Twenty four hour time and crossing midnight

When you reach for it

Bogdana, Talara and Dakota work the night shift at a fire station. The shift starts in the evening and ends the next morning. Every call, every truck check and every meal goes into the station log with the time it happened.

A clock with hands goes around twice a day, so “7{:}20” could be twenty past seven in the morning or twenty past seven in the evening. On a twelve-hour clock the label a.m. or p.m. says which half of the day you mean (Morning and afternoon on a twelve hour clock). A label is easy to forget, and in a fire station log a forgotten label could put a call twelve hours off. So the station writes times on the 24-hour clock, which gives every hour of the day its own number. Then no label is needed.

The shift raises a second question. Bogdana starts at 22{:}15 and goes home at 06{:}40. The end time looks smaller than the start time, yet she worked most of the night. Finding how long a stretch of time lasts is Elapsed time: count from the start to the end. This lesson shows how to do that when the count runs through midnight into the next day.

Reach for this lesson when a time is written on the 24-hour clock, when you need to change a time between the two clocks, or when a stretch of time crosses midnight.

The procedure

Part 1: every hour its own number

On the 24-hour clock, the hour number is the number of whole hours since midnight. The day starts at midnight, written 00{:}00. One hour later it is 01{:}00, then 02{:}00, and so on. Noon, twelve hours after midnight, is 12{:}00. The clock does not start again at 1 after noon; it keeps counting. One hour after noon is 13{:}00, two hours after noon is 14{:}00, and the last minute of the day is 23{:}59. One minute after that it is midnight again, 00{:}00 of the next day.

A 24-hour time is written with two digits for the hour and two for the minutes, so seven in the morning is 07{:}00, not 7{:}00. The minutes work the same way on both clocks: they run from 00 to 59. People often say 15{:}30 as “fifteen thirty” and 15{:}00 as “fifteen hundred”.

From a.m. and p.m. to the 24-hour clock. Think about how many hours have passed since midnight.

• A morning hour from 1 a.m. to 11 a.m. is already the number of hours since midnight. Keep it, and write it with two digits: 7{:}20 a.m. is 07{:}20.
• An afternoon or evening hour from 1 p.m. to 11 p.m. comes after the 12 hours of the morning. Add 12: 3{:}30 p.m. is 3 + 12 = 15, so it is 15{:}30.
• The hour 12 is the one to watch. 12{:}00 p.m. is noon, 12 hours after midnight, so it stays 12{:}00. 12{:}00 a.m. is midnight itself, zero hours after midnight, so it is 00{:}00. In the same way 12{:}25 a.m. is 00{:}25, twenty-five minutes into the day.

From the 24-hour clock to a.m. and p.m. Go the other way.

• Hours 00 to 11 are a.m. Hour 00 becomes 12: 00{:}40 is 12{:}40 a.m. Hours 01 to 11 keep their number: 09{:}15 is 9{:}15 a.m.
• Hours 12 to 23 are p.m. Hour 12 stays 12: 12{:}10 is 12{:}10 p.m. Hours 13 to 23 lose 12: 15{:}30 is 15 - 12 = 3, so 3{:}30 p.m.

Why 12? Because the twelve-hour clock starts its count again at noon, and the morning holds exactly 12 hours. Adding 12 puts the morning’s hours back in front of an afternoon hour; taking 12 away removes them.

There is one more good thing about the 24-hour clock. Within one day, a later time always has a larger number, so putting times in order is just comparing numbers. On the twelve-hour clock, 12{:}05 a.m. comes before 1{:}00 a.m., even though 12 is more than 1.

Some schedules write 24{:}00 for the very end of a day, the midnight that closes it. It is the same moment as 00{:}00 of the next day.

Part 2: elapsed time, and crossing midnight

Inside one day. On the 24-hour clock, the time from an earlier time to a later one on the same day is the later time take away the earlier one. Subtract the minutes, then the hours. If there are not enough minutes to subtract from, trade one hour for 60 minutes, because an hour is 60 minutes (Units of time). For example, from 13{:}40 to 16{:}15, write 16{:}15 as 15 hours 75 minutes and subtract: 2 hours 35 minutes. Counting up from the start gives the same answer, and it makes a good check.

Across midnight. Bogdana’s shift runs from 22{:}15 to 06{:}40 the next morning. Taking 22{:}15 away from 06{:}40 makes no sense, because the end belongs to a different day. So split the time at midnight, where the day changes:

• Before midnight: from 22{:}15 to midnight. Write midnight as 24{:}00, which is the same as 23 hours 60 minutes, and subtract: 24{:}00 - 22{:}15 is 1 hour 45 minutes.
• After midnight: the new day starts at 00{:}00, so a time after midnight already counts its hours since midnight. From 00{:}00 to 06{:}40 is 6 hours 40 minutes.
• Add the two pieces, regrouping 60 minutes into an hour:

Bogdana’s shift lasts 8 hours 25 minutes. The two pieces cover the whole stretch with no gap and no overlap, because they meet at midnight.

