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Module by: Denny Burzynski, Wade Ellis. 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 add whole numbers. By the end of this module, students should be able to understand the addition process, add whole numbers, and use the calculator to add one whole number to another.

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## Section Overview

• Addition Visualized on the Number Line
• Calculators

Suppose we have two collections of objects that we combine together to form a third collection. For example,

We are combining a collection of four objects with a collection of three objects to obtain a collection of seven objects.

The process of combining two or more objects (real or intuitive) to form a third, the total, is called addition.

In addition, the numbers being added are called addends or terms, and the total is called the sum. The plus symbol (+) is used to indicate addition, and the equal symbol (=) is used to represent the word "equal." For example, 4+3=74+3=7 size 12{4+3=7} {} means "four added to three equals seven."

## Addition Visualized on the Number Line

Addition is easily visualized on the number line. Let's visualize the addition of 4 and 3 using the number line.

To find 4+34+3 size 12{4+3} {},

1. Start at 0.
2. Move to the right 4 units. We are now located at 4.
3. From 4, move to the right 3 units. We are now located at 7.

Thus, 4+3=74+3=7 size 12{4+3=7} {}.

We'll study the process of addition by considering the sum of 25 and 43.

25 + 43 ̲ means 25 + 43 ̲ means

We write this as 68.

We can suggest the following procedure for adding whole numbers using this example.

### Example 1: The Process of Adding Whole Numbers

The process:

1. Write the numbers vertically, placing corresponding positions in the same column.

25 + 43 ̲ 25 + 43 ̲

2. Add the digits in each column. Start at the right (in the ones position) and move to the left, placing the sum at the bottom.

25 + 43 ̲ 68 25 + 43 ̲ 68

### Caution:

Confusion and incorrect sums can occur when the numbers are not aligned in columns properly. Avoid writing such additions as

25 + 43 ̲ 25 + 43 ̲

25 + 43 ̲ 25 + 43 ̲

### Sample Set A

#### Example 2

276 + 103 ̲ 379 6 + 3 = 9 . 7 + 0 = 7 . 2 + 1 = 3 . 276 + 103 ̲ 379 6 + 3 = 9 . 7 + 0 = 7 . 2 + 1 = 3 . alignl { stack { size 12{6+3=9 "." } {} # size 12{7+0=7 "." } {} # size 12{2+1=3 "." } {} } } {}

#### Example 3

1459 + 130 ̲ 1589 9 + 0 = 9 . 5 + 3 = 8 . 4 + 1 = 5 . 1 + 0 = 1 . 1459 + 130 ̲ 1589 9 + 0 = 9 . 5 + 3 = 8 . 4 + 1 = 5 . 1 + 0 = 1 . alignl { stack { size 12{9+0=9 "." } {} # size 12{5+3=9 "." } {} # size 12{4+1=5 "." } {} # size 12{1+0=1 "." } {} } } {}

In each of these examples, each individual sum does not exceed 9. We will examine individual sums that exceed 9 in the next section.

### Practice Set A

Perform each addition. Show the expanded form in problems 1 and 2.

88

5,527

#### Exercise 3

##### Solution

267,166

It often happens in addition that the sum of the digits in a column will exceed 9. This happens when we add 18 and 34. We show this in expanded form as follows.

Notice that when we add the 8 ones to the 4 ones we get 12 ones. We then convert the 12 ones to 1 ten and 2 ones. In vertical addition, we show this conversion by carrying the ten to the tens column. We write a 1 at the top of the tens column to indicate the carry. This same example is shown in a shorter form as follows:

8+4=128+4=12 Write 2, carry 1 ten to the top of the next column to the left.

### Sample Set B

Perform the following additions. Use the process of carrying when needed.

#### Example 4

5 + 8 = 13 Write 3, carry 1 ten. 1 + 7 + 5 = 13 Write 3, carry 1 hundred. 1 + 8 + 3 = 12 Write 2, carry 1 thousand. 1 + 1 = 2 5 + 8 = 13 Write 3, carry 1 ten. 1 + 7 + 5 = 13 Write 3, carry 1 hundred. 1 + 8 + 3 = 12 Write 2, carry 1 thousand. 1 + 1 = 2

The sum is 2233.

