1.4: Addition of Whole Numbers

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Learning Objectives

understand the addition process
be able to add whole numbers
be able to use the calculator to add one whole number to another

Addition

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

$A group of four dots is combine with a group of three dots to yield seven dots.$

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

Definition: Addition

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 = 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 + 3

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

Thus, 4 + 3 = 7

$A number line from 0 to 11. An arrow is drawn from 0 to 4, and is labeled 4. An arrow is drawn from 4 to 7 to a dot on the 7, and is labeled 3. There is a plus sign in between the two arrows.$

The Addition Process

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

$屏幕快照 2020-09-21 下午4.05.48.png$

$Vertical math. 2 tens + 5 ones over 4 tens + 3 ones = 6 tens + 8 ones$

We write this as 68.

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

The Process of Adding Whole Numbers

To add whole numbers,

The process:

Write the numbers vertically, placing corresponding positions in the same column.
$\begin{array} {r} {25} \\ {\underline{+43}} \end{array}$
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.
$\begin{array} {r} {25} \\ {\underline{+43}} \\ {68} \end{array}$

Caution

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

$\begin{array} {l} {25} \\ {\underline{+43}} \end{array}$

$\begin{array} {r} {25} \\ {\underline{+43\ \ }} \end{array}$

Sample Set A

Add 276 and 103.

Solution

$\begin{array} {r} {276} \\ {\underline{+103}} \\ {379} \end{array}$ $\begin{array} {r} {6 + 3 = 9.} \\ {7 + 0 = 7.} \\ {2 + 1 = 3.} \end{array}$

Sample Set A

Add 1459 and 130

Solution

$\begin{array} {r} {1459} \\ {\underline{+130}} \\ {1589} \end{array}$ $\begin{array} {r} {9 + 0 = 9.} \\ {5 + 3 = 8.} \\ {4 + 1 = 5.} \\ {1 + 0 = 1.} \end{array}$

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.

Add 63 and 25.

Answer

$Vertical addition. 6 tens + 3 ones, over 2 tens + 5 ones = 8 tens + 8 ones.$

Practice Set A

Add 4,026 and 1,501.

Answer

5,527

$Vertical addition. 4 thousands + 0 hundreds + 2 tens + 6 ones, over 1 thousand + 5 hundreds + 0 tens + 1 one = 5 thousands + 5 hundreds + 2 tens + 7 ones.$

Practice Set A

Add 231,045 and 36,121.

Answer: 267,166

Addition Involving Carrying

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.

$18 + 34 is separated into 1 ten + 8 ones over 3 tens + 4 ones. The sum of the ones column exceeds nine. The sum is 4 tens + 12 ones, which is separated into 4 tens + 1 ten + 2 ones. This is simplified to 5 tens + 2 ones, which is simplified to 52.$

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:

$18 + 34 = 52. Above the tens column is a carried one.$

8 + 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.

Add 1875 and 358.

Solution

$1875 + 358 = 2233. Above the tens, hundreds, and thousands columns are carried ones.$

$\begin{array} {lcl} {5 + 8 = 13} & \ \ & {\text{Write 3, carry 1 ten.}} \\ {1 + 7 + 5 = 13} & \ \ & {\text{Write 3, carry 1 hundred.}} \\ {1 + 8 + 3 = 12} & \ \ & {\text{Write 2, carry 1 thousand.}} \\ {1 + 1 = 2} & \ \ & {} \end{array}$

The sum is 2233.

Sample Set B

Add 89,208 and 4,946.

Solution

$89,208 + 4,946 = 94,154. Above the tens, thousands, and ten-thousands columns are carried ones.$

$\begin{array} {lcl} {8 + 6 = 14} & \ \ & {\text{Write 4, carry 1 ten.}} \\ {1 + 0 + 4 = 5} & \ \ & {\text{Write the 5 (nothing to carry).}} \\ {2 + 9 = 11} & \ \ & {\text{Write 1, carry one thousand.}} \\ {1 + 9 + 4 = 14} & \ \ & {\text{Write 4, carry one ten thousand}.} \\ {1 + 8 = 9} & \ \ & {} \end{array}$

The sum is 94,154.

