Time: Clocks, Calendars & Time Conversion

Time is the quantity we use most often and the one that breaks the usual rules of arithmetic. Seconds and minutes go up in sixties, hours in twenty-fours, days in sevens, and months in a ragged pattern of 28, 30 and 31. Because these units do not follow the neat tens of the metric system, time problems need their own methods. This guide covers the units of time, the 12-hour and 24-hour clocks, reading an analogue clock and finding the angle between its hands, adding and subtracting times, timetables, calendars and leap years, and time zones. It ends with the history of how our time system came about, with the uncertain parts marked as uncertain.

1.Units of Time

To change a larger unit into a smaller one, multiply. To change a smaller unit into a larger one, divide. When the division does not come out evenly, the remainder is the leftover smaller unit.

2.The 12-Hour Clock

The 12-hour clock divides the day into two halves of 12 hours. We add AM or PM to say which half. The letters come from Latin: ante meridiem means “before midday” and post meridiem means “after midday”.

• AM covers the time from midnight up to noon.
• PM covers the time from noon up to midnight.
• 12:00 AM is midnight (the start of the day). 12:00 PM is noon.

Because “12 AM” and “12 PM” confuse many people, timetables and legal documents often write 12 noon and 12 midnight instead. Midnight can also be unclear about the date, since it is both the end of one day and the start of the next. “Midnight on Friday” can mean the start or the end of Friday, so a 24-hour time with a date is safer.

3.The 24-Hour Clock

The 24-hour clock numbers the hours from 00 to 23, so no AM or PM is needed. Times are written with four digits: two for the hour and two for the minutes, such as 07:30 or 19:30. The clock face in Figure 1 shows how the two systems line up.

Converting between the two

4.Reading an Analogue Clock

The shorter hour hand goes once round the face in 12 hours. The longer minute hand goes round in 60 minutes. Each numbered position on the face is 5 minutes on the minute hand (1 means 5 minutes past, 2 means 10, 3 means 15, 6 means 30, 9 means 45). In words, we say “quarter past” (15), “half past” (30) and “quarter to” (45).

The angle between the hands

The face is a circle of 360° with 12 hours, so each hour mark is 30° apart. This gives two speeds:

• The minute hand turns 360° ÷ 60 = 6° per minute.
• The hour hand turns 30° per hour, which is 0.5° per minute. It does not jump from one number to the next. At 3:30 it is halfway between 3 and 4.

5.Adding Time and Finding Elapsed Time

Do not treat clock times as ordinary decimals. When a minutes total reaches 60, it becomes 1 hour, just like a carry.

Elapsed time: the jump method

To find how long something lasted, jump from the start time to the next whole hour, then by whole hours, then to the finish.

Mixed units: hours, minutes and seconds

6.Decimal Hours and Speed

In decimal hours the part after the decimal point is a fraction of an hour, so multiply it by 60 to get minutes. So 2.4 h = 2 h + 0.4 × 60 min = 2 h 24 min. To go the other way, divide the minutes by 60: 3 h 45 min = 3 + 45 ÷ 60 = 3.75 h, and 1 h 20 min = 1⅓ h ≈ 1.33 h.

This matters for speed problems, where the time must be a decimal when you use time = distance ÷ speed. For a 90 km journey at 72 km/h, time = 90 ÷ 72 = 1.25 h, which is 1 h 15 min (0.25 × 60 = 15). A 150 km journey at 60 km/h takes 2.5 h = 2 h 30 min.

7.Timetables

Timetables use the 24-hour clock so that a morning time and an evening time cannot be confused. Keep three things apart: the departure time, the arrival time and the duration.

8.Days, Weeks and the Calendar

The days repeat in a cycle of 7. To find a future weekday, divide the number of days by 7 and move forward by the remainder.

A memory aid is the rhyme “Thirty days hath September, April, June and November”. Another is the knuckle trick: make two fists and count along the knuckles and the gaps between them, starting with January on the first knuckle. Every knuckle is a 31-day month and every gap is a shorter month. July and August are both knuckles (one on each fist), so they are both 31.

Leap years

The Earth takes about 365.2422 days to go round the Sun. A calendar of exactly 365 days would slowly drift away from the seasons, so a day is added to February in some years. The Gregorian rule is:

• A year divisible by 4 is a leap year (2024, 2028);
• except a year divisible by 100 is not a leap year (1900, 2100);
• unless it is also divisible by 400, which is a leap year (2000, 2400).

Over 400 years this gives 97 leap years, an average year length of 365 + ¼ − 1⁄100 + 1⁄400 = 365.2425 days, very close to the true value.

Centuries

The 1st century is the years 1 to 100, so the 19th century is 1801 to 1900 and the 20th century is 1901 to 2000. The year 2000 is the last year of the 20th century, and the 21st century began in 2001.

Writing dates

The date 04/05/2026 means 4 May in many countries (day/month/year), but 5 April in the United States (month/day/year). To avoid confusion, write the month in words (4 May 2026) or use the international order year-month-day (2026-05-04).

9.Time Zones

The Earth turns through 360° in 24 hours, which is 15° each hour, so places 15° of longitude apart are about 1 hour apart in local time. Time zones are described by their offset from UTC (Coordinated Universal Time), the world’s time standard. Pakistan Standard Time is UTC+5, which means Pakistan’s clocks are 5 hours ahead of UTC.

