Problems on clocks are among the most scoring and predictable topics in Quantitative Aptitude. But many students find them confusing because a clock has moving hands, angle changes every minute, and the relative speed between the hour and minute hand creates shifting positions.

If you understand these three ideas:
✔ Movement of minute hand
✔ Movement of hour hand
✔ Relative gain between both hands

Then you can solve ANY clock question effortlessly.

This complete Problems on Clocks guide covers all formulas, concepts, shortcuts, cases, diagrams, FAQs, and exam tips, making it the perfect one-stop resource for competitive exams.

Quick Overview: Problems on Clocks Formulas

Concept / Situation	Considered (What We Measure)	Used (Speed / Idea Used)	Formula (With Meaning Explained Inside Row)
Minute Spaces	60 equal divisions on clock dial	–	1 minute space = 6° (360° ÷ 60)
Movement of Minute Hand	Full circle = 360°	6° per minute	Minute hand speed = 360° ÷ 60 = 6°/min
Movement of Hour Hand	Full circle = 360° in 12 hrs	0.5° per minute	Hour hand speed = 360° ÷ (12×60) = 0.5°/min
Relative Speed of Hands	Gap between hands	5.5° per minute	Relative speed = 6° – 0.5° = 5.5°/min
Coinciding of Hands	Gap of 360° closed	Relative speed	Time = 360° ÷ 5.5° per min
Hands at Right Angle	90° gap	Relative speed	Time = 90° ÷ 5.5° for first right angle within an hour
Hands Opposite	180° gap	Relative speed	Time = 180° ÷ 5.5° for opposite positions
Angle Between Hands	Angle at any given time	Speed × time difference	Angle =
Fast Clock	Incorrect extra minutes	Difference from real time	Fast by x min → Indicated time – correct time
Slow Clock	Loss of minutes	Difference from real time	Slow by x min → Correct time – indicated time

Formulas for Problems on Clocks

Clock problems become very simple once you understand how the hour hand and minute hand move and how much angle each hand covers in a given time. Every question in this chapter is built on relative speed, minute spaces, and angle calculation.
Before solving any problem, make sure the units (minutes, degrees, minute spaces) are consistent.

1. Minute Space & Degree Conversion Formulas

Understanding minute spaces is the first step in all clock questions. The clock dial is divided into 60 parts, forming minute spaces—each one representing a fixed angular movement.

1. Each Minute Space = 6°

Because a circular dial is 360°:

1 minute space = 360° ÷ 60 = 6°

This conversion helps convert distance between hands into degrees instantly.

2. Degree Covered by Minute Hand Per Minute

The minute hand completes a full circle in 60 minutes.

Movement = 360° in 60 min = 6°/min

Useful when finding angles at given times.

3. Degree Covered by Hour Hand Per Minute

The hour hand completes a full circle in 12 hours = 720 minutes.

Movement = 360° ÷ 720 = 0.5°/min

This makes the hour hand 12 times slower than the minute hand.

Basic Angle Formulas (Foundation of Clock Problems)

All clock questions from coincidence, right angle, opposite, to angle-at-time are built on these formulas.

Angle Formed Between Hands

At any time “H:M”

Angle = |30H – 5.5M|

Where:
H = hour
M = minutes
30H = hour hand movement (30° per hour)
5.5M = relative minute effect (minute hand minus hour hand)

Relative Speed Between Hands

Minute hand = 6°/min
Hour hand = 0.5°/min

Relative Speed = 6 – 0.5 = 5.5°/min

This tells how fast the gap closes between the hands.

2. Coincidence of Hands (Minute Hand Catches Hour Hand)

When two hands overlap, the minute hand must catch the hour hand.

Formula: Time for Coincidence

Time = 360° ÷ 5.5 = 65 5/11 minutes

Meaning:
The gap of a full circle must be closed by the 5.5°/min relative speed.

Why this works

At 12:00 the gap starts at 0°
After 65 5/11 minutes, the minute hand catches the hour hand again.
This repeats 11 times in 12 hours.

Common mistakes

– Forgetting to use relative speed
– Assuming they meet exactly at HH:00

Key Tip

Time between two coincidences = 65 5/11 minutes always.

3. Hands at Right Angles (90° or 15 Minute Spaces Apart)

The right angle between hands occurs when the angular difference is 90°.

Formula:

Time = 90° ÷ 5.5

There are two such times between every hour because hands form right angle twice.

