Clocks Concepts

1. Angle Between the Hands

• What is the question? Find the angle between the hour hand and minute hand at a given time.

• Formula: $\theta = |30H - 5.5M|$

  If the answer is greater than $180^\circ$, use:

  $360^\circ-\theta$

• Example: Find the smaller angle between the hands at 4:20.

  Here:

  H = 4
  M = 20

  $\theta = |30(4)-5.5(20)|$

  $= |120-110|$

  $=10^\circ$

  Answer: $10^\circ$

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2. Finding the Time When Hands Coincide

• What is the question? Find the exact time between two given hours when the hour and minute hands overlap.

• Formula: $M=\frac{60H}{11}$

  where $H$ is the starting hour.

• Example: At what time between 5 and 6 do the hands coincide?

  $M=\frac{60(5)}{11}$

  $=\frac{300}{11}$

  $=27\frac{3}{11}\text{ minutes}$

  Answer: 5:27 $\frac\{3\}\{11\}$ minutes

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3. Finding the Time When Hands Are at $90^\circ$

• What is the question? Find the time between two given hours when the hour and minute hands form a right angle.

• Formula:

  $|30H-5.5M|=90$

  Solve for $M$.

• Example: Find the time between 3 and 4 when the hands are at $90^\circ$.

  $|30(3)-5.5M|=90$

  $|90-5.5M|=90$

  This gives:

  90 - 5.5M = 90

  giving $M=0$, so 3:00 is one solution.

  The other solution:

  90 - 5.5M = -90

  $5.5M=180$

  $M=\frac{180}{5.5}=\frac{360}{11}$

  $=32\frac{8}{11}$

  Answer: 3:00 and 3:32 $\frac\{8\}\{11\}$

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4. Finding the Time When Hands Are Opposite

• What is the question? Find the time when the hour and minute hands are exactly $180^\circ$ apart.

• Formula:

  $|30H-5.5M|=180$

• Example: Find the time between 5 and 6 when the hands are opposite.

  $|30(5)-5.5M|=180$

  $|150-5.5M|=180$

  Taking the valid value:

  $150-5.5M=-180$

  $5.5M=330$

  $M=60$

  This reaches 6:00, so there is no interior time between 5 and 6 at exactly $180^\circ$.

  For aptitude questions, always check whether the calculated minute actually lies between $0$ and $60$.

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5. Relative Speed of Clock Hands

• What is the question? Questions asking how long it takes the hands to gain a particular angle on each other.

• Formula:

  Minute hand speed = 6°/minute
  Hour hand speed   = 0.5°/minute

  Therefore:

  $\text{Relative speed}=6-0.5=5.5^\circ/\text{minute}$

• Example: How much time does the minute hand take to gain $55^\circ$ on the hour hand?

  $\text{Time}=\frac{55}{5.5}$

  $=10\text{ minutes}$

  Answer: 10 minutes

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6. Fast and Slow Clocks

• What is the question? A clock gains or loses time at a fixed rate. Find the actual time or the time shown by the clock.

• Formula:

  $\text{Total error}=\text{Error per hour}\times\text{Number of hours}$

• Example: A clock gains 3 minutes every hour. It was correct at 6 AM. What is the actual time when the clock shows 6 PM?

  From 6 AM to the clock’s 6 PM:

  Clock time = 12 hours
  Gain       = 3 × 12 = 36 minutes

  Since the clock is 36 minutes fast:

  $12\text{ hours}-36\text{ minutes}$

  $=11\text{ h }24\text{ min}$

  Therefore:

  6 AM + 11 h 24 min
  = 5:24 PM

  Answer: 5:24 PM

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7. Clock Loses Time

• What is the question? A clock loses a fixed amount of time per hour. Find the actual time or displayed time.

• Formula:

  $\text{Total loss}=\text{Loss per hour}\times\text{Elapsed hours}$

• Example: A clock loses 5 minutes every hour. It was correct at 8 AM. What time will it show at 2 PM?

  Actual elapsed time:

  8 AM → 2 PM = 6 hours

  Loss:

  $6\times5=30\text{ minutes}$

  Therefore the clock shows:

  2:00 PM - 30 minutes
  = 1:30 PM

  Answer: 1:30 PM

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8. Clock Correct Again / Synchronization

• What is the question? A faulty clock gains or loses time. Find when it will show the correct time again.

• Formula:

  If a clock gains $g$ minutes per hour:

  $\text{Time for 12-hour error}=\frac{720}{g}\text{ hours}$

  If it loses $l$ minutes per hour:

  $\text{Time for 12-hour error}=\frac{720}{l}\text{ hours}$

• Example: A clock gains 6 minutes per hour. After how many hours will it be 12 hours ahead?

  $\frac{720}{6}=120\text{ hours}$

  $120\text{ hours}=5\text{ days}$

  Answer: 5 days

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9. Mirror Image of a Clock

• What is the question? A mirror shows a clock reading. Find the actual time.

• Formula:

  $\text{Actual time}=11:60-\text{Mirror time}$

• Example: A mirror shows 3:40. What is the actual time?

  11:60
  -3:40
  -----
   8:20

  Answer: 8:20

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10. Finding Mirror Time

• What is the question? The actual clock time is given. Find what time will appear in its mirror.

• Formula:

  $\text{Mirror time}=11:60-\text{Actual time}$

• Example: The actual time is 7:25. What will the mirror show?

  11:60
  -7:25
  -----
   4:35

  Answer: 4:35

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11. Number of Coincidences in a Day

• What is the question? Find how many times the hour and minute hands coincide in 12 or 24 hours.

• Formula:

  In 12 hours → 11 coincidences
  In 24 hours → 22 coincidences

• Example: How many times do the hands of a clock coincide in 24 hours?

  $11\times2=22$

  Answer: 22 times

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12. Number of Right Angles in a Day

• What is the question? Find how many times the hands form a $90^\circ$ angle.

• Formula:

  In 12 hours → 22 times
  In 24 hours → 44 times

• Example: How many times do the hands form a right angle in 24 hours?

  $22\times2=44$

  Answer: 44 times

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13. Number of Opposite Positions in a Day

• What is the question? Find how many times the hands are exactly $180^\circ$ apart.

• Formula:

  In 12 hours → 11 times
  In 24 hours → 22 times

• Example: How many times are the hands opposite in 24 hours?

  $11\times2=22$

  Answer: 22 times