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Grade 4 · 4.NBT.A.3 · 5.NBT.A.4

Rounding to any place, including decimals

The car’s odometer says 386{,}451 miles. Nobody needs every digit to say how far the car has gone. Which simpler number is closest, and how do you pick it for a ten, a thousand or a hundred thousand?

You need
Rounding to the nearest ten or hundredGrade 3Decimal place valueGrades 4 to 5Numbers to one millionGrade 4
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Rounding decimalsGrade 5Estimating by roundingGrade 4
Where you are 3 in Rounding
Number9
Place value15
Rounding3
Rounding to any place, including decimals
Rounding to the nearest ten or hundredRounding decimals

When you reach for it

Reza, Losana and Palesa are on a long road trip. The car’s odometer counts every mile the car has ever driven, and a trip meter counts the miles since they left home. Along the way, signs give the population of each town they pass: how many people live there. These numbers are exact, but most of the time nobody needs every digit. “We have driven about 600 miles” or “that town has about 8{,}000 people” says enough.

Rounding trades a number for a nearby number that ends in zeros from some place onward. You choose the place: the nearest ten, hundred, thousand, ten thousand or hundred thousand. The answer is the number of that kind that is closest to the one you started with. You have already met this for tens and hundreds (Rounding to the nearest ten or hundred: a number sits between two multiples of ten and goes to the closer one). This lesson does the same thing at every place, using the places of Numbers to one million.

The lesson takes three days. Part 1 rounds to the nearest ten, hundred and thousand on a number line. Part 2 rounds large numbers to the nearest ten thousand and hundred thousand, including the rounding that carries into the next place. Part 3 chooses a sensible place, says “about”, and takes a short look at a decimal.

The procedure

Part 1: two neighbors and a halfway mark

The two neighbors. To round 638 to the nearest hundred, look only at numbers that end in two zeros: 500, 600, 700, 800. The number 638 sits between 600 and 700. Those two are its neighbors for this place, and the answer is one of them, whichever is closer.

600 610 620 630 640 650 660 670 680 690 700 halfway 638
Figure 1638 sits between 630 and 640, closer to 640. It also sits between 600 and 700, short of the halfway mark 650, so it is closer to 600.

The halfway mark. Exactly between 600 and 700 is 650. Any number to the left of 650 is closer to 600, and any number to the right is closer to 700. The dot for 638 is to the left, so to the nearest hundred

638 \approx 600

The sign \approx is read “is about”. It is not an equals sign: 638 and 600 are different numbers, and the sign says that one stands in for the other.

The same number line also shows the tens. To the nearest ten, 638 sits between 630 and 640, with halfway mark 635. It is past 635, so 638 \approx 640 to the nearest ten.

Why one digit decides. Drawing a number line every time is slow, and you do not need to. Look at the halfway marks: 635 for tens, 650 for hundreds, and for thousands, between 4{,}000 and 5{,}000, the mark 4{,}500. Each one is the lower neighbor with a 5 in the place just to the right of the rounding place, and zeros after that. So comparing a number with the halfway mark comes down to that one digit.

  • If the digit just right of the rounding place is 0, 1, 2, 3 or 4, the number is below halfway. It rounds to the lower neighbor: the digit in the rounding place stays the same, and every digit to its right becomes 0.
  • If that digit is 6, 7, 8 or 9, the number is past halfway. It rounds to the upper neighbor: the digit in the rounding place goes up by one, and every digit to its right becomes 0.

The digits further right cannot change the answer. To the nearest hundred, 638 and 649 both have a 3 or 4 in the tens place, so both are below 650, whatever their ones digits are. The places to the right of the deciding digit are always worth less than one of that digit’s place, because each place is worth ten times the place to its right.

Exactly halfway. A digit of 5 followed only by zeros, as in 650, means the number sits exactly on the halfway mark, the same distance from both neighbors. Here “nearest” does not choose. So that everyone gets the same answer, there is an agreement, a convention, used in this lesson and in most schoolbooks: a number exactly halfway rounds up. With that agreement the rule is short: a deciding digit of 5 or more rounds up, and 4 or less keeps the lower neighbor. A 5 followed by any digit that is not zero, as in 651, is past halfway anyway, so it rounds up for a real reason, not by agreement.