#### Example 5

8 + 6 = 14 Write 4, carry 1 ten. 1 + 0 + 4 = 5 Write the 5 (nothing to carry). 2 + 9 = 11 Write 1, carry one thousand. 1 + 9 + 4 = 14 Write 4, carry one ten thousand. 1 + 8 = 9 8 + 6 = 14 Write 4, carry 1 ten. 1 + 0 + 4 = 5 Write the 5 (nothing to carry). 2 + 9 = 11 Write 1, carry one thousand. 1 + 9 + 4 = 14 Write 4, carry one ten thousand. 1 + 8 = 9

The sum is 94,154.

#### Example 6

8 + 5 = 13 Write 3, carry 1 ten. 1 + 3 + 9 = 13 Write 3, carry 1 hundred. 1 + 0 = 1 8 + 5 = 13 Write 3, carry 1 ten. 1 + 3 + 9 = 13 Write 3, carry 1 hundred. 1 + 0 = 1

As you proceed with the addition, it is a good idea to keep in mind what is actually happening.

The sum is 133.

#### Example 7

Find the sum 2648, 1359, and 861.

8 + 9 + 1 = 18 Write 8, carry 1 ten. 1 + 4 + 5 + 6 = 16 Write 6, carry 1 hundred. 1 + 6 + 3 + 8 = 18 Write 8, carry 1 thousand. 1 + 2 + 1 = 4 8 + 9 + 1 = 18 Write 8, carry 1 ten. 1 + 4 + 5 + 6 = 16 Write 6, carry 1 hundred. 1 + 6 + 3 + 8 = 18 Write 8, carry 1 thousand. 1 + 2 + 1 = 4

The sum is 4,868.

Numbers other than 1 can be carried as illustrated in Example 8.

#### Example 8

Find the sum of the following numbers.

6 + 5 + 1 + 7 = 19 Write 9, carry the 1. 1 + 1 + 0 + 5 + 1 = 8 Write 8. 0 + 9 + 9 + 8 = 26 Write 6, carry the 2. 2 + 8 + 9 + 8 + 6 = 33 Write 3, carry the 3. 3 + 7 + 3 + 5 = 18 Write 8, carry the 1. 1 + 8 = 9 Write 9. 6 + 5 + 1 + 7 = 19 Write 9, carry the 1. 1 + 1 + 0 + 5 + 1 = 8 Write 8. 0 + 9 + 9 + 8 = 26 Write 6, carry the 2. 2 + 8 + 9 + 8 + 6 = 33 Write 3, carry the 3. 3 + 7 + 3 + 5 = 18 Write 8, carry the 1. 1 + 8 = 9 Write 9.

The sum is 983,689.

#### Example 9

The number of students enrolled at Riemann College in the years 1984, 1985, 1986, and 1987 was 10,406, 9,289, 10,108, and 11,412, respectively. What was the total number of students en­rolled at Riemann College in the years 1985, 1986, and 1987?

We can determine the total number of students enrolled by adding 9,289, 10,108, and 11,412, the number of students enrolled in the years 1985, 1986, and 1987.

The total number of students enrolled at Riemann College in the years 1985, 1986, and 1987 was 30,809.

### Practice Set B

Perform each addition. For the next three problems, show the expanded form.

#### Exercise 4

##### Solution

87

=7tens+1ten+7ones =8tens+7ones =87 =7tens+1ten+7ones =8tens+7ones =87

#### Exercise 5

##### Solution

561

= 4 hundreds + 15 tens + 1 ten + 1 one = 4 hundreds + 16 tens + 1 one = 4 hundreds + 1 hundred + 6 tens + 1 one = 5 hundreds + 6 tens + 1 one =561 = 4 hundreds + 15 tens + 1 ten + 1 one = 4 hundreds + 16 tens + 1 one = 4 hundreds + 1 hundred + 6 tens + 1 one = 5 hundreds + 6 tens + 1 one =561

#### Exercise 6

##### Solution

115

= 10 tens + 1 ten + 5 ones = 11 tens + 5 ones = 1 hundred + 1 ten + 5 ones =115 = 10 tens + 1 ten + 5 ones = 11 tens + 5 ones = 1 hundred + 1 ten + 5 ones =115

#### Exercise 7

##### Solution

153,525

For the next three problems, find the sums.