Sample Set B

Add 38 and 95.

Solution

$38 + 95 = 133. Above the tens and the hundreds columns are carried ones.$

$\begin{array} {lcl} {8 + 5 = 13} & \ \ & {\text{Write 3, carry 1 ten.}} \\ {1 + 3 + 9 = 13} & \ \ & {\text{Write 3, carry 1 hundred.}} \\ {1 + 0 = 1} & \ \ & {} \end{array}$

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

$38 + 95, which is separated into 3 tens + 8 ones over 9 tens + 5 ones. The sum is 12 tens + 13 ones, which is equal to 12 tens + 1 ten + 3 ones, which simplifies to 13 tens + 3 ones, which is equal to 1 hundred + 3 tens + 3 ones, which equals 133.$

The sum is 133.

Sample Set B

Find the sum 2648, 1359, and 861.

Solution

$2648 + 1359 + 861 = 4868. Above the tens, hundreds, and thousands columns are carried ones.$

$\begin{array} {lcl} {8 + 9 + 1 = 18} & \ \ & {\text{Write 8, carry 1 ten.}} \\ {1 + 4 + 5 + 6 = 16} & \ \ & {\text{Write 6, carry 1 hundred.}} \\ {1 + 6 + 3 + 8 = 18} & \ \ & {\text{Write 8, carry 1 thousand.}} \\ {1 + 2 + 1 = 4} & \ \ & {} \end{array}$

The sum is 4,868.

Numbers other than 1 can be carried as illustrated in next example.

Sample Set B

Find the sum of the following numbers.

Solution

$878016 + 9905 + 38951 + 56817 = 983689. Above the tens column is a carried one. Above the thousands column is a carried 2. Above the ten-thousands column is a carried 3. Above the hundred-thousands column is a carried 1.$

$\begin{array} {lcl} {6 + 5 + 1 + 7 = 19} & \ \ & {\text{Write 9, carry the 1.}} \\ {1 + 1 + 0 + 5 + 1 = 8} & \ \ & {\text{Write 8.}} \\ {0 + 9 + 9 + 8 = 26} & \ \ & {\text{Write 6, carry the 2.}} \\ {2 + 8 + 9 + 8 + 6 = 33} & \ \ & {\text{Write 3, carry the 3}.} \\ {3 + 7 + 3 + 5 = 18} & \ \ & {\text{Write 8, carry the 1.}} \\ {1 + 8 = 9} & \ \ & {\text{Write 9.}} \end{array}$

The sum is 983,689.

Sample Set B

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?

Solution

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.

$9,289 + 10,108 + 11,412 = 30,809. Above the tens, hundreds, and ten-thousands columns are carried ones.$

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.

Add 58 and 29.

Answer: 87; $Vertical addition. 5 tens + 8 ones, over 2 tens + 9 ones = 7 tens + 17 ones.$; $\begin{array} {l} {\text{= 7 tens + 1 ten + 7 ones}} \\ {\text{= 8 tens + 7 ones}} \\ {\text{= 87}} \end{array}$

Practice Set B

Add 476 and 85.

Answer

561

$4 hundreds + 7 tens + 6 ones over 8 tens + 5 ones = 4 hundreds + 15 tens + 11 ones.$

$\begin{array} {r} {\text{= 4 hundreds + 15 tens + 1 ten + 1 one}} \\ {\text{= 4 hundreds + 16 tens + 1 one}} \\ {\text{= 4 hundreds + 1 hundred + 6 tens + 1 one}} \\ {\text{= 5 hundreds + 6 tens + 1 one}} \\ {\text{= 561}} \end{array}$

Practice Set B

Add 27 and 88.

Answer

115

$Vertical addition. 2 tens + 7 ones, over 8 tens + 8 ones = 10 tens + 15 ones.$

$\begin{array} {l} {\text{= 10 tens + 1 ten + 5 ones}} \\ {\text{= 11 tens + 5 ones}} \\ {\text{= 11 hundred + 1 ten + 5 ones}} \\ {\text{= 115}} \end{array}$

Practice Set B

Add 67,898 and 85,627.

Answer: 153,525

For the next three problems, find the sums.

