Episode 8 — Aptitude and Reasoning / 8.19 — Clocks
8.19.a Clocks - Concepts and Formulas
1. Clock Structure and Basics
Anatomy of a Clock
A standard clock (analog) has:
- 12 hour markings (1 through 12)
- 60 minute markings (small divisions)
- A minute hand (longer hand)
- An hour hand (shorter hand)
- Some clocks have a second hand
Key Measurements
| Element | Value |
|---|---|
| Total angle of clock face | 360 degrees |
| Angle between consecutive hour marks | 360 / 12 = 30 degrees |
| Angle between consecutive minute marks | 360 / 60 = 6 degrees |
| Number of minute divisions between two hour marks | 5 |
| Angle per minute division | 6 degrees |
Visualization
12
11 1
10 2
9 3
8 4
7 5
6
Each hour mark is 30 degrees apart.
From 12 to 3 = 90 degrees (right angle)
From 12 to 6 = 180 degrees (straight line)
From 12 to 9 = 270 degrees (or 90 degrees the other way)
2. Speed of Clock Hands
Minute Hand Speed
The minute hand completes one full rotation (360 degrees) in 60 minutes.
Speed of minute hand = 360 / 60 = 6 degrees per minute
In 1 minute: minute hand moves 6 degrees
In 5 minutes: minute hand moves 30 degrees (one hour mark)
In 1 hour: minute hand moves 360 degrees (full rotation)
Hour Hand Speed
The hour hand completes one full rotation (360 degrees) in 12 hours = 720 minutes.
Speed of hour hand = 360 / 720 = 0.5 degrees per minute
In 1 minute: hour hand moves 0.5 degrees
In 1 hour: hour hand moves 30 degrees (one hour mark)
In 12 hours: hour hand moves 360 degrees (full rotation)
Relative Speed
The minute hand moves FASTER than the hour hand.
Relative speed = Speed of minute hand - Speed of hour hand
= 6 - 0.5
= 5.5 degrees per minute
This means: Every minute, the minute hand gains 5.5 degrees over the hour hand.
Speed Summary Table
| Hand | Speed (degrees/min) | Speed (degrees/hour) | Full rotation |
|---|---|---|---|
| Minute hand | 6 | 360 | 60 minutes |
| Hour hand | 0.5 | 30 | 12 hours |
| Relative | 5.5 | 330 | 65 5/11 min |
3. Angle Between the Hands
The Master Formula
The angle between the hour hand and minute hand at H hours and M minutes is:
Angle = |30H - 5.5M| degrees
Where:
H= hour (use 12-hour format, values 1-12)M= minutes past the hour- Take the absolute value
- If the result > 180, subtract from 360 to get the acute/obtuse angle
Derivation of the Formula
At H hours and M minutes:
Position of hour hand from 12 o'clock:
= 30*H + 0.5*M degrees
(30 degrees per hour already passed + 0.5 degrees per minute within the hour)
Position of minute hand from 12 o'clock:
= 6*M degrees
(6 degrees per minute from the 12)
Angle between them:
= |Position of hour hand - Position of minute hand|
= |(30H + 0.5M) - 6M|
= |30H - 5.5M|
Important Convention
If the calculated angle > 180 degrees:
Actual angle = 360 - calculated angle
(We typically want the smaller angle between the two hands)
Examples
Example 1: Find the angle at 3:20
H = 3, M = 20
Angle = |30(3) - 5.5(20)|
= |90 - 110|
= |-20|
= 20 degrees
Example 2: Find the angle at 7:45
H = 7, M = 45
Angle = |30(7) - 5.5(45)|
= |210 - 247.5|
= |-37.5|
= 37.5 degrees
Example 3: Find the angle at 5:30
H = 5, M = 30
Angle = |30(5) - 5.5(30)|
= |150 - 165|
= |-15|
= 15 degrees
Example 4: Find the angle at 9:00
H = 9, M = 0
Angle = |30(9) - 5.5(0)|
= |270 - 0|
= 270 degrees
Since 270 > 180:
Actual angle = 360 - 270 = 90 degrees
4. Hands Overlapping (0 degrees)
When do the hands overlap?
