Grade 5 · 5.NBT.A.4

Rounding decimals

The scale at the cider stand reads 7.648 pounds of apples, but the price sign only has room for one digit after the point. Which shorter number is closest to the real weight?

You need

Rounding to any place, including decimalsGrade 4 Decimal place valueGrades 4 to 5

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

Bukola, Esteban and Minseo run a cider stand. Their scale shows weights to the thousandth of a pound, like 7.648 pounds of apples. Their jugs are filled from a tank whose meter reads to the hundredth of a gallon, like 13.95 gallons. Most of the time nobody needs all those digits. The price sign has room for one digit after the point, and a customer only wants to know “about how much”.

Rounding a decimal means trading it for a shorter number that is as close as possible. You say which place to keep: the nearest whole number, the nearest tenth, or the nearest hundredth. You already round whole numbers to a ten or a hundred (Rounding to any place, including decimals). Decimals work the same way, one place at a time, because each place is ten times smaller than the one to its left (Decimal place value, Thousandths: reading, writing and expanded form).

The procedure

Two neighbors and a halfway mark

Take 7.648 and round it to the nearest whole number. Look only at whole numbers. The number sits between 7 and 8, so those are its two neighbors. One of them is the answer, whichever is closer.

Figure 17.648 lies between 7 and 8, past the halfway mark 7.5. It is closer to 8.

The point exactly between the neighbors is the halfway mark, 7.5. Anything to the left of 7.5 is closer to 7. Anything to the right is closer to 8. The dot for 7.648 is to the right, so

7.648 \approx 8

to the nearest whole number. The sign \approx is read “is about”.

Now round the same number to the nearest tenth. The neighbors are the two numbers that stop at the tenths place on either side of it: 7.6 and 7.7. The halfway mark between them is 7.65.

Figure 2Zoom in to the tenths. 7.648 lies between 7.6 and 7.7, just short of the halfway mark 7.65, so it is closer to 7.6.

7.648 is just short of 7.65, so it is closer to 7.6, and 7.648 \approx 7.6 to the nearest tenth.

Why one digit decides

You do not need to draw a number line every time. The halfway mark always has a 5 in the place right after the one you keep, and zeros after that. To the nearest tenth, the halfway mark between 7.6 and 7.7 is 7.65. So comparing the number with the halfway mark comes down to the digit just after the kept place.

That digit is 0, 1, 2, 3 or 4. The number is below halfway, so keep the lower neighbor. The kept digit stays the same and the digits after it are dropped.
That digit is 5, 6, 7, 8 or 9. The number is at or past halfway, so take the upper neighbor. The kept digit goes up by one.

For 7.648 to the nearest tenth, the digit after the tenths is 4. Below halfway, so 7.6.

Digits further right can never change the result. In 7.648 the hundredths digit 4 and everything after it are worth less than 0.05, because even 7.6499 is less than 7.65. And a hundredths digit of 5 or more already puts the number at or past halfway, whatever follows.

When a number is exactly halfway

A number like 2.5 is the same distance from 2 and from 3. Neither neighbor is closer, so the choice is a convention, an agreement everyone uses: a number exactly halfway rounds up. That is why the rule above puts the digit 5 with the “go up” digits.

Worked examples

The lesson is taught in two parts. Examples 1 to 3 belong to the first: rounding to the nearest whole number and the nearest tenth, with the number line as the picture. Examples 4 to 7 belong to the second: rounding to the nearest hundredth, rounding that carries through a 9, keeping a zero at the end, and what goes wrong when you round in steps.

Example 1

Rounding to the nearest whole number

Round each amount to the nearest whole number. (a) A tank holds 12.39 gallons of cider. (b) A crate of apples weighs 0.86 pounds more than the label says.

Solution

(a) 12.39 lies between the whole numbers 12 and 13. The halfway mark is 12.5. The digit just after the ones place is the tenths digit, 3. A 3 is below 5, so 12.39 is below halfway, and

12.39 \approx 12

The hundredths digit 9 looks big, but it cannot matter: 12.39 is still less than 12.5.

