1.3: Add Whole Numbers

Learning Objectives

By the end of this section, you will be able to:

• Use addition notation
• Model addition of whole numbers
• Add whole numbers without models
• Translate word phrases to math notation
• Add whole numbers in applications

Be Prepared 1.1

Before you get started, take this readiness quiz.

What is the number modeled by the $\text{base-}$ blocks?

If you missed this problem, review [link].

Be Prepared 1.2

Write the number three hundred forty-two thousand six using digits?
If you missed this problem, review [link].

Use Addition Notation

A college student has a part-time job. Last week he worked $3$ hours on Monday and $4$ hours on Friday. To find the total number of hours he worked last week, he added $3$ and $4.$

The operation of addition combines numbers to get a sum. The notation we use to find the sum of $3$ and $4$ is:

$$3+4$$

We read this as three plus four and the result is the sum of three and four. The numbers $3$ and $4$ are called the addends. A math statement that includes numbers and operations is called an expression.

Addition Notation

To describe addition, we can use symbols and words.

Operation	Notation	Expression	Read as	Result
Addition	$+$	$3+4$	three plus four	the sum of $3$ and $4$

Example 1.11

Translate from math notation to words:

1. ⓐ $7+1$
2. ⓑ $12+14$

Answer

• ⓐ The expression consists of a plus symbol connecting the addends 7 and 1. We read this as seven plus one. The result is the sum of seven and one.
• ⓑ The expression consists of a plus symbol connecting the addends 12 and 14. We read this as twelve plus fourteen. The result is the sum of twelve and fourteen.

Try It 1.21

Translate from math notation to words:

1. ⓐ $8+4$
2. ⓑ $18+11$

Try It 1.22

Translate from math notation to words:

1. ⓐ $21+16$
2. ⓑ $100+200$

Model Addition of Whole Numbers

Addition is really just counting. We will model addition with $\text{base-10}$ blocks. Remember, a block represents $1$ and a rod represents $10.$ Let’s start by modeling the addition expression we just considered, $3+4.$

Each addend is less than $10,$ so we can use ones blocks.

We start by modeling the first number with 3 blocks.
Then we model the second number with 4 blocks.
Count the total number of blocks.

There are $7$ blocks in all. We use an equal sign $\text{(=)}$ to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. $3+4=7.$

Manipulative Mathematics

Example 1.12

Model the addition $2+6.$

Answer

$2+6$ means the sum of $2$ and $6$

Each addend is less than 10, so we can use ones blocks.

Model the first number with 2 blocks.
Model the second number with 6 blocks.
Count the total number of blocks	There are $8$ blocks in all, so $2+6=8.$

Try It 1.23

Model: $3+6.$

Try It 1.24

Model: $5+1.$

When the result is $10$ or more ones blocks, we will exchange the $10$ blocks for one rod.

Example 1.13

Model the addition $5+8.$

Answer

$5+8$ means the sum of $5$ and $8.$

Each addend is less than 10, se we can use ones blocks.
Model the first number with 5 blocks.
Model the second number with 8 blocks.
Count the result. There are more than 10 blocks so we exchange 10 ones blocks for 1 tens rod.
Now we have 1 ten and 3 ones, which is 13.	5 + 8 = 13

Notice that we can describe the models as ones blocks and tens rods, or we can simply say ones and tens. From now on, we will use the shorter version but keep in mind that they mean the same thing.

Try It 1.25

Model the addition: $5+7.$

Try It 1.26

Model the addition: $6+8.$

Next we will model adding two digit numbers.

Example 1.14

Model the addition: $17+26.$

Answer

$17+26$ means the sum of 17 and 26.

Model the 17.	1 ten and 7 ones
Model the 26.	2 tens and 6 ones
Combine.	3 tens and 13 ones
Exchange 10 ones for 1 ten.	4 tens and 3 ones $40+3=43$
We have shown that $17+26=43$

Try It 1.27

Model each addition: $15+27.$

Try It 1.28

Model each addition: $16+29.$

Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.

Imagine filling in Table 1.1 by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

Table 1.1

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.