Part 2: large numbers, and rounding that carries

Bigger places, same move. Nothing new is needed for the big places. To round to the nearest ten thousand, the neighbors are numbers that end in four zeros, such as 380{,}000 and 390{,}000. The halfway mark has a 5 in the thousands place: 385{,}000. The deciding digit is the thousands digit, the one just right of the ten thousands place.

380,000 385,000 390,000 halfway 386,451
Figure 2Between 380{,}000 and 390{,}000, the number 386{,}451 is past the halfway mark 385{,}000, so it rounds to 390{,}000.

Every digit to the left of the rounding place is kept exactly as it is, because both neighbors share those digits. Every digit to the right becomes 0, because the neighbors end in zeros from the rounding place onward. Only the rounding digit itself may change, and only by going up one.

When the rounding digit is a 9. Going up one is easy unless the digit is already 9. Then it becomes ten, and a place cannot hold ten. Trade the ten for one in the next place to the left, the same trade that makes 999{,}999 + 1 a million. The digit in the rounding place becomes 0 and the digit to its left goes up by one. If that digit is a 9 too, the trade runs on.

390,000 395,000 400,000 halfway 396,500 rounds up
Figure 3396{,}500 is past the halfway mark 395{,}000, so to the nearest ten thousand it rounds up to 400{,}000, and the carry changes the hundred thousands digit from 3 to 4.

The number line keeps this from being a surprise. The upper neighbor of 396{,}500 for ten thousands is simply the next number after 390{,}000 that ends in four zeros, and that number is 400{,}000.

Part 3: choosing the place, and saying “about”

Which place? Rounding always needs a named place: “round 386{,}451” alone has no single answer. The place is chosen to fit the job. A rounded number keeps the big places and drops the small ones, so rounding to a big place gives a simple number that may be far off, and rounding to a small place stays close but keeps more digits. Usually the right place is one that keeps the digits that matter for the question and drops the rest.

Saying “about”. A rounded number is said with “about”, “around” or “roughly”, and written with \approx. A road sign saying a town has 8{,}472 people is exact; saying the town has about 8{,}500 people is a rounded description of the same count.

Rounding is not estimating. Rounding changes one number. Rounding several numbers first and then adding or multiplying them to see whether an answer makes sense is estimation, and it has its own lessons: Estimating by rounding, and for picking numbers that work well together, Friendly numbers and deliberate over and under estimates.

A short look at a decimal. The places to the right of the ones place follow the same ten-times pattern (Decimal place value), so a decimal rounds the same way: two neighbors, a halfway mark, and one deciding digit. Example 7 rounds a trip meter reading to the nearest whole mile. Rounding decimals to the nearest tenth or hundredth is taught in full in Rounding decimals.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3, 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Example 1

Miles on the first day

At the end of the first day, the trip meter shows 638 miles. Round 638 to the nearest ten and to the nearest hundred.

Solution

To the nearest ten, the multiples of ten on either side of 638 are 630 and 640. The halfway mark between them is 635. The ones digit is 8, so 638 is past 635. It is 8 away from 630 and only 2 away from 640, so

638 \approx 640 \text{ to the nearest ten}

To the nearest hundred, the neighbors are 600 and 700, and the halfway mark is 650. Now the deciding digit is the tens digit, 3. A 3 in the tens place means 638 is below 650: it is 38 away from 600 and 62 away from 700. So

638 \approx 600 \text{ to the nearest hundred}

The two answers are not the same. Rounding to a different place asks a different question, so it can give a different answer.

Example 2

Exactly on the halfway mark

When the trip begins, the odometer reads 52{,}750 miles. Round this to the nearest hundred and to the nearest thousand.

