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Elapsed time and the 24-hour clock

Measurement · level 3, lesson 3 of 6

A question to start

A bus leaves at 9:45 and arrives at 11:20. How long is the trip?

Sam types \(11.20 - 9.45\) into a calculator and gets 1.75. "One hour and 75 minutes," he says. But an hour only has 60 minutes, so "75 minutes" is more than an hour on its own. Something has gone wrong. The real answer is 1 hour 35 minutes, and this lesson shows a way to find it that never goes wrong: counting on along a time line.

The idea

Elapsed time

Elapsed time means "how much time has passed" between a start and an end. Finding it is like finding the distance between two places on a road: you measure the gap.

The tricky part is that time does not go in tens. You learned these in level 2:

  • 60 seconds = 1 minute
  • 60 minutes = 1 hour
  • 24 hours = 1 day

Because an hour is 60 minutes and not 100, ordinary subtraction of the numbers on a clock gives wrong answers. The calculator treated 11.20 and 9.45 as decimals, as if each hour had 100 minutes.

Why 60?

The habit of counting in sixties is very old. The ancient Babylonians, who lived in what is now Iraq about 4000 years ago, wrote numbers in groups of 60. Later astronomers kept their sixtieths for splitting circles and hours, and the habit has lasted until today.

There is a good reason it lasted. 60 can be shared fairly in lots of ways: by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. Half an hour is 30 minutes, a third is 20, a quarter is 15, a fifth is 12, a sixth is 10. With 100 minutes in an hour, a third of an hour would be 33 and a bit, which is awkward.

Counting on with a time line

To find the time from 9:45 to 11:20, do not subtract. Count on, in easy jumps:

  1. Jump from the start to the next whole hour.
  2. Jump in whole hours as far as you can.
  3. Jump the last few minutes.
9:4510:0011:0011:20 15 min1 h20 min

Add the jumps: 15 min + 1 h + 20 min = 1 h 35 min. Each jump is easy, and the whole hours stay separate from the minutes, so there is no mix-up.

Why does this work when subtracting the clock numbers does not? Because each jump only ever uses one kind of unit. The first jump fills up an hour (from 45 minutes to 60). After that, everything lines up with whole hours, so you can count hours plainly, and the last jump is just minutes.

When the minutes add past 60

Sometimes you add times and get more than 60 minutes. Trade 60 minutes for 1 hour, just as you trade 10 ones for 1 ten when you add numbers.

For example, 2 h 40 min + 1 h 35 min = 3 h 75 min. But 75 minutes is 60 minutes + 15 minutes, which is 1 h 15 min. So the total is 4 h 15 min.

The 24-hour clock

A normal clock face shows 12 hours, so every time happens twice a day: 7:00 in the morning and 7:00 in the evening. We add a.m. (before noon) or p.m. (after noon) to tell them apart.

The 24-hour clock solves this a different way: it keeps counting after noon instead of starting again at 1. Timetables for trains, planes and buses use it, and so do hospitals, the army and many phones. Nobody can mix up the morning train with the evening one.

midnight6 a.m.noon 6 p.m.11 p.m. 00:0006:0012:00 18:0023:00 a.m.: same numbers p.m.: add 12 to the hour

The rules:

  • Morning (a.m.): the hour stays the same, written with two digits. 7:30 a.m. is 07:30.
  • Afternoon and evening (p.m.): add 12 to the hour. 3:15 p.m. is \(3 + 12 = 15\), so 15:15.
  • Noon is 12:00, and midnight is 00:00, the start of a new day. One minute after midnight is 00:01.
  • To go back from 24-hour time, if the hour is more than 12, subtract 12 and say p.m. 20:45 is 8:45 p.m.

Why add 12? By noon, 12 hours of the day have already passed. So 3 p.m. is 3 hours after noon, which is \(12 + 3 = 15\) hours after midnight. A 24-hour time simply says how many hours and minutes since midnight.

That also makes elapsed times easier: from 08:20 to 15:20 is just \(15 - 8 = 7\) hours, with no a.m. and p.m. to worry about.

Crossing midnight

From 22:30 at night to 06:15 the next morning, count on in two parts: up to midnight, then on from midnight.

  • 22:30 to 24:00 (midnight) is 1 h 30 min.
  • 00:00 to 06:15 is 6 h 15 min.
  • Total: 1 h 30 min + 6 h 15 min = 7 h 45 min.