(b) 0.86 lies between 0 and 1. The tenths digit is 8, which is 5 or more, so the number is past the halfway mark 0.5 and rounds up: 0.86 \approx 1. A number less than one can round to 1 or to 0, and here it is 1.

Example 2

Rounding to the nearest tenth

Esteban pours 3.072 gallons of cider into a cooler and Minseo pours 4.256 gallons into another. Round each to the nearest tenth.

Solution

3.072 has tenths digit 0, so it sits between 3.0 and 3.1. Its halfway mark is 3.05. The digit after the tenths is the hundredths digit, 7, which is 5 or more, so 3.072 is past 3.05 and rounds up:

3.072 \approx 3.1

4.256 sits between 4.2 and 4.3. The hundredths digit is 5, so the number is at or past the halfway mark 4.25. In fact it is past it, since 4.256 is more than 4.250. It rounds up: 4.256 \approx 4.3. The thousandths digit 6 was never needed.

Example 3

Which numbers round to 5.3?

Bukola tells a customer the jug holds “5.3 gallons, to the nearest tenth”. The meter shows hundredths. What are the smallest and largest readings it could have shown?

Solution

The readings that round to 5.3 are the ones closer to 5.3 than to either neighbor tenth, 5.2 or 5.4. The halfway mark below 5.3 is 5.25 and the halfway mark above is 5.35.

5.25 is exactly halfway between 5.2 and 5.3, and halfway rounds up, so 5.25 rounds to 5.3. That makes 5.25 the smallest reading. One hundredth less, 5.24, rounds to 5.2.

5.35 is exactly halfway between 5.3 and 5.4, and rounds up to 5.4, so it is not allowed. The largest reading in hundredths is one hundredth less, 5.34. Every reading from 5.25 up to 5.34 rounds to 5.3.

The stretch that rounds to 5.3 runs half a tenth on each side of it. Only the convention for halfway numbers decides which end belongs.

Example 4

Rounding to the nearest hundredth

Round to the nearest hundredth. (a) 7.648 pounds of apples. (b) 2.037 gallons of cider.

Solution

(a) Keeping hundredths, the neighbors of 7.648 are 7.64 and 7.65, with halfway mark 7.645. The digit after the hundredths is the thousandths digit, 8, so the number is past halfway and rounds up:

7.648 \approx 7.65

The same weight rounds to 8, to 7.6 or to 7.65, depending on the place you keep. Each answer is the closest number of its kind.

(b) 2.037 sits between 2.03 and 2.04. The thousandths digit 7 is 5 or more, so 2.037 \approx 2.04.

Example 5

Rounding that carries through a 9

The tank meter reads 13.95 gallons. Round it to the nearest tenth.

Solution

The tenths digit of 13.95 is 9, so the lower neighbor is 13.9. One tenth more than 13.9 is 14.0, because ten tenths make a one: the 9 tenths become 0 tenths and the ones digit 3 becomes 4. So the neighbors are 13.9 and 14.0.

Figure 313.95 sits exactly halfway between 13.9 and 14.0. By the usual rule a halfway number rounds up, to 14.0.

The hundredths digit is 5, and nothing follows it, so 13.95 is exactly halfway. A halfway number rounds up:

13.95 \approx 14.0

Write the answer as 14.0, not 14. The two are equal as numbers, but “to the nearest tenth” asks for a tenths digit, and 14.0 shows it. It tells the customer the amount is closer to 14 gallons than to 13.9 or 14.1. Writing 14 would only say it is closer to 14 than to 13 or 15, which is less than you know.

Example 6

One number, three places

Minseo’s scale reads 4.599 pounds. Round it to the nearest whole number, tenth and hundredth.

Solution

Whole number: the tenths digit is 5, so 4.599 is past halfway between 4 and 5 and rounds to 5.

Tenth: the neighbors are 4.5 and 4.6. The hundredths digit is 9, so the number rounds up to 4.6.