The hands overlap when the angle between them is 0 degrees:
30H - 5.5M = 0
M = 30H / 5.5
M = 60H / 11
Overlap Times for Each Hour
| Between | Overlap at | Exact time |
|---|---|---|
| 12 and 1 | 12:00:00 | H=12, M=0 (starting point) |
| 1 and 2 | 1:05:27.27 | M = 60/11 = 5 + 5/11 min |
| 2 and 3 | 2:10:54.55 | M = 120/11 = 10 + 10/11 min |
| 3 and 4 | 3:16:21.82 | M = 180/11 = 16 + 4/11 min |
| 4 and 5 | 4:21:49.09 | M = 240/11 = 21 + 9/11 min |
| 5 and 6 | 5:27:16.36 | M = 300/11 = 27 + 3/11 min |
| 6 and 7 | 6:32:43.64 | M = 360/11 = 32 + 8/11 min |
| 7 and 8 | 7:38:10.91 | M = 420/11 = 38 + 2/11 min |
| 8 and 9 | 8:43:38.18 | M = 480/11 = 43 + 7/11 min |
| 9 and 10 | 9:49:05.45 | M = 540/11 = 49 + 1/11 min |
| 10 and 11 | 10:54:32.73 | M = 600/11 = 54 + 6/11 min |
| 11 and 12 | 12:00:00 | Coincides with next cycle |
Key Facts About Overlaps
- In 12 hours, the hands overlap 11 times (NOT 12)
- In 24 hours, the hands overlap 22 times
- Time between consecutive overlaps = 12*60/11 = 720/11 = 65 + 5/11 minutes
- Between 11 and 1 (crossing 12), there is only ONE overlap at 12:00
5. Hands at Right Angle (90 degrees)
When are the hands at 90 degrees?
|30H - 5.5M| = 90
Case 1: 30H - 5.5M = 90 -> M = (30H - 90) / 5.5 = (60H - 180) / 11
Case 2: 30H - 5.5M = -90 -> M = (30H + 90) / 5.5 = (60H + 180) / 11
Key Facts About Right Angles
- In 12 hours, the hands are at right angles 22 times
- In 24 hours, the hands are at right angles 44 times
- Time between consecutive right angles = 720/22 = 360/11 = 32 + 8/11 minutes
Right Angle Times (First Few)
| Between | First 90-degree | Second 90-degree |
|---|---|---|
| 12 and 1 | 12:16:21.8 | 12:49:05.5 |
| 1 and 2 | 1:21:49.1 | 1:54:32.7 |
| 2 and 3 | 2:27:16.4 | (none - only 1 right angle between 2 and 4) |
| 3 and 4 | 3:00:00 | 3:32:43.6 |
| ... | ... | ... |
Special case: Between 3-4 and 8-9, one of the two right angles actually belongs to the adjacent hour.
6. Hands in Straight Line (180 degrees)
When are the hands at 180 degrees (opposite)?
|30H - 5.5M| = 180
30H - 5.5M = 180 -> M = (30H - 180) / 5.5 = (60H - 360) / 11
Key Facts About Straight Lines
- In 12 hours, the hands are in a straight line (180 degrees) 11 times
- In 24 hours, the hands are in a straight line 22 times
- Time between consecutive straight lines = 720/11 = 65 + 5/11 minutes
Straight Line (180 degrees) Times
| Between | Time of 180 degrees |
|---|---|
| 12 and 1 | 12:32:43.6 |
| 1 and 2 | 1:38:10.9 |
| 2 and 3 | 2:43:38.2 |
| 3 and 4 | 3:49:05.5 |
| 4 and 5 | 4:54:32.7 |
| 5 and 6 | 6:00:00 |
| 6 and 7 | 6:05:27.3 (actually past the overlap at 6) |
| 7 and 8 | 7:05:27.3 ... (recalculate) |
Note: Between 5 and 7, there is only ONE occurrence of 180 degrees (at 6:00), not two.
7. Summary of Hand Positions in 12 Hours
| Position | Angle | Occurrences in 12 hrs | Occurrences in 24 hrs |
|---|---|---|---|
| Overlap (coincide) | 0 degrees | 11 | 22 |
| Right angle | 90 degrees | 22 | 44 |
| Straight line (opposite) | 180 degrees | 11 | 22 |
| Straight line (0 or 180) | 0 or 180 degrees | 22 | 44 |
Time gap between consecutive same events:
| Event | Time gap |
|---|---|
| Consecutive overlaps | 720/11 = 65 + 5/11 min |
| Consecutive 180-degree | 720/11 = 65 + 5/11 min |
| Consecutive right angles | 360/11 = 32 + 8/11 min |
8. Gaining and Losing Time (Faulty Clocks)
Concept
A faulty clock may run faster or slower than a correct clock.
If a clock GAINS x minutes per hour:
In 1 hour of real time, the faulty clock shows 60 + x minutes.
If a clock LOSES x minutes per hour:
In 1 hour of real time, the faulty clock shows 60 - x minutes.