A day on Mars

A day on Mars, called a sol, is 24 h 39 min 35 s long. That is 39 minutes 35 seconds longer than a day on Earth, nearly 40 minutes. People who drive rovers sometimes live on "Mars time" for a while, so their work day starts about 40 minutes later every Earth day, to keep pace with the rover's mornings.

Worked examples

1. The bus. 9:45 to 11:20: 15 min to 10:00, 1 h to 11:00, 20 min to 11:20. Total 1 h 35 min.

2. A film. A film starts at 19:40 and lasts 1 h 55 min. When does it end?

Count on from the start. 19:40 + 1 h = 20:40. Then 55 more minutes: 20 min takes you to 21:00, and 35 min more takes you to 21:35. (That is 9:35 p.m.)

3. Working backwards. School starts at 08:50 and the walk takes 35 minutes. When must Mia leave?

Count back 35 minutes from 08:50. Back 50 minutes would be 08:00, which is too far. So: back 35 minutes from 08:50 is \(50 - 35 = 15\), giving 08:15.

4. Adding durations. A train trip has two parts: 3 h 50 min and 2 h 25 min. How long in all?

3 h + 2 h = 5 h. 50 min + 25 min = 75 min = 1 h 15 min. Total: 5 h + 1 h 15 min = 6 h 15 min.

5. Seconds to minutes. A song lasts 210 seconds. How long is that in minutes and seconds?

\(3 \times 60 = 180\), and \(210 - 180 = 30\). So 3 min 30 s. (Not 2.1 minutes!)

6. The sol. How much longer than an Earth day are 3 sols?

Each sol is 39 min 35 s longer. Three of them: 117 min 105 s. Trade 60 s for 1 min: 105 s is 1 min 45 s, giving 118 min 45 s. And 118 min is 1 h 58 min. So 3 sols are 1 h 58 min 45 s longer than 3 Earth days: almost 2 hours.

Try it

  1. How long is it from 2:35 p.m. to 4:10 p.m.?
  2. Write in 24-hour time: (a) 6:05 a.m.; (b) 4:30 p.m.; (c) 11:55 p.m.; (d) 12:20 a.m. (twenty past midnight).
  3. Write in 12-hour time with a.m. or p.m.: (a) 09:10; (b) 13:45; (c) 21:00.
  4. A swim lesson starts at 16:50 and lasts 45 minutes. When does it end?
  5. How many seconds in 4 minutes 20 seconds? How many minutes in 2.5 hours?
  6. Add: 1 h 45 min + 2 h 30 min + 50 min.
  7. A night flight leaves at 23:15 and lands at 05:40. How long is the flight?
  8. A rover team's work day starts at 08:00. Each Earth day they start 40 minutes later, to follow the sol. When do they start after 3 days? After 6 days?
Answers
  1. 2:35 to 3:00 is 25 min; 3:00 to 4:00 is 1 h; 4:00 to 4:10 is 10 min. Total 1 h 35 min.
  2. (a) 06:05. (b) \(4 + 12 = 16\), so 16:30. (c) \(11 + 12 = 23\), so 23:55. (d) Twenty past midnight is 20 minutes after 00:00, so 00:20.
  3. (a) 9:10 a.m. (b) \(13 - 12 = 1\), so 1:45 p.m. (c) \(21 - 12 = 9\), so 9:00 p.m.
  4. 16:50 + 10 min = 17:00, then 35 more minutes: 17:35.
  5. \(4 \times 60 = 240\), and \(240 + 20 = 260\) seconds. 2.5 hours is 2 hours and half an hour: \(120 + 30 = 150\) minutes.
  6. Hours: \(1 + 2 = 3\) h. Minutes: \(45 + 30 + 50 = 125\) min = 2 h 5 min. Total 5 h 5 min.
  7. 23:15 to midnight is 45 min; midnight to 05:40 is 5 h 40 min. Total: 5 h 85 min = 6 h 25 min.
  8. After 3 days: \(3 \times 40 = 120\) min = 2 h later, so 10:00. After 6 days: \(6 \times 40 = 240\) min = 4 h later, so 12:00. (A real sol is about 39.6 minutes longer, so the real times would be a tiny bit earlier.)

Common mistake

Subtracting clock times like decimals. \(11.20 - 9.45 = 1.75\) is wrong, because the calculator thinks an hour has 100 minutes. Count on with a time line instead, or change both times into minutes after midnight first (\(11 \times 60 + 20 = 680\) and \(9 \times 60 + 45 = 585\), difference 95 minutes = 1 h 35 min).

Reading 1.5 hours as 1 h 50 min. 1.5 hours means one and a half hours: 1 h 30 min. Half of 60 minutes is 30.

Where it leads