#### Exercise 8

57 26   84 ̲ 57 26   84 ̲

167

#### Exercise 9

847 825   796 ̲ 847 825   796 ̲

2,468

#### Exercise 10

16,945 8,472 387,721 21,059            629 ̲ 16,945 8,472 387,721 21,059            629 ̲

434,826

## Calculators

Calculators provide a very simple and quick way to find sums of whole numbers. For the two problems in Sample Set C, assume the use of a calculator that does not require the use of an ENTER key (such as many Hewlett-Packard calculators).

### Sample Set C

Use a calculator to find each sum.

#### Example 10

 34 + 21 34 + 21 size 12{"34"+"21"} {} Display Reads Type 34 34 Press + 34 Type 21 21 Press = 55

The sum is 55.

#### Example 11

 106 + 85 + 322 + 406 106 + 85 + 322 + 406 size 12{"106"+"85"+"322"+"406"} {} Display Reads Type 106 106 The calculator keeps a running subtotal Press + 106 Type 85 85 Press = 191 ← 106+85106+85 size 12{"106"+"85"} {} Type 322 322 Press + 513 ← 191+322191+322 size 12{"191"+"322"} {} Type 406 406 Press = 919 ← 513+406513+406 size 12{"513"+"406"} {}

The sum is 919.

### Practice Set C

Use a calculator to find the following sums.

#### Exercise 11

62+81+1262+81+12 size 12{"62"+"81"+"12"} {}

155

#### Exercise 12

9,261+8,543+884+1,0629,261+8,543+884+1,062 size 12{9,"261"+8,"543"+"884"+1,"062"} {}

19,750

#### Exercise 13

10,221+9,016+11,44510,221+9,016+11,445 size 12{"10","221"+0,"016"+"11","445"} {}

30,682

## Exercises

For the following problems, perform the additions. If you can, check each sum with a calculator.

### Exercise 14

14 + 5 14 + 5 size 12{"14"+5} {}

19

### Exercise 15

12 + 7 12 + 7 size 12{"12"+7} {}

### Exercise 16

46 + 2 46 + 2 size 12{"46"+2} {}

48

### Exercise 17

83 + 16 83 + 16 size 12{"83"+"16"} {}

### Exercise 18

77 + 21 77 + 21 size 12{"77"+"21"} {}

98

### Exercise 19

321 +   42 ̲ 321 +   42 ̲

### Exercise 20

916 +   62 ̲ 916 +   62 ̲

978

### Exercise 21

104 + 561 ̲ 104 + 561 ̲

### Exercise 22

265 + 103 ̲ 265 + 103 ̲

368

### Exercise 23

552 + 237 552 + 237 size 12{"552"+"237"} {}

### Exercise 24

8, 521 + 4, 256 8, 521 + 4, 256 size 12{8,"521"+4,"256"} {}

12,777

### Exercise 25

16,408 +   3,101 ̲ 16,408 +   3,101 ̲

### Exercise 26

16,515 + 42,223 ̲ 16,515 + 42,223 ̲

58,738

### Exercise 27

616 , 702 + 101 , 161 616 , 702 + 101 , 161 size 12{"616","702"+"101","161"} {}

### Exercise 28

43 , 156 , 219 + 2, 013 , 520 43 , 156 , 219 + 2, 013 , 520 size 12{"43","156","219"+2,"013","520"} {}

