Clocks: Complete Master Guide for UPSC/MPSC CSAT (English)

Introduction and Importance of Clocks in CSAT

The “Clocks” chapter is a crucial segment of the quantitative aptitude and logical reasoning sections of the CSAT paper. This topic tests a candidate’s spatial reasoning, mathematical aptitude, and understanding of angular concepts. Clock problems revolve around the movement of the minute hand and the hour hand, and the relative angles they form with each other at various times of the day. Because the relationships between the hands are constant and predictable, clock problems can be solved with high accuracy using specific formulas and logical deductions. For UPSC and MPSC aspirants, mastering this topic ensures quick, reliable points that can make a significant difference in the final score.

Core Concepts, Formulas, and Tricks

A clock is essentially a circle divided into 12 equal parts (hours) and 60 equal smaller parts (minutes). The total angle of a circle is 360°.

1. Speed of the Hands

• Minute Hand: Completes a full circle (360°) in 60 minutes.
  Speed = 360° / 60 = 6° per minute.
• Hour Hand: Completes a full circle (360°) in 12 hours (720 minutes).
  Speed = 360° / 720 = 0.5° per minute.
• Relative Speed: The minute hand moves faster than the hour hand.
  Relative speed = 6° – 0.5° = 5.5° per minute (or 11/2 ° per minute).

2. Angle Between Hands Formula

To find the angle (θ) between the hour and minute hands at a given time (H hours and M minutes), use the universal formula:

θ = | 30H – 5.5M |

Note: If the angle exceeds 180°, subtract it from 360° to find the reflex angle, depending on what the question asks.

3. Special Positions of Hands

The hands of a clock assume specific relative positions over a 12-hour period:

• Coincide (0°): The hands overlap exactly 11 times in 12 hours (22 times in 24 hours). They do not overlap between 11:00 and 1:00 except exactly at 12:00.
• Opposite (180°): The hands form a straight line opposite each other 11 times in 12 hours (22 times in 24 hours). They do not form 180° between 5:00 and 7:00 except exactly at 6:00.
• Right Angle (90°): The hands are perpendicular to each other 22 times in 12 hours (44 times in 24 hours).

4. Time of Coincidence Formula

To find the exact time between hour H and H+1 when the hands coincide, are opposite, or make an angle θ, you can use:
Time = (5H ± (θ / 6)) × (12/11) minutes past H.
Or simpler, for finding when they coincide between H and H+1, the formula is: (60/11) × H minutes past H.

Solved Examples with Step-by-Step Explanations

Example 1: Finding the angle at a specific time

Question: What is the angle between the hour hand and the minute hand at 3:40?

Step 1: Identify H (hours) and M (minutes). Here, H = 3 and M = 40.

Step 2: Apply the formula θ = | 30H – 5.5M |.

Step 3: Calculate: θ = | 30(3) – 5.5(40) | = | 90 – 220 | = | -130 | = 130°.

Answer: The angle between the hands at 3:40 is 130°.

Example 2: Finding the exact time of coincidence

Question: At what exact time between 4 and 5 o’clock will the hands of a clock overlap?

Step 1: Use the coincidence formula: (60/11) × H minutes past H.

Step 2: Here H = 4. Calculation: (60/11) × 4 = 240/11 minutes.

Step 3: Convert fraction to mixed number: 240 ÷ 11 = 21 with a remainder of 9. So, it is 21 9/11 minutes.

Answer: The hands will overlap at exactly 21 9/11 minutes past 4.

Example 3: Finding right angles

Question: At what time between 7 and 8 o’clock will the hands form a straight line but not be together?

Step 1: “Straight line but not together” means they are 180° apart. We use the formula θ = | 30H – 5.5M | and set θ = 180.

Step 2: H = 7. 180 = | 30(7) – 5.5M |

Step 3: 180 = | 210 – 5.5M |. This gives two equations:
210 – 5.5M = 180 => 5.5M = 30 => M = 30 / 5.5 = 60 / 11 = 5 5/11 minutes.
210 – 5.5M = -180 => 5.5M = 390 => M = 390 / 5.5 = 780 / 11 = 70 10/11 minutes (which is past 8 o’clock, so invalid).

Answer: The hands form a straight line at 5 5/11 minutes past 7.

Pro-Tips to Avoid Common Mistakes

• Hour Hand Movement: A very common mistake is assuming the hour hand stays perfectly still on the number while the minute hand moves. Remember, the hour hand moves 0.5° every minute. The formula θ = | 30H – 5.5M | automatically accounts for this movement.
• Reflex Angles: Read the question carefully. If the formula gives you an angle of 200° and the options are all under 180°, subtract 200° from 360° to get 160°. A clock has two angles between the hands: an interior angle and a reflex angle.
• Counting Frequencies: Do not blindly multiply. Remember the exceptions. Hands overlap 22 times a day, NOT 24 times. They are at 180° 22 times a day, NOT 24 times. They are at 90° 44 times a day, NOT 48 times.
• 12 O’clock Starting Point: When dealing with minutes passed, conceptually treating 12 o’clock as zero degrees helps visualize the movement before relying on formulas.