Clock and Calendar
Clock and Calendar is a scoring topic of the reasoning paper of ADRE Grade IV. Clock questions deal with the hands of a clock, the angle between them, and the number of times the hands are together or in a straight line. Calendar questions deal with days, dates, weeks and years, and ask for the day of the week on a particular date. Both parts are based on a small number of fixed facts and formulae, so a candidate who learns them well can answer these questions very quickly and with complete confidence.
Basic Facts About a Clock
A clock has a round face divided into twelve equal parts of thirty degrees each. The minute hand completes one full round of 360 degrees in sixty minutes, so it moves six degrees in one minute. The hour hand completes a full round in twelve hours, so it moves thirty degrees in an hour and half a degree in one minute. The difference between the two movements is 5.5 degrees in a minute, and this single figure solves most angle questions. The hands overlap twenty-two times in a day and form a straight line forty-four times in a day.
Angle Between the Hands
To find the angle between the hands at a given time, use the formula: angle = |30H − 5.5M|, where H is the hour and M is the minute. For example, at 3:00 the angle is 30 × 3 = 90 degrees. At 3:30 the angle is |90 − 165| = 75 degrees. If the answer comes out greater than 180 degrees, subtract it from 360 to get the smaller angle. The hands are at right angles when the angle is 90 degrees, opposite when it is 180 degrees, and together when it is zero.
Basic Facts About a Calendar
Every ordinary year has 365 days, which is fifty-two weeks and one extra day. This extra day is called the odd day, and it advances the day of the week by one. A leap year has 366 days, which is fifty-two weeks and two odd days, and it occurs when the year is divisible by four but not by one hundred, or when it is divisible by four hundred. A century has one hundred years with twenty-four leap years, giving five odd days. One hundred days contain fourteen weeks and two odd days, and four hundred years contain zero odd days, which is why the calendar repeats itself every four hundred years.
Finding the Day of the Week
To find the day of the week for a date, count the number of odd days from a known reference. For a given date, add the odd days of the completed years, the odd days of the completed months of the current year, and the number of days of the date itself, then divide the total by seven. The remainder gives the day, where one means Sunday, two means Monday and so on. This method works for any century once the base odd days of that century are known.
Table of Odd Days
- Ordinary year — 1 odd day; leap year — 2 odd days.
- January — 3 odd days (31 days); February — 0 or 1 odd day.
- March — 3; April — 2; May — 3; June — 2; July — 3; August — 3.
- September — 2; October — 3; November — 2; December — 3.
- One hundred years — 5 odd days; four hundred years — 0 odd days.
How This Topic Appears in the Examination
Clock and calendar questions usually appear as two to four objective questions of one mark each. Clock questions ask for the angle between the hands, the time when the hands are together or opposite, or the number of overlaps in a day. Calendar questions ask for the day of the week on a given date, or the day after or before a certain number of days. Because the formulae are fixed, a practised candidate can solve these questions in a few seconds, which makes this topic very valuable for scoring well.
Memory Tips
Memorise the figure 5.5 degrees per minute, since it is the heart of most clock problems. Learn the odd days of each month by heart so that you do not have to recalculate them. Remember the fixed facts: 22 overlaps and 44 straight lines in a day. For calendars, always start from a century and work forward. Write the odd-day table on rough paper before solving so that the calculation stays neat.
Common Mistakes to Avoid
- Forgetting to subtract the angle from 360 when it is greater than 180.
- Confusing the odd days of an ordinary year and a leap year.
- Making a mistake in identifying whether a year is a leap year.
- Forgetting that a century has twenty-four leap years, not twenty-five.
- Using the wrong reference day when counting forward or backward.
How to Prepare This Topic
- Learn the clock formulae for the angle between the hands.
- Memorise the odd-day values of the months and the years.
- Solve at least ten clock and ten calendar questions every day.
- Note the fixed facts about overlaps and straight lines of the hands.
- Practise with a watch in hand to check your answers.
Frequently Asked Questions
How many degrees does the minute hand move in one minute?
The minute hand moves 6 degrees in one minute, and the hour hand moves half a degree. The relative speed between the hands is therefore 5.5 degrees per minute.
What is the formula for the angle between the clock hands?
The angle is |30H − 5.5M|, where H is the hour and M is the minute. If the result is more than 180 degrees, subtract it from 360 to get the smaller angle.
How many times do the clock hands overlap in a day?
The hands overlap 22 times in a day and form a straight line 44 times in a day. They are opposite each other 22 times in a day.
How do I know whether a year is a leap year?
A year is a leap year if it is divisible by four but not by one hundred, or if it is divisible by four hundred. A century is a leap year only when it is divisible by four hundred.
What is an odd day in the calendar?
An odd day is the extra day left after dividing the total number of days by seven. An ordinary year has one odd day and a leap year has two odd days.