Clocks Questions and Answers

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

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

Opposite directions means \( \theta = 180 \), with \( H = 4 \):

\[ \left| 120 - 5.5M \right| = 180 \]

\[ 5.5M = 300 \quad\Longrightarrow\quad M = \frac{600}{11} = 54\tfrac{6}{11} \]

The other root, \( 5.5M = -60 \), is negative and discarded.

Answer: D

The minute hand covers \( 360^{\circ} \) in sixty minutes:

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

\[ 6 \times 6 = 36^{\circ} \]

The 6° option is the rate rather than the total, and 30° is the hour hand's hourly figure.

Answer: C

In each twelve-hour period the hands are at right angles 22 times - twice in most hours, but the crossings near 3 and 9 o'clock overlap so that two are lost.

\[ 22 \times 2 = 44 \text{ times in 24 hours} \]

The count is not 48, because between 2–3 and 3–4 (and again around 9) one right angle is shared rather than occurring twice.

Answer: D

At noon on Sunday watch gains 2 Minutes.
And at 2PM on following Sunday it is 4 Minutes 48 Seconds fast.
So, it gains 6 Minutes 48 Seconds i.e. 2 Minutes + 4 Minutes 48 Seconds (total 408 Seconds) in 170 Hours (Hours from Sunday Noon (12PM) to following Sunday 2PM)
Therefore, watch gains 120 Seconds (2 Minutes) in 170*120/408 = 50 Hours
50 Hours = 2 Days 2 Hours
Thus, Watch was correct after 2 days 2 hours from Sunday noon that is Tuesday 2PM.

Answer: D

Correct answer: 9:10 a.m.

From one Monday to the next is exactly 7 days.

\[\text{Total gain} = 7 \times 10 = 70 \text{ minutes} = 1 \text{ hour } 10 \text{ minutes}\]

Running ahead by that amount, the clock shows 9:10 a.m. when the true time is 8 a.m.

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.

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°

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.

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

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

\[ \frac{2}{0.04} = 50 \text{ hours} \]

\[ \text{noon Monday} + 50 \text{ h} = 2 \text{ p.m. on Wednesday} \]