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Grade 3 · 3.MD.A.1 · 4.MD.A.2

Elapsed time

Story hour starts at a quarter past ten and ends at twenty to twelve. Neither clock reading says how long it lasted, so how do you get from two times of day to a length of time?

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▴ 1 aboveMorning and afternoon on a twelve hour clockTelling time to the nearest five minutes
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When you reach for it

A clock answers one question: what time is it right now? Reading it to the minute is the work of Telling time. But many everyday questions are about a different kind of time. How long was story hour? If the bus leaves at ten to three and the ride takes 45 minutes, when does it get there? The sleigh ride ended at ten past four after 55 minutes, so when did it start?

Each of these joins two kinds of quantity. A time of day, such as 10{:}15 a.m., names one moment. A length of time, such as 45 minutes, says how much time passes between two moments. The length of time between a start and an end is called the elapsed time. There are three questions, depending on which of the three things is missing:

You knowYou wantThe question
start and endthe lengthHow long did it take?
start and lengththe endWhen does it end?
end and lengththe startWhen did it start?

All three are solved the same way, on a number line for the day, by jumping to the next hour mark. You reach for this whenever a plan has a start, an end, or a length in it: a library schedule, a school bus timetable, a ride that has to be back before dark.

The procedure

The lesson runs over four days. Part 1 separates a time of day from a length of time and puts both on a number line. Part 2 finds how long. Part 3 finds when something ends. Part 4 finds when something started, and reads a schedule, where all three questions turn up together.

Part 1: a time of day and a length of time

Picture the day as a The number line running left to right, earliest to latest. A time of day is a point on that line. A length of time is a stretch of the line, from one point to another, in the same way that the distance between two numbers is a stretch in Length on a number line. The point 10{:}15 a.m. tells you where you are. The stretch “45 minutes” tells you how far you go, and it can start anywhere.

That difference decides what you may add and subtract:

  • a time of day plus a length of time is a later time of day (2{:}50 p.m. and then 45 minutes more is a time);
  • a later time of day minus an earlier one is a length of time;
  • a length plus a length is a length (45 minutes and 40 minutes make 85 minutes);
  • two times of day are never added. 3{:}00 p.m. “plus” 2{:}00 p.m. names no moment and no length, just as adding two house numbers on a street gives no house and no distance.

The hour marks, 7{:}00, 8{:}00, 12{:}00 and so on, are the landmarks of the line. From one hour mark to the next is 1 hour, which is 60 minutes. A time such as 7{:}35 sits 35 minutes past the mark 7{:}00. Because the whole hour is 60 minutes, the same point is 60 - 35 = 25 minutes before the next mark, 8{:}00. So the jump from any time to the next hour mark is

60 - \text{(the minutes in the time)}.

This is why the hour mark is the place to aim for. Clock readings do not count like ordinary numbers. After 10{:}59 comes 11{:}00, not 10{:}60, because the minutes start over every 60, not every 10 or every 100. Inside one hour the minutes subtract as usual. Across an hour mark they do not, so the hour mark is where the jumps change step.

Lengths of time are given in minutes or in hours and minutes. Since 1 hour is 60 minutes, 1 hour 25 minutes is 60 + 25 = 85 minutes, and 2 hours is 60 + 60 = 120 minutes. Going the other way, take out one 60 for each whole hour: 100 minutes is 60 + 40, which is 1 hour 40 minutes.

Part 2: how long? Jump from the start to the end

To find how long something lasted, draw an open number line: a plain line with no ruler marks, where only the points you need are drawn, in order. Put the start at the left and the end at the right, and the hour marks that fall between them. Then jump:

  • from the start to the first hour mark, 60 minus the start’s minutes;
  • from hour mark to hour mark, 1 hour each;
  • from the last hour mark to the end, which is just the end’s minutes, since the end is that many minutes past its hour.

Every jump begins or ends on an hour mark, so each one is easy to find. The jumps together cover the whole stretch from start to end without gaps or overlaps, so the elapsed time is their sum. Add the minutes jumps, then the hours, and trade any 60 minutes for 1 hour if you want the answer in hours and minutes.

If the start and end fall in the same hour, there is no hour mark between them and one jump does it: subtract the minutes. From 9{:}10 to 9{:}40 is 40 - 10 = 30 minutes.

