Clocks Questions and Answers

Q: The minute-hand and second-hand of a clock cross each other _______ times between 09:15:00 AM and 09:45:00 AM on a day.

The hands cross 30 times.The second hand turns at 6° per second and the minute hand at 0.1° per second, so the second hand gains 5.9° per second. At 09:15:00 the minute hand points at 3 (90°) and the second hand at 12 (0°), so the second hand starts 90° behind. It overtakes the minute hand each time it has gained 90° + 360°k:\[ \begin{aligned} t_{k} &= \frac{90 + 360k}{5.9} \le 1800 \\ 90 + 360k &\le 10620 \Rightarrow k \le 29.25 \end{aligned} \]So k = 0, 1, ..., 29: 30 crossings, the first about 15 s after 09:15:00 and the last about 15 s before 09:45:00. Over 30 minutes the second hand gains 29.5 laps, and 29 is that figure rounded down, missing the crossing that the 90° starting gap adds. 31 counts one lap too many, and 15 is half the true count.

Q: Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?

Rohit spent \(65\frac{5}{11}\) minutes at the restaurant.The minute hand moves 6° per minute and the hour hand 0.5° per minute, so the minute hand gains 5.5° per minute. To coincide again it must gain a full 360°:\[ \begin{aligned} t &= \frac{360}{5.5} = \frac{720}{11} \\ &= 65\frac{5}{11}\ \text{min} \end{aligned} \]He arrived when the hands met just after 1 o'clock (at about 1:05) and left when they next met (at about 2:11), so the time between is one full coincidence interval.The other mixed numbers do not equal 720/11; the ones with 13 in the denominator are not based on the hands' relative speed at all.

Q: A clock is set right at 5 a.m. The clock looses 16 minutes in 24 hours. Then the true time when the clock indicates 10 p.m. on 4th day will be:

The true time will be 11 p.m.From 5 a.m. on the first day to 10 p.m. on the fourth day, the clock shows 3 days and 17 hours, which is 89 hours. The clock loses 16 minutes a day, so it shows 1424 minutes for every 1440 true minutes.\[ \begin{aligned} \text{true} &= 89 \times \frac{1440}{1424} \\ &= 89 \times \frac{90}{89} = 90\ \text{hours} \end{aligned} \]90 true hours after 5 a.m. on day 1 is 11 p.m. on day 4, an hour later than the clock shows. A losing clock always shows less time than has passed, so the true time is later than 10 p.m., which rules out 8 p.m. and 9 p.m.; 12 p.m. overstates the loss.

Q: Given below are two statements:Statement I : An accurate clock shows the time as 3:00 pm. After the hour hand has moved 135°, the time would be 7:30 pm.Statement II : The angle between the minute hand and hour hand when the clock shows 4:25 pm. is 17.5°.In the light of the above statements, choose the correct answer from the options given below:

Both Statement I and Statement II are true.The hour hand moves 30° per hour, so 135° is \(135 / 30 = 4.5\) hours. From 3:00 pm that is 7:30 pm. Statement I is true.For Statement II use the clock-angle formula \(|30H - 5.5M|\):\[ |30 \times 4 - 5.5 \times 25| = |120 - 137.5| = 17.5^\circ \]Statement II is true. The trap is to forget that the hour hand moves on from the 4 during the 25 minutes (by 12.5°).

Q: At what time between 6 and 7 o'clock will the hands of a watch coincides with each other?

The hands coincide at \(32\frac{8}{11}\) minutes past 6.At 6:00 the minute hand is 30 minute-spaces behind the hour hand. It gains 55 minute-spaces every 60 minutes, so it catches up after\[ \begin{aligned} 30 \times \tfrac{60}{55} &= \tfrac{360}{11} \\ &= 32\tfrac{8}{11}\ \text{min} \end{aligned} \]The general rule is \(\frac{60}{11} \times 5h\) minutes past \(h\) o'clock. The other options correspond to other hours or arithmetic slips.

Q: What is the angle between the hour hand and the minute hand of a clock when the clock shows 10 hours 10 minutes ?

The angle is 245°, the reflex angle.The angle between the hands at \(H\) hours \(M\) minutes is \(|30H - 5.5M|\).\[ \begin{aligned} |30 \times 10 - 5.5 \times 10| &= |300 - 55| \\ &= 245^\circ \end{aligned} \]The smaller angle between the hands is \(360^\circ - 245^\circ = 115^\circ\), which is not among the options, so the keyed answer is the reflex angle of 245°. 240° (or 120°) is what you get by leaving the hour hand fixed on the 10, forgetting that it moves 5° past it in 10 minutes.

