At the county fair gate, people came through in the first hour. Is that nearer or , nearer or , and how do you decide when a number sits exactly in the middle?

Grade 3 · 3.NBT.A.1 · 4.NBT.A.3

Rounding to the nearest ten or hundred

At the county fair gate, 347 people came through in the first hour. Is that nearer 340 or 350, nearer 300 or 400, and how do you decide when a number sits exactly in the middle?

You need

Number lines with different scalesGrade 2

Unlocks

Rounding to any place, including decimalsGrade 4

Where you are 3 in Rounding

Number9

Place value15

Rounding3

Rounding to any place, including decimals

Rounding to the nearest ten or hundred

Rounding decimals

When you reach for it

Aiyetoro counts people at the gate of the county fair. In the first hour, 347 people come through. When the fair manager asks how the morning is going, nobody needs to hear “three hundred forty-seven.” “About 350” says the same thing in a way that is easy to hear, remember and work with. “About 300” is rougher, but still true enough for a sign on the barn wall.

Each of those is rounding: replacing a number with a nearby number that is easier to use. The nearby numbers we round to here are the tens (10, 20, 30, and so on) and the hundreds (100, 200, 300, and so on). Rounding to the nearest ten asks which ten is closest. Rounding to the nearest hundred asks which hundred is closest.

You reach for it whenever an exact number is more detail than the moment needs: telling someone a count, writing a quick note, or, in the next lesson, estimating a sum or a difference before working it out. The whole move rests on one picture, the number line from Number lines with different scales, with its tick marks stepping by ones, tens or hundreds. This lesson uses whole numbers up to 1{,}000.

The lesson runs over three days. Part 1 rounds to the nearest ten. Part 2 rounds to the nearest hundred. Part 3 handles the edges: rounding that carries into the next hundred, small numbers, and working backward from a rounded number to the numbers it could have come from.

The procedure

Part 1: the nearest ten

Find the two tens. Every whole number that is not itself a multiple of ten sits between two tens that are next to each other. For 347, look at the Place value: 347 is 3 hundreds, 4 tens and 7 ones, which is the same as 34 tens and 7 ones. Thirty-four tens is 340, and one more ten is 350. So 347 sits between 340 and 350. The tens digit names the lower ten, and the upper ten is ten more.

Put the number on a line. Draw a number line from 340 to 350 with a tick at every one. The number 347 is the seventh tick past 340.

Figure 1347 is 7 past 340 and only 3 short of 350. It is past the halfway mark 345, so its nearest ten is 350.

Compare the two distances. From 340 up to 347 is 7 steps. From 347 up to 350 is 3 steps. Three is less than seven, so 350 is nearer. That is the whole meaning of the phrase: 347 rounded to the nearest ten is 350.

The halfway mark. Halfway between 340 and 350 is 345, five steps from each end. Every number past the halfway mark is nearer the upper ten, and every number before it is nearer the lower ten. So instead of measuring both distances each time, you can ask one question: is the number before or after halfway?

Why the ones digit is enough. The halfway mark between two tens always has 5 ones. A number between 340 and 350 is 340 plus its ones digit. If the ones digit is 1, 2, 3 or 4, the number is before 345, so it rounds down to 340. If the ones digit is 6, 7, 8 or 9, the number is after 345, so it rounds up to 350. The hundreds and tens digits only say which pair of tens you are between; the ones digit says where you are inside that pair.

Ones digit	1, 2, 3, 4	5	6, 7, 8, 9
Where the number sits	before halfway	exactly halfway	after halfway
Nearest ten	the lower ten	the rule below decides	the upper ten

A ones digit of 0 means the number already is a ten. 290 is 29 tens and no ones, so it is its own nearest ten: rounding leaves it alone.

Exactly halfway is a choice. The number 345 is 5 from 340 and 5 from 350. Neither ten is nearer, and no amount of measuring will make one of them nearer. Something outside the numbers has to decide. The rule used in school mathematics, and in this lesson, is a convention, an agreement people have made so that everyone gets the same answer:

> A number exactly halfway between two tens rounds up, to the greater ten.

So 345 rounds to 350. The rule is a choice, not a fact about numbers. Some other fields have agreed on a different rule; for example, some computer programs send an exact halfway number to whichever neighbor has an even digit in that place. Each rule is fair inside its own agreement. What matters is that everyone working together uses the same one.

Part 2: the nearest hundred

Rounding to the nearest hundred is the same idea on a coarser line. The steps are now hundreds: 300, 400, 500, and so on.

