1.8: Rounding Whole Numbers

Last updated

Oct 24, 2022
Save as PDF
- 1.7.4: Using Fractions, Decimals and Percents to Describe Charts
- 1.9: Scientific Notation

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Learning Objectives

understand that rounding is a method of approximation
be able to round a whole number to a specified position

Rounding as an Approximation

A primary use of whole numbers is to keep count of how many objects there are in a collection. Sometimes we're only interested in the approximate number of objects in the collection rather than the precise number. For example, there are approximately 20 symbols in the collection below.

$An arrangement of symbols.$

The precise number of symbols in the above collection is 18.

Definition: Rounding

We often approximate the number of objects in a collection by mentally seeing the collection as occurring in groups of tens, hundreds, thousands, etc. This process of approximation is called rounding. Rounding is very useful in estimation. We will study estimation in Chapter 8.

When we think of a collection as occurring in groups of tens, we say we're rounding to the nearest ten. When we think of a collection as occurring in groups of hundreds, we say we're rounding to the nearest hundred. This idea of rounding continues through thousands, ten thousands, hundred thousands, millions, etc.

The process of rounding whole numbers is illustrated in the following examples.

Example $\PageIndex{1}$

Round 67 to the nearest ten.

Solution

On the number line, 67 is more than halfway from 60 to 70. The digit immediately to the right of the tens digit, the round-off digit, is the indicator for this.

$A number line from 0 to 70. At the dash for the number sixty is a label, 6 tens. At the dash for 70 is a label, 7 tens. In between the two dashes is a dot on the number 67. Below, is a statement. 67 is closer to 7 tens than it is to 6 tens.$

Thus, 67, rounded to the nearest ten, is 70.

Example $\PageIndex{2}$

Round 4,329 to the nearest hundred.

Solution

On the number line, 4,329 is less than halfway from 4,300 to 4,400. The digit to the immediate right of the hundreds digit, the round-off digit, is the indicator.

$A number line from 0 to 4,400. The mark for 4,300 is labeled, 3 hundreds. The mark for 4,400 is labeled, 4 hundreds. A dot on the number 4,329 is in between the two marks. Below the number line is a statement. 4,329 is closer to 43 hundreds than it is to 44 hundreds.$

Thus, 4,329, rounded to the nearest hundred is 4,300.

Example $\PageIndex{3}$

Round 16,500 to the nearest thousand.

Solution

On the number line, 16,500 is exactly halfway from 16,000 to 17,000.

$A number line from 0 to 17,000. The 16,000 mark is labeled, 6 thousands. The 17,000 mark is labeled, 7 thousands. In between the two marks is a dot on the number 16,500.$

By convention, when the number to be rounded is exactly halfway between two numbers, it is rounded to the higher number.

Thus, 16,500, rounded to the nearest thousand, is 17,000.

Example $\PageIndex{4}$

A person whose salary is $41,450 per year might tell a friend that she makes $41,000 per year. She has rounded 41,450 to the nearest thousand. The number 41,450 is closer to 41,000 than it is to 42,000.

The Method of Rounding Whole Numbers

From the observations made in the preceding examples, we can use the following method to round a whole number to a particular position.

Mark the position of the round-off digit.
Note the digit to the immediate right of the round-off digit.
1. If it is less than 5, replace it and all the digits to its right with zeros. Leave the round-off digit unchanged.
2. If it is 5 or larger, replace it and all the digits to its right with zeros. Increase the round-off digit by 1.

Sample Set A

Use the method of rounding whole numbers to solve the following problems.

Round 3,426 to the nearest ten.

Solution

We are rounding to the tens position. Mark the digit in the tens position
$3,426, with the 2 labeled, tens position.$
Observe the digit immediately to the right of the tens position. It is 6. Since 6 is greater than 5, we round up by replacing 6 with 0 and adding 1 to the digit in the tens position (the round-off position): 2+1=32+1=3 .
3,430

Thus, 3,426 rounded to the nearest ten is 3,430.

Sample Set A

Round 9,614,018,007 to the nearest ten million.

Solution

We are rounding to the nearest ten million.
$9,614,018,007, with the first 1 labeled, ten millions position.$
Observe the digit immediately to the right of the ten millions position. It is 4. Since 4 is less than 5, we round down by replacing 4 and all the digits to its right with zeros.
9,610,000,000

Thus, 9,614,018,007 rounded to the nearest ten million is 9,610,000,000.

Sample Set A

Round 148,422 to the nearest million.

Solution

Since we are rounding to the nearest million, we'll have to imagine a digit in the millions position. We'll write 148,422 as 0,148,422.
$0,148,422, with the 0 labeled, millions position.$
The digit immediately to the right is 1. Since 1 is less than 5, we'll round down by replacing it and all the digits to its right with zeros.
0,000,000
This number is 0.

Thus, 148,422 rounded to the nearest million is 0.

Sample Set A

Round 397,000 to the nearest ten thousand.

Solution

We are rounding to the nearest ten thousand.
$397,000, with the 9 labeled, ten thousand position.$
The digit immediately to the right of the ten thousand position is 7. Since 7 is greater than 5, we round up by replacing 7 and all the digits to its right with zeros and adding 1 to the digit in the ten thousands position. But 9+1=109+1=10 and we must carry the 1 to the next (the hundred thousands) position.
400,000

Thus, 397,000 rounded to the nearest ten thousand is 400,000.

Practice Set A

Use the method of rounding whole numbers to solve each problem.

Round 3387 to the nearest hundred.

Answer: 3400

Practice Set A

Round 26,515 to the nearest thousand.

Answer: 27,000

Practice Set A

Round 30,852,900 to the nearest million.

Answer: 31,000,000

Practice Set A

Round 39 to the nearest hundred.

