Clock and Calendar

The hour and minute spaces on the dial, the speeds of the two hands and the three basic facts the whole chapter rests on.

The round dial of a clock is one full turn of 360°. On it are 12 big numbers (the hours) and 60 small spaces (the minute marks). Between two big numbers there are 5 spaces. The short, thick hand is the hour hand; the long, thin hand is the minute hand.

This table is the foundation of the whole chapter. Three things in it deserve special attention:

1. The hour hand does not stand still. At 9:30 it is not on 9; it has moved 15° past 9 (30 minutes × 0.5°). This is the point most often missed.
2. In 12 hours the minute hand makes 12 laps and the hour hand 1. So the minute hand catches the hour hand 11 times in 12 hours — about once every hour, except that between 11 and 1 it happens only once, exactly at 12.
3. Every minute the minute hand gains 5.5° on the hour hand. The solution of every question on meeting hands and the angle between them comes from this 5.5°.

Converting between 12-hour and 24-hour time and finding the time between two clock times.

The dial shows 12 hours while a day has 24. So the time is written with am (from midnight to noon) or pm (from noon to midnight). Railways, the army and many exams use the 24-hour clock, in which the hours run from 00 to 23 and the time is written in four digits (such as 1430 h).

The safest way to find the time between two clock times is to write both in 24-hour form. If the second looks smaller (such as from 10 pm to 6 am) it belongs to the next day, so add 24 hours to it.

At what angle from 12 each hand stands at any time, and how to find the (smaller) angle between them.

At any time h hours m minutes the position of each hand is measured from 12 (clockwise):

You need not memorise the formula — every time write the position of each hand separately and then subtract. The formula comes from the table by itself: |(30h + 0.5m) − 6m| = |30h − 5.5m|.

The smaller angle: two hands make two angles (say 100° and 260°) which add up to 360°. When “the angle” is asked, the smaller one (not more than 180°) is usually wanted. So if the gap is more than 180°, subtract it from 360°.

When the hands make a fixed angle between n and n+1 o’clock — one method, four cases.

Counting how often the hands meet, are opposite or are at right angles in a day or in a stretch of time.

Converting between the time of a clock that gains or loses x minutes in 24 hours and the true time.

In a correct clock the hands meet every $65\frac{5}{11}$ minutes — using this to spot a faulty clock; and a clock that passes the right time at a uniform pace.

The rule for the time of a clock in a mirror, the reverse question (the real time from the mirror) and the times that look the same in a mirror.

The leap-year rule, the odd days of ordinary and leap years, the table of months and the odd days of whole centuries.

The four steps of the odd-days method, the remainder code of the days and a check; the short way from a nearby known date.

“Yesterday” and “the day after tomorrow” questions, the weekday of two dates of the same month and counting days between two dates.

Two conditions for getting the same calendar, how often a weekday occurs in a month, months that begin on the same weekday and the weekdays of dates from Indian history.

The first step of every type of question at a glance.

The commonest reasons for losing marks in clock and calendar questions and how to avoid them.

Simple angles of the hands, the time of meeting, a mirror and straight calendar questions.

The standard level of every type: angles, meetings, fast and slow clocks, a mirror, weekday from a date and repeating calendars.

Hard questions: hands x° apart, counting over a stretch, the right time of a faulty clock, weekdays of distant dates and repeating calendars.