Clocks

Angle between hands, time calculations

Clock problems are a common topic in aptitude tests that involve calculating angles between clock hands, determining times when hands are in specific positions (overlap, opposite, perpendicular), and solving problems related to fast or slow clocks. Understanding the movement rates of hour and minute hands is essential for solving these problems efficiently.

Key Formulas / Rules

\text{Angle traced by hour hand in 12 hours = 360 deg, so per hour = 30 deg}

\text{Angle traced by hour hand per minute = 0.5 deg}

\text{Angle traced by minute hand per minute = 6 deg}

\text{Relative speed of minute hand over hour hand = 5.5 deg per minute}

\text{Angle between hands at H hours M minutes = |30H - 5.5M|}

\text{If angle > 180 deg, subtract from 360 deg to get the smaller angle}

\text{Hands overlap every 65 5/11 minutes (12 times in 12 hours)}

\text{Hands are opposite every 32 8/11 minutes (11 times in 12 hours)}

\text{Hands are perpendicular 22 times in 12 hours (44 times in 24 hours)}

Key Concepts

Understanding Clock Hand Movements

The hour hand completes one full rotation (360 deg) in 12 hours, moving at 30 deg per hour or 0.5 deg per minute. The minute hand completes one full rotation in 60 minutes, moving at 6 deg per minute. The second hand moves at 6 deg per second. The relative speed between minute and hour hands is crucial - the minute hand gains 5.5 deg per minute over the hour hand.

Calculating Angle Between Hands

To find the angle at H hours M minutes, use the formula: |30H - 5.5M|. This gives the angle measured clockwise from the hour hand to the minute hand. If the result exceeds 180 deg, subtract it from 360 deg to get the smaller angle between the two hands. For example, at 3:40, the angle is |30x3 - 5.5x40| = |90 - 220| = 130 deg.

Finding Times When Hands Are in Special Positions

Hands overlap when the angle is 0 deg, occur 11 times in 12 hours (every 65 5/11 minutes). They are opposite (180 deg apart) 11 times in 12 hours. They are perpendicular (90 deg apart) 22 times in 12 hours. To find when hands form a specific angle , use: (30H +/- ) / 5.5 = minutes past H o'clock. The +/- depends on whether the minute hand is ahead (+) or behind (-) the hour hand.

Fast and Slow Clock Problems

A fast clock gains time, a slow clock loses time. If a clock gains/loses x minutes per hour, it shows incorrect time. To find correct time: Calculate total time elapsed, determine total gain/loss, add or subtract from shown time. For example, if a clock gains 5 minutes per hour and shows 4:00 after 6 real hours, the actual time is 3:30 (gained 30 minutes).

Mirror Image of Clock Time

The mirror image of a clock time can be found by subtracting the time from 12:00 (or 11:60 for easier calculation). For time H:M, mirror time = (11 - H):(60 - M). If minutes exceed 60, borrow 1 hour. For example, mirror of 3:20 is 8:40. If the result is 0 hours, use 12. This works because the clock face is symmetric about the vertical 12-6 line.

Solved Examples

Problem 1:

What is the angle between the hour hand and minute hand at 4:40?

Solution:

Using the formula: Angle = |30H - 5.5M|
Step 1: H = 4, M = 40
Step 2: Angle = |30x4 - 5.5x40| = |120 - 220| = |-100| = 100 deg
The angle between the hands at 4:40 is 100 deg.

Note: Since 100 deg < 180 deg, this is the smaller angle.

Problem 2:

At what time between 3 and 4 o'clock will the hands of a clock be together (overlap)?

Solution:

At 3:00, the hour hand is at 90 deg (3 x 30 deg), minute hand at 0 deg.
The minute hand needs to catch up 90 deg at a relative speed of 5.5 deg/min.
Time = Distance/Speed = 90 deg / 5.5 deg per minute
Time = 90 / 5.5 = 90 x 2/11 = 180/11 = 16 4/11 minutes

Answer: The hands overlap at 3:16 4/11 (or approximately 3:16:22).

Problem 3:

A watch gains 5 minutes per hour. It was set correctly at 9 AM. What is the correct time when the watch shows 6:30 PM?

Solution:

Time shown on watch: 9:00 AM to 6:30 PM = 9 hours 30 minutes = 570 minutes
Gain per hour: 5 minutes
Total gain in 570 watch minutes:
Since the watch runs fast, 65 minutes watch time = 60 minutes real time
Real time = (570 x 60) / 65 = 34200/65 = 526.15 minutes ~= 8 hours 46 minutes

Correct time = 9:00 AM + 8 hours 46 minutes = 5:46 PM (approximately).
Alternatively: 570 minutes shown, gain = (570/65) x 5 = 43.8 minutes
Actual time elapsed = 570 - 43.8 = 526.2 minutes = 8h 46m
Correct time: 5:46 PM

Problem 4:

What is the mirror image of 7:25 on a clock?

Solution:

To find mirror image, subtract from 11:60 (or 12:00 with adjustment).
Method: (11 - H):(60 - M)
Step 1: Hours = 11 - 7 = 4
Step 2: Minutes = 60 - 25 = 35

Answer: The mirror image is 4:35.

Verification: The sum of original and mirror time equals 11:60 when both have same AM/PM. 7:25 + 4:35 = 12:00 (or 11:60).

Tips & Strategies

Always draw a rough sketch of the clock face to visualize the positions of hour and minute hands - this helps avoid sign errors in calculations.
Remember that the relative speed of 5.5 deg/min comes from (6 deg - 0.5 deg) - the minute hand gains on the hour hand.
For angle problems, if your calculation gives more than 180 deg, subtract from 360 deg to get the smaller angle between the two hands.
When finding specific times (overlap, opposite, perpendicular), note that these occur at regular intervals - hands overlap every 65 5/11 minutes starting from 12:00.
For fast/slow clock problems, use the ratio method: if a clock gains x min per hour, then (60+x) minutes of wrong time = 60 minutes of correct time.
In mirror image problems, remember that left becomes right and vice versa - the sum of the original time and mirror time equals 12:00 (or 11:60 for easier calculation).
For hands perpendicular problems, there are two solutions in each hour (except when the angle condition is at the hour boundary).

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