Clocks Questions and Answers

Answer: A

Correct answer: 22

The hands point in opposite directions once for every full \(360°\) the minute hand gains on the hour hand. That gain happens 11 times in 12 hours, so the hands are opposite 11 times in 12 hours and 22 times in 24 hours.

The same reasoning gives 22 coincidences in 24 hours, since each relative revolution produces exactly one of each.

Answer: A

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

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

"Same straight line but not together" means \( \theta = 180 \), with \( H = 7 \):

\[ \left| 210 - 5.5M \right| = 180 \]

\[ 5.5M = 30 \quad\Longrightarrow\quad M = \frac{60}{11} = 5\tfrac{5}{11} \]

The other root, \( 5.5M = 390 \), gives \( M = 70.9 \), beyond the hour.

Answer: A

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

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

Together means \( \theta = 0 \), with \( H = 2 \):

\[ 5.5M = 60 \quad\Longrightarrow\quad M = \frac{120}{11} = 10\tfrac{10}{11} \]

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

Answer: B

Correct answer: 95°

Measure both hands from the 12 o’clock mark.

\[\text{Hour hand} = 5 \times 30 + 10 \times 0.5 = 150 + 5 = 155°\]

\[\text{Minute hand} = 10 \times 6 = 60°\]

\[\text{Angle} = 155 - 60 = 95°\]

Answering \(90°\) ignores the \(5°\) the hour hand has moved past the 5.

Answer: D

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: D

The dial is divided into 60 minute-marks around \( 360^{\circ} \), so each mark spans

\[ \frac{360}{60} = 6^{\circ} \]

Being 25 minute-marks apart:

\[ 25 \times 6 = 150^{\circ} \]

The hands named do not matter here - the separation is given directly in minute-divisions.

Answer: A

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

The clock runs fast, so it reaches 11 am before the true time does.

From Sunday 1 am to Monday 11 am is 34 hours on the clock.

\[ \text{gain} = 20 \times \frac{34}{24} = \frac{85}{3} = 28\tfrac{1}{3} \text{ min} = 28 \text{ min } 20 \text{ s} \]

The true time is that much behind the clock:

\[ 11{:}00 - 28{:}20 = 10{:}31{:}40 \text{ am} \]

So the alarm sounded 31 minutes 40 seconds past 10 am, true time.

Answer: A

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

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

Opposite directions means a straight line with \( \theta = 180 \), and \( H = 2 \):

\[ \left| 60 - 5.5M \right| = 180 \]

\[ 5.5M = 240 \quad\Longrightarrow\quad M = \frac{480}{11} = 43\tfrac{7}{11} \]

The other root, \( 5.5M = -120 \), is negative and so discarded.