On a twelve-hour clock the hour after 12 is 1. Noon is the hour mark 12{:}00, and the times after it are p.m.: 11{:}00 a.m., then 12{:}00 p.m., then 1{:}00 p.m. The jump from 12{:}00 to 1{:}00 is 1 hour like any other. Why the label changes at noon is the business of Morning and afternoon on a twelve hour clock.

Part 3: when does it end? Jump forward

Now the start and the length are known. Start at the starting point and spend the length in jumps to the right. The first jump goes to the next hour mark, if the length is long enough to reach it. Subtract that jump from the length to see how many minutes are left. Then jump whole hours while at least 60 minutes are left, and finally jump the minutes that remain. The point where the last jump lands is the end time.

Each step keeps a running total honest: what has been spent plus what is left is always the whole length. When nothing is left, the whole length has been spent, so the landing point is the end. If a jump passes 12{:}00, the time changes from a.m. to p.m.

Part 4: when did it start? Jump backward; reading a schedule

When the end and the length are known, run the same jumps the other way. Start at the end and jump to the left. The first jump goes back to the hour mark at or just before the end, which is the end’s minutes. Then jump back whole hours, then the minutes still left. Jumping back some minutes from an hour mark lands at 60 minus those minutes, in the hour before: 45 minutes back from 4{:}00 is 3{:}15, because 60 - 45 = 15.

A backward answer is checked by a forward question. If the start you found is right, then the time from that start to the given end is the given length.

A schedule is a table of times of day: when each activity starts and ends, or when a bus reaches each stop. Every row with a start and an end is a “how long” question. The space between one row’s end and the next row’s start is a length of time too. And a plan such as “be there by 10{:}00 a.m.” is a “when must I start” question, solved by jumping backward from the deadline.

Worked examples

Example 1

Part 1: the bus at 7:35

Qillaq’s school bus stops at her corner at 7{:}35 a.m. The ride to school takes 25 minutes, and the bus pulls up at school at 8{:}00 a.m. Which of these three are times of day, and which is a length of time? Show them on a number line from 7{:}00 to 8{:}00, and check that they agree.

7:00 7:35 8:00 25 min
Figure 1The times of day are points. The 25 minutes is the stretch between two of them.
Solution

7{:}35 a.m. and 8{:}00 a.m. are times of day. Each names one moment, a place the clock hands can point to. The 25 minutes is a length of time: it says how much time passes, and it does not say when.

On the number line the hour from 7{:}00 to 8{:}00 is cut into 12 equal steps of 5 minutes, since 12 \times 5 = 60. The point 7{:}35 is 35 minutes past 7{:}00, which is 7 steps of 5. The two times of day are points, and the ride is the stretch between them.

To check, measure that stretch. Counting on by fives from 7{:}35 gives 7{:}40, 7{:}45, 7{:}50, 7{:}55, 8{:}00, which is 5 fives, or 25 minutes. The shortcut says the same: 7{:}35 is 35 minutes past the hour, so it is 60 - 35 = 25 minutes before 8{:}00. The three facts agree.

Example 2

Part 1: two sleigh rides

Ujarak’s sleigh ride lasted 1 hour 15 minutes. Aputi’s lasted 80 minutes. Whose ride was longer, and by how much? Write Aputi’s ride in hours and minutes as well.

Solution

The two lengths are in different units, so make them alike before comparing. One hour is 60 minutes, so Ujarak’s ride was

\begin{aligned} 1 \text{ hour } 15 \text{ minutes} &= 60 + 15 \text{ minutes} \\ &= 75 \text{ minutes}. \end{aligned}

Now both are in minutes. Aputi’s 80 minutes is more than 75 minutes, so her ride was longer, by 80 - 75 = 5 minutes.

For hours and minutes, take one 60 out of 80: 80 = 60 + 20, so Aputi’s ride was 1 hour 20 minutes. Compared that way, both rides have 1 hour, and 20 minutes is 5 more than 15 minutes. Either form gives the same answer, which is a good check.

Example 3

Part 2: story hour

Story hour at the library starts at 10{:}15 a.m. and ends at 11{:}40 a.m. How long is it?

10:15 11:00 11:40 45 min 40 min
Figure 2Story hour, 10{:}15 a.m. to 11{:}40 a.m. Jump to the hour mark, then to the end.
Solution

Draw an open number line with 10{:}15 at the left, 11{:}40 at the right, and the one hour mark between them, 11{:}00.

The first jump goes from 10{:}15 to 11{:}00. The start is 15 minutes past the hour, so this jump is 60 - 15 = 45 minutes. The second jump goes from 11{:}00 to 11{:}40, which is 40 minutes, because 11{:}40 is 40 minutes past its hour mark.