Q: What is the reflex angle between the hands of a clock at 4:30 ?

Correct answer: 315°First find the ordinary angle between the hands.\[\text{Hour hand} = 4 \times 30 + 30 \times 0.5 = 120 + 15 = 135°\]\[\text{Minute hand} = 30 \times 6 = 180°\]The hands are \(180 - 135 = 45°\) apart, so the reflex angle is\[360 - 45 = 315°\]

Important Instructions

The minute-hand and second-hand of a clock cross each other _______ times between 09:15:00 AM and 09:45:00 AM on a day.

A.30
B.15
C.29
D.31

Exam Year(s)

GATE ME - 2023,

Answer: A

The hands cross 30 times.

The second hand turns at 6° per second and the minute hand at 0.1° per second, so the second hand gains 5.9° per second. At 09:15:00 the minute hand points at 3 (90°) and the second hand at 12 (0°), so the second hand starts 90° behind. It overtakes the minute hand each time it has gained 90° + 360°k:

\[ \begin{aligned} t_{k} &= \frac{90 + 360k}{5.9} \le 1800 \\ 90 + 360k &\le 10620 \Rightarrow k \le 29.25 \end{aligned} \]

So k = 0, 1, ..., 29: 30 crossings, the first about 15 s after 09:15:00 and the last about 15 s before 09:45:00. Over 30 minutes the second hand gains 29.5 laps, and 29 is that figure rounded down, missing the crossing that the 90° starting gap adds. 31 counts one lap too many, and 15 is half the true count.

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Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?

A.\(64\frac{6}{11}\)
B.\(66\frac{5}{13}\)
C.\(65\frac{5}{11}\)
D.\(66\frac{6}{13}\)

Exam Year(s)

GATE Physics - 2025,

Answer: C

Rohit spent \(65\frac{5}{11}\) minutes at the restaurant.

The minute hand moves 6° per minute and the hour hand 0.5° per minute, so the minute hand gains 5.5° per minute. To coincide again it must gain a full 360°:

\[ \begin{aligned} t &= \frac{360}{5.5} = \frac{720}{11} \\ &= 65\frac{5}{11}\ \text{min} \end{aligned} \]

He arrived when the hands met just after 1 o'clock (at about 1:05) and left when they next met (at about 2:11), so the time between is one full coincidence interval.

The other mixed numbers do not equal 720/11; the ones with 13 in the denominator are not based on the hands' relative speed at all.

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A clock is set right at 5 a.m. The clock looses 16 minutes in 24 hours. Then the true time when the clock indicates 10 p.m. on 4th day will be:

A.11 p.m.
B.8 p.m.
C.9 p.m.
D.12 p.m.

Exam Year(s)

HPPSC HPAS - 2021,

Answer: A

The true time will be 11 p.m.

From 5 a.m. on the first day to 10 p.m. on the fourth day, the clock shows 3 days and 17 hours, which is 89 hours. The clock loses 16 minutes a day, so it shows 1424 minutes for every 1440 true minutes.

\[ \begin{aligned} \text{true} &= 89 \times \frac{1440}{1424} \\ &= 89 \times \frac{90}{89} = 90\ \text{hours} \end{aligned} \]

90 true hours after 5 a.m. on day 1 is 11 p.m. on day 4, an hour later than the clock shows. A losing clock always shows less time than has passed, so the true time is later than 10 p.m., which rules out 8 p.m. and 9 p.m.; 12 p.m. overstates the loss.

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Given below are two statements:

Statement I : An accurate clock shows the time as 3:00 pm. After the hour hand has moved 135°, the time would be 7:30 pm.
Statement II : The angle between the minute hand and hour hand when the clock shows 4:25 pm. is 17.5°.

In the light of the above statements, choose the correct answer from the options given below:

A.Both Statement I and Statement II are true.
B.Both Statement I and Statement II are false.
C.Statement I is true but Statement II is false.
D.Statement I is false but Statement II is true.

Exam Year(s)

Haryana Civil Services - 2026, HPSC - 2026,

Answer: A

Both Statement I and Statement II are true.

The hour hand moves 30° per hour, so 135° is \(135 / 30 = 4.5\) hours. From 3:00 pm that is 7:30 pm. Statement I is true.

For Statement II use the clock-angle formula \(|30H - 5.5M|\):

\[ |30 \times 4 - 5.5 \times 25| = |120 - 137.5| = 17.5^\circ \]

Statement II is true. The trap is to forget that the hour hand moves on from the 4 during the 25 minutes (by 12.5°).

