Clocks Questions and Answers

Answer: B

Correct answer: 11 p.m.

While the true time advances 1440 minutes, this clock advances only \(1440 - 16 = 1424\) minutes.

From 5 a.m. on the first day to 10 p.m. on the fourth is 3 days and 17 hours, or 89 hours, as shown by the clock. Scaling to true time:

\[89 \times \frac{1440}{1424} = 89 \times \frac{90}{89} = 90 \text{ hours}\]

Adding 90 hours to 5 a.m. gives 3 days and 18 hours later, which is 11 p.m. on the fourth day.

Answer: A

At h o' clock, the minutes hand is 5 h minute spaces
behind the hour hand.
The minute hand gains 55 min spaces in 60 mins.

∴ The minute hand will gain 5h minute spaces
in 60/55 * 5h = 60h/11 minutes.

∴ The two hands will be together between
'h' and 'h+1' o' clock at (60/11)h minutes h o' clock.

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.