# Clocks and Calendars

**Clocks and Calendars**

Clocks and calendars are common topics in quantitative aptitude exams, including the TCS National Qualifier Test (NQT). Let's go over some important concepts and problem-solving techniques related to clocks and calendars that you may encounter in the exam.

__Clocks:__

**Analogue Clock**: An analogue clock consists of an hour hand, a minute hand, and sometimes a second hand. Remember the following facts:

- In 12 hours, the hour hand completes one full revolution.
- In 60 minutes, the minute hand completes one full revolution.
- In 60 seconds, the second hand completes one full revolution.

**Time Calculation:** To calculate the time between two positions of the clock hands, you can use the following formulas:

- Angle covered by the hour hand = (30 × hours) + (0.5 × minutes)
- Angle covered by the minute hand = (6 × minutes)
- Angle between the hour and minute hands =| Angle covered by the hour hand - Angle covered by the minute hand |

Problems involving finding angles, time intervals, or the next occurrence of a specific time are common in clock problems.

**Examples:**

**Example 1**: Find the angle between the hour and minute hands of a clock at 3:20.

**Solution:**

Angle covered by the hour hand = (30 × 3) + (0.5 × 20) = 90 + 10 = 100 degrees

Angle covered by the minute hand = (6 × 20) = 120 degrees

Angle between the hour and minute hands = |100 - 120| = 20 degrees

**Example 2**: At what time between 3 and 4 o'clock will the hour and minute hands of a clock coincide?

**Solution:**

The hour hand moves 30 degrees per hour, while the minute hand moves 360 degrees per hour. Therefore, the hour and minute hands coincide when the hour hand moves (30 × t) degrees, where t is the number of hours since 3 o'clock. The minute hand moves (360 × t) degrees.

Setting up the equation: 30t = 360t

Simplifying, we get: 330t = 0

Therefore, t = 0

The hour and minute hands coincide at 3 o'clock.

__Calendars:__

**Leap Years:** A leap year is a year that is divisible by 4, except for years that are divisible by 100 but not divisible by 400. For example, the year 2000 was a leap year.

**Finding the Day**: You can find the day of the week for any given date using various methods like the Doomsday Algorithm or Zeller's Congruence.

**Counting Days:** To find the number of days between two given dates, you can use the following steps:

- Count the number of days in the years between the two dates.
- Add the number of leap years between the two dates.
- Count the number of days in the months between the two dates.
- Add the remaining days in the target month.

Problems involving finding the day of the week, counting days, or finding the next occurrence of a specific date are common in calendar problems.

**Examples:**

**Example 1**: What day of the week was January 1, 2022?

**Solution:** We can use the Zeller's Congruence formula to find the day of the week. Zeller's Congruence: h = (q + [(13(m + 1)) / 5] + K + [(K/4)] + [(J/4)] - (2J)) mod 7

In this formula:

h is the day of the week (0 for Saturday, 1 for Sunday, and so on)

q is the day of the month (1 for January 1st)

m is the month number (January is 13, February is 14, and so on)

K is the year of the century (2022 % 100 = 22)

J is the zero-based century (2022 / 100 = 20)

Substituting the values into the formula: h = (1 + [(13 * 14) / 5] + 22 + [22/4] + [20/4] - (2*20)) mod 7 Simplifying, we get: h = 6

Therefore, January 1, 2022, was a Saturday.

**Example 2**: How many odd days are there in 600 years?

**Solution:** To find the number of odd days in a given number of years, we divide the years by 400 (leap year cycle) and find the remainder.

600 divided by 400 leaves a remainder of 200. Since 200 is not divisible by 7, there are 200 odd days in 600 years.

Therefore, there are 200 odd days in 600 years.

**Tips for Problem Solving:**

- Read the problem carefully and understand the given information.
- Sketch the clock or calendar if necessary to visualise the problem.
- Identify what is asked in the problem and plan your approach accordingly.
- Use formulas, rules, or algorithms related to clocks and calendars to solve the problem.

Practise solving different types of problems to improve your speed and accuracy