1.8: Rounding Whole Numbers
- Page ID
- 116749
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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.
The precise number of symbols in the above collection is 18.
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.
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.
Thus, 67, rounded to the nearest ten, is 70.
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.
Thus, 4,329, rounded to the nearest hundred is 4,300.
Round 16,500 to the nearest thousand.
Solution
On the number line, 16,500 is exactly halfway from 16,000 to 17,000.
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.
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.
- If it is less than 5, replace it and all the digits to its right with zeros. Leave the round-off digit unchanged.
- If it is 5 or larger, replace it and all the digits to its right with zeros. Increase the round-off digit by 1.
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
- 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.
Round 9,614,018,007 to the nearest ten million.
Solution
- We are rounding to the nearest ten million.
- 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.
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.
- 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.
Round 397,000 to the nearest ten thousand.
Solution
- We are rounding to the nearest ten thousand.
- 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