Clock Reasoning Questions: Concepts, Formulas & Practice

Example:
What is the angle between the hands at 4:20?
Hour hand at 4:20: 4 × 30 + (20/60) × 30 = 120 + 10 = 130°
Minute hand at 20 minutes: 20 × 6 = 120°
Angle = |130° – 120°| = 10°

2. Coincidence of Hands: When Do the Hands Meet?

These questions ask at what time the hour and minute hands coincide or overlap. They are popular in exams because they require both logic and quick calculation.

• The hands coincide 11 times in 12 hours, or 22 times in 24 hours.
• Use the formula: t = (60/11) × H minutes past H o’clock to find the exact time of coincidence.
• These questions can also be reversed—given a time, determine if the hands coincide.

Example:
At what time between 2 and 3 o’clock will the hands be together?
t = (60/11) × 2 = 120/11 ≈ 10 10/11 minutes past 2.

3. Right Angle and Straight Line: Finding Perpendicular and Opposite Hands

Some questions ask when the hands form a right angle (90°) or a straight line (180°). These are variations of angle-based questions but have specific patterns that make them easier to solve with practice.

• The hands form a right angle 22 times in 12 hours (44 times in 24 hours).
• They form a straight line (180°) 11 times in 12 hours (22 times in 24 hours).
• Use the angle formula and solve for the required angle.

Example:
At what time between 3 and 4 o’clock are the hands at right angles?
Set θ = 90° and solve for M using the angle formula.

4. Mirror and Water Image Questions: Reading Reflections Correctly

Mirror and water image questions assess your ability to use your spatial skills to visualize clock faces in reverse. These types of questions are quite common in the reasoning part of the test and can be answered by a simple subtraction.

• For mirror images: Subtract the time given from 12:00.
• For water images: The method may differ depending on the question, but it often involves subtracting from 6:00 or 18:00.

Example:
If a clock shows 8:40, what will its mirror image show?
Mirror image = 12:00 – 8:40 = 3:20

5. Faulty or Defective Clocks: Handling Clocks That Gain or Lose Time

Some questions involve clocks that do not keep accurate time, either gaining or losing seconds, minutes, or hours. These require you to calculate the true time or the time shown by the faulty clock after a certain period.

• Determine the gain or loss per hour or minute.
• Scale up to the total time period in question.
• Adjust the final answer according to the gain or loss.

Example:
A clock gains 4 seconds in 3 minutes and is set right at 8 AM. What time will it show at 10 PM?
From 8 AM to 10 PM = 14 hours = 840 minutes.
Gains in 840 minutes: (4/3) × 840 = 1120 seconds = 18 minutes 40 seconds.
Clock will show 10:18:40 PM when it is actually 10:00 PM.

6. Special Cases: Coincidence, Opposite, and Right Angle Patterns

Some questions may require the number of occurrences of the hands meeting, forming a straight line, or being at right angles within a certain time period. The patterns above save time and prevent unnecessary calculations.

• Hands meet 11 times in 12 hours.
• Hands are opposite (180° apart) 11 times in 12 hours.
• Hands are at right angles 22 times in 12 hours.

Quick Note: Each type has its patterns—master them and you’ll solve questions faster and more accurately.

Solving clock reasoning questions efficiently requires a systematic approach. By following these steps, you can maximize accuracy and minimize time spent per question.

1. Identify the Type of Question:
   Quickly determine if it’s about angles, coincidence, mirror images, or faulty clocks.
2. Recall the Relevant Formula:
   Use the correct formula for the situation—angle, time for coincidence, or mirror image.
3. Plug in the Values:
   Substitute the hour and minute values or the time period as needed.
4. Calculate Step-by-Step:
   Break down the calculation into smaller parts to avoid mistakes.
5. Double-Check Units:
   Ensure your answer is in the correct unit (degrees, minutes, etc.).
6. Review for Reasonableness:
   Does the answer make sense? For example, the angle should not exceed 180° unless specifically asked.

Even experienced candidates can fall into certain traps when solving clock problems. Here are some pitfalls to watch out for:

• Confusing Hour and Minute Hand Movements:
  Remember that the hour hand moves slower than the minute hand.
• Ignoring the Effect of Minutes on the Hour Hand:
  The hour hand also moves as minutes pass—don’t forget to account for this.
• Wrongly Applying Mirror Image Logic:
  Always subtract from 12:00 for mirror images, unless specified otherwise.
• Overlooking the Absolute Value in Angle Calculations:
  Always take the positive difference between the two hands.
• Not Scaling Gain/Loss Correctly in Faulty Clocks:
  Carefully calculate the total gain or loss over the entire period.

