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Logical Reasoning: Complete Study Material Clocks
Clocks
Table of Contents
- 1: Foundation & Basic Concepts
- 2: Angle Between Hands
- 3: Coincidence & Opposite Positions
- 4: Time Gained or Lost (Faulty Clocks)
- 5: Mirror Images & Reflections
- 6: Advanced Puzzles & Data Sufficiency
1: Foundation & Basic Concepts
1.1 The Clock Dial
A standard analog clock has:
· 12 hour markings (1 to 12) equally spaced at 30° each (since 360°/12 = 30°).
· 60 minute markings (each minute mark is 6°, because 360°/60 = 6°).
· Two hands: hour hand and minute hand; sometimes a second hand (but rarely used).
Key Observations:
· In 1 hour (60 minutes), the minute hand completes one full revolution (360°).
· In 1 hour, the hour hand moves 30° (from one hour mark to the next).
· In 1 minute, the minute hand moves 6° (360°/60).
· In 1 minute, the hour hand moves 0.5° (30°/60 = 0.5°).
1.2 Speeds of the Hands
· Minute hand speed: 360° per hour = 6° per minute.
· Hour hand speed: 30° per hour = 0.5° per minute.
· Relative speed of the minute hand with respect to the hour hand:
6° - 0.5° = 5.5° per minute.
This means the minute hand gains 5.5° on the hour hand every minute.
1.3 Concept of Angle and Time
The angle between the two hands is a function of the time. For a given time H hours M minutes (using 12‑hour format, H from 0 to 12), we can compute:
· Angle of minute hand from 12 o’clock = 6M degrees.
· Angle of hour hand from 12 o’clock = 30H + 0.5M degrees (since hour hand moves 0.5° per minute past the hour).
The absolute difference between these two angles is the angle between the hands. Sometimes we take the smaller angle (≤ 180°).
Formula:
$$\theta = | 30H - 5.5M | \text{ degrees}$$
[!NOTE]
Why $360^\circ - \theta$?
The formula calculates the angle from the 12 o'clock mark. Since a clock is a circle of $360^\circ$, there are always two angles between the hands: an interior angle and a reflex (exterior) angle.
- If your result $\theta$ is $> 180^\circ$, you have found the reflex angle.
- To find the smaller internal angle (which is usually what exams ask for), subtract it from $360^\circ$: Smaller Angle = $360^\circ - \theta$.
1.4 Overlap and Coincidence
When the hands coincide (overlap), the angle between them is 0° (or 360°). This occurs when:
$$30H - 5.5M = 0$$ (or multiples of 360, but within 12 hours we only get 11 overlaps).
The times of overlap can be derived: for a given hour H (0 to 10), the minute at overlap is $M = \frac{60}{11} H$.
They overlap 11 times in 12 hours (since at 12:00 they overlap, then next at about 1:05, 2:10, … up to 10:55, then again at 12:00). So in a day (24 hours), they overlap 22 times.
1.5 Straight Line (Opposite) Positions
The hands form a straight line (180° apart) when the angle between them is 180°. This occurs 11 times in 12 hours as well (including at 6:00 exactly, and other times). The formula: $30H - 5.5M = 180$ (or $\pm180$). The number of such positions in 12 hours is 11; in 24 hours, 22 times.
1.6 Right Angle Positions
Right angles (90° apart) occur when the angle between the hands is 90° (or 270°). The formula: $30H - 5.5M = \pm90$. There are 22 right‑angle positions in 12 hours (i.e., 44 in 24 hours), because for each hour there are two such positions (except near the ends of the 12‑hour cycle where one might fall on the hour boundary).
1.7 Worked Examples – Foundation Level
Example 1 – Speeds and Angles
Question: How many degrees does the minute hand move in 20 minutes?
Step 1: Minute hand speed = 6° per minute.
Step 2: In 20 minutes, movement = $20 \times 6 = 120°$.
Answer: 120°
Example 2 – Hour Hand Movement
Question: How many degrees does the hour hand move from 2:00 to 2:30?
Step 1: Hour hand moves 0.5° per minute.
Step 2: Time elapsed = 30 minutes.
Step 3: Movement = $30 \times 0.5 = 15°$.
Answer: 15°
Example 3 – Angle at a Given Time
Question: Find the angle between the hour and minute hands at 3:20.