Finding Correct Time
When a clock gains:
In (60 + x) minutes of faulty clock time, real time = 60 minutes
If faulty clock shows T minutes have passed:
Real time passed = T * 60 / (60 + x)
When a clock loses:
In (60 - x) minutes of faulty clock time, real time = 60 minutes
If faulty clock shows T minutes have passed:
Real time passed = T * 60 / (60 - x)
When Two Clocks Show Correct Time Again
A clock that gains x minutes per day will show correct time again when it has gained exactly 12 hours (720 minutes):
Days to show correct time = 720 / x days
A clock that loses x minutes per day will show correct time again when it has lost exactly 12 hours:
Days to show correct time = 720 / x days
Note: This is for 12-hour clocks. For 24-hour clocks, use 1440 minutes instead of 720.
Example: A clock gains 5 minutes every hour
Gain per hour = 5 minutes
Gain per day = 5 * 24 = 120 minutes
Days to show correct time = 720 / 120 = 6 days
The clock will show correct time again after 6 days.
Two Faulty Clocks
If one clock gains a minutes/hour and another loses b minutes/hour:
They will show the same time when the total difference = 12 hours
Difference per hour = (a + b) minutes
Hours until same time = 720 / (a + b) hours
Days until same time = 720 / [24 * (a + b)] days
9. Mirror Image of a Clock
Concept
When you see a clock in a mirror, the image is laterally inverted (left-right reversal). The time shown in the mirror is different from the actual time.
Formula to Find Actual Time from Mirror Image
Actual time = 11:60 - Mirror time (when mirror time < 12:00)
Actual time = 23:60 - Mirror time (when using 24-hour format)
Simplified rule:
Actual time = 12:00 - Mirror time
But since minutes can't be negative, we use:
Actual time = 11:60 - Mirror time
Examples
Example 1: Mirror shows 2:30
Actual time = 11:60 - 2:30
= 9:30
Example 2: Mirror shows 8:15
Actual time = 11:60 - 8:15
= 3:45
Example 3: Mirror shows 6:00
Actual time = 11:60 - 6:00
= 5:60
= 6:00
(6:00 looks the same in the mirror!)
Example 4: Mirror shows 12:35
Actual time = 11:60 - 12:35
Since 11:60 < 12:35, subtract from 23:60:
Actual time = 23:60 - 12:35
= 11:25
Times That Look the Same in a Mirror
These times look identical when seen in a mirror:
12:00, 6:00 (approximately)
1:05 and 10:55
2:10 and 9:50
3:15 and 8:45
4:20 and 7:40
5:25 and 6:35
10. Angle Traced by Clock Hands
Angle Traced by Hour Hand
In t minutes, the hour hand traces:
Angle = 0.5 * t degrees
In t hours, the hour hand traces:
Angle = 30 * t degrees
Angle Traced by Minute Hand
In t minutes, the minute hand traces:
Angle = 6 * t degrees
Examples
Example 1: Angle traced by minute hand in 25 minutes
Angle = 6 * 25 = 150 degrees
Example 2: Angle traced by hour hand in 3 hours 40 minutes
t = 3 * 60 + 40 = 220 minutes
Angle = 0.5 * 220 = 110 degrees
11. Finding Time When Angle is Given
Problem Type: At what time between H and (H+1) is the angle X degrees?
|30H - 5.5M| = X
Case 1: 30H - 5.5M = X -> M = (30H - X) / 5.5
Case 2: 30H - 5.5M = -X -> M = (30H + X) / 5.5
Both values of M must satisfy: 0 <= M < 60
Typically, one or both values are valid.
Example: At what time between 4 and 5 are the hands at 60 degrees?
H = 4, X = 60
Case 1: M = (30*4 - 60) / 5.5 = (120 - 60) / 5.5 = 60/5.5 = 120/11 = 10 + 10/11 min
Case 2: M = (30*4 + 60) / 5.5 = (120 + 60) / 5.5 = 180/5.5 = 360/11 = 32 + 8/11 min
Both are valid (between 0 and 60).
The hands are at 60 degrees at:
4:10:10/11 and 4:32:8/11
12. Summary of All Formulas
| Concept | Formula |
|---|---|
| Minute hand speed | 6 degrees/minute |
| Hour hand speed | 0.5 degrees/minute |
| Relative speed | 5.5 degrees/minute |
| Angle at H:M | ` |
| Hands overlap | M = 60H/11 |
| Hands at 90 degrees | M = (60H +/- 180)/11 |
| Hands at 180 degrees | M = (60H - 360)/11 |
| Overlaps in 12 hours | 11 times |
| Right angles in 12 hours | 22 times |
| 180-degree in 12 hours | 11 times |
| Gap between overlaps | 65 + 5/11 minutes |
| Gap between right angles | 32 + 8/11 minutes |
| Mirror image time | 11:60 - displayed time |
| Faulty clock (gains x/hr) | Shows correct after 720/x_per_day days |
| Angle by minute hand in t min | 6t degrees |
| Angle by hour hand in t min | 0.5t degrees |