Now, let’s apply the concepts to a set of practice questions, each with a solution. This will help you reinforce your understanding of the concepts and familiarize yourself with the different ways in which clock problems.

Question 1: Find the angle between the hour and minute hands at 7:30.

Solution:

• Hour hand at 7:30: 7 × 30 + (30/60) × 30 = 210 + 15 = 225°
• Minute hand at 30 minutes: 30 × 6 = 180°
• Angle = |225° – 180°| = 45°

Question 2: At what time between 2 and 3 o’clock will the hands coincide?

Solution:

• t = (60/11) × 2 = 120/11 ≈ 10 10/11 minutes past 2.

Question 3: How many times do the hands coincide in a day?

Solution:

• 11 times in 12 hours, so 22 times in 24 hours.

Question 4: If a clock shows 5:40, what is its mirror image?

Solution:

• Mirror image = 12:00 – 5:40 = 6:20

Question 5: A clock gains 4 seconds in 3 minutes and is set right at 8 AM. What time will it show at 10 PM?

Solution:

• From 8 AM to 10 PM: 14 hours = 840 minutes.
• Gains in 840 minutes: (4/3) × 840 = 1120 seconds = 18 minutes 40 seconds.
• Clock will show 10:18:40 PM when it is actually 10:00 PM.

Question 6: At what time between 3 and 4 o’clock are the hands at right angles?

Solution:

• Set θ = 90° and solve for M using the formula |30H – 5.5M| = 90.
• For H = 3:
  
  • 30 × 3 – 5.5M = 90 → 90 – 5.5M = 90 → 5.5M = 0 → M = 0 (first solution)
  • 30 × 3 – 5.5M = –90 → 90 – 5.5M = –90 → 5.5M = 180 → M = 32.727 (approx 32 8/11)
• So, hands are at right angles at 3:00 and approximately 3:32:44.

Question 7: How many times in a day are the hands at a right angle?

Solution:

• 22 times in 12 hours, so 44 times in 24 hours.

Question 8: If a clock is 5 minutes fast every hour, how much time will it gain in 6 hours?

Solution:

• 5 minutes per hour × 6 hours = 30 minutes fast in 6 hours.

Question 9: At what time between 4 and 5 o’clock will the hands of a clock be at a right angle?

Solution:

Let the time after 4 o’clock be M minutes.

Using the formula for right angles:
|30H – 5.5M| = 90
H = 4

So,
|120 – 5.5M| = 90
Case 1: 120 – 5.5M = 90 ⇒ 5.5M = 30 ⇒ M = 30/5.5 ≈ 5.45 minutes
Case 2: 120 – 5.5M = -90 ⇒ 120 + 90 = 5.5M ⇒ 210 = 5.5M ⇒ M = 38.18 minutes

Answer:
The hands form a right angle at approximately 4:05:27 and 4:38:11.

Question 10: At what time between 8 and 9 o’clock will the hands be together?

Solution:

Time after 8 o’clock = (60/11) × 8 = 480/11 ≈ 43.64 minutes

Answer:
At 8:43:35, the hands will be together.

Question 11: How many times in 24 hours do the hands of a clock form a straight line?

Solution:

In 12 hours, the hands are in a straight line 11 times.
So, in 24 hours: 11 × 2 = 22 times.

Answer:
22 times in 24 hours.

Question 12: If the minute hand is at 12 and the hour hand is at 3, what is the angle between them?

Solution:

Each hour division = 30°
Angle = 3 × 30° = 90°

Answer:
90°

Question 13: A clock is set right at 5 am. It loses 10 minutes in 24 hours. What will be the correct time when it shows 5 pm the next day?

Solution:

From 5 am one day to 5 pm the next day = 36 hours (on the correct clock).
For every 24 hours, clock loses 10 minutes.
So, in 36 hours: (10/24) × 36 = 15 minutes lost.

When the faulty clock shows 5 pm, the correct time is 5 pm + 15 minutes = 5:15 pm.

Answer:
5:15 pm

Question 14: At what time between 1 and 2 o’clock will the hands of a clock be at a right angle?

Solution:

|30H – 5.5M| = 90
H = 1

Case 1: 30 – 5.5M = 90 ⇒ 5.5M = -60 ⇒ M = -10.91 (not possible)
Case 2: 30 – 5.5M = -90 ⇒ 30 + 90 = 5.5M ⇒ 120 = 5.5M ⇒ M = 21.82

Answer:
At 1:21:49, the hands are at a right angle.

