Competitive Exams / CUET UG / Logical Reasoning / Clocks & Calendars

Clocks & Calendars for CUET UG

Master the angle formula, coincidence and mirror rules, odd-day calendar logic, and data sufficiency for clocks and calendars with original notes and timed practice inside the Learn at My Place competitive flow.

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Overview

Why This Chapter Matters in CUET

Clocks and Calendars is a high-accuracy chapter because every question reduces to one or two formulas. Once the angle formula and odd-day system are internalised, even complex questions become one-step calculations.

The chapter also tests data sufficiency, which rewards a disciplined two-step habit: test each statement in isolation before combining them. That habit alone eliminates the most common wrong answers.

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Section A

Notes & Concept Builder

Formula first, answer second

1. Clock Basics — Dial and Hand Speeds 2. Angle Formula — θ = (11/2)m − 30h 3. Coincidence, Straight Line, and Right Angle 4. Gain / Lose — Fast and Slow Clock Logic 5. Mirror Image of Clock 6. Calendar Basics — Odd Days 7. Century Odd Days — 100, 200, 300, 400 Years 8. Data Sufficiency — Testing Statements

1. Clock Basics — Dial and Hand Speeds

A clock dial has 12 hours and is divided into 60 minute spaces. The hour hand completes one full revolution in 12 hours, while the minute hand completes one full revolution every 60 minutes.

Speed facts: Minute hand covers 360° in 60 min → 6° per minute. Hour hand covers 360° in 720 min → 0.5° per minute. Relative speed of minute hand over hour hand = 5.5° per minute.

In one hour, the minute hand gains exactly 55 minute spaces over the hour hand. That gain is the engine behind every angle and coincidence calculation.

2. Angle Formula — θ = (11/2)m − 30h

The standard formula for the angle between the two hands when the time shown is h hours and m minutes is:

$\theta = \left|\frac{11}{2}m - 30h\right|$

If the result exceeds 180°, subtract it from 360° to get the smaller (reflex) angle. Always take the absolute value before comparing.

Quick example: at 3:40 →

\theta = |\frac{11}{2} \times 40 - 30 \times 3| = |220 - 90| = 130°

This single formula replaces all ad-hoc reasoning for every clock angle question in CUET.

3. Coincidence, Straight Line, and Right Angle

Hands coincide (0°) 11 times in 12 hours and 22 times in 24 hours. Hands are in a straight line (180° apart) 11 times in 12 hours and 22 times in 24 hours. Hands are at a right angle (90°) 22 times in 12 hours and 44 times in 24 hours.

Common trap: students say hands coincide 12 times in 12 hours, but the 11 o'clock and 12 o'clock coincidence merge into one at exactly 12:00, so the correct count is 11.

For any target angle $\theta$ , use the formula $m = \frac{2}{11}(30h \pm \theta)$ to find the exact minute.

4. Gain / Lose — Fast and Slow Clock Logic

A fast clock gains time: it shows a time ahead of the correct time. A slow clock loses time: it shows a time behind the correct time.

In a correct clock the minute hand and hour hand coincide every $65\frac{5}{11}$ minutes. If a clock's hands coincide more often, it is losing time; if less often, it is gaining time.

Formula for gain/lose in T hours:

\text{Gain or loss} = \frac{720}{11} \times \frac{\text{difference in coincidence interval}}{\text{stated interval}}

minutes per 12 hours.

Always identify whether the clock is ahead or behind before calculating the net error over the required duration.

5. Mirror Image of Clock

When a clock is seen in a mirror, the image shows the complement of the real time. The rule depends on whether the visible time is before or after 12 o'clock:

If the mirror time is between 1:00 and 11:59 → Real time = $11:60 - \text{mirror time}$ .

If the mirror time shows 12:00 → Real time is also 12:00.

Example: mirror shows 4:50 → real time =

11:60 - 4:50 = 7:10

. Mirror shows 8:15 → real time =

11:60 - 8:15 = 3:45

The subtraction is always from 11:60 (treat it as 12:00 minus 1 minute carry) for times that do not land exactly on 12:00.

6. Calendar Basics — Odd Days

An odd day is the remainder when the number of days in a period is divided by 7. Odd days determine which day of the week a date falls on.

An ordinary year has 365 days = 52 weeks + 1 day → 1 odd day.

A leap year has 366 days = 52 weeks + 2 days → 2 odd days.

Leap year rule: divisible by 4, except century years which must be divisible by 400. So 1900 is not a leap year, but 2000 is.

Once you have the total odd days of a period, map the remainder (0–6) to Sun, Mon, Tue, Wed, Thu, Fri, Sat to get the day.

