Episode 8 — Aptitude and Reasoning / 8.19 — Clocks
8.19.c Clocks - Solved Examples
Example 1: Basic Angle Calculation
Problem: Find the angle between the hands of a clock at 3:20.
Solution:
H = 3, M = 20
Angle = |30H - 5.5M|
= |30(3) - 5.5(20)|
= |90 - 110|
= |-20|
= 20 degrees
The angle at 3:20 is 20 degrees.
Example 2: Angle at Half Hour
Problem: Find the angle between the hands at 7:30.
Solution:
H = 7, M = 30
Angle = |30(7) - 5.5(30)|
= |210 - 165|
= 45 degrees
Example 3: Angle Greater Than 180
Problem: Find the angle between the hands at 8:10.
Solution:
H = 8, M = 10
Angle = |30(8) - 5.5(10)|
= |240 - 55|
= 185 degrees
Since 185 > 180, the acute/obtuse angle = 360 - 185 = 175 degrees
The angle is 175 degrees.
Example 4: Angle at Exact Hour
Problem: Find the angle at 5:00.
Solution:
H = 5, M = 0
Angle = |30(5) - 5.5(0)|
= |150 - 0|
= 150 degrees
Example 5: Finding Time of Overlap
Problem: At what time between 4 and 5 do the hands of a clock overlap?
Solution:
H = 4, angle = 0
M = 60H/11 = 60(4)/11 = 240/11 = 21 + 9/11 minutes
The hands overlap at 4:21 and 9/11 minutes.
That is approximately 4:21:49.
Example 6: Finding Time of Right Angle
Problem: At what time between 3 and 4 are the hands at right angles?
Solution:
H = 3, angle = 90
Case 1: 30H - 5.5M = 90
90 - 5.5M = 90
5.5M = 0
M = 0
Time = 3:00 (this is the starting point of the interval)
Case 2: 30H - 5.5M = -90
90 - 5.5M = -90
5.5M = 180
M = 180/5.5 = 360/11 = 32 + 8/11 minutes
Time = 3:32 and 8/11 minutes
The hands are at right angles at 3:00 and at 3:32 and 8/11 minutes.
Example 7: Straight Line (180 degrees)
Problem: At what time between 2 and 3 are the hands in a straight line but not coinciding?
Solution:
H = 2, angle = 180
30H - 5.5M = 180 or 30H - 5.5M = -180
Case 1: 60 - 5.5M = 180
5.5M = -120
M = -120/5.5 (negative, so invalid)
Case 2: 60 - 5.5M = -180
5.5M = 240
M = 240/5.5 = 480/11 = 43 + 7/11 minutes
The hands are at 180 degrees at 2:43 and 7/11 minutes.
Example 8: Specific Angle Between Hands
Problem: At what times between 5 and 6 is the angle between the hands 60 degrees?
Solution:
H = 5, angle = 60
Case 1: 30(5) - 5.5M = 60
150 - 5.5M = 60
5.5M = 90
M = 90/5.5 = 180/11 = 16 + 4/11 minutes
Case 2: 30(5) - 5.5M = -60
150 - 5.5M = -60
5.5M = 210
M = 210/5.5 = 420/11 = 38 + 2/11 minutes
The angle is 60 degrees at:
5:16 and 4/11 minutes (~5:16:22)
5:38 and 2/11 minutes (~5:38:11)
Example 9: How Many Times Do Hands Overlap in a Day?
Problem: How many times do the minute and hour hands overlap in a day?
Solution:
In 12 hours, the hands overlap 11 times.
(Not 12, because between 11 and 1 there is only one overlap at 12:00)
In 24 hours = 2 * 12 hours:
Overlaps = 2 * 11 = 22 times per day.
Example 10: Right Angles in 12 Hours
Problem: How many times are the hands at right angles in 12 hours?
Solution:
In every hour, the hands are at right angles twice (approximately).
In 12 hours, that would be 24 times.
BUT: Between 2-3 and 3-4, one right angle is shared (at 3:00).
Similarly between 8-9 and 9-10, one is shared (at 9:00).
Wait, the correct reasoning:
The hands make a right angle every 360/11 minutes.
In 12 hours (720 minutes): 720 / (360/11) = 720 * 11/360 = 22 times.
The hands are at right angles 22 times in 12 hours.
Example 11: Mirror Image Problem
Problem: A clock shows 3:45 in a mirror. What is the actual time?
Solution:
Actual time = 11:60 - Mirror time
= 11:60 - 3:45
= 8:15
The actual time is 8:15.
Example 12: Mirror Image (Afternoon)
Problem: Looking at a clock in a mirror, you see 9:20. What is the actual time?
Solution:
Actual time = 11:60 - 9:20
= 2:40
The actual time is 2:40.
Example 13: Mirror Image (Near 12)
Problem: A mirror shows 12:25. What is the real time?
Solution:
12:25 is past 11:60, so we use:
Actual time = 23:60 - 12:25 = 11:35
The actual time is 11:35.