Question 15: How many times do the hands of a clock coincide between 1 and 12 o’clock?

Solution:

In 12 hours, hands coincide 11 times.

Answer:
11 times.

Question 16: If the time is 6:15, what is the angle between the hour and minute hands?

Solution:

Hour hand: 6 × 30 + (15/60) × 30 = 180 + 7.5 = 187.5°
Minute hand: 15 × 6 = 90°
Difference = |187.5 – 90| = 97.5°

Answer:
97.5°

Question 17: At what time between 7 and 8 o’clock will the hands be at a right angle for the second time?

Solution:

|30H – 5.5M| = 90, H = 7

Case 1: 210 – 5.5M = 90 ⇒ 5.5M = 120 ⇒ M = 21.82
Case 2: 210 – 5.5M = -90 ⇒ 210 + 90 = 5.5M ⇒ 300 = 5.5M ⇒ M = 54.55

Answer:
At 7:21:49 and 7:54:33

Question 18: If a clock is 7 minutes fast at noon and gains 2 minutes every hour, what will be the correct time when the clock shows 7 pm?

Solution:

From 12 pm to 7 pm = 7 hours
Gain in 7 hours = 2 × 7 = 14 minutes
Total fastness = 7 + 14 = 21 minutes

When the clock shows 7 pm, the correct time is 7 pm – 21 min = 6:39 pm

Answer:
6:39 pm

Question 19: At what time between 5 and 6 o’clock do the hands of a clock coincide?

Solution:

t = (60/11) × 5 = 300/11 ≈ 27.27 minutes

Answer:
At 5:27:16

Question 20: What is the angle between the hands at 9:45?

Solution:

Hour hand: 9 × 30 + (45/60) × 30 = 270 + 22.5 = 292.5°
Minute hand: 45 × 6 = 270°
Difference = |292.5 – 270| = 22.5°

Answer:
22.5°

Question 21: At what time between 10 and 11 o’clock will the hands be at a right angle?

Solution:

|30H – 5.5M| = 90, H = 10

Case 1: 300 – 5.5M = 90 ⇒ 5.5M = 210 ⇒ M = 38.18
Case 2: 300 – 5.5M = -90 ⇒ 300 + 90 = 5.5M ⇒ 390 = 5.5M ⇒ M = 70.91 (not possible, as > 60)

Answer:
At 10:38:11

Question 22: The hands of a clock are 180° apart at what time between 6 and 7 o’clock?

Solution:

|30H – 5.5M| = 180, H = 6

Case 1: 180 – 5.5M = 180 ⇒ 5.5M = 0 ⇒ M = 0
Case 2: 180 – 5.5M = -180 ⇒ 180 + 180 = 5.5M ⇒ 360 = 5.5M ⇒ M = 65.45 (not possible)

Answer:
At exactly 6:00

Question 23: A watch is 10 minutes slow at 6 am and gains 2 minutes every hour. What is the correct time when the watch shows 12 noon?

Solution:

From 6 am to 12 noon = 6 hours
Gain in 6 hours = 2 × 6 = 12 minutes
Net error = -10 + 12 = +2 minutes

When the watch shows 12:00, actual time is 12:00 – 2 = 11:58

Answer:
11:58 am

Question 24: If the time is 2:24, what is the angle between the hour and minute hands?

Solution:

Hour hand: 2 × 30 + (24/60) × 30 = 60 + 12 = 72°
Minute hand: 24 × 6 = 144°
Difference = |144 – 72| = 72°

Answer:
72°

Question 25: At what time between 11 and 12 o’clock will the hands be together?

Solution:

t = (60/11) × 11 = 60 minutes (i.e., at 12:00)

Answer:
At exactly 12:00

Question 26: How many times in a day are the hands in a straight line but not overlapping?

Solution:

Hands are in a straight line (180°) 22 times in 24 hours. Out of these, 2 times (12:00 am and 12:00 pm), they overlap.
So, 22 – 2 = 20 times

Answer:
20 times

Question 27: A clock gains 6 minutes every hour. If it is set right at 10 am, what will be the correct time when it shows 4 pm?

Solution:

From 10 am to 4 pm = 6 hours (by the correct clock)
In 6 hours, clock gains 6 × 6 = 36 minutes
So, when it shows 4:00 pm, actual time is 4:00 – 0:36 = 3:24 pm

Answer:
3:24 pm

Question 28: At what time between 12 and 1 o’clock will the hands be at a right angle?