Solution

To the nearest hundred, the neighbors are numbers that end in two zeros. The digits to the left of the hundreds place, 52{,}7, stay, so the neighbors are 52{,}700 and 52{,}800. The halfway mark is 52{,}750, and that is the number itself. It is 50 from each neighbor, so “nearest” does not decide. By the halfway convention it rounds up:

52{,}750 \approx 52{,}800 \text{ to the nearest hundred}

To the nearest thousand, the neighbors are 52{,}000 and 53{,}000, and the halfway mark is 52{,}500. The deciding digit is the hundreds digit, 7, so 52{,}750 is past halfway, and no convention is needed:

52{,}750 \approx 53{,}000 \text{ to the nearest thousand}
Example 3

One odometer, three places

Later in the trip the odometer reads 386{,}451 miles. Round this to the nearest thousand, the nearest ten thousand and the nearest hundred thousand. How far is each rounded number from the real reading?

Solution

Each place has its own neighbors and its own deciding digit.

nearesthalfway markresult
thousand386{,}500386{,}000
ten thousand385{,}000390{,}000
hundred thousand350{,}000400{,}000

The neighbors are 386{,}000 and 387{,}000, then 380{,}000 and 390{,}000, then 300{,}000 and 400{,}000. To the nearest thousand, the hundreds digit 4 is below 5, so the thousands digit stays 6 and the digits after it become zeros: 386{,}451 \approx 386{,}000.

To the nearest ten thousand, the thousands digit 6 is 5 or more, so the ten thousands digit goes from 8 to 9: 386{,}451 \approx 390{,}000. The number line for this place is the second figure in the procedure.

To the nearest hundred thousand, the ten thousands digit 8 is 5 or more, so the hundred thousands digit goes from 3 to 4: 386{,}451 \approx 400{,}000. The hundred thousands digit of the reading is 3, yet it rounds to 400{,}000, because 386{,}451 is past the halfway mark 350{,}000.

The distances are

\begin{aligned} 386{,}451 - 386{,}000 &= 451 \\ 390{,}000 - 386{,}451 &= 3{,}549 \\ 400{,}000 - 386{,}451 &= 13{,}549 \end{aligned}

The bigger the place, the simpler the rounded number and the farther it can be from the real one.

Example 4

A rounding that carries

Palesa’s guidebook gives the population of a city they drive through as 396{,}500. Round it to the nearest thousand and to the nearest ten thousand.

Solution

To the nearest thousand, the neighbors are 396{,}000 and 397{,}000, with halfway mark 396{,}500. The population is exactly on the halfway mark, so by the convention it rounds up: 396{,}500 \approx 397{,}000.

To the nearest ten thousand, the deciding digit is the thousands digit, 6, so the number rounds up. The ten thousands digit is 9, and going up one makes it ten. Ten ten thousands are one hundred thousand, so the ten thousands digit becomes 0 and the hundred thousands digit goes from 3 to 4:

396{,}500 \approx 400{,}000 \text{ to the nearest ten thousand}

The number line in Part 2 shows the same answer without any trading: the neighbors are 390{,}000 and 400{,}000, and 396{,}500 is past the halfway mark 395{,}000. Notice that the 3 changed, even though it sits to the left of the rounding place. Rounding can change a digit to the left of the rounding place, but only through this carry.

Example 5

Working backward from a rounded number

Losana’s road map lists the town of Willow Creek as having about 24{,}000 people, rounded to the nearest thousand. What is the least the real population could be, and what is the greatest?

Solution

A whole number rounds to 24{,}000 when 24{,}000 is its nearer neighbor among the thousands. Two halfway marks fence it in. Below is the mark between 23{,}000 and 24{,}000, which is 23{,}500. Above is the mark between 24{,}000 and 25{,}000, which is 24{,}500.

At the low end, 23{,}500 is exactly halfway and rounds up, to 24{,}000, so it counts. The number below it, 23{,}499, has hundreds digit 4 and rounds down to 23{,}000. The least is 23{,}500.