Step 1: Use formula $\theta = |30H - 5.5M|$.
Here H = 3, M = 20.
$\theta = |30 \times 3 - 5.5 \times 20| = |90 - 110| = 20°$.
Answer: 20°
Example 4 – Relative Speed
Question: At what rate does the minute hand gain on the hour hand per minute?
Step 1: Minute hand speed = 6°/min, hour hand = 0.5°/min.
Step 2: Relative speed = $6 - 0.5 = 5.5°/min$.
Answer: 5.5° per minute
1.8 Common Mistakes
| Mistake | Prevention |
|---|---|
| Using 24‑hour format directly in formula | Convert to 12‑hour format (e.g., 15:00 → 3:00). |
| Forgetting that hour hand moves continuously | The hour hand is not fixed at the hour; it moves 0.5° per minute. |
| Taking the larger angle without specifying | Usually we take the smaller angle; if >180°, subtract from 360°. |
| Misinterpreting “opposite direction” | Straight line = 180° apart, not necessarily overlapping. |
1.9 Pro Tips
· Memorize key speeds: minute hand = 6°/min, hour hand = 0.5°/min, relative speed = 5.5°/min.
· For angle calculation, the formula $\theta = |30H - 5.5M|$ is fast. Use it after converting H to 12‑hour format.
· For overlap, right angle, and straight line counts, remember: overlaps = 11 in 12 hours; straight line = 11; right angle = 22 in 12 hours.
· When a time is given as “past H o’clock”, it’s already in H:M format.
1.10 Quick Practice – Foundation Level
- How many degrees does the minute hand move in 45 minutes?
- How many degrees does the hour hand move in 2 hours 30 minutes?
- Find the angle between the hands at 6:15.
- At what time between 4 and 5 will the hands coincide? (Leave as fraction of minutes.)
- How many times do the hands form a straight line in 12 hours?
Answers:
- $45 \times 6 = 270°$
- 2.5 hours = 150 minutes → $150 \times 0.5 = 75°$
- H=6, M=15 → $|180 - 82.5| = 97.5°$ (or 262.5°, smaller = 97.5°)
- Overlap formula: $M = \frac{60}{11} H = \frac{60}{11} \times 4 = \frac{240}{11} \approx 21.82$ minutes past 4.
- 11 times (in 12 hours).
Summary of Subtopic 1
| Concept | Key Points |
|---|---|
| Dial markings | 12 hours (30° each), 60 minutes (6° each). |
| Speeds | Minute hand 6°/min, hour hand 0.5°/min, relative speed 5.5°/min. |
| Angle formula | $\theta = |
| Overlap | 11 times in 12 hours; formula $M = \frac{60}{11}H$. |
| Straight line | 11 times in 12 hours; right angle 22 times in 12 hours. |
2: Angle Between Hands
2.1 The Standard Formula
At H hours and M minutes (using 12‑hour format, H from 0 to 12), the positions of the hands are:
· Minute hand angle from 12 o’clock = 6M degrees (since 360° in 60 minutes → 6° per minute).
· Hour hand angle from 12 o’clock = 30H + 0.5M degrees (since 30° per hour plus 0.5° per minute).
The absolute difference between these two angles is the angle between the hands:
$$\theta = | 30H - 5.5M | \text{ degrees}$$
If $\theta > 180^\circ$, the smaller (interior) angle is calculated as:
$$\text{Smaller Angle} = 360^\circ - \theta$$
Derivation:
Difference = $|(30H + 0.5M) - 6M| = |30H - 5.5M|$.
2.2 Using the Formula – Examples
Example 1 – Basic Calculation
Question: Find the angle between the hands at 4:20.
Step 1: $H = 4$, $M = 20$.
Step 2: $\theta = |30 \times 4 - 5.5 \times 20| = |120 - 110| = 10°$.
Answer: 10°
Example 2 – Larger Angle
Question: Find the angle at 9:15.
Step 1: $H = 9$, $M = 15$.
Step 2: $\theta = |270 - 82.5| = 187.5°$.
Since $187.5 > 180$, the smaller angle = $360 - 187.5 = 172.5°$.
Answer: 172.5°
Example 3 – Exact Hours
Question: Angle at 6:00.
Step 1: $H = 6$, $M = 0$.
