What is the angle between the hands of a clock at 3:00?

90°. At 3:00 the minute hand is at 12 and the hour hand at 3, three hour-marks apart: 3 30^ = 90^ .

How many degrees does the minute hand of a clock move in 20 minutes?

120°. The minute hand moves 6^ per minute, so in 20 minutes it moves 6 20 = 120^ .

What is the angle between the hands of a clock at 4:20?

10°. Hour hand = 30(4) + 0.5(20) = 130^ ; minute hand = 6(20) = 120^ . The angle between them is 130 - 120 = 10^ .

At what time between 2 and 3 o'clock do the hands of a clock coincide?

2:10 1011. At 2:00 the hour hand leads the minute hand by 60^ . The minute hand gains 5.5^ per minute, so it closes the gap in 605.5 = 12011 = 10 1011 minutes past 2.

A clock gains 5 minutes every hour. If it is set right at 12 noon, what time will it show when the correct time is 6 p.m. the same day?

6:30 p.m.. In the 6 hours from noon to 6 p.m. the clock gains 5 6 = 30 minutes, so it shows 6:30 p.m.

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Clocks — Questions & Answers

The minute hand moves 6° per minute, the hour hand 0.5° per minute, so the minute hand gains 5.5° on the hour hand each minute. Most clock questions are one application of that gain.

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Key formulas

Angle of hour hand from 12: $30H + 0.5M$ degrees; minute hand: $6M$ degrees
Angle between hands $= |30H - 5.5M|$ degrees (take $360 -$ that if it exceeds 180)
The hands coincide every $65\tfrac{5}{11}$ minutes

5 solved Clocks questions

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01
What is the angle between the hands of a clock at 3:00?
4 steps · reveal one at a time
1. Step 1: Hand Positions
  First, let's identify where the hands are at 3:00. The minute hand will be pointing straight up at the 12. The hour hand will be pointing directly at the 3.
2. Step 2: Angle per Mark
  A full circle on a clock face is $360^\circ$ . There are 12 hour marks around the clock. This means the angle between any two consecutive hour marks is $360^\circ / 12 = 30^\circ$ .
3. Step 3: Calculate Total Angle
  At 3:00, the minute hand is at 12 and the hour hand is at 3. These positions are 3 hour marks apart (from 12 to 1, 1 to 2, and 2 to 3). So, the total angle is $3 \times 30^\circ = 90^\circ$ .
4. Step 4: The answer
  So the answer is (C) 90°. The hands of a clock at 3:00 form a right angle, which is $90^\circ$ .
02
How many degrees does the minute hand of a clock move in 20 minutes?
4 steps · reveal one at a time
1. Step 1: Clock face basics
  A clock face is a circle, which means it covers $360^\circ$ . The minute hand completes one full rotation around the clock in 60 minutes.
2. Step 2: Minute hand speed
  To find out how many degrees the minute hand moves per minute, we divide the total degrees by the total minutes: $360^\circ \div 60$ minutes. This gives us $6^\circ$ per minute.
3. Step 3: Calculate total movement
  Now we know the minute hand moves $6^\circ$ every minute. We need to find out how much it moves in 20 minutes. We multiply the degrees per minute by the number of minutes: $6^\circ \times 20$ minutes.
4. Step 4: The answer
  So the answer is (B) 120°. The minute hand moves $6^\circ$ per minute, so in 20 minutes it moves $6 \times 20 = 120^\circ$ .
03
What is the angle between the hands of a clock at 4:20?
3 steps · reveal one at a time
1. Step 1: Calculate Hour Hand
  First, let's find the angle of the hour hand from the 12 o'clock mark. The hour hand moves $30^\circ$ per hour and $0.5^\circ$ per minute. At 4:20, its angle is $30(4) + 0.5(20) = 120 + 10 = 130^\circ$ .
2. Step 2: Calculate Minute Hand
  Now, let's find the angle of the minute hand from the 12 o'clock mark. The minute hand moves $6^\circ$ per minute. At 20 minutes past the hour, its angle is $6(20) = 120^\circ$ .
3. Step 3: The answer
  So the answer is (D) 10°. The angle between the hands is the absolute difference between their positions: $130^\circ - 120^\circ = 10^\circ$ .
04
At what time between 2 and 3 o'clock do the hands of a clock coincide?
4 steps · reveal one at a time
1. Step 1: Initial hand positions
  First, let's find the positions of the hands at 2 o'clock. The minute hand points at 12, which is $0^\circ$ . The hour hand points at 2. Since each hour mark is $30^\circ$ , the hour hand is at $2 \times 30^\circ = 60^\circ$ . The hour hand leads by $60^\circ$ .
2. Step 2: Relative speed
  Next, we find how fast the minute hand gains on the hour hand. The minute hand moves $6^\circ$ per minute, and the hour hand moves $0.5^\circ$ per minute. So, the minute hand gains $6^\circ - 0.5^\circ = 5.5^\circ$ on the hour hand every minute.
3. Step 3: Time to coincide
  To coincide, the minute hand must cover the initial $60^\circ$ lead of the hour hand. Using the relative speed, the time taken is $\text{distance} / \text{speed} = 60^\circ / 5.5^\circ \text{ per minute}$ . This simplifies to $60 / (11/2) = 120/11$ minutes.
4. Step 4: The answer
  So the answer is (C) 2:10 $\tfrac{10}{11}$ . The time is $120/11$ minutes past 2 o'clock, which is $10\tfrac{10}{11}$ minutes past 2.
05
A clock gains 5 minutes every hour. If it is set right at 12 noon, what time will it show when the correct time is 6 p.m. the same day?
3 steps · reveal one at a time
1. Step 1: Calculate total time
  The clock is set correctly at 12 noon. We need to find out what time it shows when the correct time is 6 p.m. This means a total of $6$ hours have passed.
2. Step 2: Find total gain
  The problem states the clock gains $5$ minutes every hour. Since $6$ hours have passed, the total time gained by the faulty clock will be $5 \times 6 = 30$ minutes.
3. Step 3: The answer
  So the answer is (B) 6:30 p.m. The clock will show the correct time of 6 p.m. plus the $30$ minutes it gained, resulting in 6:30 p.m.

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