The two jumps cover the whole of story hour, so it lasts

\begin{aligned} 45 + 40 &= 85 \text{ minutes} \\ &= 60 + 25 \text{ minutes} \\ &= 1 \text{ hour } 25 \text{ minutes}. \end{aligned}

Check another way. From 10{:}15 to 11{:}15 is exactly 1 hour, since the hour went up by one and the minutes stayed the same. From 11{:}15 to 11{:}40 is 40 - 15 = 25 more minutes. That is 1 hour 25 minutes again.

Example 4

Part 2: across noon at the library

Ujarak goes into the library at 11{:}35 a.m. and comes out at 1{:}20 p.m. How long was he there, in hours and minutes and in minutes?

11:35 12:00 1:00 1:20 25 min 1 hour 20 min
Figure 3From 11{:}35 a.m. to 1{:}20 p.m. Noon is an hour mark like any other.
Solution

The visit starts in the morning and ends in the afternoon, so it passes noon. The hour marks between the start and the end are 12{:}00 and 1{:}00. On a twelve-hour clock the hour after 12 is 1.

  • From 11{:}35 a.m. to 12{:}00 noon is 60 - 35 = 25 minutes.
  • From 12{:}00 to 1{:}00 p.m. is 1 hour.
  • From 1{:}00 p.m. to 1{:}20 p.m. is 20 minutes.

The minutes jumps add to 25 + 20 = 45 minutes, and there is 1 hour besides. Ujarak was in the library for 1 hour 45 minutes, which is 60 + 45 = 105 minutes.

Subtracting the clock numbers would not work here. The end, 1{:}20, has a smaller hour than the start, 11{:}35, because the clock face started over at noon. The number line keeps the times in their real order, so the jumps do not mind.

Example 5

Part 3: the bus ride home

Aputi’s school bus leaves at 2{:}50 p.m., and the ride takes 45 minutes. When does she get off?

2:50 3:00 3:35 10 min 35 min
Figure 4The 45-minute bus ride, spent as 10 minutes and then 35 minutes.
Solution

Start at 2{:}50 and jump forward. The next hour mark is 3{:}00, which is 60 - 50 = 10 minutes away. The ride is longer than that, so take the jump.

That spends 10 of the 45 minutes, leaving 45 - 10 = 35 minutes. That is less than an hour, so the last jump is 35 minutes past 3{:}00, which lands on 3{:}35. No jump passed noon or midnight, so the time is still p.m. Aputi gets off at 3{:}35 p.m.

Check by asking the “how long” question: from 2{:}50 to 3{:}00 is 10 minutes, and from 3{:}00 to 3{:}35 is 35, and 10 + 35 = 45 minutes, the length of the ride.

Example 6

Part 3: a long sleigh ride past noon

A sleigh ride starts at 11{:}40 a.m. and lasts 1 hour 35 minutes. When does it end?

11:40 12:00 1:00 1:15 20 min 1 hour 15 min
Figure 5A ride of 1 hour 35 minutes that starts before noon and ends after it.
Solution

Write the length in minutes first, so it can be spent jump by jump: 1 hour 35 minutes is 60 + 35 = 95 minutes.

  • From 11{:}40 a.m. the next hour mark is 12{:}00 noon, 60 - 40 = 20 minutes away. Left: 95 - 20 = 75 minutes.
  • 75 minutes is at least an hour, so jump 1 hour, from 12{:}00 to 1{:}00 p.m. Left: 75 - 60 = 15 minutes.
  • Jump the last 15 minutes, from 1{:}00 to 1{:}15.

The ride passed noon, so the end is in the afternoon: it ends at 1{:}15 p.m.

Another order of jumps works too, because the length is the same however it is split. Jump the whole hour first: 11{:}40 a.m. plus 1 hour is 12{:}40 p.m. Then 35 minutes: 20 of them reach 1{:}00, and the other 15 reach 1{:}15 p.m. Same answer.

Example 7

Part 4: when did the sleigh ride start?

A sleigh ride ended at 4{:}10 p.m. It lasted 55 minutes. When did it start?

3:15 4:00 4:10 10 min 45 min
Figure 6Jumping backward from the end of the sleigh ride to find its start.
Solution

Start at the end, 4{:}10, and jump backward. The hour mark at or just before 4{:}10 is 4{:}00, and 4{:}10 is 10 minutes past it, so the first jump back is 10 minutes. Left: 55 - 10 = 45 minutes.