Solution:

|30H – 5.5M| = 90, H = 12

Case 1: 360 – 5.5M = 90 ⇒ 5.5M = 270 ⇒ M = 49.09
Case 2: 360 – 5.5M = -90 ⇒ 360 + 90 = 5.5M ⇒ 450 = 5.5M ⇒ M = 81.82 (not possible)

Answer:
At 12:49:05

Question 29: If a clock shows 3:15, what is the angle between the hour and minute hands?

Solution:

Hour hand: 3 × 30 + (15/60) × 30 = 90 + 7.5 = 97.5°
Minute hand: 15 × 6 = 90°
Difference = |97.5 – 90| = 7.5°

Answer:
7.5°

Question 30: How many times in 24 hours do the hands of a clock form a right angle?

Solution:

22 times in 12 hours, so 44 times in 24 hours.

Answer:
44 times

Key Takeaways so Far:

• Practicing reasoning clock questions is the best way to improve your understanding.
• Step-by-step solutions clarify where you might go wrong.
• Exposure to varied problems prepares you for any exam scenario.

As you advance, you will encounter more difficult problems that involve multiple concepts or twists. Here are some advanced variations to look out for:

• Multiple Clocks: Problems that involve two or more clocks with different gains or losses.
• Reverse Calculation: Given the angle, determine the possible times when the hands are in that angle.
• Fractional Minutes and Seconds: Some problems may require answers to be in seconds or in fractions of a minute.
• Unusual Time Formats: Some problems may use 24-hour time or require the use of AM/PM.
• Comparative Problems:
  Comparing the time shown by a faulty clock with a correct clock at a certain moment.

Example:
A clock in the kitchen is 5 minutes slow per hour, while the bedroom clock is 5 minutes fast per hour. Both show 12:00 at the start. When will they show the same time again?

• The difference per hour = 5 + 5 = 10 minutes.
• To make up 12 hours difference: 12 × 6 = 72 hours.
• Both clocks will show the same time after 72 hours.

Scoring high in clock reasoning is not just about knowledge—it’s about strategy. Here’s how to make the most of your preparation and exam time:

1. Practice Regularly: Regular practice will enable you to identify patterns and remember formulas easily. Clock reasoning mock tests should be taken regularly on the online platforms.
2. Time Yourself: Practice clock reasoning mock tests under timed conditions, just like you would in an actual test.
3. Categorize Questions: Practice questions on clock reasoning should be categorized (angle, coincidence, mirror image, and so on) to develop expertise.
4. Review Mistakes: Review mistakes to identify where you went wrong and not repeat those errors.
5. Use Shortcuts Wisely: Important results (such as the number of coincidences or right angles) should be memorized to save time.
6. Stay Calm and Move On: If a question seems tricky, don’t get stuck—move on and return if time allows.

What we learned so far:

• Smart exam strategies save both time and nerves.
• Reviewing mistakes is as important as practicing new questions.
• Time management is crucial in the reasoning section

Clock reasoning questions are a high-yield topic for any competitive exam. By mastering the core concepts, practicing a wide variety of problems, and learning to avoid common mistakes, you can confidently tackle any clock-related question you encounter. Remember to keep your formulas handy, practice regularly, and approach each question methodically for the best results.

Why Clock Reasoning Matters

Clock reasoning questions are an essential part of many competitive exams, offering a quick way to earn marks while testing your logical thinking and calculation skills. Mastering them can give you a real edge in tight exam situations.

Practical Advice for Learners

• Memorize and understand key formulas for angles, coincidences, and mirror images.
• Practice all question types, from basics to advanced variations.
• Time yourself regularly to simulate exam pressure.
• Review your mistakes to avoid repeating them.
• Use a systematic, stepwise approach for every problem.
• Stay calm and confident—clock reasoning rewards preparation and composure.

1. What is the most important formula for clock reasoning questions?

The formula θ = |30H – 5.5M| is vital for finding the angle between the hour and minute hands.

2. How many times do the hour and minute hands coincide in 24 hours?

22 times.

3. How do I solve mirror image clock questions?

Subtract the given time from 12:00 to get the mirror image time.

4. What should I do if a clock gains or loses time?

Calculate the total gain or loss over the period and adjust the final time accordingly.

5. Are clock reasoning questions difficult?

With practice and a clear understanding of the concepts, these questions become some of the easiest to score in reasoning sections.