Step 2: $\theta = |180 - 0| = 180°$.
Answer: 180° (straight line)
2.3 Finding Time Given an Angle
Sometimes we are given the angle and need to find the time(s) between two given hours (e.g., between 3 and 4). For a given hour H, the minute hand position M at which a specific angle occurs can be found by solving:
$$30H - 5.5M = \pm \theta$$
where $\theta$ is the desired angle (we consider both possibilities because the minute hand could be ahead or behind the hour hand). The solutions are:
$$M = \frac{30H \mp \theta}{5.5}$$
(Remember to take the sign corresponding to $\theta$ positive or negative inside the absolute value).
We then take values of M between 0 and 60 that satisfy the time range.
Important: For angles like 0° (coincidence) and 180° (straight line), there are specific formulas, but we can use the same approach.
Example 4 – Time When Hands Are at a Given Angle
Question: At what time between 3 and 4 o’clock will the hands be at 30°?
Step 1: Here $H = 3$. The formula: $30H - 5.5M = \pm 30$.
So $90 - 5.5M = 30$ or $90 - 5.5M = -30$.
Case 1: $90 - 5.5M = 30 \implies 5.5M = 60 \implies M = 60 / 5.5 = 600/55 = 120/11 \approx 10.91$ minutes.
That is 3:10:54.
Case 2: $90 - 5.5M = -30 \implies 5.5M = 120 \implies M = 120 / 5.5 = 240/11 \approx 21.82$ minutes.
That is 3:21:49.
Answer: 3: $\frac{120}{11}$ minutes and 3: $\frac{240}{11}$ minutes.
Example 5 – Time When Hands Coincide (Angle 0°)
Question: At what time between 5 and 6 do the hands coincide?
Step 1: For $H = 5$, set $30 \times 5 - 5.5M = 0 \implies 150 - 5.5M = 0 \implies M = 150 / 5.5 = 300/11 \approx 27.27$ minutes.
So time = 5: $\frac{300}{11}$ minutes (≈ 5:27:16).
Answer: 5: $\frac{300}{11}$
2.4 General Formula for Overlap Time
For overlap, $30H - 5.5M = 0 \implies M = \frac{60}{11} H$.
For $H = 0$ to 10, we get the 11 overlaps in 12 hours.
2.5 Formula for Right Angles (90°)
For a given hour H, the times when the hands are at 90° are obtained from:
$$30H - 5.5M = 90 \quad \text{or} \quad 30H - 5.5M = -90$$
This gives two times per hour (except for the hour boundaries where one solution might be 60, which is the next hour). Over 12 hours, there are 22 right‑angle occurrences.
2.6 Formula for Straight Line (180°)
Similarly, for 180°:
$$30H - 5.5M = 180 \quad \text{or} \quad 30H - 5.5M = -180$$
There are 11 such positions in 12 hours (including 6:00 exactly).
2.7 Common Mistakes
| Mistake | Prevention |
|---|---|
| Using H in 24‑hour format | Convert to 12‑hour format (e.g., 14:30 → $H=2, M=30$). |
| Forgetting absolute value | Always compute $\theta = |
| Solving with only one sign | Usually two solutions per hour (except for 0° and 180°). |
| Misinterpreting "between H and H+1" | The minute M must be $0 \le M < 60$. |
2.8 Pro Tips
· Memorize factor 5.5: It appears in all time‑finding formulas ($1/5.5 = 2/11$).
· Mental Math: Compute $(30H \mp \theta)$, multiply by 2, then divide by 11.
· Right Angles: Two times per hour (careful near 3 and 9).
· Smallest Angle: Always take the smaller angle unless the "reflex angle" is asked.
2.9 Practice Set – Angle Between Hands
- Find the angle between the hands at 2:30.
- Find the angle at 7:20.
- At what time between 8 and 9 will the hands be at 60°?
- At what time between 10 and 11 will the hands be opposite (180°)?
- How many times between 4 and 5 do the hands form a 90° angle?
- What is the angle at 12:15?
- Find the time(s) between 2 and 3 when the angle is 120°.
Answers:
- $H=2, M=30 \implies |60 - 165| = 105°$.
- $H=7, M=20 \implies |210 - 110| = 100°$.
- $H=8$, solve $240 - 5.5M = \pm 60$.