Identity Property of Addition

The sum of any number $a$ and $0$ is the number.

$$\begin{array}{}\\ a+0=a\\ 0+a=a\end{array}$$

Example 1.15

Find each sum:

1. ⓐ $0+11$
2. ⓑ $42+0$

Answer

ⓐ The first addend is zero. The sum of any number and zero is the number.	$0+11=11$
ⓑ The second addend is zero. The sum of any number and zero is the number.	$42+0=42$

Try It 1.29

Find each sum:

1. ⓐ $0+19$
2. ⓑ $39+0$

Try It 1.30

Find each sum:

1. ⓐ $0+24$
2. ⓑ $57+0$

Look at the pairs of sums.

$2+3=5$	$3+2=5$
$4+7=11$	$7+4=11$
$8+9=17$	$9+8=17$

Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.

Commutative Property of Addition

Changing the order of the addends $a$ and $b$ does not change their sum.

$$a+b=b+a$$

Example 1.16

Add:

1. ⓐ $8+7$
2. ⓑ $7+8$

Answer

• ⓐ
  Add.	$8+7$
  	$15$

• ⓑ
  Add.	$7+8$
  	$15$

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from part ⓑ just by recognizing that the addends were the same as in part ⓐ, but in the reverse order. As a result, both sums are the same.

Try It 1.31

Add: $9+7$ and $7+9.$

Try It 1.32

Add: $8+6$ and $6+8.$

Example 1.17

Add: $28+61.$

Answer

To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.

Write the numbers so the ones and tens digits line up vertically.	$\begin{array}{c}\hfill 28\\ \\ \hfill \underset{\text{\_\_\_\_}}{+61}\end{array}$
Then add the digits in each place value. Add the ones: $8+1=9$ Add the tens: $2+6=8$	$\begin{array}{c}\hfill 28\\ \\ \hfill \underset{\text{\_\_\_\_}}{+61}\\ \hfill 89\end{array}$

Try It 1.33

Add: $32+54.$

Try It 1.34

Add: $25+74.$

In the previous example, the sum of the ones and the sum of the tens were both less than $10.$

Figure 1.10

When we add the ones, $7+6,$ we get $13$ ones. Because we have more than $10$ ones, we can exchange $10$ of the ones for $1$ ten. Now we have $4$ tens and $3$ ones. Without using the model, we show this as a small red $1$ above the digits in the tens place.

When the sum in a place value column is greater than $9,$ we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, $10$ ones for $1$ ten or $10$ tens for $1$ hundred.

How To

Add whole numbers.

1. Step 1. Write the numbers so each place value lines up vertically.
2. Step 2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than $9,$ carry to the next place value.
3. Step 3. Continue adding each place value from right to left, adding each place value and carrying if needed.

Example 1.18

Add: $43+69.$

Answer

Write the numbers so the digits line up vertically.	$\begin{array}{c}\hfill 43\\ \\ \hfill \underset{\text{\_\_\_\_}}{+69}\end{array}$
Add the digits in each place. Add the ones: $3+9=12$
Write the $2$ in the ones place in the sum. Add the $1$ ten to the tens place.	$\begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{\_\_\_\_}}{+69}\\ \hfill 2\end{array}$
Now add the tens: $1+4+6=11$ Write the 11 in the sum.	$\begin{array}{c}\hfill \stackrel{1}{4}3\\ \hfill \underset{\text{\_\_\_\_}}{+69}\\ \hfill 112\end{array}$

Try It 1.35

Add: $35+98.$

Try It 1.36

Add: $72+89.$

Example 1.19

Add: $324+586.$

Answer

Write the numbers so the digits line up vertically.
Add the digits in each place value. Add the ones: $4+6=10$ Write the $0$ in the ones place in the sum and carry the $1$ ten to the tens place.
Add the tens: $1+2+8=11$ Write the $1$ in the tens place in the sum and carry the $1$ hundred to the hundreds
Add the hundreds: $1+3+5=9$ Write the $9$ in the hundreds place.