45,169,739

### Exercise 29

17 + 6 17 + 6 size 12{"17"+6} {}

### Exercise 30

25 + 8 25 + 8 size 12{"25"+8} {}

33

### Exercise 31

84 +   7 ̲ 84 +   7 ̲

### Exercise 32

75 +   6 ̲ 75 +   6 ̲

81

### Exercise 33

36 + 48 36 + 48 size 12{"36"+"48"} {}

### Exercise 34

74 + 17 74 + 17 size 12{"74"+"17"} {}

91

### Exercise 35

486 + 58 486 + 58 size 12{"486"+"58"} {}

### Exercise 36

743 + 66 743 + 66 size 12{"743"+"66"} {}

809

### Exercise 37

381 + 88 381 + 88 size 12{"381"+"88"} {}

### Exercise 38

687 + 175 ̲ 687 + 175 ̲

862

### Exercise 39

931 + 853 ̲ 931 + 853 ̲

### Exercise 40

1, 428 + 893 1, 428 + 893 size 12{1,"428"+"893"} {}

2,321

### Exercise 41

12 , 898 + 11 , 925 12 , 898 + 11 , 925 size 12{"12","898"+"11","925"} {}

### Exercise 42

631,464 + 509,740 ̲ 631,464 + 509,740 ̲

1,141,204

### Exercise 43

805,996 +   98,516 ̲ 805,996 +   98,516 ̲

### Exercise 44

38,428,106 + 522,936,005 ̲ 38,428,106 + 522,936,005 ̲

562,364,111

### Exercise 45

5, 288 , 423 , 100 + 16 , 934 , 785 , 995 5, 288 , 423 , 100 + 16 , 934 , 785 , 995 size 12{5,"288","423","100"+"16","934","785","995"} {}

### Exercise 46

98 , 876 , 678 , 521 , 402 + 843 , 425 , 685 , 685 , 658 98 , 876 , 678 , 521 , 402 + 843 , 425 , 685 , 685 , 658 size 12{"98","876","678","521","402"+"843","425","685","685","658"} {}

#### Solution

942,302,364,207,060

### Exercise 47

41 + 61 + 85 + 62 41 + 61 + 85 + 62 size 12{"41"+"61"+"85"+"62"} {}

### Exercise 48

21 + 85 + 104 + 9 + 15 21 + 85 + 104 + 9 + 15 size 12{"21"+"85"+"104"+9+"15"} {}

234

### Exercise 49

116 27 110 110 +     8 ̲ 116 27 110 110 +     8 ̲

### Exercise 50

75,206 4,152 + 16,007 ̲ 75,206 4,152 + 16,007 ̲

95,365

### Exercise 51

8,226 143 92,015 8 487,553       5,218 ̲ 8,226 143 92,015 8 487,553       5,218 ̲

### Exercise 52

50,006 1,005 100,300 20,008 1,000,009       800,800 ̲ 50,006 1,005 100,300 20,008 1,000,009       800,800 ̲

1,972,128

### Exercise 53

616 42,018 1,687 225 8,623,418 12,506,508 19 2,121       195,643 ̲ 616 42,018 1,687 225 8,623,418 12,506,508 19 2,121       195,643 ̲

For the following problems, perform the additions and round to the nearest hundred.

### Exercise 54

1,468   2,183 ̲ 1,468   2,183 ̲

3,700

### Exercise 55

928,725      15,685 ̲ 928,725      15,685 ̲

### Exercise 56

82,006   3,019,528 ̲ 82,006   3,019,528 ̲

3,101,500

### Exercise 57

18,621     5,059 ̲ 18,621     5,059 ̲

### Exercise 58

92   48 ̲ 92   48 ̲

100

### Exercise 59

16   37 ̲ 16   37 ̲

### Exercise 60

21   16 ̲ 21   16 ̲

0

### Exercise 61

11,172 22,749   12,248 ̲ 11,172 22,749   12,248 ̲

### Exercise 62

240 280 210   310 ̲ 240 280 210   310 ̲

1,000

### Exercise 63

9,573 101,279   122,581 ̲ 9,573 101,279   122,581 ̲

For the next five problems, replace the letter mm with the whole number that will make the addition true.

### Exercise 64

62 +     m ̲ 67 62 +     m ̲ 67

5

### Exercise 65

106 +     m ̲ 113 106 +     m ̲ 113

### Exercise 66

432 +     m ̲ 451 432 +     m ̲ 451

19

### Exercise 67

803 +     m ̲ 830 803 +     m ̲ 830

### Exercise 68

1,893 +       m ̲ 1,981 1,893 +       m ̲ 1,981

88

### Exercise 69

The number of nursing and related care facilities in the United States in 1971 was 22,004. In 1978, the number was 18,722. What was the total num­ber of facilities for both 1971 and 1978?

### Exercise 70

The number of persons on food stamps in 1975, 1979, and 1980 was 19,179,000, 19,309,000, and 22,023,000, respectively. What was the total number of people on food stamps for the years 1975, 1979, and 1980?