At the high end, 24{,}500 is exactly halfway and rounds up, to 25{,}000, so it does not count. The number just below it, 24{,}499, has hundreds digit 4 and rounds down to 24{,}000. The greatest is 24{,}499.

So the population is somewhere from 23{,}500 to 24{,}499. The map’s “about 24{,}000” could mean any of those 1{,}000 whole numbers.

Example 6

Choosing a sensible place

Reza writes down the populations of three towns they will pass: Ashford 8{,}472, Brookline 12{,}905 and Cedar Hill 7{,}618. He wants rounded numbers that still show which town is biggest and which is smallest. Round all three to the nearest ten thousand, to the nearest thousand, and to the nearest hundred, and decide which place does the job.

Solution

Round each number at each place, using the deciding digit. To the nearest ten thousand all three give 10{,}000; the other two places give this table.

townto the thousandto the hundred
Ashford8{,}0008{,}500
Brookline13{,}00012{,}900
Cedar Hill8{,}0007{,}600

To the nearest ten thousand, 8{,}472 and 7{,}618 sit between 0 and 10{,}000, past the halfway mark 5{,}000, so both round to 10{,}000. And 12{,}905 has thousands digit 2, so it rounds down to 10{,}000. All three towns become “about 10{,}000”, and the difference Reza cares about is gone.

To the nearest thousand, Brookline stands out as the biggest, but Ashford and Cedar Hill both become 8{,}000, so the smallest cannot be told.

To the nearest hundred, the tens digits 7, 0 and 1 decide: 8{,}500, 12{,}900 and 7{,}600. These are all different and in the same order as the real populations. The nearest hundred is the place that does Reza’s job: Brookline is about 12{,}900, Ashford about 8{,}500 and Cedar Hill about 7{,}600.

Example 7

A trip meter with a decimal

Palesa’s trip meter shows tenths of a mile: 248.7 miles since lunch. Round this to the nearest whole mile.

Solution

The whole numbers on either side are 248 and 249. The halfway mark between them is 248.5, which has a 5 in the tenths place, just right of the ones place. The tenths digit of 248.7 is 7, so it is past halfway:

248.7 \approx 249 \text{ miles}

It is the same move as before, one place to the right of the ones: the ones digit goes up by one and the digit after it is dropped. Rounding decimals carries this on to tenths and hundredths.

Boundaries

  • A place must be named. “Round 386{,}451” has no answer until the place is given, and different places give different answers, as Example 3 shows.
  • A place bigger than the number gives 0 or one unit of that place. 4{,}382 to the nearest hundred thousand lies between 0 and 100{,}000 and is short of halfway, so it rounds to 0. That is correct, and tells you nothing. A useful place is no bigger than the number’s own leading place.
  • Rounding a whole number to the ones place changes nothing. A whole number already ends at the ones place, so it is its own nearest whole number.
  • A rounded number can have one more digit. Through carrying, 9{,}960 to the nearest hundred is 10{,}000, a five-digit number. It can never have fewer digits than the original, except when it rounds to 0.
  • This lesson rounds whole numbers, 0, 1, 2, 3 and so on. For numbers below zero, “rounds up” no longer means “moves away from zero”, and the halfway agreement has to be said again with more care.

Where this goes next

Rounding to any place is the first tool for talking about size without the clutter of every digit. Estimating by rounding rounds the numbers in a sum or product before working it, to check that an exact answer makes sense, and Friendly numbers and deliberate over and under estimates chooses nearby numbers that are easy to work with together. The same two neighbors and halfway mark round decimals in Rounding decimals, which uses the decimal places of Decimal place value. All of these sit inside the wider idea of Rounding: replacing a number with a nearby one that is easier to use.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, 17 to 21 with Part 3, and 22 to 25 ask you to explain.