Try It 1.37

Add: $456+376.$

Try It 1.38

Add: $269+578.$

Example 1.20

Add: $\mathrm{1,683}+479.$

Answer

Write the numbers so the digits line up vertically.	$\begin{array}{c}\hfill \mathrm{1,683}\\ \\ \hfill \underset{\text{\_\_\_\_\_\_}}{+479}\end{array}$
Add the digits in each place value.
Add the ones: $3+9=12.$ Write the $2$ in the ones place of the sum and carry the $1$ ten to the tens place.	$\begin{array}{c}\hfill \mathrm{1,6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{\_\_\_\_\_\_}}{+479}\\ \hfill 2\end{array}$
Add the tens: $1+7+8=16$ Write the $6$ in the tens place and carry the $1$ hundred to the hundreds place.	$\begin{array}{c}\hfill \mathrm{1,}\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{\_\_\_\_\_\_}}{+479}\\ \hfill 62\end{array}$
Add the hundreds: $1+6+4=11$ Write the $1$ in the hundreds place and carry the $1$ thousand to the thousands place.	$\begin{array}{c}\hfill \mathrm{1,}\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{\_\_\_\_\_\_}}{+479}\\ \hfill 162\end{array}$
Add the thousands $1+1=2$. Write the $2$ in the thousands place of the sum.	$\begin{array}{c}\hfill \stackrel{1}{\mathrm{1,}}\stackrel{1}{6}\stackrel{1}{8}3\\ \\ \hfill \underset{\text{\_\_\_\_\_\_}}{+479}\\ \hfill \mathrm{2,162}\end{array}$

When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.

Try It 1.39

Add: $\mathrm{4,597}+685.$

Try It 1.40

Add: $\mathrm{5,837}+695.$

Example 1.21

Add: $\mathrm{21,357}+861+\mathrm{8,596}.$

Answer

Write the numbers so the place values line up vertically.	$\begin{array}{c}\hfill \mathrm{21,357}\\ \hfill 861\\ \\ \hfill \underset{\text{\_\_\_\_\_\_\_}}{+\mathrm{8,596}}\end{array}$
Add the digits in each place value.
Add the ones: $7+1+6=14$ Write the $4$ in the ones place of the sum and carry the $1$ to the tens place.	$\begin{array}{c}\hfill \mathrm{21,3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{\_\_\_\_\_\_\_}}{+\mathrm{8,596}}\\ \hfill 4\end{array}$
Add the tens: $1+5+6+9=21$ Write the $1$ in the tens place and carry the $2$ to the hundreds place.	$\begin{array}{c}\hfill \mathrm{21,}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{\_\_\_\_\_\_\_}}{+\mathrm{8,596}}\\ \hfill 14\end{array}$
Add the hundreds: $2+3+8+5=18$ Write the $8$ in the hundreds place and carry the $1$ to the thousands place.	$\begin{array}{c}\hfill 2\stackrel{1}{\mathrm{1,}}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{\_\_\_\_\_\_\_}}{+\mathrm{8,596}}\\ \hfill 814\end{array}$
Add the thousands $1+1+8=10$. Write the $0$ in the thousands place and carry the $1$ to the ten thousands place.	$\begin{array}{c}\hfill \stackrel{1}{2}\stackrel{1}{\mathrm{1,}}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{\_\_\_\_\_\_\_}}{+\mathrm{8,596}}\\ \hfill 0814\end{array}$
Add the ten-thousands $1+2=3$. Write the $3$ in the ten thousands place in the sum.	$\begin{array}{c}\hfill \stackrel{1}{2}\stackrel{1}{\mathrm{1,}}\stackrel{2}{3}\stackrel{1}{5}7\\ \hfill 861\\ \\ \hfill \underset{\text{\_\_\_\_\_\_\_}}{+\mathrm{8,596}}\\ \hfill \mathrm{30,814}\end{array}$

This example had three addends. We can add any number of addends using the same process as long as we are careful to line up the place values correctly.

Try It 1.41

Add: $\mathrm{46,195}+397+\mathrm{6,281}.$

Try It 1.42

Add: $\mathrm{53,762}+196+\mathrm{7,458}.$

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in Table 1.2.

Operation	Words	Example	Expression
Addition	plus sum increased by more than total of added to	$1$ plus $2$ the sum of $3$ and $4$ $5$ increased by $6$ $8$ more than $7$ the total of $9$ and $5$ $6$ added to $4$	$1+2$ $3+4$ $5+6$ $7+8$ $9+5$ $4+6$

Table 1.2

Example 1.22

Translate and simplify: the sum of $19$ and $23.$

Answer

The word sum tells us to add. The words of $19$ and $23$ tell us the addends.