Hundredth: the neighbors are 4.59 and 4.60, since one hundredth more than 4.59 carries: ten hundredths make a tenth. The thousandths digit is 9, so 4.599 rounds up to 4.60. The zero stays, because it is the hundredths digit the question asked for.

4.599 \approx 5, \qquad 4.599 \approx 4.6, \qquad 4.599 \approx 4.60

Example 7

Rounding once, not in steps

Bukola has 6.448 pounds of apples and wants the weight to the nearest tenth. Esteban suggests rounding to the hundredth first, then to the tenth. Do the two ways agree?

Solution

Rounding once: the neighbor tenths are 6.4 and 6.5, with halfway mark 6.45. The hundredths digit is 4, so 6.448 is below halfway and 6.448 \approx 6.4.

In steps: to the nearest hundredth, the thousandths digit 8 rounds 6.448 up to 6.45. Then 6.45 is exactly halfway between 6.4 and 6.5, and rounds up to 6.5.

The answers differ, and rounding once is right. 6.448 really is closer to 6.4: its distance to 6.4 is 0.048, and its distance to 6.5 is 0.052. The first step moved the number from just below the halfway mark onto it, and the second step treated it as if it had always been there. Always round from the original number.

Boundaries

Exactly halfway is a convention. A number such as 2.5, 13.95 or 41.005 is the same distance from both neighbors, so “nearest” does not pick one. The rule used here rounds it up. Some calculators and computer programs use a different agreement for halfway numbers, so two tools can disagree on halfway numbers.
Round from the original number. Rounding in steps can give a different answer from rounding once, as Example 7 shows. Steps can never do better than rounding once, because rounding once already gives the closest number.
A rounded number is a different number. 7.6 is close to 7.648 but not equal to it. Write \approx, never =, between a number and its rounded value.
The zero at the end carries meaning. 14.0 and 14 are equal, but as answers to “the nearest tenth” only 14.0 says what was asked. The same goes for 4.60 to the nearest hundredth.
A number with no digits past the kept place is already rounded. 3.7 to the nearest tenth is 3.7. To the nearest hundredth it is 3.70: write a zero to show the hundredths place.

Where this goes next

Rounding to the nearest whole number, tenth or hundredth is the decimal part of Rounding to any place, including decimals, and it sits inside the wider idea of Rounding. The next lessons in this unit move the decimal point by multiplying and dividing by powers of ten, and rounding is a quick way to see whether such an answer has the right size. Later, rounding is one of the tools for estimating with Friendly numbers and deliberate over and under estimates. Money is rounded to the nearest hundredth all the time, because a cent is a hundredth of a dollar.

Exercises

Part 1: the nearest whole number

Round 4.71 to the nearest whole number.
Round 18.293 to the nearest whole number.
Round 0.502 to the nearest whole number.
Round 29.6 to the nearest whole number.

Part 1: the nearest tenth

Round 3.84 to the nearest tenth.
Round 12.367 to the nearest tenth.
Round 0.05 to the nearest tenth.
Round 6.719 to the nearest tenth.
Round 25.081 to the nearest tenth.

Part 1: at the stand

A bag of apples weighs 9.38 pounds. Name its two neighbor whole numbers and round it to the nearest whole number. Then name its two neighbor tenths and round it to the nearest tenth.
Which of these round to 2.7 to the nearest tenth: 2.65, 2.749, 2.75, 2.709, 2.6?
Esteban says a jug holds 6 gallons, to the nearest whole number. The meter shows tenths. What are the smallest and largest readings it could have shown?

Part 2: the nearest hundredth

Round 5.276 to the nearest hundredth.
Round 0.634 to the nearest hundredth.
Round 41.005 to the nearest hundredth.
Round 8.199 to the nearest hundredth.

Part 2: nines and zeros

Round 2.96 to the nearest tenth.
Round 19.98 to the nearest tenth.
Round 0.997 to the nearest hundredth.
Round 9.951 to the nearest whole number, to the nearest tenth and to the nearest hundredth.
Round 3.604 to the nearest hundredth.