Find the two hundreds. 347 has 3 hundreds, so it sits between 300 and 400. The hundreds digit names the lower hundred, and the upper hundred is one hundred more.

Put the number on a line. This time the tick marks step by ten, because a line from 300 to 400 with a tick at every one would have a hundred ticks. The number 347 sits between the ticks for 340 and 350, just before 350.

Figure 2On a line marked in tens, the same 347 sits just before the halfway mark 350. It is 47 from 300 and 53 from 400, so its nearest hundred is 300.

Compare the distances. From 300 to 347 is 47. From 347 to 400 is 53, because 347 + 53 = 400. Forty-seven is less, so 300 is nearer: 347 rounded to the nearest hundred is 300.

The halfway mark is 50 past the hundred. Halfway between 300 and 400 is 350. A number before 350 rounds down to 300; a number after 350 rounds up to 400; and 350 itself is exactly halfway, so the same convention as in Part 1 sends it up, to 400.

Why the tens digit is enough. A number between 300 and 400 is 300 plus a part less than 100, and that part is made of the tens digit and the ones digit together. For 347 the part is 47. Compare it with 50:

If the tens digit is 0, 1, 2, 3 or 4, the part is at most 49, which is less than 50. Whatever the ones digit is, it adds at most 9, never enough to reach the next ten, so the number is before halfway. Round down.
If the tens digit is 5, 6, 7, 8 or 9, the part is at least 50. The number is at halfway or past it, and halfway rounds up too. Round up.

So for the nearest hundred, the tens digit decides and the ones digit cannot change the answer.

Two answers for one number. The same 347 rounds to 350 to the nearest ten and to 300 to the nearest hundred. That is not a contradiction. The two questions ask about different lines. On the line of tens, the closest mark is 350. On the line of hundreds, 350 is not a mark at all, and the closest mark is 300. Rounding to the nearest ten keeps more detail; rounding to the nearest hundred keeps less and is easier to say.

Part 3: across a hundred, and working backward

When rounding up carries. The ten after 990 is 1{,}000, the same way the ten after 340 is 350. A number like 995 has 99 tens and 5 ones. Rounding up adds one more ten, and 99 tens plus one ten is 100 tens, which is 1{,}000. On paper the tens digit cannot go from 9 to “10”, so the extra ten is traded for one more hundred, just as in regrouping in addition. The same thing happens at every hundred: 296 rounds up to 300, because the ten after 290 is 300.

Figure 3The ten after 990 is 1{,}000. 995 sits exactly halfway, and the round-up rule sends it to 1{,}000.

Small numbers can round to zero. A number less than 10 sits between 0 and 10. If it is before 5, its nearest ten is 0. Likewise a number less than 100 sits between 0 and 100, and if it is before 50, its nearest hundred is 0. Zero is a perfectly good multiple of ten and of a hundred.

Working backward. Sometimes you see only the rounded number, like a sign that says “about 400 cups sold,” and you want to know what the true number could have been. Think of the two halfway marks on either side of 400. To the nearest hundred, the halfway mark below 400 is 350, and 350 rounds up, so it belongs to 400. The halfway mark above is 450, and 450 also rounds up, to 500, so it does not belong. The whole numbers that round to 400 run from 350 up to 449.

Figure 4The whole numbers that round to 400 to the nearest hundred run from the halfway mark 350 up to 449. At 450 the next hundred takes over.

The same reasoning works for tens. The numbers that round to 350 to the nearest ten start at the halfway mark 345 and stop just before the next halfway mark 355: they are 345 to 354. One end of each range is a halfway mark and is included; the other end is the number just before the next halfway mark.

Worked examples

Examples 1 and 2 belong to Part 1, Examples 3 and 4 to Part 2, and Examples 5, 6 and 7 to Part 3.

Example 1

The first hour at the gate

Aiyetoro’s count for the first hour at the county fair gate is 347 people. Round 347 to the nearest ten.

Solution

First find the two tens. 347 is 34 tens and 7 ones, so it lies between 34 tens, which is 340, and 35 tens, which is 350.

Next, compare distances on the number line. From 340 to 347 is 7 steps; from 347 to 350 is 3 steps. The number is nearer 350.

The halfway mark gives the same answer faster: halfway between 340 and 350 is 345, and 347 is past it, because its ones digit 7 is more than 5.

So 347 rounded to the nearest ten is 350. A quick check: 350 - 347 = 3, and 3 is less than 5, so 350 is within half a ten of the count, as the nearest ten must be.

Example 2

Three counts, one of them halfway

Bolanle keeps the entry book for the animal barns. There are 62 goats, 238 rabbits and 415 chickens entered. Round each number to the nearest ten.