	The sum of $19$ and $23$
Translate.	$19+23$
Add.	$42$
	The sum of $19$ and $23$ is $42.$

Try It 1.43

Translate and simplify: the sum of $17$ and $26.$

Try It 1.44

Translate and simplify: the sum of $28$ and $14.$

Example 1.23

Translate and simplify: $28$ increased by $31.$

Answer

The words increased by tell us to add. The numbers given are the addends.

	$28$ increased by $31.$
Translate.	$28+31$
Add.	$59$
	So $28$ increased by $31$ is $59.$

Try It 1.45

Translate and simplify: $29$ increased by $76.$

Try It 1.46

Translate and simplify: $37$ increased by $69.$

Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.

Example 1.24

Hao earned grades of $87,93,68,95,\text{and}89$ on the five tests of the semester. What is the total number of points he earned on the five tests?

Answer

We are asked to find the total number of points on the tests.

Write a phrase.	the sum of points on the tests
Translate to math notation.	$87+93+68+95+89$
Then we simplify by adding.
Since there are several numbers, we will write them vertically.	$\begin{array}{}\\ \\ \\ \stackrel{3}{8}7\hfill \\ 93\hfill \\ 68\hfill \\ 95\hfill \\ \underset{\text{\_\_\_\_}}{+89}\hfill \\ 432\hfill \end{array}$
Write a sentence to answer the question.	Hao earned a total of 432 points.

Notice that we added points, so the sum is $432$ points. It is important to include the appropriate units in all answers to applications problems.

Try It 1.47

Mark is training for a bicycle race. Last week he rode $18$ miles on Monday, $15$ miles on Wednesday, $26$ miles on Friday, $49$ miles on Saturday, and $32$ miles on Sunday. What is the total number of miles he rode last week?

Try It 1.48

Lincoln Middle School has three grades. The number of students in each grade is $230,165,\text{and}325.$ What is the total number of students?

Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.

Example 1.25

Find the perimeter of the patio shown.

Answer

We are asked to find the perimeter.
Write a phrase.	the sum of the sides
Translate to math notation.	$4+6+2+3+2+9$
Simplify by adding.	$26$
Write a sentence to answer the question.
We added feet, so the sum is $26$ feet.	The perimeter of the patio is $26$ feet.

Try It 1.49

Find the perimeter of each figure. All lengths are in inches.

Try It 1.50

Find the perimeter of each figure. All lengths are in inches.

Media

ACCESS ADDITIONAL ONLINE RESOURCES

• Adding Two-Digit Numbers with $\text{base-10}$ blocks
• Adding Three-Digit Numbers with $\text{base-10}$ blocks
• Adding Whole Numbers

Section 1.2 Exercises

Practice Makes Perfect

Use Addition Notation

In the following exercises, translate the following from math expressions to words.

59.

$5+2$

60.

$6+3$

61.

$13+18$

62.

$15+16$

63.

$214+642$

64.

$438+113$

Model Addition of Whole Numbers

In the following exercises, model the addition.

65.

$2+4$

66.

$5+3$

67.

$8+4$

68.

$5+9$

69.

$14+75$

70.

$15+63$

71.

$16+25$

72.

$14+27$

Add Whole Numbers

In the following exercises, fill in the missing values in each chart.

73.

74.

75.

76.

77.

78.

In the following exercises, add.

79.

1. ⓐ $0+13$
2. ⓑ $13+0$

80.

1. ⓐ $0+\mathrm{5,280}$
2. ⓑ $\mathrm{5,280}+0$

81.

1. ⓐ $8+3$
2. ⓑ $3+8$

82.

1. ⓐ $7+5$
2. ⓑ $5+7$

83.

$45+33$

84.

$37+22$

85.

$71+28$

86.

$43+53$

87.

$26+59$

88.

$38+17$

89.

$64+78$

90.

$92+39$

91.

$168+325$

92.

$247+149$

93.

$584+277$

94.

$175+648$

95.

$832+199$

96.

$775+369$

97.

$\mathrm{6,358}+492$

98.

$\mathrm{9,184}+578$

99.

$\mathrm{3,740}+\mathrm{18,593}$

100.

$\mathrm{6,118}+\mathrm{15,990}$

101.

$\mathrm{485,012}+\mathrm{619,848}$

102.