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At what time between 6 and 7 o'clock will the hands of a watch coincides with each other?

A.\(23\frac{2}{11}\) minutes past 6
B.\(27\frac{1}{11}\) minutes past 6
C.\(32\frac{8}{11}\) minutes past 6
D.\(33\frac{1}{11}\) minutes past 6

Exam Year(s)

ISRO Assistant - 2023,

Answer: C

The hands coincide at \(32\frac{8}{11}\) minutes past 6.

At 6:00 the minute hand is 30 minute-spaces behind the hour hand. It gains 55 minute-spaces every 60 minutes, so it catches up after

\[ \begin{aligned} 30 \times \tfrac{60}{55} &= \tfrac{360}{11} \\ &= 32\tfrac{8}{11}\ \text{min} \end{aligned} \]

The general rule is \(\frac{60}{11} \times 5h\) minutes past \(h\) o'clock. The other options correspond to other hours or arithmetic slips.

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What is the angle between the hour hand and the minute hand of a clock when the clock shows 10 hours 10 minutes ?

A.240°
B.130°
C.245°
D.120°

Exam Year(s)

Kerala PSC KAS - 2025,

Answer: C

The angle is 245°, the reflex angle.

The angle between the hands at \(H\) hours \(M\) minutes is \(|30H - 5.5M|\).

\[ \begin{aligned} |30 \times 10 - 5.5 \times 10| &= |300 - 55| \\ &= 245^\circ \end{aligned} \]

The smaller angle between the hands is \(360^\circ - 245^\circ = 115^\circ\), which is not among the options, so the keyed answer is the reflex angle of 245°. 240° (or 120°) is what you get by leaving the hour hand fixed on the 10, forgetting that it moves 5° past it in 10 minutes.

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At what time between 3 and 4 o'clock will the hands of a watch point in opposite directions ?

A.\(49\frac{1}{11}\) minutes past 3
B.50 minutes past 3
C.\(49\frac{5}{11}\) minutes past 3
D.\(49\frac{10}{11}\) minutes past 3

Exam Year(s)

KPSC KAS - 2020,

Answer: A

The hands point in opposite directions at \(49\frac{1}{11}\) minutes past 3.

At 3 o'clock the hour hand is 15 minute-divisions ahead of the minute hand. In m minutes the minute hand moves m divisions and the hour hand \(\frac{m}{12}\). For the hands to be opposite, the minute hand must be 30 divisions ahead of the hour hand:

\[ \begin{aligned} m &= 15 + \frac{m}{12} + 30 \\ \frac{11m}{12} &= 45 \\ m &= \frac{540}{11} = 49\frac{1}{11} \end{aligned} \]

The minute hand gains \(5\frac{1}{2}\) degrees a minute and must gain 90 + 180 = 270 degrees, which gives the same \(\frac{270}{5.5} = 49\frac{1}{11}\) minutes. 50 minutes ignores the hour hand's movement, and the other fractions come from arithmetic slips.

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A Clock is started at Noon, by 10 minutes past 7, the hour hand has turn through:

A.215
B.180
C.210
D.195

Answer: A

In a clock, 12 hours form 360° degree. So one hour forms 30°. Therefore 7 hours form 210° (30° X 7) degree. In a minute, hour hand moves 0.5° degree, so in 10 minutes it moves 0.5° X 10 = 5°
Thus, by 10 Minutes past 7, hour hand has turn through 210° + 5° = 215°

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At what time between 7.02 and 8.02 will the hands of a watch be in the straight line but no together?

A.5 (3/11) min past 7
B.5 (5/11) min past 7
C.5 (3/11) min past 7
D.43 min past 7

Answer: B

When the hands of the clock are in the same straight line but not
together, they are 30 minute spaces apart.
At 7 o'clock, they are 25 min. spaces apart.
Minute hand must gain only 5 min. space
55 min. spaces are gained in 60 min.
5 min. spaces are gained in (60/55 × 5) min = 5 (5/11) min.
Required time = 5 (5/11) min. past 7.

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10.

What is the reflex angle between the hands of a clock at 4:30 ?

A.300°
B.305°
C.310°
D.315°

Answer: B

Correct answer: 315°

First find the ordinary angle between the hands.

\[\text{Hour hand} = 4 \times 30 + 30 \times 0.5 = 120 + 15 = 135°\]

\[\text{Minute hand} = 30 \times 6 = 180°\]

The hands are \(180 - 135 = 45°\) apart, so the reflex angle is

\[360 - 45 = 315°\]

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