Practice Set B

$\begin{array} {r} {57} \\ {26} \\ {\underline{\ \ 84}} \end{array}$

Answer: 167

Practice Set B

$\begin{array} {r} {847} \\ {825} \\ {\underline{\ \ 796}} \end{array}$

Answer: 2,468

Practice Set B

$\begin{array} {r} {16,945} \\ {8,472} \\ {387,721} \\ {21,059} \\ {\underline{\ \ \ \ \ \ \ \ 629}} \end{array}$

Answer: 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.

34 + 21		Display Reads
Type	34	34
Press	+	34
Type	21	21
Press	=	55

Solution

The sum is 55.

Sample Set C

106 + 85 + 322 + 406		Display Reads
Type	106	106	The calculator keeps a running subtotal
Press	+	106
Type	85	85
Press	=	191	$\leftarrow$ 106 + 85
Type	322	322
Press	+	513	$\leftarrow$ 191 + 322
Type	406	406
Press	=	919	$\leftarrow$ 513 + 406

Answer: The sum is 919.

Practice Set C

Use a calculator to find the following sums.

62 + 81 + 12

Answer: 155

Practice Set C

9,261 + 8,543 + 884 + 1,062

Answer: 19,750

Practice Set C

10,221 + 9,016 + 11,445

Answer: 30,682

Exercises

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

Exercise $\PageIndex{1}$

14 + 5

Answer: 19

Exercise $\PageIndex{2}$

12 + 7

Exercise $\PageIndex{3}$

46 + 2

Answer: 48

Exercise $\PageIndex{4}$

83 + 16

Exercise $\PageIndex{5}$

77 + 21

Answer: 98

Exercise $\PageIndex{6}$

$\begin{array} {r} {321} \\ {\underline{+\ \ 84}} \end{array}$

Exercise $\PageIndex{7}$

$\begin{array} {r} {916} \\ {\underline{+\ \ 62}} \end{array}$

Answer: 978

Exercise $\PageIndex{8}$

$\begin{array} {r} {104} \\ {\underline{+561}} \end{array}$

Exercise $\PageIndex{9}$

$\begin{array} {r} {265} \\ {\underline{+103}} \end{array}$

Answer: 368

Exercise $\PageIndex{10}$

552 + 237

Exercise $\PageIndex{11}$

8,521 + 4,256

Answer: 12,777

Exercise $\PageIndex{12}$

$\begin{array} {r} {16,408} \\ {\underline{+\ \ 3,101}} \end{array}$

Exercise $\PageIndex{13}$

$\begin{array} {r} {16,515} \\ {\underline{+42,223}} \end{array}$

Answer: 58,738

Exercise $\PageIndex{14}$

616,702 + 101,161

Exercise $\PageIndex{15}$

43,156,219 + 2,013,520

Answer: 45,169,739

Exercise $\PageIndex{16}$

17 + 6

Exercise $\PageIndex{17}$

25 + 8

Answer: 33

Exercise $\PageIndex{18}$

$\begin{array} {r} {84} \\ {\underline{+\ \ 7}} \end{array}$

Exercise $\PageIndex{19}$

$\begin{array} {r} {75} \\ {\underline{+\ \ 6}} \end{array}$

Answer: 81

Exercise $\PageIndex{20}$

36 + 48

Exercise $\PageIndex{21}$

74 + 17

Answer: 91

Exercise $\PageIndex{22}$

486 + 58

Exercise $\PageIndex{23}$

743 + 66

Answer: 809

Exercise $\PageIndex{24}$

381 + 88

Exercise $\PageIndex{25}$

$\begin{array} {r} {687} \\ {\underline{+175}} \end{array}$

Answer: 862

Exercise $\PageIndex{26}$

$\begin{array} {r} {931} \\ {\underline{+853}} \end{array}$

Exercise $\PageIndex{27}$

1,428 + 893

Answer: 2,321

Exercise $\PageIndex{28}$

12,898 + 11,925

Exercise $\PageIndex{29}$

$\begin{array} {r} {631,464} \\ {\underline{+509,740}} \end{array}$

Answer: 1,141,204

Exercise $\PageIndex{30}$

$\begin{array} {r} {805,996} \\ {\underline{+\ \ 98,516}} \end{array}$

