Clocks Questions and Answers

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

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

The hands are together when \( \theta = 0 \), with \( H = 7 \):

\[ 30(7) = 5.5M \quad\Longrightarrow\quad 210 = 5.5M \]

\[ M = \frac{210}{5.5} = \frac{420}{11} = 38\tfrac{2}{11} \]

So they meet at \( 38\tfrac{2}{11} \) minutes past 7. The hands coincide once in each hour except between 11 and 12, which is why there are 22 coincidences a day and not 24.

Answer: B

The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. (Because between 5 and 7 they point in opposite directions at 6 o'clcok only).

So, in a day, the hands point in the opposite directions 22 times.

Answer: B

Correct answer: 100°

The hour hand covers \(360°\) in 12 hours, so it moves \(30°\) per hour, or \(0.5°\) per minute.

\[3 \times 30 + 20 \times 0.5 = 90 + 10 = 100°\]

The minute hand, by contrast, would sweep \(1200°\) in the same interval.

Answer: B

At 8 o’ clock, the hour hand is at 240 degrees from the vertical.
In 30 minutes,
Hour hand = 240 + 30 x (0.5) = 240 + 15 = 255 (The hour hand moves at 0.5 dpm)
Minute hand = 30 x (6) = 180 (The minute hand moves at 6 dpm)
Difference or angle between the hands = 255 - 180 = 75 degrees

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

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

\[ \left| 240 - 5.5M \right| = 180 \]

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

That is \( 10\tfrac{10}{11} \) minutes past 8. The other root, \( 5.5M = 420 \), gives \( M = 76.4 \), beyond the hour.

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.