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
