Clocks Questions and Answers

Answer: C

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

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

\[ \theta = \left| 30(3) - 5.5(40) \right| = \left| 90 - 220 \right| = 130^{\circ} \]

Taking \( 30H \) as the hour hand's position from 12 automatically includes its drift during the 40 minutes, which is the step most often forgotten.

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: 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: 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: C

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock).

AM

12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55
 

PM

12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

The hands overlap about every 65 minutes, not every 60 minutes.

 The hands coincide 22 times in a day.

Answer: C

In 12 hours, the hands coincide or are in opposite direction 22 times.

 In 24 hours, the hands coincide or are in opposite direction 44 times a day.

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

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

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

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

At 4 o' clock, the minute hand is 20 minute spaces behind the hour hand.
Now, when the two hands are at right angles, they are fifteen min spaces
apart. So, they are at right angles in following two cases.

Case I ? When the minute hand is 15 min spaces behind the hour hand.
In this case, min hand will have to gain (20-15) = 5 min spaces.
= (60/55)*5 = 60/11
∴ They are at right angle at 60/11 minutes past 4.
Case II ? When the minute hand is 15 minute spaces ahead of the hour hand.
In this case the minute hand will have to gain (20+15) = 35 min spaces.
= (60/55)*35 = 420/11 min
∴ They are at right angles at 420/11 min past 4.

Answer: C

Time from 8 am on Sunday to 8 pm on following Sunday = 7 days 12 hours
= 180 hours

∴ The watch gains (5 + 29/5) min or 54/5 min in 180 hours.

Now 54/5 minutes are gained in 180 hours.

∴ 5 minutes are gained in [180*(5/54)*5] hour = 83 hr 20 min
= 3 days 11 hours 20 minutes
Watch is correct 3 days 11 hours 20 minutes
after 8 AM of Sunday i.e. 7:20PM on Wednesday.