Meaning

15 minute spaces × 6° = 90°

Keywords to identify

– “right angle”
– “at 90°”
– “perpendicular hands”

Common mistakes

– Using 90 instead of ±90 (two times each hour)

4. Hands in Opposite Directions (Straight Line, 180° Apart)

This happens when the two hands face opposite directions.

Formula:

Time = 180° ÷ 5.5

Meaning

30 minute spaces × 6° = 180°

Why this happens

Gap = half circle (opposite)

Keywords

– “straight line”
– “hands opposite”
– “180° apart”

5. Angle at Any Given Time (Most Common Formula)

To find angle at H hours M minutes:

Formula:

Angle = |30H – 5.5M|

Meaning

– Hour hand moves 0.5°/min
– Minute hand moves 6°/min
– Difference gives exact angle

When to use

– “Find angle at 5:24?”
– “At what time is angle 100°?”
– “Clock shows 3:20, what’s the angle?”

6. Fast and Slow Clocks (Error in Time Keeping)

If a clock does not show correct time:

Fast Clock

Clock shows time ahead.

Slow Clock

Clock shows time behind.

Formula:

Fast by x minutes → Indicated time – x = Correct time
Slow by x minutes → Indicated time + x = Correct time

Examples

Clock shows 8:15, actual is 8:00 → 15 minutes fast
Clock shows 7:45, actual is 8:00 → 15 minutes slow

7. Relative Speed Formulas (Core of All Clock Problems)

These behave similar to trains, but instead of distances, angles are covered.

Relative Speed = 6 – 0.5 = 5.5° per min

Used whenever two hands move relative to each other:

✔ Coinciding
✔ Right angle
✔ Opposite
✔ Meeting again
✔ Angle formation

8. Time for a Required Angle Between Hands

If you want the hands to have angle X°:

Formula:

Time = X° ÷ 5.5

Where X = angle (90°, 180°, etc.)

This works because relative speed is 5.5°/min.

9. Number of Coincidences, Right Angles & Opposites

Coincidences in 12 hours = 11 times

Right angles in 12 hours = 22 times

Opposites in 12 hours = 11 times

These follow from relative positions.

Smart Tips and Practical Tricks for Clock Problems

Mastering clock questions becomes simple when you understand how the hands move and how angles form. Most mistakes come from not understanding relative speed or angle formation.

1. Remember Degree Movements Clearly

Minute hand = 6° per minute
Hour hand = 0.5° per minute
Difference = 5.5° per minute

2. Convert Everything to Degrees or Minute Spaces

Never mix both.
Use:
1 minute space = 6°.

3. Use Relative Speed for Coincidence Questions

Relative speed = 5.5° per minute
Time to coincide = 360° ÷ 5.5.

4. Use Line Diagrams

Sketching positions quickly helps visualize overlaps, right angles, and opposite directions.

5. Identify Patterns

Almost all questions fall into:

• Coincidence
• Right angle
• Opposite direction
• Angle formation
• Fast/slow error
  Recognizing these saves time.

FAQs About Clock Problems 

Q1. What is a minute space in a clock? 

A minute space is one of the 60 equal divisions of a clock dial.
Example: From 12 to 1 there are 5 minute spaces. Total = 60.

Q2. How many degrees does one minute space represent?

One minute space = 6° because 360° ÷ 60 = 6°.
Example:
10 minute spaces = 10 × 6° = 60°.

Q3. How fast does the minute hand move per minute?

Minute hand moves 6 degrees per minute.
Calculation: 360° ÷ 60 min = 6° per min.

Q4. How fast does the hour hand move per minute?

Hour hand moves 0.5° per minute.
Example: In 20 minutes → 20 × 0.5° = 10°.

Q5. Why do the clock hands coincide? 

They coincide because the minute hand gains on the hour hand.
Gain per minute = 6° – 0.5° = 5.5°.
Formula: Time = 360° ÷ 5.5 = 65 5/11 minutes.

Q6. How many times do the hands coincide in 12 hours?

They coincide 11 times, not 12, because the first coincidence happens after a little more than 1 hour.

Q7. What is the angle when the hands are at a right angle? 

Right angle = 90° = 15 minute spaces.
Example:
Angle = 15 × 6° = 90°.

Q8. What is the angle when the hands are opposite?

Opposite direction angle = 180°.
Example: 180° ÷ 6° = 30 minute spaces.

Q9. What is a fast clock?

A fast clock shows time ahead of correct time.
Example:
Clock shows 8:15 but correct time = 8:00 → clock is 15 minutes fast.

Q10. What is a slow clock?

A slow clock shows time behind correct time.
Example:
Clock shows 7:45 but correct time = 8:00 → clock is 15 minutes slow.