Clocks Questions and Answers

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.

Answer: B

The hour hand sweeps the full \( 360^{\circ} \) in twelve hours, so it turns \( 30^{\circ} \) an hour.

\[ 6 \text{ hours} \times 30^{\circ} = 180^{\circ} \]

Only the hour hand is asked about; the minute hand would have turned \( 6 \times 360^{\circ} \) over the same period.

Answer: B

From 8 am Sunday to 8 pm the following Sunday is

\[ 7 \times 24 + 12 = 180 \text{ hours} \]

\[ \text{total gain} = 5 + 5\tfrac{4}{5} = 10.8 \text{ minutes} \]

\[ \text{rate} = \frac{10.8}{180} = 0.06 \text{ min per hour} \]

Starting 5 minutes slow, it becomes correct after

\[ \frac{5}{0.06} = 83\tfrac{1}{3} \text{ h} = 83 \text{ h } 20 \text{ min} \]

\[ 8 \text{ a.m. Sunday} + 83 \text{ h } 20 \text{ min} = 7{:}20 \text{ p.m. on Wednesday} \]

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

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

With \( H = 8 \) and \( \theta = 90 \):

\[ \left| 240 - 5.5M \right| = 90 \]

\[ 5.5M = 150 \quad\Longrightarrow\quad M = \frac{300}{11} = 27\tfrac{3}{11} \]

The other root gives \( M = 60 \), which is 9 o'clock exactly and so outside the hour asked about.

Answer: A

Time from 5am on a day to 10 pm on fourth day= 89 hours
Presently 23hrs. 44min. of this clock= 24 hours of right clock.
∴ 356/15 hrs of this clock= (24*15/356*89) hrs. Of right clock.
= 90 hrs of right clock.
Along these lines, the right time is 11 pm.

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: C

Correct answer: 120°

The minute hand completes \(360°\) in 60 minutes, so it turns \(6°\) each minute.

\[20 \times 6 = 120°\]

This is a third of the dial, matching the fact that 20 minutes is a third of an hour.