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Addition and Subtraction of Whole Numbers: Rounding Whole Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to round whole numbers. By the end of the module students should be able to understand that rounding is a method of approximation and round a whole number to a specified position.

Section Overview

  • Rounding as an Approximation
  • The Method of Rounding Numbers

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.

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 1

Round 67 to the nearest ten.

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 2

Round 4,329 to the nearest hundred.

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 3

Round 16,500 to the nearest thousand.

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 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.

  1. Mark the position of the round-off digit.
  2. 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.

Example 5

Round 3,426 to the nearest ten.

  1. We are rounding to the tens position. Mark the digit in the tens position
     3,426, with the 2 labeled, tens position.
  2. 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 = 3 2+1=3 .
    3,430

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

Example 6

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

  1. We are rounding to the nearest ten million.
    9,614,018,007, with the first 1 labeled, ten millions position.
  2. 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.

Example 7

Round 148,422 to the nearest million.

  1. 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.
  2. 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.

Example 8

Round 397,000 to the nearest ten thousand.

  1. We are rounding to the nearest ten thousand.
    397,000, with the 9 labeled, ten thousand position.
  2. 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 = 10 9+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.

Exercise 1

Round 3387 to the nearest hundred.

Solution

3400

Exercise 2

Round 26,515 to the nearest thousand.

Solution

27,000

Exercise 3

Round 30,852,900 to the nearest million.

Solution

31,000,000

Exercise 4

Round 39 to the nearest hundred.

Solution

0

Exercise 5

Round 59,600 to the nearest thousand.

Solution

60,000

Exercises

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

Exercise 6

1,642

Table 1
hundred thousand ten thousand million
       

Solution

Table 2
hundred thousand ten thousand million
1,600 2000 0 0

Exercise 7

5,221

Table 3
hundred thousand ten thousand million
       

Exercise 8

91,803

Table 4
Hundred thousand ten thousand million
       

Solution

Table 5
Hundred thousand ten thousand million
91,800 92,000 90,000 0

Exercise 9

106,007

Table 6
hundred thousand ten thousand million
       

Exercise 10

208

Table 7
hundred thousand ten thousand million
       

Solution

Table 8
hundred thousand ten thousand million
200 0 0 0

Exercise 11

199

Table 9
hundred thousand ten thousand million
       

Exercise 12

863

Table 10
hundred thousand ten thousand million
       

Solution

Table 11
hundred thousand ten thousand million
900 1,000 0 0

Exercise 13

794

Table 12
hundred thousand ten thousand million
       

Exercise 14

925

Table 13
hundred thousand ten thousand million
       

Solution

Table 14
hundred thousand ten thousand million
900 1,000 0 0

Exercise 15

909

Table 15
hundred thousand ten thousand million
       

Exercise 16

981

Table 16
hundred thousand ten thousand million
       

Solution

Table 17
hundred thousand ten thousand million
1,000 1,000 0 0

Exercise 17

965

Table 18
hundred thousand ten thousand million
       

Exercise 18

551,061,285

Table 19
hundred thousand ten thousand million
       

Solution

Table 20
hundred thousand ten thousand million
551,061,300 551,061,000 551,060,000 551,000,000

Exercise 19

23,047,991,521

Table 21
hundred thousand ten thousand million
       

Exercise 20

106,999,413,206

Table 22
Hundred thousand ten thousand million
       

Solution

Table 23
hundred thousand ten thousand million
106,999,413,200 106,999,413,000 106,999,410,000 106,999,000,000

Exercise 21

5,000,000

Table 24
hundred thousand ten thousand million
       

Exercise 22

8,006,001

Table 25
hundred thousand ten thousand million
       

Solution

Table 26
Hundred Thousand ten thousand Million
8,006,000 8,006,000 8,010,000 8,000,000

Exercise 23

94,312

Table 27
hundred thousand ten thousand million
       

Exercise 24

33,486

Table 28
hundred thousand ten thousand million
       

Solution

Table 29
hundred thousand ten thousand million
33,500 33,000 30,000 0

Exercise 25

560,669

Table 30
hundred thousand ten thousand million
       

Exercise 26

388,551

Table 31
hundred thousand ten thousand million
       

Solution

Table 32
hundred thousand ten thousand million
388,600 389,000 390,000 0

Exercise 27

4,752

Table 33
hundred thousand ten thousand million
       

Exercise 28

8,209

Table 34
hundred thousand ten thousand million
       

Solution

Table 35
hundred thousand ten thousand million
8,200 8,000 10,000 0

Exercise 29

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

Exercise 30

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

Solution

19,310,000

Exercise 31

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

Exercise 32

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

Solution

29,000,000

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

Exercise 33

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

Exercise 34

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

Solution

70% or 75%

Exercise 35

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

Exercise 36

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

Solution

$5,500,000,000

Exercise 37

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

Exercise 38

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

Solution

230,000

Exercise 39

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

Exercise 40

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

Solution

5,400,000

Exercise 41

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

Exercises for Review

Exercise 42

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

Solution

graphing

Exercise 43

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

Exercise 44

((Reference)) Write 42,109 as you would read it.

Solution

Forty-two thousand, one hundred nine

Exercise 45

((Reference)) Write "six hundred twelve" using digits.

Exercise 46

((Reference)) Write "four billion eight" using digits.

Solution

4,000,000,008

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