A clock is a circle (360°) divided into 12 hours (30° per hour) and 60 minutes (6° per minute).
Formulas:
- Minute Hand: Moves 6° per minute.
- Hour Hand: Moves 0.5° per minute (30° per hour).
- Angle Between Hands (θ) =∣30H−5.5M∣
where,
H = hour
M = minutes
Clock questions and answers are provided below for you to learn and practice.
Question 1: Find the angle between the hands of a clock at 3:20 PM.
Solution:
Movement of Clock Hands
Minute Hand
- Completes a full circle (360°) in 60 minutes
- Moves 6° per minute
Hour Hand
- Completes a full circle (360°) in 12 hours (720 minutes)
- Moves 30° per hour
- Also moves continuously: 0.5° per minute
Position at 3:20
Minute Hand at 20 minutes
- Movement = 20×6 = 120∘
- Position = 120° from 12 o’clock
Hour Hand at 3:00
- Position = 3×30 = 90∘
Hour Hand movement in 20 minutes
- Movement = 20×0.5 = 10∘
- New position = 90 + 10 = 100∘
Angle Between the Hands: Angle = 120∘ − 100∘ = 20∘
Question 2: At what time between 3 PM and 4 PM would the two hands of the clock be together?
Solution:
At 3:00 PM:
Hour hand: At 3 × 30° = 90° (from 12 o'clock).
Minute hand: At 0°.
Time After 3:00 PM When Hands Overlap:Let t = minutes after 3:00 PM when the hands overlap.
Minute hand position: 6t degrees.
Hour hand position: 90+0.5t degrees.At overlap:
6t = 90 + 0.5t
6t − 0.5t = 90
5.5t = 90
t = \frac{90}{5.5} = \frac{180}{11}\ \text{minutes} \approx 16 \tfrac{4}{11}\ \text{minutes} The hour and minute hands overlap at: approximately 3:16:22 PM
Question 3: How many times in a day the two hands of a clock coincide?
Solution:
Between 11 to 1, the hands of the clock coincide only once, i.e., at 12. At 12:00 AM and 12:00 PM, the hour hand and the minute hand do not coincide with each other So, every 12 hours, they coincide 11 times. Therefore, the two hands of the clock coincide 22 times in a day.
Question 4: At what time between 5 and 6 o'clock, do the minute and hour hands make an angle of 34 degree with each other.
Solution:
The angle between the minute hand and the hour hand at 5 o'clock is 150 degrees.
The angle between the hands becomes 34 degrees when the angle changes by 116 degrees and 184 degrees, i.e. (150-34) and (150+34).
The angle changes by 5.5 degrees in 1 min.
The angle changes by 116 degrees in 1/5.5 x 116 = 21 1/11 min.
The angle changes by 184 degrees in 1/5.5 x 184 = 33 5/11 min.
Therefore the angle between the two hands is 34 degrees when the time is 5 hr 21 1/11 min, and again at 5 hr 33 5/11 min.
Question 5: How many times do the hands of the clock coincide between 2 and 3 o'clock?
Solution:
Between 2:00 and 3:00, the hands of the clock will coincide once.
The time for the hands to coincide after 2:00 is approximately 2 hours and 10 minutes.
The general rule is that the hands coincide 11 times in 12 hours. Therefore, between 2:00 and 3:00, the hands will coincide once, as they do between every other hour.
Thus, the hands of the clock coincide once between 2:00 and 3:00.
Question 6: At what time between 12 PM and 1 PM would the two hands of the clock be together?
Solution:
At 12:00 PM:
Hour hand: At 0°0°0° (since it's 12:00).
Minute hand: At 0°0°0° (since it's 12:00).
Time after 12:00 PM when the hands overlap:
Let t = minutes after 12:00 PM when the hands overlap.Minute hand position: 6t degrees.
Hour hand position: 0+0.5t degrees.
At overlap: 6t = 0.5t
Simplifying:6t − 0.5t = 0
5.5t = 0t = \frac{0}{5.5} = 0 Therefore, the hands coincide exactly at 12:00 PM.
Question 7: How many times do the hands of a clock coincide between 4 o'clock and 5 o'clock?
Solution:
Between 4:00 and 5:00, the hands of the clock will coincide once.
The time for the hands to coincide after 4:00 is approximately 4 hours and 21 minutes.The general rule is that the hands coincide 11 times in 12 hours. Therefore, between 4:00 and 5:00, the hands will coincide once.
Thus, the hands of the clock coincide once between 4:00 and 5:00.
Question 8: At what times do the hands of a clock coincide between 6 o'clock and 7 o'clock?
Solution:
At 6:00, the hands are at different positions:
The minute hand is at 0° (on the 12).
The hour hand is at 180° (on the 6).
The minute hand moves 6° per minute, and the hour hand moves 0.5° per minute.
Let the time after 6:00 be ttt minutes when the hands coincide.
The position of the minute hand after ttt minutes is: 6t
The position of the hour hand after ttt minutes is: 180+0.5t
At the point of coincidence, the two hands are at the same position, so:
t6t = 180 + 0.5t 6t − 0.5t = 180
5.5t = 180
t=\frac{180}{5.5}=32 \frac{8}{11}minutes Thus, the hands of the clock coincide at approximately 6:32 and 8/11 minutes (or around 6:32:43).
So, the hands of the clock will coincide at approximately 6:32:43.