A second way gives the same answer. Keep counting hours from the start of the first day, past 24. Then 06{:}40 the next morning is 24 + 6 = 30 hours 40 minutes after the first midnight, and 30{:}40 - 22{:}15 = 8{:}25. Adding 24 to the end hour says “this time is on the next day”.

The same two ideas find an end time. Add the hours and minutes to the start time. If the hour reaches 24 or more, the clock has passed midnight: take 24 away and move to the next day. To go back from an end time to a start time before midnight, add 24 to the end hour first and then subtract. Keeping track of which day it is, and what that day’s date is, is Calendars and dates.

Worked examples

Examples 1, 2 and 3 go with Part 1, and Examples 4 to 7 with Part 2.

Boundaries

• Hours 00 to 23, minutes 00 to 59. A 24-hour time has no hour 24 or more, except that 24{:}00 is sometimes written for the end of a day. An hour of 24 or more in the middle of a sum means midnight was passed: take 24 away and move to the next day.
• Exactly at twelve. 12{:}00 p.m. is noon and 12{:}00 a.m. is midnight. Many schedules avoid the doubt by writing “noon” and “midnight”, or by using the 24-hour clock.
• A time does not name the day. 06{:}40 happens every day. Splitting at midnight works when the stretch is shorter than a whole day and you know it ends on the next day. If a stretch could last more than 24 hours, the two times alone cannot tell how long it was; you also need the dates.
• The same clock for both ends. The subtraction only makes sense when both times are read on the same clock in the same place. Clocks in places that read different hours at the same moment are the business of Time zones.

Where this goes next

The 24-hour clock is how train and bus timetables, hospitals and many countries write the time, and it is the natural clock for Time zones, where a time in one city can land after midnight in another and the date changes. Counting days, weeks and dates when a stretch runs longer than one night is Calendars and dates. The trading of an hour for 60 minutes used here comes back in Multi-step problems with unit conversions.

Exercises

Part 1: every hour its own number

1. Write 14{:}00 on the twelve-hour clock.
2. Write 08{:}25 on the twelve-hour clock.
3. Write 21{:}10 on the twelve-hour clock.
4. Write 00{:}45 on the twelve-hour clock.
5. Write 12{:}20 on the twelve-hour clock.
6. Write 4{:}35 p.m. on the 24-hour clock.
7. Write 11{:}50 p.m. on the 24-hour clock.
8. Write 12{:}15 a.m. on the 24-hour clock.
9. Write 10{:}00 a.m. and 10{:}00 p.m. on the 24-hour clock. How many hours apart are they?
10. Put these times from one day in order, earliest first: 8{:}30 p.m., 12{:}40 p.m., 3{:}20 a.m., 12{:}20 a.m.

Part 2: elapsed time, and crossing midnight

11. How long is it from 09{:}20 to 11{:}45 on the same day?
12. How long is it from 14{:}50 to 17{:}10 on the same day?
13. Dinner at the station is cooked from 18{:}35 to 23{:}05, a slow stew. How long does it cook?
14. A shift runs from 21{:}00 to 05{:}00 the next morning. How long is it?
15. A call starts at 23{:}20 and ends at 02{:}10 the next morning. How long is it?
16. Talara works from 20{:}45 to 07{:}30 the next morning. How long is her shift?
17. A training drill starts at 22{:}40 and lasts 3 hours 35 minutes. When does it end?
18. A call starts at 19{:}55 and the crew is out for 5 hours 20 minutes. When do they get back?
19. A truck repair ended at 01{:}20 after taking 2 hours 45 minutes. When did it start?
20. A call lasted from 10{:}50 p.m. to 1{:}25 a.m. the next morning. How long was it?
21. Talara goes to bed in the station bunk room at 23{:}30, and the alarm rings for her shift change at 06{:}15. A call takes her out of bed from 01{:}40 to 03{:}05. How long was she in bed?

Reasoning

22. Explain why you add 12 to change 5{:}00 p.m. to the 24-hour clock, but do not add 12 to change 12{:}00 p.m.
23. A shift runs from 21{:}30 to 04{:}50 the next morning. Dakota subtracts 21{:}30 - 04{:}50 and gets 16 hours 40 minutes. What did he really find? How long is the shift?
24. Bogdana writes a time of 24{:}30 in the log. Explain why that is not a time on the 24-hour clock, and write the time she means.
25. One shift runs from 19{:}00 to 07{:}00 the next morning, and the day shift runs from 07{:}00 to 19{:}00. Show that each lasts 12 hours and that together they cover a whole day. Then explain why, for any two different times, the time from the first to the second plus the time from the second back around to the first is always 24 hours.