Solution

Take each number in turn and find its two tens.

62 is between 60 and 70. Its ones digit is 2, so it is 2 past 60 and 8 short of 70. It is before the halfway mark 65, so it rounds down: 62 rounds to 60.

238 is between 230 and 240. Its ones digit is 8, so it is 8 past 230 and only 2 short of 240. It is after the halfway mark 235, so it rounds up: 238 rounds to 240.

415 is between 410 and 420. Its ones digit is 5, so it sits exactly on the halfway mark: 5 from 410 and 5 from 420. Neither ten is nearer. The convention decides: a number exactly halfway rounds up. So 415 rounds to 420.

The goats round to 60, the rabbits to 240 and the chickens to 420. Only the last answer depended on the halfway rule; the first two follow from distance alone.

Example 3

The same count, to the nearest hundred

The fair manager wants the first-hour count of 347 people for the sign on the barn wall, rounded to the nearest hundred.

Solution

347 has 3 hundreds, so it lies between 300 and 400.

On a number line from 300 to 400 marked in tens, 347 sits between 340 and 350, just before the halfway mark 350. Measuring confirms it: 347 - 300 = 47, and 400 - 347 = 53. The number is 47 from 300 and 53 from 400, so 300 is nearer.

The tens digit says the same thing at a glance. It is 4, so the part past 300 is in the forties, less than 50, and the number rounds down.

So 347 rounded to the nearest hundred is 300. In Example 1 the same count rounded to 350 to the nearest ten. Both are right: the nearest mark on a line of tens is 350, and the nearest mark on a line of hundreds is 300.

Example 4

A pumpkin exactly halfway

Bolanle’s pumpkin wins a ribbon at the county fair. The scale says it weighs 650 pounds. Round the weight to the nearest hundred, and then to the nearest ten.

Solution

To the nearest hundred: 650 has 6 hundreds, so it lies between 600 and 700. The halfway mark between them is 650, which is the weight itself.

Figure 5650 is exactly 50 from 600 and 50 from 700. Neither hundred is nearer, so an agreed rule has to decide.

From 600 to 650 is 50, and from 650 to 700 is also 50. The pumpkin’s weight is exactly as close to 600 as to 700. Measuring cannot settle it, so the convention does: a number exactly halfway rounds up. To the nearest hundred, 650 rounds to 700 pounds.

This answer rests on the agreement, not on distance. Any weight even a little above 650 is truly nearer 700, and any weight a little below is truly nearer 600; only 650 itself needs the rule.

To the nearest ten: 650 is 65 tens and 0 ones. It already is a ten, so it rounds to itself, 650 pounds. The question “which ten is nearest?” has the number itself as its answer.

Example 5

The last hour of the day

Ebele sells lemonade at the fair. By closing time she has sold 995 cups. Round 995 to the nearest ten, and then to the nearest hundred.

Solution

To the nearest ten: 995 is 99 tens and 5 ones. So it lies between 99 tens, which is 990, and 100 tens, which is 1{,}000. The ten after 990 is 1{,}000.

The ones digit is 5, so 995 is exactly halfway: 5 from 990 and 5 from 1{,}000. By the convention, a number exactly halfway rounds up. So 995 rounds to 1{,}000 to the nearest ten.

Notice what happened to the digits. Adding one ten to 99 tens makes 100 tens. The tens digit cannot hold 10, so ten tens are traded for one hundred, and ten hundreds for one thousand. Every digit changes, and the result has four digits even though 995 has three.

To the nearest hundred: 995 has 9 hundreds, so it lies between 900 and 1{,}000, with halfway mark 950. The tens digit is 9, so 995 is well past halfway: it is 95 from 900 and only 5 from 1{,}000. It rounds up to 1{,}000.

Both roundings give 1{,}000 cups, which is the right thing to say: Ebele sold about a thousand cups.

Example 6

What could the true count be?

Ebele’s sign at the end of the fair says “about 400 cups of iced tea, to the nearest hundred.” What are the least and the greatest whole numbers of cups she could have sold?

Solution

Work outward from 400 to the halfway marks on each side. The hundreds next to 400 are 300 and 500, so the halfway marks are 350 and 450.

Below 400: every number from 351 to 399 is nearer 400 than 300. The halfway mark 350 itself is exactly as close to 300 as to 400, and the convention rounds it up, to 400. So 350 counts, and 349 does not, since 349 is before halfway and rounds down to 300. The least possible number is 350.