Case +60: $M = 180/5.5 = 360/11 \approx 32.73$ min.
Case -60: $M = 300/5.5 = 600/11 \approx 54.55$ min. - $H=10$, $300 - 5.5M = 180$. $M = 120/5.5 = 240/11 \approx 21.82$ min.
- $H=4$, solve $120 - 5.5M = \pm 90$. Two valid solutions ($\approx 5.45$ min and $\approx 38.18$ min).
- $H=12$ (0), $M=15 \implies |0 - 82.5| = 82.5°$.
- $H=2, 60 - 5.5M = -120 \implies M = 180/5.5 = 360/11 \approx 32.73$ min.
Summary of Subtopic 2
| Concept | Key Points |
|---|---|
| Angle formula | $\theta = |
| Finding time | $M = \frac{30H \mp \theta}{5.5}$ |
| Overlap | $M = \frac{60}{11}H$ (11 times in 12h). |
| Right angles | 2 times per hour (usually). |
3: Coincidence & Opposite Positions
3.1 Overview of Special Positions
The minute hand gains 5.5° per minute on the hour hand. This relative motion leads to:
· Coincidence (Overlap): Angle = 0° (or 360°).
· Opposite (Straight Line): Angle = 180°.
· Right Angles: Angle = 90° (or 270°).
3.2 Coincidence (Overlap)
Overlap occurs when the minute hand gains exactly 360° on the hour hand.
Time interval = $\frac{360}{5.5} = \frac{720}{11} \approx 65.45$ minutes ($1\text{h } 5\text{m } 27\text{s}$).
In 12 hours, they overlap 11 times.
Formula: $M = \frac{60}{11} \times H$ (for $H = 0, 1, \dots, 10$).
Approximate Times:
- 12:00
- 1:05:27
- 2:10:55
- 3:16:22
- 4:21:49
- 5:27:16
- 6:32:43
- 7:38:11
- 8:43:38
- 9:49:05
- 10:54:33
In 24 hours (a full day), they coincide 22 times.
3.3 Opposite Positions (180° Apart)
Hands are opposite when the angle is 180°. Successive opposite positions occur every $\frac{720}{11}$ minutes.
In 12 hours, they are opposite 11 times (including 6:00).
Formula: $30H - 5.5M = \pm 180$.
Simpler Way: Opposite times are offset by $\frac{360}{11} \approx 32.73$ minutes from overlap times.
Approximate Times:
- 6:00 (exact)
- 7:05:27
- 8:10:55
- 9:16:22
- 10:21:49
- 11:27:16
- 12:32:43
- 1:38:11
- 2:43:38
- 3:49:05
- 4:54:33
In 24 hours, they are opposite 22 times.
3.4 Right Angle Positions (90°)
Hands are at right angles when the angle is 90°. Time between right angles is $\frac{180}{11} \approx 16.36$ minutes.
In 12 hours, there are 22 right angles.
In 24 hours, there are 44 right angles.
3.5 Key Formulas Summary
| Event | Angle | Count (12h) | Interval | Time Formula |
|---|---|---|---|---|
| Coincidence | 0° | 11 | $720/11$ min | $M = \frac{60}{11}H$ |
| Opposite | 180° | 11 | $720/11$ min | $M = \frac{30H \pm 180}{5.5}$ |
| Right Angle | 90° | 22 | $180/11$ min | $30H - 5.5M = \pm 90$ |
3.6 Worked Examples
Example 1 – Coincidence Time
Question: At what time between 8 and 9 do the hands coincide?
Step 1: $H = 8$. $M = \frac{60}{11} \times 8 = \frac{480}{11} \approx 43.64$ min.
Answer: 8:43:38
Example 2 – Opposite Time
Question: At what time between 5 and 6 are the hands opposite?
Step 1: $150 - 5.5M = -180 \implies 5.5M = 330 \implies M = 60$.
Answer: 6:00
Example 3 – Right Angle Count
Question: How many times between 2:00 and 4:00 are the hands at right angles?
Step 1: Intervals from 2 to 4. Solve for $H=2$ ($|60-5.5M|=90$) and $H=3$ ($|90-5.5M|=90$).
Result: 1 time from 2:00-3:00 ($2:27:16$) and 2 times from 3:00-4:00 ($3:00$ and $3:32:44$).
