How to Solve: CUET Reasoning – Clock and Calendar

How to Solve: CUET Reasoning – Clock and Calendar

Introduction

"If you can’t calculate the day of the week for your birthday in 2025 or figure out when the clock hands overlap, you’re leaving easy marks on the CUET. Let’s fix that—fast."

What You Need To Know First

1. Basic arithmetic (addition, subtraction, division with remainders).
2. Modular arithmetic (working with cycles, e.g., 12-hour clock, 7-day week).
3. Leap year rules (divisible by 4, but not by 100 unless also by 400).

Key Vocabulary

Formulas To Know

1. Angle Between Clock Hands

Formula: θ = |30H - 5.5M| - H = Hour (0 to 12) - M = Minutes (0 to 59) - θ = Angle in degrees (always ≤ 180°)

Memorise This.

Example: At 3:30, θ = |30(3) - 5.5(30)| = |90 - 165| = 75°.

2. Odd Days Calculation

Formula: Total odd days = (Total days) % 7 - % = Modulo (remainder after division by 7).

Memorise This.

Example: 15 days = 2 weeks + 1 odd day → 15 % 7 = 1.

3. Leap Year Check

Rules: 1. If year is not divisible by 4 → Not a leap year. 2. If year is divisible by 4 but not by 100 → Leap year. 3. If year is divisible by 100 and 400 → Leap year. 4. Else → Not a leap year.

Memorise This.

Example: - 2024 → Divisible by 4, not by 100 → Leap year. - 1900 → Divisible by 100, not by 400 → Not a leap year.

4. Zeller’s Congruence (Day of the Week)

Formula: h = (q + [13(m+1)/5] + K + [K/4] + [J/4] + 5J) % 7 - h = Day of the week (0=Saturday, 1=Sunday, 2=Monday, ..., 6=Friday) - q = Day of the month - m = Month (3=March, 4=April, ..., 14=February) - K = Year of the century (year % 100) - J = Zero-based century (year / 100)

Given on exam sheet (but practice it!).

Example: Find the day for 15 August 1947. - q = 15, m = 8 (August), year = 1947 → K = 47, J = 19. - Plug into formula → h = 5 → Friday.

Step-by-Step Method

For Clock Problems:

1. Identify the time (hour and minutes).
2. Plug into angle formula: θ = |30H - 5.5M|.
3. Check if angle > 180°: If yes, subtract from 360°.
4. Simplify (e.g., 270° → 90° is the smaller angle).

Example: Find the angle at 7:20. 1. H = 7, M = 20. 2. θ = |30(7) - 5.5(20)| = |210 - 110| = 100°. 3. 100° < 180° → Answer = 100°.

For Calendar Problems:

1. Break the period into years, months, days.
2. Calculate odd days for each part:
3. Years: (Number of leap years) × 2 + (Normal years) × 1.
4. Months: Use odd days per month (e.g., January = 3, February = 0/1).
5. Days: Total days % 7.
6. Sum all odd days.
7. Take modulo 7 of the total.
8. Map to day of the week (0=Sunday, 1=Monday, ..., 6=Saturday).

Example: Find the day on 1 January 2025 if 1 January 2024 was a Monday. 1. 2024 is a leap year → 366 days = 2 odd days. 2. Total odd days = 2. 3. Monday + 2 days = Wednesday.

Worked Examples

Example 1 - Basic (Clock Angle)

Question: What is the angle between the clock hands at 4:40?

Solution: 1. H = 4, M = 40. 2. θ = |30(4) - 5.5(40)| = |120 - 220| = 100°. 3. 100° < 180° → Answer = 100°.

What we did and why: Used the angle formula directly. No need to adjust since 100° is the smaller angle.

Example 2 - Medium (Mirror Time)

Question: If the time is 10:10, what was the time 3 hours and 20 minutes ago?

Solution: 1. Subtract 3 hours: 10:10 → 7:10. 2. Subtract 20 minutes: 7:10 → 6:50. 3. Check for mirror time: 6:50 is not a mirror time (mirror of 6:50 is 5:10). 4. Answer = 6:50.

What we did and why: Subtracted time directly. Mirror time is a distractor here.

Example 3 - Exam Style (Calendar Problem)

Question: If 15 August 1947 was a Friday, what day was 26 January 1950?

Solution: 1. Break into periods:
- 15 Aug 1947 → 31 Dec 1947
- 1948 (leap year)
- 1949
- 1 Jan 1950 → 26 Jan 1950 2. Calculate odd days:
- 1947 (Aug-Dec): 16 (Aug) + 30 (Sep) + 31 (Oct) + 30 (Nov) + 31 (Dec) = 138 days → 138 % 7 = 5.
- 1948 (leap year): 366 days → 366 % 7 = 2.
- 1949: 365 days → 365 % 7 = 1.
- 1950 (Jan 1-26): 26 days → 26 % 7 = 5. 3. Total odd days = 5 + 2 + 1 + 5 = 13 → 13 % 7 = 6. 4. Friday + 6 days = Thursday.

What we did and why: Broke the problem into manageable chunks. Calculated odd days for each period and summed them up.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. For clock problems, use θ = |30H - 5.5M|—always. If the angle is over 180°, subtract from 360°. For calendars, break the problem into years, months, and days. Calculate odd days for each, sum them up, and take modulo 7. Leap years add 2 odd days, normal years add 1. Watch out for century years—they’re tricky. And if the question mentions mirror time, subtract from 12:00. That’s it. Practice 3-4 problems tonight, and you’ll own this topic. Good luck!

Final Note for Teachers: - Pacing: Spend 50% of time on calendar problems (more marks). - Visuals: Use a clock diagram for angle problems. - Common Pitfall: Students forget to adjust for leap years in century years. Drill this!