Exercise $\PageIndex{31}$

$\begin{array} {r} {38,428,106} \\ {\underline{+522,936,005}} \end{array}$

Answer: 561,364,111

Exercise $\PageIndex{32}$

5,288,423,100 + 16,934,785,995

Exercise $\PageIndex{33}$

98,876,678,521,402 + 843,425,685,685,658

Answer: 942,302,364,207,060

Exercise $\PageIndex{34}$

41 + 61 + 85 + 62

Exercise $\PageIndex{35}$

21 + 85 + 104 + 9 + 15

Answer: 234

Exercise $\PageIndex{36}$

$\begin{array} {r} {116} \\ {27} \\ {110} \\ {110} \\ {\underline{+\ \ \ \ 8}} \end{array}$

Exercise $\PageIndex{37}$

$\begin{array} {r} {75,206} \\ {4,152} \\ {\underline{+16,007}} \end{array}$

Answer: 95,365

Exercise $\PageIndex{38}$

$\begin{array} {r} {8,226} \\ {143} \\ {92,015} \\ {8} \\ {487,553} \\ {\underline{+\ \ \ \ 5,218}} \end{array}$

Exercise $\PageIndex{39}$

$\begin{array} {r} {50,006} \\ {1,005} \\ {100,300} \\ {20,008} \\ {1,000,009} \\ {\underline{+\ \ \ 800,800}} \end{array}$

Answer: 1,972,128

Exercise $\PageIndex{40}$

$\begin{array} {r} {616} \\ {42,018} \\ {1,687} \\ {225} \\ {8,623,418} \\ {12,506,508} \\ {19} \\ {2,121} \\ {\underline{\ \ \ \ \ \ 195,643}} \end{array}$

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

Exercise $\PageIndex{41}$

$\begin{array} {r} {1,468} \\ {\underline{2,183}} \end{array}$

Answer: 3,700

Exercise $\PageIndex{42}$

$\begin{array} {r} {928,725} \\ {\underline{\ \ \ \ 15,685}} \end{array}$

Exercise $\PageIndex{43}$

$\begin{array} {r} {82,006} \\ {\underline{\ \ 3,019,528}} \end{array}$

Answer: 3,101,500

Exercise $\PageIndex{44}$

$\begin{array} {r} {18,621} \\ {\underline{\ \ \ \ 5,059}} \end{array}$

Exercise $\PageIndex{45}$

$\begin{array} {r} {92} \\ {\underline{\ \ 48}} \end{array}$

Answer: 100

Exercise $\PageIndex{46}$

$\begin{array} {r} {16} \\ {\underline{\ \ 37}} \end{array}$

Exercise $\PageIndex{47}$

$\begin{array} {r} {21} \\ {\underline{\ \ 16}} \end{array}$

Answer: 0

Exercise $\PageIndex{48}$

$\begin{array} {r} {11,171} \\ {22,749} \\ {\underline{\ \ 12,248}} \end{array}$

Exercise $\PageIndex{49}$

$\begin{array} {r} {240} \\ {280} \\ {210} \\ {\underline{\ \ 310}} \end{array}$

Answer: 1000

Exercise $\PageIndex{50}$

$\begin{array} {r} {9,573} \\ {101,279} \\ {\underline{\ \ 122,581}} \end{array}$

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

Exercise $\PageIndex{51}$

$\begin{array} {r} {62} \\ {\underline{+\ \ \ \ m}} \\ {67} \end{array}$

Answer: 5

Exercise $\PageIndex{52}$

$\begin{array} {r} {106} \\ {\underline{+\ \ \ \ m}} \\ {113} \end{array}$

Exercise $\PageIndex{53}$

$\begin{array} {r} {432} \\ {\underline{+\ \ \ \ m}} \\ {451} \end{array}$

Answer: 19

Exercise $\PageIndex{54}$

$\begin{array} {r} {803} \\ {\underline{+\ \ \ \ m}} \\ {830} \end{array}$

Exercise $\PageIndex{55}$

$\begin{array} {r} {1,893} \\ {\underline{+\ \ \ \ \ \ m}} \\ {1,981} \end{array}$

Answer: 88

Exercise $\PageIndex{56}$

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 $\PageIndex{57}$

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?

