Calculation of the time for which hands of clock meet

1

How many times do the hands of a clock coincide in a period of 12 hours?

A. 10 B. 11 C. 12 D. 22

Show Answer

✔ Correct: B

The hands coincide 11 times in 12 hours because they do not coincide between 11 and 1 o'clock except exactly at 12.

2

What is the angle between the hands of a clock at 3:15?

A. 0 degrees B. 7.5 degrees C. 10 degrees D. 12.5 degrees

Show Answer

✔ Correct: B

Angle = |(30 * H) - (5.5 * M)|. Here H=3, M=15. Angle = |90 - 82.5| = 7.5 degrees.

3

At what time between 4 and 5 o'clock do the hands of a clock coincide?

A. 4:21 9/11 B. 4:22 8/11 C. 4:20 D. 4:25

Show Answer

✔ Correct: A

The formula is T = (60/11) * H. For H=4, T = 240/11 = 21 9/11 minutes past 4.

1

Assertion (A): The hands of a clock coincide exactly 22 times in a day.
Reason (R): The hands of a clock meet once every 65 5/11 minutes.

Directions: Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both Assertion and Reason are true and the Reason is the correct explanation.

2

Assertion (A): The angle between the minute hand and hour hand at 6 o'clock is 180 degrees.
Reason (R): The hour hand moves 0.5 degrees per minute.

Directions: Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: B

Both are true, but the reason does not directly explain why the angle is 180 degrees at 6 o'clock.

1

Calculate the time between 2 and 3 o'clock when the hands are at right angles.

Show Answer & Solution

At 2 o'clock, the gap is 10 min spaces. For right angle, gap must be 15 or 25 min spaces. Case 1: 10 + x = 15 => x = 5 min. Time = 5 * 12/11 = 5 5/11 min past 2. Case 2: 10 + x = 25 => x = 15 min. Time = 15 * 12/11 = 16 4/11 min past 2.

2

Find the angle traced by the minute hand in 15 minutes.

Show Answer & Solution

The minute hand moves 360 degrees in 60 minutes. Therefore, it moves 360/60 = 6 degrees per minute. In 15 minutes, angle = 15 * 6 = 90 degrees.

1

Derive the formula for the time at which the hands of a clock coincide between H and (H+1) o'clock.

Show Answer & Solution

Step 1: At H o'clock, the minute hand is at 12 and hour hand is at H. The distance between them is 5H minute spaces. Step 2: To coincide, the minute hand must gain 5H spaces. Step 3: Relative speed is 55/60 = 11/12 spaces/min. Step 4: Time T = Distance / Relative Speed = 5H / (11/12) = 60H/11 minutes past H.

2

Calculate the exact time between 8 and 9 o'clock when the hands of a clock are opposite to each other (180 degrees).

Show Answer & Solution

Step 1: For the hands to be opposite, the difference between their positions must be 30 minute spaces. Step 2: At 8 o'clock, the minute hand is at 0 and the hour hand is at 40 (measured from 12). Step 3: To be opposite, the minute hand must cover the 40-minute gap plus 30 more to reach the 10 position, totaling 10 minutes past 12 position relative to the hour hand. Calculation: M = (60/11) * (H + 6) for H < 6, or (60/11) * (H-6) for H >= 6. Here H=8, M = (60/11) * (8-6) = 120/11 = 10 10/11. Time is 10 10/11 minutes past 8.

1

Case Study: A clock is a circular dial with 60 minute marks. The minute hand moves 6 degrees per minute, and the hour hand moves 0.5 degrees per minute. The hands of a clock meet at regular intervals, known as the gain time period.

Sub-question 1: What is the relative speed of the minute hand with respect to the hour hand?

Show Solution

Relative speed = 6 degrees/min - 0.5 degrees/min = 5.5 degrees/min.

Sub-question 2: How many minutes does it take for the hands to meet again once they have coincided?

Show Solution

The hands coincide when the minute hand covers 360 degrees relative to the hour hand. Time = 360 / 5.5 = 720/11 = 65 5/11 minutes.

Relative Angular Velocity of Clock Hands

General Formula for Coincidence

Section A: Multiple Choice Questions (MCQs)

Section B: Assertion-Reason Questions

Section C: Short Answer Questions

Section D: Long Answer Questions

Section E: Case-Based Questions

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