Jumping back 45 minutes from 4{:}00 goes into the hour before, the hour of 3. It lands 45 minutes before 4{:}00, which is 60 - 45 = 15 minutes past 3{:}00: the point 3{:}15. The ride started at 3{:}15 p.m.

Check forward: from 3{:}15 to 4{:}00 is 60 - 15 = 45 minutes, and from 4{:}00 to 4{:}10 is 10 more. 45 + 10 = 55 minutes, the length we were given.

Example 8

Part 4: a Saturday at the library

Here is the library’s Saturday afternoon schedule.

ActivityStartsEnds
Puppet show1{:}20 p.m.2{:}05 p.m.
Book club2{:}15 p.m.3{:}30 p.m.
Craft table3{:}40 p.m.4{:}35 p.m.

Ujarak stays for all three.

  1. Which activity is longest?
  2. The bus home leaves the stop outside the library at 4{:}50 p.m., and packing up and walking to the stop takes him 10 minutes. Can he stay to the end of the craft table?
  3. How long is he at the library, from the start of the puppet show to the end of the craft table?
Solution

Each row is a “how long” question.

  • Puppet show: 1{:}20 to 2{:}00 is 60 - 20 = 40 minutes, and 2{:}00 to 2{:}05 is 5 more, so 45 minutes.
  • Book club: 2{:}15 to 3{:}00 is 45 minutes, and 3{:}00 to 3{:}30 is 30, so 75 minutes, which is 1 hour 15 minutes.
  • Craft table: 3{:}40 to 4{:}00 is 20 minutes, and 4{:}00 to 4{:}35 is 35, so 55 minutes.

The book club, at 75 minutes, is the longest.

For the bus, this is a “when does it end” question. The craft table ends at 4{:}35 p.m., and 10 minutes later is 4{:}45 p.m., with no hour mark in between. That is before 4{:}50 p.m., so he can stay to the end and still reach the stop with 50 - 45 = 5 minutes to spare.

For the whole visit, jump from 1{:}20 p.m. to 4{:}35 p.m.: 40 minutes to 2{:}00, then 2 hours to 4{:}00, then 35 minutes to 4{:}35. The minutes add to 40 + 35 = 75, which is 1 hour 15 minutes, so the total is 3 hours 15 minutes, or 3 \times 60 + 15 = 195 minutes.

Check by adding up the pieces of the afternoon. The breaks between activities are 2{:}05 to 2{:}15 and 3{:}30 to 3{:}40, 10 minutes each. The activities and breaks together fill the afternoon with no gaps, and

45 + 10 + 75 + 10 + 55 = 195 \text{ minutes}.

The two ways agree.

Boundaries

  • Within one day, on the twelve-hour clock. Every time here carries a.m. or p.m. Without the label, 3{:}15 could be in the morning or the afternoon, and the elapsed time from 11{:}00 to 3{:}15 could be 4 hours 15 minutes or 16 hours 15 minutes. A stretch that runs past midnight into the next day is handled in Twenty four hour time and crossing midnight.
  • The minutes start over every 60. A clock reading is two counts, hours and minutes, written side by side; it is not a single base-ten number, and regrouping in time is by 60, not by 10 or 100. So subtraction by place value works only inside one hour. From 10{:}50 a.m. to 11{:}20 a.m. is 10 + 20 = 30 minutes, while the base-ten numbers 1120 and 1050 differ by 70: the 100 between 1050 and 1150 stands for one hour, which is only 60 minutes. That is the reason for jumping through the hour mark.
  • Later minus earlier. An elapsed time is a length, so it is never negative. The start has to come before the end in the day; if it seems not to, as when the hour number drops from 11 to 1, check whether noon was passed.
  • To the minute. The answers are as exact as the clock readings. A bus that leaves at “2{:}50” may leave some seconds after, and a time read to the minute ignores them.

Where this goes next

The same jumps work with longer units once their sizes are known: 24 hours in a day, 7 days in a week, as in Units of time. The twenty-four-hour clock, Twenty four hour time and crossing midnight, gives every hour of the day its own number, so a stretch across noon needs no change of label, and it handles a stretch across midnight. Time zones adds a shift between clocks in different places.

The open number line with friendly landmarks is not only for clocks. It is the same move as counting up to a round number to subtract, as in Counting up to find a difference, with the hour mark playing the part of the round ten or hundred. And reading a schedule is a first taste of Word problems with distance, time, volume, mass and money, where several lengths and times are fitted together.