Total: 3 times.
Example 4 – General Counting
Question: How many times do the hands form a straight line (180°) in a day?
Answer: 22 times.
3.7 Common Mistakes
| Mistake | Prevention |
|---|---|
| Counting 12:00 as opposite | At 12:00, they overlap. At 6:00, they are opposite. |
| Counting 12 overlaps in 12h | There are only 11. |
| Assuming 2 right angles per hour | Boundary hours (like 3:00 and 9:00) reduce the count. |
3.8 Pro Tips
· Coincidence period: Memorize $65\text{m } 27\text{s}$.
· Count Right Angles: Approx $\frac{11T}{180}$ for duration T, but check boundaries.
· Midway point: Opposite times are exactly midway (angularly) between overlaps.
3.9 Practice Set – Coincidence & Opposite Positions
- At what time between 2 and 3 will the hands coincide?
- At what time between 9 and 10 are the hands opposite?
- How many times between 1:00 and 2:00 are the hands at right angles?
- How many times in a day (24 hours) do the hands form a 90° angle?
- At what time between 10 and 11 do the hands form a straight line?
- The hands coincide at 12:00. What is the next time they coincide?
- How many times between 4:00 and 5:00 are the hands at 90°?
Answers:
- $2:10:55$ ($M = 120/11$).
- $9:16:22$ ($M = 180/11$).
- 1 time ($1:21:49$).
- 44 times.
- $10:21:49$ ($M = 240/11$).
- $1:05:27$.
- 2 times.
Summary of Subtopic 3
| Concept | Key Points |
|---|---|
| Coincidence | $M = \frac{60}{11}H$; 11 times in 12h. |
| Opposite | 11 times in 12h (offset by 32.73m from overlap). |
| Right Angle | 22 times in 12h. |
| Interval | $720/11$ min for $0°/180°$, $180/11$ min for $90°$. |
4: Time Gained or Lost (Faulty Clocks)
4.1 Core Concepts
A faulty clock is one that does not keep accurate time. It may gain or lose a certain number of minutes per hour (or per day).
· Gaining time: The clock runs fast. For every actual hour, it shows more than 60 minutes.
· Losing time: The clock runs slow. For every actual hour, it shows less than 60 minutes.
Key idea: The faulty clock’s movement is proportional to actual time. The ratio (k) of the faulty clock’s time to correct time is consistent:
$$k = \frac{\text{Faulty time elapsed}}{\text{Correct time elapsed}}$$
4.2 Types of Faulty Clock Problems
- Constant Gain/Loss: Finding actual time from faulty time or vice versa.
- Time After Duration: What time will it show after X hours?
- Correct Time Retrieval: Given faulty reading, solve for true time.
- Rate Calculation: Finding gain/loss rate per hour from an observed error.
4.3 Step-by-Step Methodology
- Calculate the Ratio (k):
- "Gains 5 min/h": ratio = $\frac{65}{60} = \frac{13}{12}$.
- "Loses 5 min/h": ratio = $\frac{55}{60} = \frac{11}{12}$.
- "Gains 10 min/day": ratio = $\frac{1450}{1440} = \frac{145}{144}$.
- Setup Proportion:
Let $C$ = Correct time elapsed, $F$ = Faulty time elapsed.
$$F = k \times C$$ - Handle Offsets: If the clock wasn't set correctly initially, account for the starting difference.
4.4 Worked Examples
Example 1 – Simple Gain per Hour
Question: A clock gains 5 minutes per hour. It is set correctly at 12:00 noon. What time will it show at 6:00 PM correct time?
Step 1: Correct elapsed = 6 hours = 360 minutes.
Step 2: Ratio $k = \frac{13}{12}$.
Step 3: Faulty elapsed = $\frac{13}{12} \times 360 = 390$ minutes (6h 30m).
Answer: 6:30 PM
Example 2 – Loss per Day
Question: A clock loses 10 minutes per day. It is set correctly at 8:00 AM on Monday. What time will it show at 8:00 AM on Wednesday (correct time)?
Step 1: Actual elapsed = 48 hours.
Step 2: Total loss = $10 \times 2 = 20$ minutes.
Step 3: Faulty elapsed = $48\text{h} - 20\text{m} = 47\text{h } 40\text{m}$.
