Clocks Questions and Answers

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.

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 \( \tfrac{1440}{720/11} = 22 \).

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.

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.

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.

Answer: D

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

\[ \theta = \left| 30H - 5.5M \right| \]

\[ \theta = \left| 300 - 137.5 \right| = 162.5^{\circ} \]

The reflex angle is what is left of the full turn:

\[ 360 - 162.5 = 197.5^{\circ} = 197\tfrac{1}{2}^{\circ} \]

A reflex angle is by definition greater than \( 180^{\circ} \), so the smaller 162.5 cannot be the answer.

Answer: D

Correct answer: 22

In each hour the hands are normally at right angles twice, which would suggest 24 in twelve hours. But between 2 and 3, and again between 8 and 9, one of those positions falls outside the hour, so two are lost, giving 22.

Equivalently, the minute hand gains \(360°\) on the hour hand 11 times in 12 hours, and each gain produces two right angles: \(11 \times 2 = 22\).

Answer: B

"Straight" covers both cases: the hands together and the hands opposite.

\[ 22 \text{ coincidences} + 22 \text{ oppositions} = 44 \text{ times a day} \]

Coincidentally this equals the number of right angles, but for a different reason.

Answer: C

At 3 o'clock, the hands of the watch are 15 min. space apart.
To be in opposite directions, they must be 30 min. spaces apart.
Minute hand must gain 45 min. space (15 + 30 min)
55 min spaces are gained in 60 min.
45 min. spaces would be gained in (60/55 × 45) min.
Required time = 49 1/11 min. past 3.

Answer: A

Correct answer: 10 (10/11) min past 2

At 2 o’clock the hour hand is \(60°\) ahead of the minute hand. The minute hand gains \(6 - 0.5 = 5.5°\) each minute, so it closes the gap in

\[t = \frac{60}{5.5} = \frac{120}{11} = 10\tfrac{10}{11}\ \text{minutes}\]

So the hands coincide at \(10\tfrac{10}{11}\) minutes past 2.