Part 1: tens, hundreds and thousands

  1. Round 274 to the nearest ten.
  2. Round 865 to the nearest ten and to the nearest hundred.
  3. The trip meter shows 1{,}438 miles. Round this to the nearest hundred.
  4. Round 6{,}719 to the nearest thousand.
  5. Round 3{,}250 to the nearest hundred and to the nearest thousand.
  6. Name the two multiples of 100 that 4{,}982 sits between, and the halfway mark between them. Then round 4{,}982 to the nearest hundred.
  7. An odometer reads 29{,}064 miles. Round this to the nearest ten, the nearest hundred and the nearest thousand.
  8. Which of these numbers round to 500 to the nearest hundred: 449, 450, 512, 549, 551, 455?

Part 2: large numbers and carrying

  1. Round 573{,}208 to the nearest ten thousand and to the nearest hundred thousand.
  2. A town has 48{,}615 people. Round this to the nearest thousand and to the nearest ten thousand.
  3. Round 729{,}950 to the nearest hundred.
  4. Round 998{,}400 to the nearest thousand and to the nearest ten thousand.
  5. Round 250{,}000 to the nearest hundred thousand.
  6. A town has 61{,}047 people. Round this to the nearest ten thousand, the nearest thousand and the nearest hundred.
  7. A town’s population, rounded to the nearest hundred, is 7{,}300. What is the least whole number it could be, and the greatest?
  8. One digit is missing: 45\square{,}812. Which digits can go in the box so that the number rounds to 460{,}000 to the nearest ten thousand?

Part 3: choosing a place, and a decimal

  1. A road sign gives a town’s population as 3{,}862. Round it to the nearest hundred and to the nearest thousand, and write a sentence about the town using each.
  2. Round 186{,}529 to the nearest ten thousand, and write the result with the sign \approx.
  3. Three towns have 14{,}380, 14{,}920 and 15{,}210 people. Round each to the nearest thousand and to the nearest hundred. Which place tells all three towns apart?
  4. The trip meter shows 63.4 miles on one day and 87.6 miles the next. Round each to the nearest whole mile.
  5. Reza says the day’s drive was about 1{,}200 miles, rounded to the nearest hundred. Could the real distance have been 1{,}149 miles? 1{,}250 miles? What whole numbers of miles could it have been?