Answer: 60,511,000

Exercise $\PageIndex{58}$

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 $\PageIndex{59}$

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?

Answer: 5,682,651 square miles

Exercise $\PageIndex{60}$

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 $\PageIndex{61}$

Find the total number of scientists employed in 1974.

$A graph entitled employment status of mathematical scientists - 1974. On the graph are histograms with scientific field titles, and a labeled number of the scientists holding the titles. There are 266,000 life scientists, 248,000 physical scientists, 170,000 computer scientists, 217,000 social scientists, 109,000 psychologists, 101,000, mathematicians, and 79,000 environmental scientists.$

Answer: 1,190,000

Exercise $\PageIndex{62}$

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

$A graph entitled sales for space vehicle systems, 1965-1980. The histograms for each year are plotted along the horizontal axis, and net sales along the vertical axis. In ascending succession, the sales were 2,449,000,000, 1,956,000,000, 1,725,000,000, 1,656,000,000, 1,562,000,000, 1,751,000,000, 2,119,000,000, 2,002,000,000, 1,870,000,000, 2,324,000,000, 2,539,000,000, and 3,254,000,000.$

Exercise $\PageIndex{63}$

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

$A graph entitled baseball attendance 1960-1980, with the histograms for each year plotted on the horizontal axis, and the attendance on the vertical axis. In ascending succession, the years had the following attendances, 20,261, 22,806, 29,191, 30,373, 41,402, 44,262, and 43,746.$

Answer: 271,564,000

Exercise $\PageIndex{64}$

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

$A graph entitled prosecutions of federal officials 1970-1980, with histograms of the years on the horizontal axis, and number of prosecutions on the vertical axis. The years in ascending succession had the following number of prosecutions, 9, 58, 58, 60, 59, 53, 111, 129, 133, 114, 123.$

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

Exercise $\PageIndex{65}$

$\begin{array} {r} {5} \\ {5} \\ {3} \\ {\underline{\ \ 7}} \end{array}$

Answer: 20

Exercise $\PageIndex{66}$

$\begin{array} {r} {8} \\ {2} \\ {6} \\ {\underline{\ \ 4}} \end{array}$

Exercise $\PageIndex{67}$

$\begin{array} {r} {9} \\ {1} \\ {8} \\ {5} \\ {\underline{\ \ 2}} \end{array}$

Answer: 25

Exercise $\PageIndex{68}$

$\begin{array} {r} {5} \\ {2} \\ {5} \\ {8} \\ {3} \\ {\underline{\ \ 7}} \end{array}$

Exercise $\PageIndex{69}$

$\begin{array} {r} {6} \\ {4} \\ {3} \\ {1} \\ {6} \\ {7} \\ {9} \\ {\underline{\ \ 4}} \end{array}$

Answer: 40

Exercise $\PageIndex{70}$

$\begin{array} {r} {20} \\ {\underline{\ \ 30}} \end{array}$

Exercise $\PageIndex{71}$

$\begin{array} {r} {15} \\ {\underline{\ \ 35}} \end{array}$

Answer: 50

Exercise $\PageIndex{72}$

$\begin{array} {r} {16} \\ {\underline{\ \ 14}} \end{array}$

Exercise $\PageIndex{73}$

$\begin{array} {r} {23} \\ {\underline{\ \ 27}} \end{array}$

Answer: 50

Exercise $\PageIndex{74}$

$\begin{array} {r} {82} \\ {\underline{\ \ 18}} \end{array}$

Exercise $\PageIndex{75}$

$\begin{array} {r} {36} \\ {\underline{\ \ 14}} \end{array}$

Answer: 50

Exercises for Review (link)

Exercise $\PageIndex{76}$

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

Exercise $\PageIndex{77}$

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

Answer: 0

Exercise $\PageIndex{78}$

Write 8,840 as you would read it.

Exercise $\PageIndex{79}$

Round 6,842 to the nearest hundred.

Answer: 6,800

Exercise $\PageIndex{80}$

Round 431,046 to the nearest million.