Exercises

Part 1: times of day and lengths of time

  1. Say whether each is a time of day or a length of time. (a) The sleigh leaves at 3{:}20 p.m. (b) The ride lasts 40 minutes. (c) Story hour ends at 11{:}00 a.m. (d) Aputi read for 1 hour. (e) The bus comes at a quarter past seven.
  2. How many minutes is it from 9{:}25 a.m. to 10{:}00 a.m.?
  3. How many minutes is it from 6{:}48 p.m. to 7{:}00 p.m.?
  4. Write 1 hour 50 minutes as a number of minutes.
  5. Write 130 minutes in hours and minutes.
  6. Which is longer, a bus ride of 1 hour 5 minutes or a bus ride of 70 minutes? By how much?

Part 2: how long?

  1. Qillaq’s bus leaves at 7{:}40 a.m. and reaches school at 8{:}05 a.m. How long is the ride?
  2. A library puppet show runs from 9{:}10 a.m. to 9{:}57 a.m. How long is it?
  3. A sleigh ride runs from 1{:}35 p.m. to 2{:}50 p.m. How long is it, in minutes and in hours and minutes?
  4. Aputi is at the library from 10{:}45 a.m. to 12{:}20 p.m. How long is she there?
  5. Ujarak reads at the library from 8{:}52 a.m. to 11{:}06 a.m. How long is that, in hours and minutes and in minutes?

Part 3: when does it end?

  1. The bus leaves the library at 3{:}25 p.m. The ride takes 30 minutes. When does it arrive?
  2. A sleigh ride starts at 10{:}40 a.m. and lasts 35 minutes. When does it end?
  3. Story hour on Tuesday starts at 9{:}45 a.m. and lasts 1 hour 10 minutes. When does it end?
  4. A bus trip to the museum starts at 11{:}28 a.m. and takes 47 minutes. When does it arrive?
  5. A puppet show begins at 2{:}55 p.m. and lasts 1 hour 25 minutes. When does it end?

Part 4: when did it start? Schedules

  1. Qillaq’s bus reaches school at 8{:}15 a.m. after a 35-minute ride. When did it leave?
  2. A sleigh ride ended at 5{:}05 p.m. It lasted 50 minutes. When did it start?
  3. A reading party at the library ended at 12{:}20 p.m. It lasted 1 hour 30 minutes. When did it start?
  4. The morning school bus schedule lists four stops: Harbor Road at 7{:}42 a.m., Library at 7{:}58 a.m., Hill Street at 8{:}09 a.m. and School at 8{:}21 a.m. (a) How long is the ride from the Library stop to School? (b) How long is the whole route, from Harbor Road to School? (c) Which two stops in a row are farthest apart in time?
  5. Story hour starts at 10{:}30 a.m. Aputi wants to arrive 10 minutes early to choose books, and the walk to the library takes 25 minutes. When must she leave home?