Reasoning

  1. When you round to the nearest thousand, explain why only the hundreds digit needs to be looked at, and why the tens and ones digits can never change the answer.
  2. The hundred thousands digit of 281{,}000 is 2. Explain why it rounds to 300{,}000 to the nearest hundred thousand.
  3. Losana rounds 4{,}382 to the nearest hundred thousand and gets 0. Is she right? Is the answer useful?
  4. Explain how rounding a six-digit number can give a seven-digit number. Which six-digit whole numbers round to a seven-digit number to the nearest thousand?
Answers
  1. 270. The neighbors are 270 and 280; the ones digit 4 is below 5, so 274 is short of the halfway mark 275.
  2. 870 to the nearest ten and 900 to the nearest hundred. The ones digit 5 puts 865 exactly halfway between 860 and 870, and halfway rounds up by the convention. For hundreds, the tens digit 6 puts it past 850.
  3. 1{,}400. The neighbors are 1{,}400 and 1{,}500; the tens digit 3 means 1{,}438 is below the halfway mark 1{,}450.
  4. 7{,}000. The neighbors are 6{,}000 and 7{,}000; the hundreds digit 7 puts 6{,}719 past 6{,}500.
  5. 3{,}300 to the nearest hundred and 3{,}000 to the nearest thousand. 3{,}250 is exactly halfway between 3{,}200 and 3{,}300, so it rounds up. For thousands, the hundreds digit 2 is below 5.
  6. Between 4{,}900 and 5{,}000, with halfway mark 4{,}950. 4{,}982 is past the mark, so it rounds to 5{,}000. The hundreds digit 9 goes up to ten, which carries into the thousands.
  7. 29{,}060, 29{,}100 and 29{,}000. The ones digit 4 keeps the tens; the tens digit 6 raises the hundreds from 0 to 1; the hundreds digit 0 keeps the thousands.
  8. 450, 512, 549 and 455. These lie from the halfway mark 450 up to just below 550. 449 rounds to 400 and 551 to 600.
  9. 570{,}000 and 600{,}000. For ten thousands the thousands digit 3 is below 5; for hundred thousands the ten thousands digit 7 is 5 or more, so the 5 becomes 6.
  10. 49{,}000 and 50{,}000. The hundreds digit 6 raises the thousands digit from 8 to 9. For ten thousands, the thousands digit 8 raises the ten thousands digit from 4 to 5.
  11. 730{,}000. The neighbors are 729{,}900 and 730{,}000, and 729{,}950 is exactly halfway, so it rounds up. The hundreds digit 9 becomes ten and carries through the thousands and ten thousands.
  12. 998{,}000 to the nearest thousand and 1{,}000{,}000 to the nearest ten thousand. The hundreds digit 4 keeps the thousands. For ten thousands, the thousands digit 8 rounds up, and the ten thousands digit 9 becomes ten, which carries into a hundred thousands digit 9, which carries again to one million.
  13. 300{,}000. It is exactly halfway between 200{,}000 and 300{,}000, so by the convention it rounds up.
  14. 60{,}000, 61{,}000 and 61{,}000. The thousands digit 1 keeps the ten thousands; the hundreds digit 0 keeps the thousands; the tens digit 4 keeps the hundreds, which is 0.
  15. From 7{,}250 to 7{,}349. 7{,}250 is exactly halfway between 7{,}200 and 7{,}300 and rounds up to 7{,}300; 7{,}350 would round up to 7{,}400, so 7{,}349 is the greatest.
  16. 5, 6, 7, 8 or 9. The number is between 450{,}000 and 460{,}000, so it rounds to 460{,}000 when the thousands digit is 5 or more. A digit of 4 or less rounds down to 450{,}000.
  17. 3{,}900 and 4{,}000. For example: “The town has about 3{,}900 people,” and “The town has about 4{,}000 people.” The tens digit 6 raises the hundreds; the hundreds digit 8 raises the thousands.
  18. 186{,}529 \approx 190{,}000. The thousands digit 6 raises the ten thousands digit from 8 to 9.
  19. To the nearest thousand: 14{,}000, 15{,}000 and 15{,}000. To the nearest hundred: 14{,}400, 14{,}900 and 15{,}200. Only the nearest hundred tells all three apart.
  20. 63 and 88. The tenths digit 4 is below 5, so 63.4 is short of 63.5; the tenths digit 6 puts 87.6 past 87.5.
  21. No and no. 1{,}149 rounds to 1{,}100, since its tens digit is 4, and 1{,}250 is exactly halfway and rounds up to 1{,}300. The distance could be any whole number from 1{,}150 to 1{,}249.
  22. The two neighbors are thousands, and the halfway mark between them is the lower neighbor plus 500. Whether a number is below or past that mark depends on its hundreds digit: 4 or less means below, 5 or more means at or past it. The tens and ones together are at most 99, which is less than one hundred, so they can never carry the number past the next hundred and change which side of the mark it is on.
  23. To the nearest hundred thousand, 281{,}000 sits between 200{,}000 and 300{,}000. The halfway mark is 250{,}000, and 281{,}000 is past it, because its ten thousands digit 8 is 5 or more. So it is closer to 300{,}000 (19{,}000 away) than to 200{,}000 (81{,}000 away). The hundred thousands digit of the original only says which two neighbors to look between; the next digit decides which one wins.
  24. She is right: 4{,}382 is between 0 and 100{,}000, and short of the halfway mark 50{,}000, so it rounds to 0. But the answer is not useful, because it hides that the town has any people at all. A place no bigger than the leading place of 4{,}382, such as the nearest thousand, 4{,}000, says something useful.
  25. When a number rounds up and every digit from the rounding place to the left end is a 9, the carry runs past the leftmost place and makes a new one. To the nearest thousand, the numbers that do this are 999{,}500 to 999{,}999: they are at or past the halfway mark 999{,}500 between 999{,}000 and 1{,}000{,}000, so they round to 1{,}000{,}000.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0