Example 14: Angle Traced by Minute Hand
Problem: Find the angle traced by the minute hand in 35 minutes.
Solution:
Angle = 6 * t (degrees)
= 6 * 35
= 210 degrees
Example 15: Angle Traced by Hour Hand
Problem: Find the angle traced by the hour hand between 2:00 PM and 5:30 PM.
Solution:
Time elapsed = 3 hours 30 minutes = 210 minutes
Angle = 0.5 * t
= 0.5 * 210
= 105 degrees
Example 16: Faulty Clock - Gaining Time
Problem: A clock gains 5 minutes every hour. If the clock was set right at 12:00 noon, what time does the clock show when the actual time is 6:00 PM?
Solution:
Real time elapsed = 6 hours = 360 minutes
Clock gains 5 min per hour, so in 1 real hour the clock shows 65 minutes.
Clock time for 360 real minutes:
Clock shows = 360 * 65/60 = 390 minutes = 6 hours 30 minutes
The clock shows 6:30 PM when the actual time is 6:00 PM.
Example 17: Faulty Clock - Losing Time
Problem: A clock loses 3 minutes every hour. If the clock shows 8:00 AM, and it was set correctly at midnight, what is the correct time?
Solution:
Clock time from midnight = 8 hours = 480 clock-minutes
The clock shows 57 minutes for every 60 real minutes.
(It loses 3 minutes per hour, so it shows 57 min for every real 60 min.)
Real time = 480 * 60/57 = 28800/57 = 505.26 minutes
= 8 hours 25.26 minutes
= approximately 8:25 AM and 16 seconds
The correct time is approximately 8:25 AM.
Example 18: When Will a Faulty Clock Show Correct Time?
Problem: A clock gains 10 minutes per day. After how many days will it show the correct time again?
Solution:
The clock needs to gain exactly 12 hours = 720 minutes to show correct time.
Gain per day = 10 minutes
Days = 720/10 = 72 days
The clock will show the correct time after 72 days.
Example 19: Two Faulty Clocks
Problem: Clock A gains 2 minutes per hour and Clock B loses 3 minutes per hour. If both are set correctly at 12:00 noon, when will they show the same time again?
Solution:
Difference per hour = 2 + 3 = 5 minutes
They show the same time when the total difference = 12 hours = 720 minutes.
Hours until same time = 720/5 = 144 hours = 6 days
They will show the same time after 6 days, i.e., at 12:00 noon on the 7th day.
Example 20: Angle at a Quarter Past
Problem: What is the angle at 10:15?
Solution:
H = 10, M = 15
Angle = |30(10) - 5.5(15)|
= |300 - 82.5|
= 217.5 degrees
Since 217.5 > 180:
Angle = 360 - 217.5 = 142.5 degrees
Example 21: Time When Angle is Known
Problem: At what time between 7 and 8 is the angle between the hands 120 degrees?
Solution:
H = 7, X = 120
Case 1: M = (30H - X)/5.5 = (210 - 120)/5.5 = 90/5.5 = 180/11 = 16 + 4/11 min
Case 2: M = (30H + X)/5.5 = (210 + 120)/5.5 = 330/5.5 = 660/11 = 60 min
Case 2 gives M = 60, which means 8:00, not between 7 and 8.
So the angle is 120 degrees at 7:16 and 4/11 minutes only.
Example 22: Overlap After a Given Time
Problem: The hands of a clock overlap at 12:00 noon. When is the next overlap?
Solution:
Time between consecutive overlaps = 720/11 minutes
= 65 + 5/11 minutes
= 65 minutes and 27.27 seconds
Next overlap at: 12:00 + 65 min 27 sec = 1:05:27 (approximately)
More precisely: 1 hour, 5 minutes, and 5/11 minutes past 12:00.
Example 23: Reflex Angle
Problem: Find the reflex angle between the hands at 4:40.
Solution:
H = 4, M = 40
Angle = |30(4) - 5.5(40)|
= |120 - 220|
= 100 degrees
This is the acute/obtuse angle.
Reflex angle = 360 - 100 = 260 degrees
Example 24: Combined Problem - Angle and Direction
Problem: At 9:50, is the minute hand ahead of or behind the hour hand? And what is the angle?
Solution:
H = 9, M = 50
Position of hour hand from 12: 30(9) + 0.5(50) = 270 + 25 = 295 degrees
Position of minute hand from 12: 6(50) = 300 degrees
Minute hand (300) is ahead of hour hand (295) by 5 degrees.
Angle = |30(9) - 5.5(50)| = |270 - 275| = 5 degrees
The minute hand is 5 degrees ahead of the hour hand.
Example 25: Clock Problem with Time Duration
Problem: A clock strikes once at 1 o'clock, twice at 2 o'clock, and so on. How many times does it strike in 12 hours?
Solution:
Total strikes = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
= 12 * 13 / 2
= 78
The clock strikes 78 times in 12 hours.
Next: 8.19 - Practice MCQs