Answer: 7:40 AM Wednesday
Example 3 – Finding Correct Time from Faulty Time
Question: A clock gains 5 min/h. It was set correctly at 6:00 AM. When the faulty clock shows 9:00 AM, what is the correct time?
Step 1: Ratio $k = \frac{13}{12}$. Faulty elapsed = 3 hours (180m).
Step 2: $180 = \frac{13}{12} \times C \implies C = 180 \times \frac{12}{13} \approx 166.15$ min ($\approx 2\text{h } 46\text{m } 9\text{s}$).
Answer: 8:46:09 AM (approx)
4.5 Common Mistakes
| Mistake | Prevention |
|---|---|
| Direct addition of gain | Use proportional ratios ($k$) for long durations. |
| Confusing Faulty vs Correct elapsed | Always verify which time you are calculating. |
| Ignoring initial offsets | Check if the clock was already fast/slow at the start. |
4.6 Pro Tips
· Ratio Trick: If it gains x min/h, $k = \frac{60+x}{60}$. Multiply by 2/11 logic for angle conversion doesn't apply here; just use simple fractions.
· Always convert time to minutes for precision.
4.7 Practice Set – Faulty Clocks
- A clock gains 2 minutes per hour. It is set correctly at 8:00 AM. What time will it show at 2:00 PM correct time?
- A clock loses 5 minutes per hour. It is set correctly at 6:00 AM. What is the correct time when it shows 9:00 AM?
- A clock gains 1 minute per hour. It is set correctly at 10:00 AM. After how many actual hours will it show the correct time again?
Answers:
- $2:12$ PM.
- $9:16:22$ AM.
- 720 hours (30 days). (Cumulative gain must reach 12 hours = 720 minutes).
5: Mirror Images & Reflections
5.1 Core Concepts
A vertical mirror reverses left and right. The reflection of time H:M is its complement within the 12-hour cycle.
Vertical Mirror Formula:
$$\text{Mirror Time} = 12:00 - \text{Actual Time}$$
(Use 11:60 for easier subtraction).
5.2 The Mirror Formula
- If $M > 0$: $(11 - H) : (60 - M)$
- If $M = 0$: $(12 - H) : 00$
Note: If H=12, treat it as 0 for subtraction.
5.3 Horizontal Reflection (Water Reflection)
Image is upside-down.
Horizontal Formula:
$$\text{Reflected Time} = 6:30 - \text{Actual Time}$$
(Note: Some conventions use 18:30 for afternoon times.)
5.4 Worked Examples
Example 1 – Basic Mirror Image
Question: A clock shows 4:25. What time will it appear in a vertical mirror?
Step: $11:60 - 4:25 = 7:35$.
Answer: 7:35
Example 2 – Water Reflection
Question: A clock shows 3:30. Water reflection?
Step: $6:30 - 3:30 = 3:00$.
Answer: 3:00
5.5 Practice Set – Mirror Images
- Mirror image of 9:40? Ans: 2:20
- Mirror image of 12:10? Ans: 11:50 (11:60 - 0:10)
- Water reflection of 1:10? Ans: 5:20 (6:30 - 1:10)
6: Advanced Puzzles & Data Sufficiency
6.1 What Are Advanced Clock Puzzles?
These involve multiple constraints (e.g., faulty clock + mirror) or finding specific intervals between conditions.
6.2 Worked Examples
Example 1 – Time Interval Between Right Angles
Question: Between 4:00 and 5:00, what is the gap between the two right-angle positions?
Step 1: H=4. $|120 - 5.5M| = 90 \implies M = 30/5.5$ and $210/5.5$.
Step 2: Difference = $180/5.5 = \frac{360}{11} \approx 32.73$ minutes.
Answer: 32m 44s approx.
Example 2 – Data Sufficiency
Question: What is the angle between hands?
Statement I: Time is 8:30.
Statement II: Hour hand is at 255° from 12.
Analysis: I alone is sufficient ($|240-165|=75°$). II alone is sufficient ($30H+0.5M=255 \implies 8:30$).
Answer: D
6.3 Practice Set – Advanced Puzzles
- How many times in 24h are hands at 90°? Ans: 44
- Faulty clock gains 3m/h. Set at 6:00 AM. Correct time when it shows 12:00? Ans: 11:42:51