Answer: 0

Practice Set A

Round 59,600 to the nearest thousand.

Answer: 60,000

Exercises

For the following problems, complete the table by rounding each number to the indicated positions.

Exercise $\PageIndex{1}$

1,642

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
1,600	2000	0	0

Exercise $\PageIndex{2}$

5,221

hundred	thousand	ten thousand	million

Exercise $\PageIndex{3}$

91,803

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
91,800	92,000	90,000	0

Exercise $\PageIndex{4}$

106,007

hundred	thousand	ten thousand	million

Exercise $\PageIndex{5}$

208

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
200	0	0	0

Exercise $\PageIndex{6}$

199

hundred	thousand	ten thousand	million

Exercise $\PageIndex{7}$

863

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
900	1,000	0	0

Exercise $\PageIndex{8}$

794

hundred	thousand	ten thousand	million

Exercise $\PageIndex{9}$

925

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
900	1,000	0	0

Exercise $\PageIndex{10}$

909

hundred	thousand	ten thousand	million

Exercise $\PageIndex{11}$

981

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
1,000	1,000	0	0

Exercise $\PageIndex{12}$

965

hundred	thousand	ten thousand	million

Exercise $\PageIndex{13}$

551,061,285

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
551,061,300	551,061,000	551,060,000	551,000,000

Exercise $\PageIndex{14}$

23,047,991,521

hundred	thousand	ten thousand	million

Exercise $\PageIndex{15}$

106,999,413,206

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
106,999,413,200	106,999,413,000	106,999,410,000	106,999,000,000

Exercise $\PageIndex{16}$

5,000,000

hundred	thousand	ten thousand	million

Exercise $\PageIndex{17}$

8,006,001

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
8,006,000	8,006,000	8,010,000	8,000,000

Exercise $\PageIndex{18}$

94,312

hundred	thousand	ten thousand	million

Exercise $\PageIndex{19}$

33,486

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
33,500	33,000	30,000	0

Exercise $\PageIndex{20}$

560,669

hundred	thousand	ten thousand	million

Exercise $\PageIndex{21}$

388,551

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
388,600	389,000	390,000	0

Exercise $\PageIndex{22}$

4,752

hundred	thousand	ten thousand	million

Exercise $\PageIndex{23}$

8,209

hundred	thousand	ten thousand	million

Answer

hundred	thousand	ten thousand	million
8,200	8,000	10,000	0

Exercise $\PageIndex{24}$

In 1950, there were 5,796 cases of diphtheria reported in the United States. Round to the nearest hundred.

Exercise $\PageIndex{25}$

In 1979, 19,309,000 people in the United States received federal food stamps. Round to the nearest ten thousand.

Answer: 19,310,000

Exercise $\PageIndex{26}$

In 1980, there were 1,105,000 people between 30 and 34 years old enrolled in school. Round to the nearest million.

Exercise $\PageIndex{27}$

In 1980, there were 29,100,000 reports of aggravated assaults in the United States. Round to the nearest million.

Answer: 29,000,000

For the following problems, round the numbers to the position you think is most reasonable for the situation.

Exercise $\PageIndex{28}$

In 1980, for a city of one million or more, the average annual salary of police and firefighters was $16,096.

Exercise $\PageIndex{29}$

The average percentage of possible sunshine in San Francisco, California, in June is 73%.

Answer: 70% or 75%

Exercise $\PageIndex{30}$

In 1980, in the state of Connecticut, $3,777,000,000 in defense contract payroll was awarded.

Exercise $\PageIndex{31}$

In 1980, the federal government paid $5,463,000,000 to Viet Nam veterans and dependants.

Answer: $5,500,000,000

Exercise $\PageIndex{32}$

In 1980, there were 3,377,000 salespeople employed in the United States.

Exercise $\PageIndex{33}$

In 1948, in New Hampshire, 231,000 popular votes were cast for the president.

Answer: 230,000

Exercise $\PageIndex{34}$

In 1970, the world production of cigarettes was 2,688,000,000,000.

Exercise $\PageIndex{35}$

In 1979, the total number of motor vehicle registrations in Florida was 5,395,000.

Answer: 5,400,000

Exercise $\PageIndex{36}$

In 1980, there were 1,302,000 registered nurses the United States.

Exercises for Review

Exercise $\PageIndex{37}$

There is a term that describes the visual displaying of a number. What is the term?

Answer: graphing

Exercise $\PageIndex{38}$

What is the value of 5 in 26,518,206?

Exercise $\PageIndex{39}$

Write 42,109 as you would read it.

Answer: Forty-two thousand, one hundred nine

Exercise $\PageIndex{40}$

Write "six hundred twelve" using digits.

Exercise $\PageIndex{41}$

Write "four billion eight" using digits.

Answer: 4,000,000,008

Search

Text Color

Text Size

Margin Size

Font Type

Learning Objectives

Rounding as an Approximation

Definition: Rounding

Example $\PageIndex{1}$

Example $\PageIndex{2}$

Example $\PageIndex{3}$

Example $\PageIndex{4}$

The Method of Rounding Whole Numbers

Sample Set A

Sample Set A

Sample Set A

Sample Set A

Exercises

Exercises for Review

Learning Objectives

Rounding as an Approximation

Definition: Rounding

Example 1.8.1\PageIndex{1}

Example 1.8.2\PageIndex{2}

Example 1.8.3\PageIndex{3}

Example 1.8.4\PageIndex{4}

The Method of Rounding Whole Numbers

Sample Set A

Sample Set A

Sample Set A

Sample Set A

Exercises

Exercises for Review

Support Center

How can we help?

Example $\PageIndex{1}$

Example $\PageIndex{2}$

Example $\PageIndex{3}$

Example $\PageIndex{4}$