Above 400: every number from 401 to 449 is nearer 400 than 500. The halfway mark 450 rounds up, but up from 450 is 500, not 400. So 450 does not count. The greatest possible number is 449.

Ebele sold at least 350 cups and at most 449 cups. Check both ends: 350 has tens digit 5 and rounds up to 400; 449 has tens digit 4 and rounds down to 400.

Example 7

Two clues together

Aiyetoro gives the gate count for the second hour twice. To Bolanle he says “250, to the nearest ten.” To Ebele he says “300, to the nearest hundred.” Both are true. What could the count be?

Solution

Use each clue to find a range, then look for the numbers in both.

First clue: the numbers that round to 250 to the nearest ten. The halfway mark below 250 is 245, which rounds up to 250, so it is included. The halfway mark above is 255, which rounds up to 260, so it is not. The first clue allows 245 to 254.

Second clue: the numbers that round to 300 to the nearest hundred. The halfway mark below is 250, which rounds up to 300, so it is included. The halfway mark above is 350, which rounds up to 400, so it is not. The second clue allows 250 to 349.

Now put the ranges side by side. The first runs from 245 to 254. The second starts at 250. The numbers 245 to 249 fail the second clue, since they round down to 200 to the nearest hundred. The numbers in both ranges are 250, 251, 252, 253 and 254.

So the count was one of those five numbers. Check one: 252 is 2 past 250, so its nearest ten is 250; its tens digit is 5, so its nearest hundred is 300. Both clues hold.

Boundaries

Rounding gives a nearby number, not an equal one. 347 rounded to the nearest ten is 350, but 347 and 350 are different numbers. A rounded number is used when “about” is good enough, and the exact number is still there when it is needed.
Exactly halfway has no nearest. Every whole number has one nearest ten except a number with ones digit 5, which is equally close to two tens; for hundreds, the same holds for a number whose last two digits are 50. In those cases the answer comes from the round-up convention, an agreement, not from distance. A different agreement would give a different answer, and both would be correct within their own rule.
The answer depends on the place. Rounding to the nearest ten and rounding to the nearest hundred are different questions and can have different answers, as 347 shows. A rounded number should always say which place it was rounded to.
Rounding in two steps is not rounding once. Rounding 347 to the nearest ten gives 350, and rounding 350 to the nearest hundred gives 400. But 347 itself is nearer 300. To round to the nearest hundred, start from the original number every time.
The answer can leave the range. Rounding a number up to 1{,}000 can give 1{,}000, a four-digit number, as 995 does. And a number below 5 rounds to 0 to the nearest ten.
Whole numbers only, here. This lesson rounds whole numbers to tens and hundreds. Rounding to other places, and rounding numbers with decimal parts, uses the same picture and comes later.

Where this goes next

The next lesson, Estimating sums and differences, puts rounding to work: round the numbers in a sum or a difference first, add or subtract the easy rounded numbers, and use the result to judge whether an exact answer is the right size. Later, Estimating by rounding does the same for larger numbers and for products, and Friendly numbers and deliberate over and under estimates shows when to bend a number toward a friendly neighbor on purpose instead of to its nearest one.

The number line picture also carries straight on. Rounding to any place, including decimals rounds to the nearest thousand, ten thousand or any other place, always by finding the two neighbors and the halfway mark between them. Rounding decimals uses the same picture on a line marked in tenths or hundredths. The idea of Rounding stays the same at every size: the nearest mark on the line you chose.

Exercises

Part 1: the nearest ten

Round 73 to the nearest ten.
Round 128 to the nearest ten.
Round 461 to the nearest ten.
Round 806 to the nearest ten.
Round 535 to the nearest ten.
Round 290 to the nearest ten.
Bolanle counts 184 jars of jam entered in the county fair jam contest. Round the number of jars to the nearest ten.
Name the two tens that 726 lies between, the halfway mark between them, and the nearest ten to 726.

Part 2: the nearest hundred

Round 218 to the nearest hundred.
Round 482 to the nearest hundred.
Round 750 to the nearest hundred.
Round 67 to the nearest hundred.
Round 803 to the nearest hundred.
Round 264 to the nearest ten and to the nearest hundred.
On the last evening of the fair, 925 people watch the tractor pull. Round this number to the nearest hundred and to the nearest ten.
Round 149 to the nearest ten and to the nearest hundred. How far is 149 from 100, and how far from 200?

Part 3: across a hundred, and working backward

Round 997 to the nearest ten.
Round 96 to the nearest ten.
Round 4 to the nearest ten.
Round 960 to the nearest hundred.
A whole number rounds to 80 to the nearest ten. What are the least and greatest numbers it could be?
The fair’s sign says “about 700 ribbons given out, to the nearest hundred.” What are the least and greatest whole numbers of ribbons it could be?