7. Century Odd Days — 100, 200, 300, 400 Years

Odd days in century blocks follow a fixed pattern because of the leap year exception for centuries:

$100 \text{ years} = 5 \text{ odd days}$

$200 \text{ years} = 3 \text{ odd days}$

$300 \text{ years} = 1 \text{ odd day}$

$400 \text{ years} = 0 \text{ odd days}$

Cycle repeats every 400 years: because 400 years = 0 odd days, the calendar repeats exactly after every 400-year block. Use this to find what day of the week any date falls on.

For a full date calculation: separate years into century block + remaining years + remaining months + remaining days, then sum all odd days and take mod 7.

8. Data Sufficiency — Testing Statements

Data sufficiency questions give a question and two statements. You must decide which statement(s) are needed to answer the question definitively.

Rule: always test Statement I alone first, then Statement II alone, then both together.

Standard answer codes: (A) Statement I alone is sufficient. (B) Statement II alone is sufficient. (C) Both statements together are needed. (D) Neither statement is sufficient even together.

Never let one statement influence your reading of the other during the individual tests. The most common mistake is combining both statements while testing one of them in isolation.

Solved Practice

Solved Examples

Apply the formula before opening

Example 1: Find the angle between the hands of a clock at 6:30.

Using the formula $\theta = |\frac{11}{2}m - 30h|$ :

$\theta = |\frac{11}{2} \times 30 - 30 \times 6| = |165 - 180| = 15°$ .

The angle between the hands at 6:30 is 15°.

Example 2: At what time between 4 and 5 o'clock are the hands of a clock coincident?

For coincidence, $\theta = 0$ , so $m = \frac{2}{11} \times 30h = \frac{2}{11} \times 120 = \frac{240}{11} = 21\frac{9}{11}$ minutes.

The hands coincide at $4:21\frac{9}{11}$ , i.e., approximately 4 hours 21 minutes 49 seconds.

Example 3: At what time between 7 and 8 o'clock are the hands at right angles?

For 90°: $m = \frac{2}{11}(30h \pm 90)$ .

$m = \frac{2}{11}(210 + 90) = \frac{600}{11} = 54\frac{6}{11}$ min → time is $7:54\frac{6}{11}$ .

$m = \frac{2}{11}(210 - 90) = \frac{240}{11} = 21\frac{9}{11}$ min → time is $7:21\frac{9}{11}$ .

Both $7:21\frac{9}{11}$ and $7:54\frac{6}{11}$ are valid answers.

Example 4: A clock shows 7:20 in a mirror. What is the actual time?

Apply mirror rule: Real time = $11:60 - 7:20$ .

$11:60 - 7:20 = 4:40$ .

The actual time is 4:40.

Example 5: A clock gains 5 minutes in every 24 hours. If it shows 10:00 AM on Monday, what time does it show at 10:00 AM on Wednesday?

Duration = 48 hours = 2 days.

Gain in 48 hours = $5 \times 2 = 10$ minutes.

The clock will show $10:00 + 10 \text{ min} =$ 10:10 AM on Wednesday.

Example 6: What day of the week is 1 January 2000?

1600 years = 0 odd days. 300 years = 1 odd day. 99 years (1901–1999) = 24 leap years + 75 ordinary years = $24 \times 2 + 75 \times 1 = 48 + 75 = 123$ odd days = $123 \mod 7 = 4$ odd days. Jan 1 = 1 day = 1 odd day.

Total = $0 + 1 + 4 + 1 = 6$ odd days → Saturday.

1 January 2000 was a Saturday.

Example 7: How many odd days are there in 200 years?

In 200 years there are 48 leap years and 152 ordinary years.

Total days = $48 \times 366 + 152 \times 365 = 17568 + 55480 = 73048$ .

$73048 \div 7 = 10435$ weeks exactly + $\mathbf{3}$ days.

200 years have 3 odd days.

Example 8: If 15 August 1947 was a Friday, what day of the week was 15 August 1948?

1948 is a leap year (divisible by 4), so 1947–48 contributes 2 odd days.

Friday + 2 days = Sunday.

Example 9: Statement I: A is older than B. Statement II: B was born in 1990. Is A born before 1990? Which statement(s) are sufficient?

Statement I alone: A is older than B but we do not know when B was born → not sufficient.

Statement II alone: gives B's birth year but says nothing about A → not sufficient.

Both together: A is older than B who was born in 1990, so A was born before 1990 → sufficient.

Answer: Both statements together are needed (C).

Example 10: How many times do the hands of a clock coincide between 12:00 noon and 12:00 midnight?

The hands coincide 11 times in every 12-hour period.

Between 12:00 noon and 12:00 midnight is exactly 12 hours.

However, the coincidence at 12:00 midnight is the boundary, so within the open interval they meet 11 times.

Next Step

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