60,511,000

### Exercise 71

The enrollment in public and nonpublic schools in the years 1965, 1970, 1975, and 1984 was 54,394,000, 59,899,000, 61,063,000, and 55,122,000, respectively. What was the total en­rollment for those years?

### Exercise 72

The area of New England is 3,618,770 square miles. The area of the Mountain states is 863,563 square miles. The area of the South Atlantic is 278,926 square miles. The area of the Pacific states is 921,392 square miles. What is the total area of these regions?

#### Solution

5,682,651 square miles

### Exercise 73

In 1960, the IRS received 1,188,000 corporate income tax returns. In 1965, 1,490,000 returns were received. In 1970, 1,747,000 returns were received. In 1972 —1977, 1,890,000; 1,981,000; 2,043,000; 2,100,000; 2,159,000; and 2,329,000 re­turns were received, respectively. What was the total number of corporate tax returns received by the IRS during the years 1960, 1965, 1970, 1972 —1977?

### Exercise 74

Find the total number of scientists employed in 1974.

1,190,000

### Exercise 75

Find the total number of sales for space vehicle systems for the years 1965-1980.

### Exercise 76

Find the total baseball attendance for the years 1960-1980.

271,564,000

### Exercise 77

Find the number of prosecutions of federal officials for 1970-1980.

For the following problems, try to add the numbers mentally.

### Exercise 78

5 5 3   7 ̲ 5 5 3   7 ̲ alignl { stack { size 12{5} {} # size 12{5} {} # size 12{3} {} # size 12{ {underline {7}} } {} } } {}

20

### Exercise 79

8 2 6    4 ̲ 8 2 6    4 ̲ alignl { stack { size 12{8} {} # size 12{2} {} # size 12{6} {} # size 12{ {underline {4}} } {} } } {}

### Exercise 80

9 1 8 5   2 ̲ 9 1 8 5   2 ̲ alignl { stack { size 12{9} {} # size 12{1} {} # size 12{8} {} # size 12{5} {} # size 12{ {underline {2}} } {} } } {}

25

### Exercise 81

5 2 5 8 3   7 ̲ 5 2 5 8 3   7 ̲ alignl { stack { size 12{5} {} # size 12{2} {} # size 12{5} {} # size 12{8} {} # size 12{3} {} # size 12{ {underline {7}} } {} } } {}

### Exercise 82

6 4 3 1 6 7 9   4 ̲ 6 4 3 1 6 7 9   4 ̲ alignl { stack { size 12{6} {} # size 12{4} {} # size 12{3} {} # size 12{1} {} # size 12{6} {} # size 12{7} {} # size 12{9} {} # size 12{ {underline {4}} } {} } } {}

40

### Exercise 83

20   30 ̲ 20   30 ̲ alignl { stack { size 12{"20"} {} # size 12{ {underline {"30"}} } {} } } {}

### Exercise 84

15   35 ̲ 15   35 ̲ alignl { stack { size 12{"15"} {} # size 12{ {underline {"35"}} } {} } } {}

50

### Exercise 85

16   14 ̲ 16   14 ̲ alignl { stack { size 12{"16"} {} # size 12{ {underline {"14"}} } {} } } {}

### Exercise 86

23   27 ̲ 23   27 ̲ alignl { stack { size 12{"23"} {} # size 12{ {underline {"27"}} } {} } } {}

50

### Exercise 87

82   18 ̲ 82   18 ̲ alignl { stack { size 12{"82"} {} # size 12{ {underline {"18"}} } {} } } {}

### Exercise 88

36   14 ̲ 36   14 ̲ alignl { stack { size 12{"36"} {} # size 12{ {underline {"14"}} } {} } } {}

50

### Exercises for Review

#### Exercise 89

((Reference)) Each period of numbers has its own name. From right to left, what is the name of the fourth period?

#### Exercise 90

((Reference)) In the number 610,467, how many thousands are there?

0

#### Exercise 91

((Reference)) Write 8,840 as you would read it.

#### Exercise 92

((Reference)) Round 6,842 to the nearest hundred.

6,800

#### Exercise 93

((Reference)) Round 431,046 to the nearest million.

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