Explain your thinking

Ebele says her number rounds to 500 to the nearest hundred. Could the same number round to 440 to the nearest ten? Explain.
345 is exactly as far from 340 as from 350. Explain why measuring cannot decide its nearest ten, and why people still need an agreed rule.
Is rounding a number to the nearest ten and then rounding that answer to the nearest hundred always the same as rounding the number straight to the nearest hundred? Give an example, and say which numbers give different answers.

Answers

70. 73 lies between 70 and 80; it is 3 past 70 and 7 short of 80, so it is before the halfway mark 75.
130. 128 lies between 120 and 130; it is 8 past 120 and 2 short of 130, so it is after the halfway mark 125.
460. 461 lies between 460 and 470, and its ones digit 1 is less than 5.
810. 806 lies between 800 and 810, and its ones digit 6 is more than 5.
540. 535 lies between 530 and 540, exactly on the halfway mark, 5 from each. By the convention, halfway rounds up.
290. 290 is 29 tens and no ones, so it already is a ten and rounds to itself.
180 jars. 184 lies between 180 and 190, and its ones digit 4 is less than 5, so it is before the halfway mark 185.
726 lies between 720 and 730; the halfway mark is 725; the nearest ten is 730, because 726 is past 725 (4 from 730, 6 from 720).
200. 218 lies between 200 and 300, and its tens digit 1 puts it before the halfway mark 250.
500. 482 lies between 400 and 500; its tens digit is 8, so it is past the halfway mark 450 (18 from 500, 82 from 400).
800. 750 lies between 700 and 800, exactly on the halfway mark, 50 from each. By the convention, halfway rounds up.
100. 67 has no hundreds, so it lies between 0 and 100. It is past the halfway mark 50: it is 33 from 100 and 67 from 0.
800. 803 lies between 800 and 900, and its tens digit 0 puts it just past 800.
To the nearest ten, 260: 264 lies between 260 and 270, with ones digit 4. To the nearest hundred, 300: 264 lies between 200 and 300, with tens digit 6, past the halfway mark 250.
To the nearest hundred, 900: 925 lies between 900 and 1{,}000, and its tens digit 2 puts it before the halfway mark 950. To the nearest ten, 930: 925 lies between 920 and 930, exactly halfway, and halfway rounds up.
To the nearest ten, 150, since 149 is 1 short of 150. To the nearest hundred, 100, since the tens digit is 4. 149 is 49 from 100 and 51 from 200, so 100 really is nearer, even though the nearest ten is 150.
1{,}000. 997 is 99 tens and 7 ones, so it lies between 990 and 1{,}000, past the halfway mark 995. It is 3 from 1{,}000.
100. 96 lies between 90 and 100; it is 6 past 90 and 4 short of 100, so it rounds up. Nine tens plus one ten is ten tens, which is 100.
0. 4 lies between 0 and 10, before the halfway mark 5, so its nearest ten is 0.
1{,}000. 960 lies between 900 and 1{,}000, and its tens digit 6 puts it past the halfway mark 950. It is 40 from 1{,}000 and 60 from 900.
Least 75, greatest 84. The halfway mark 75 rounds up to 80, so it is included; 85 rounds up to 90, so the numbers stop at 84.
Least 650, greatest 749. The halfway mark 650 rounds up to 700, so it is included; 750 rounds up to 800, so the numbers stop at 749.
No. The numbers that round to 440 to the nearest ten are 435 to 444. Every one of them has tens digit 3 or 4, so each is before the halfway mark 450 and rounds to 400 to the nearest hundred. The numbers that round to 500 start at 450, so no number fits both.
345 - 340 = 5 and 350 - 345 = 5, so the two distances are equal and neither ten is nearer. Distance is the only thing “nearest” can mean, so the numbers alone give no answer. Without a rule, one person could write 340 and another 350 for the same number, and their results would not match. The round-up rule is an agreement that makes everyone’s answer the same; it is a choice people made, not something the numbers force.
No. For example, 148 rounds to 150 to the nearest ten, and 150 rounds to 200 to the nearest hundred. But 148 is 48 from 100 and 52 from 200, so straight to the nearest hundred it rounds to 100. The two ways differ exactly for the numbers whose last two digits are 45, 46, 47, 48 or 49: rounding to the ten first pushes them up to a halfway mark like 150, which then rounds up, while the number itself is before halfway and rounds down.