, how much time will the watch have gained?
Answer: C
Time from 9 a.m. to 4 p.m. = 7 hrs
The clock gains 5 secs in 3 min
In 60 min, it will gain, (60 x 5) / 3 = 100 sec
i.e. the clock gains 100 seconds in one hour
In 7 hours, it will gain 700 sec.
700/60 mins = (11)2/3 mins
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A watch which gains uniformly is 2 minutes low at noon on Monday and is 4 min. 48 sec fast at 2 p.m. on the following Monday. When was it correct?
Answer: B
Find the total gain and the total time, then work out when the watch passed through correctness.
From noon Monday to 2 pm the following Monday is
\[ 7 \times 24 + 2 = 170 \text{ hours} \]
\[ \text{total gain} = 2 + 4\tfrac{4}{5} = 6.8 \text{ minutes} \]
\[ \text{rate} = \frac{6.8}{170} = 0.04 \text{ min per hour} \]
It starts 2 minutes slow, so it becomes correct after
\[ \frac{2}{0.04} = 50 \text{ hours} \]
\[ \text{noon Monday} + 50 \text{ h} = 2 \text{ p.m. on Wednesday} \]
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A clock loses 5 minutes every hour and was set right at 11AM on a Monday. When will it show the correct time again?
Answer: A
As mentioned in Key Point O, faulty clock will show correct time when it loses or gains 12 Hours.
In the given problem, clock loses 5 Minutes in an hour. So, 1 Minute lost in every 12 minutes (60 minutes / 5 minutes = 12 minutes).
Now, for losing 12 hours i.e. 720 Minutes(12 X 60 = 720) it will take 720 X 12 = 8640 Minutes = 144 Hours = 6 Days.
So, Clock will show correct time after 6 days from 11AM Monday.
Thus, Answer is 11AM on Sunday.
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How many times do the hands of a clock coincide in a day?
Answer: B
In twelve hours the hands coincide eleven times, not twelve: between 11 and 12 they meet only at 12 o'clock itself, so that hour contributes no separate coincidence.
\[ 11 \times 2 = 22 \text{ times in 24 hours} \]
Equivalently, the minute hand laps the hour hand once every \( \tfrac{720}{11} = 65\tfrac{5}{11} \) minutes, and \( \tfrac{1440}{720/11} = 22 \).
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The angle between the minute hand and the hour hand of a clock when the time is 8.30, is:
Answer: B
The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is
\[ \theta = \left| 30H - 5.5M \right| \]
\[ \theta = \left| 30(8) - 5.5(30) \right| = \left| 240 - 165 \right| = 75^{\circ} \]
At half past the hour the hour hand has moved \( 15^{\circ} \) beyond the 8, which is why the answer is not the round \( 90^{\circ} \) between the 6 and the 8.
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At what time between 4 and 5 o’clock will the hands of a clock first be at right angles ?
Answer: D
Correct answer: 5 (5/11) min past 4
At 4 o’clock the hour hand leads by \(120°\). For the first right angle the minute hand must reduce that lead to \(90°\), a relative gain of \(30°\).
\[t = \frac{30}{5.5} = \frac{60}{11} = 5\tfrac{5}{11}\ \text{minutes}\]
The second right angle in that hour comes when the minute hand has gained \(210°\), at \(38\tfrac{2}{11}\) minutes past 4.
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A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours will be the true time when the clock indicates 1 p.m. on the following day?
Answer: A
The clock gains, so it reaches 1 pm before the true time does.
From 8 am to 1 pm the next day is 29 hours on the clock. The clock's 24 h 10 min corresponds to 24 true hours:
\[ \text{true time} = 29 \times 60 \times \frac{1440}{1450} = 1740 \times \frac{144}{145} \]
\[ = 1728 \text{ minutes} = 28 \text{ h } 48 \text{ min} \]
\[ 8 \text{ a.m.} + 28 \text{ h } 48 \text{ min} = 12{:}48 \text{ p.m.} \]
So the true time is 48 minutes past 12.
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