Part 2: at the stand

A customer’s apples come to 27.395 dollars by the price per pound. A cent is a hundredth of a dollar. Round the cost to the nearest cent.
Bukola’s scale shows 11.954 pounds. The sign shows weights to the nearest tenth. What does it show?

Reasoning

Minseo rounds 4.499 to the nearest whole number in steps: hundredth, then tenth, then whole number. What does she get, and what is the correct answer? Explain which one is right using the distance to each neighbor.
Esteban rounds 6.796 to the nearest hundredth and writes 6.8. Bukola writes 6.80. Both numbers are equal. Explain why Bukola’s answer is the better one.

Answers

5. The tenths digit is 7, so 4.71 is past the halfway mark 4.5.
18. The tenths digit is 2, so 18.293 is below 18.5.
1. The tenths digit is 5 and 0.502 is past 0.5, so it rounds up.
30. The tenths digit is 6, so 29.6 rounds up, and one more than 29 is 30.
3.8. The neighbors are 3.8 and 3.9, and the hundredths digit 4 puts 3.84 below 3.85.
12.4. The hundredths digit is 6, so 12.367 is past 12.35 and rounds up from 12.3.
0.1. 0.05 is exactly halfway between 0.0 and 0.1, and a halfway number rounds up.
6.7. The hundredths digit is 1, so 6.719 is below 6.75.
25.1. The hundredths digit is 8, so 25.081 is past 25.05 and rounds up from 25.0.
Whole numbers 9 and 10: the tenths digit 3 gives 9.38 \approx 9. Tenths 9.3 and 9.4: the hundredths digit 8 gives 9.38 \approx 9.4.
2.65, 2.749 and 2.709. 2.65 is halfway and rounds up to 2.7; 2.749 and 2.709 have hundredths digits 4 and 0. 2.75 rounds to 2.8 and 2.6 is already 2.6.
Smallest 5.5, largest 6.4. 5.5 is halfway between 5 and 6 and rounds up to 6; 6.5 would round up to 7, so the largest reading in tenths is 6.4.
5.28. The thousandths digit is 6, so 5.276 rounds up from 5.27.
0.63. The thousandths digit is 4, so 0.634 is below 0.635.
41.01. 41.005 is exactly halfway between 41.00 and 41.01, and rounds up.
8.20. The thousandths digit 9 rounds up from 8.19, and ten hundredths make a tenth, so the next hundredth is 8.20. The zero stays.
3.0. The neighbors are 2.9 and 3.0; the hundredths digit 6 rounds up.
20.0. The neighbors are 19.9 and 20.0; the hundredths digit 8 rounds up, carrying through both nines.
1.00. The neighbors are 0.99 and 1.00; the thousandths digit 7 rounds up.
10, 10.0 and 9.95. Whole: the tenths digit is 9. Tenth: 9.951 is past 9.95, so it rounds up from 9.9 to 10.0. Hundredth: the thousandths digit 1 keeps 9.95.
3.60. The thousandths digit is 4, so 3.604 is below 3.605; keep the zero, it is the hundredths digit.
27.40 dollars. 27.395 is exactly halfway between 27.39 and 27.40, and rounds up.
12.0 pounds. The neighbors are 11.9 and 12.0, with halfway mark 11.95. 11.954 is past it, so it rounds up, carrying through the 9.
In steps she gets 4.50, then 4.5, then 5. The correct answer is 4. 4.499 is 0.499 from 4 and 0.501 from 5, so 4 is closer. Rounding to the hundredth first moved the number up onto the halfway mark 4.5, which then rounded up.
Both are equal to the rounded value 6.80, but the question asked for the nearest hundredth, and only 6.80 shows a hundredths digit. It says 6.796 is closer to 6.80 than to 6.79 or 6.81. Writing 6.8 says only that it is closer to 6.8 than to 6.7 or 6.9.