Some places use daylight saving time (“summer time”), putting their clocks forward by an hour in summer. The dates differ from country to country. For example, the UK changes on the last Sundays of March and October, and the United States on different dates. A few places use offsets that are not whole hours, such as India at UTC+5:30 and Nepal at UTC+5:45. Pakistan does not currently use daylight saving time. Rules can change, so for real travel use an official source for the date in question.

Other calendars exist too. The Islamic (Hijri) calendar is based on the Moon: 12 lunar months last about 354 days (12 × 29.53 days), so its year is roughly 11 days shorter than the solar year, and its months move through the seasons.

10.The History of Telling Time

Our use of 60 seconds and 60 minutes comes from the base-60 number system of ancient Mesopotamia, and is linked to the number-base ideas in our number base article. The English words “minute” and “second” come from Latin phrases meaning the “first small part” and the “second small part” of an hour or a degree.

Ancient Egyptians divided the daytime and the night each into 12 parts, so the hours were not the same length in every season. Greek astronomers later used hours of equal length. Mechanical clocks appeared in Europe in the late 13th and 14th centuries. As railways spread in the 19th century, towns had to agree on a common time: British railways used London time from the 1840s. In 1884 an international conference chose the Greenwich meridian as the world’s prime meridian. Today the second is defined by the vibrations of a caesium atom (9 192 631 770 per second, since 1967), and UTC is kept by atomic clocks.

The calendar we use, the Gregorian calendar, was introduced by Pope Gregory XIII in 1582 to correct the drift in the Julian calendar. Different countries adopted it at different times. Britain and its colonies changed in 1752, when Wednesday 2 September was followed by Thursday 14 September, so 11 dates were skipped.

11.Common Mistakes

• Using 100 instead of 60. There are 60 minutes in an hour, not 100.
• Reading decimal hours as minutes. 1.5 h is 1 h 30 min, and 2.35 h is 2 h 21 min.
• Adding 12 to 12 PM, or forgetting that 12 AM is 00:00.
• Forgetting to cross midnight. A duration may go into the next day.
• Subtracting clock times as decimals. 10:50 − 09:35 is 1 h 15 min, not 1.15 hours.
• Ignoring leap years, or treating every century year as a leap year.
• Assuming a month has 30 days. Use the true month lengths.
• Putting the hour hand on the number. It moves 0.5° each minute.
• Forgetting units. 90 minutes and 90 hours are very different.

12.Try It Yourself

1. Write 11:45 PM and 12:05 AM in 24-hour time.
2. How long is it from 08:50 to 14:20?
3. Today is Wednesday. What day will it be in 50 days?

13.Exam-Style Questions

1. [3 marks] Write (a) 7:05 PM in 24-hour time, (b) 00:40 in 12-hour time, (c) 12:30 AM in 24-hour time.
2. [3 marks] A train leaves at 22:50 and arrives at 03:25 the next day. How long is the journey?
3. [3 marks] Add 3 h 48 min 50 s and 2 h 25 min 40 s.
4. [3 marks] (a) Write 3 h 20 min as a decimal number of hours (to 2 decimal places). (b) Write 2.35 hours in hours and minutes.
5. [4 marks] Find the angle between the hands of a clock at (a) 4:00, (b) 2:20.
6. [4 marks] A flight leaves Islamabad (UTC+5) at 09:30 local time and takes 6 h 45 min. Find the local time in London (UTC+0) when it lands.
7. [4 marks] How many days are there from 5 March to 2 May, not counting 5 March, in a year that is not a leap year? If 5 March is a Thursday, what day is 2 May?

14.Key Points

• Time units go in 60s, 24s and 7s. To go to a smaller unit, multiply. To go to a larger unit, divide.
• 12 AM is 00:00 and 12 PM is 12:00. For 1 PM to 11 PM, add 12 to the hour.
• The minute hand turns 6° per minute and the hour hand 0.5° per minute.
• For elapsed time, jump to the next whole hour first. Watch for crossing midnight.
• Decimal hours: the digits after the point are a fraction of 60 minutes. 2.35 h = 2 h 21 min.
• Weekday problems: divide by 7 and use the remainder. A common year moves a date forward 1 weekday, and a leap year 2.
• A year is a leap year if it is divisible by 4, except century years not divisible by 400.
• For time zones: convert to UTC, calculate, then convert to the new zone.

15.For Teachers and Parents

A real analogue clock with movable hands is the best teaching tool for this topic, because learners can see that the hour hand creeps as the minute hand goes round. Ask them to set the clock to a time and say the 24-hour version. For elapsed time, use a number line and the jump method, which avoids borrowing. For calendars, give each learner a real calendar and ask them to find the weekday of their birthday in future years, using the remainder method to predict first. The Clock Calculator and Date Calculator on this page let learners check their own answers, including times that cross midnight. Insist on units in every line of working, and on writing times with four digits in 24-hour form.

16.Related Pages and Further Reading

• Number Base Systems (why 60?)
• Measurement: Metre, Litre & Gram
• Factors & Multiples Speed Race
• Practice Test: Time, Area & Volume

For further reading, the International Bureau of Weights and Measures (BIPM) explains how the second and UTC are defined, and the website timeanddate.com lists current time zone and daylight saving rules. This article is original EDUSAMBAM writing. Examination requirements differ between boards, so compare your practice with the specification of your own board.