Reasoning

  1. Explain why the jump from any time of day to the next hour mark is 60 minus the minutes in the time. Use it to find how many minutes it is from 6{:}07 p.m. to 7{:}00 p.m.
  2. A sleigh ride starts at a time ending in :15 and ends later the same day at another time ending in :15. What can you say for sure about how long it lasted? Explain.
  3. Does a ride that lasts 50 minutes always end in the next hour after it starts? Give an example of each kind, and say exactly which starting times make it end in the next hour.
  4. The bus leaves at 9{:}00 a.m. and comes back at 4{:}00 p.m. Which of these makes sense: 9{:}00 a.m. plus 4 hours, 9{:}00 a.m. plus 4{:}00 p.m., or 4{:}00 p.m. minus 9{:}00 a.m.? Say what each one that makes sense gives, and explain why the other does not.
Answers
  1. (a) A time of day. (b) A length of time. (c) A time of day. (d) A length of time. (e) A time of day, 7{:}15. A time of day names one moment; a length says how much time passes.
  2. 35 minutes. 9{:}25 is 25 minutes past the hour, so it is 60 - 25 = 35 minutes before 10{:}00.
  3. 12 minutes. 60 - 48 = 12.
  4. 110 minutes. 60 + 50 = 110.
  5. 2 hours 10 minutes. 130 = 60 + 60 + 10.
  6. The 70-minute ride, by 5 minutes. 1 hour 5 minutes is 60 + 5 = 65 minutes, and 70 - 65 = 5.
  7. 25 minutes. 7{:}40 to 8{:}00 is 20 minutes, and 8{:}00 to 8{:}05 is 5; 20 + 5 = 25.
  8. 47 minutes. Both times are in the hour of 9, so subtract the minutes: 57 - 10 = 47.
  9. 75 minutes, which is 1 hour 15 minutes. 1{:}35 to 2{:}00 is 25 minutes, and 2{:}00 to 2{:}50 is 50; 25 + 50 = 75 = 60 + 15.
  10. 1 hour 35 minutes, or 95 minutes. 10{:}45 to 11{:}00 is 15 minutes, 11{:}00 to 12{:}00 is 1 hour, and 12{:}00 to 12{:}20 is 20 minutes; 15 + 20 = 35.
  11. 2 hours 14 minutes, or 134 minutes. 8{:}52 to 9{:}00 is 8 minutes, 9{:}00 to 11{:}00 is 2 hours, and 11{:}00 to 11{:}06 is 6 minutes; 8 + 6 = 14, and 120 + 14 = 134.
  12. 3{:}55 p.m. No hour mark is passed: 25 + 30 = 55 minutes past 3.
  13. 11{:}15 a.m. 20 minutes reach 11{:}00, leaving 35 - 20 = 15 minutes.
  14. 10{:}55 a.m. The length is 70 minutes. 15 minutes reach 10{:}00, leaving 70 - 15 = 55 minutes.
  15. 12{:}15 p.m. 32 minutes reach 12{:}00 noon, leaving 47 - 32 = 15 minutes. Passing noon makes it p.m.
  16. 4{:}20 p.m. The length is 85 minutes. 5 minutes reach 3{:}00, leaving 80 minutes; 1 hour reaches 4{:}00, leaving 20 minutes.
  17. 7{:}40 a.m. Back 15 minutes to 8{:}00, leaving 35 - 15 = 20; back 20 from 8{:}00 lands at 60 - 20 = 40 minutes past 7.
  18. 4{:}15 p.m. Back 5 minutes to 5{:}00, leaving 45; back 45 from 5{:}00 lands at 60 - 45 = 15 minutes past 4.
  19. 10{:}50 a.m. The length is 90 minutes. Back 20 to 12{:}00, leaving 70; back 1 hour to 11{:}00 a.m., leaving 10; back 10 from 11{:}00 lands at 60 - 10 = 50 minutes past 10.
  20. (a) 23 minutes: 2 minutes to 8{:}00 and 21 more. (b) 39 minutes: 18 minutes to 8{:}00 and 21 more. (c) Harbor Road to Library, 16 minutes. The other gaps are 11 minutes (Library to Hill Street) and 12 minutes (Hill Street to School), and 16 + 11 + 12 = 39 matches part (b).
  21. 9{:}55 a.m. She must arrive by 10{:}20 a.m. Then 25 minutes back: 20 reach 10{:}00, and 5 more from 10{:}00 lands at 60 - 5 = 55 minutes past 9.
  22. 53 minutes. A time with m minutes is m minutes past its hour mark, and the whole hour to the next mark is 60 minutes, so the part still to go is 60 - m. Here 60 - 7 = 53.
  23. It lasted a whole number of hours. From one time ending in :15, each jump of 1 hour lands on another time ending in :15, and any length that is not whole hours would leave the minutes different from 15. So the length is 1 hour, 2 hours, and so on.
  24. No. Starting at 3{:}05 p.m., the ride ends at 3{:}55 p.m., in the same hour. Starting at 3{:}20 p.m., it ends at 4{:}10 p.m., in the next hour. The jump to the next hour mark is 60 minus the start’s minutes, and the ride reaches the next hour when that jump is 50 minutes or less: when the start is 10 or more minutes past the hour. (Starting exactly at :10, it ends exactly on the next hour mark.)
  25. 9{:}00 a.m. plus 4 hours makes sense: a time of day plus a length of time is a later time of day, 1{:}00 p.m. (3 hours reach noon, 1 more reaches 1{:}00). 4{:}00 p.m. minus 9{:}00 a.m. makes sense too: a later time minus an earlier one is the length between them, 3 hours to noon and 4 more to 4{:}00, so 7 hours. 9{:}00 a.m. plus 4{:}00 p.m. does not: both are points on the day’s number line, neither is a stretch, so there is no length to move by, and the sum names no moment and no length.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0