$\mathrm{368,911}+\mathrm{857,289}$

103.

$\mathrm{24,731}+592+\mathrm{3,868}$

104.

$\mathrm{28,925}+817+\mathrm{4,593}$

105.

$\mathrm{8,015}+\mathrm{76,946}+\mathrm{16,570}$

106.

$\mathrm{6,291}+\mathrm{54,107}+\mathrm{28,635}$

Translate Word Phrases to Math Notation

In the following exercises, translate each phrase into math notation and then simplify.

107.

the sum of $13$ and $18$

108.

the sum of $12$ and $19$

109.

the sum of $90$ and $65$

110.

the sum of $70$ and $38$

111.

$33$ increased by $49$

112.

$68$ increased by $25$

113.

$250$ more than $599$

114.

$115$ more than $286$

115.

the total of $628$ and $77$

116.

the total of $593$ and $79$

117.

$\mathrm{1,482}$ added to $915$

118.

$\mathrm{2,719}$ added to $682$

Add Whole Numbers in Applications

In the following exercises, solve the problem.

119.

Home remodeling Sophia remodeled her kitchen and bought a new range, microwave, and dishwasher. The range cost $\text{\$1,100},$ the microwave cost $\text{\$250},$ and the dishwasher cost $\text{\$525}.$ What was the total cost of these three appliances?

120.

Sports equipment Aiden bought a baseball bat, helmet, and glove. The bat cost $\text{\$299},$ the helmet cost $\text{\$35},$ and the glove cost $\text{\$68}.$ What was the total cost of Aiden’s sports equipment?

121.

Bike riding Ethan rode his bike $14$ miles on Monday, $19$ miles on Tuesday, $12$ miles on Wednesday, $25$ miles on Friday, and $68$ miles on Saturday. What was the total number of miles Ethan rode?

122.

Business Chloe has a flower shop. Last week she made $19$ floral arrangements on Monday, $12$ on Tuesday, $23$ on Wednesday, $29$ on Thursday, and $44$ on Friday. What was the total number of floral arrangements Chloe made?

123.

Apartment size Jackson lives in a $7$ room apartment. The number of square feet in each room is $238,120,156,196,100,132,$ and $225.$ What is the total number of square feet in all $7$ rooms?

124.

Weight Seven men rented a fishing boat. The weights of the men were $175,192,148,169,205,181,$ and $225$ pounds. What was the total weight of the seven men?

125.

Salary Last year Natalie’s salary was $\text{\$82,572}.$ Two years ago, her salary was $\text{\$79,316},$ and three years ago it was $\text{\$75,298}.$ What is the total amount of Natalie’s salary for the past three years?

126.

Home sales Emma is a realtor. Last month, she sold three houses. The selling prices of the houses were $\text{\$292,540},\text{\$505,875},$ and $\mathrm{\$423,699}.$ What was the total of the three selling prices?

In the following exercises, find the perimeter of each figure.

127.

128.

129.

130.

131.

132.

133.

134.

Everyday Math

135.

Calories Paulette had a grilled chicken salad, ranch dressing, and a $\text{16-ounce}$ drink for lunch. On the restaurant’s nutrition chart, she saw that each item had the following number of calories:

Grilled chicken salad – $320$ calories
Ranch dressing – $170$ calories
$\text{16-ounce}$ drink – $150$ calories

What was the total number of calories of Paulette’s lunch?

136.

Calories Fred had a grilled chicken sandwich, a small order of fries, and a $\text{12-oz}$ chocolate shake for dinner. The restaurant’s nutrition chart lists the following calories for each item:

Grilled chicken sandwich – $420$ calories
Small fries – $230$ calories
$\text{12-oz}$ chocolate shake – $580$ calories

What was the total number of calories of Fred’s dinner?

137.

Test scores A student needs a total of $400$ points on five tests to pass a course. The student scored $82,91,75,88,\text{and}70.$ Did the student pass the course?

138.

Elevators The maximum weight capacity of an elevator is $1150$ pounds. Six men are in the elevator. Their weights are $210,145,183,230,159,\text{and}164$ pounds. Is the total weight below the elevator’s maximum capacity?

Writing Exercises

139.

How confident do you feel about your knowledge of the addition facts? If you are not fully confident, what will you do to improve your skills?

140.

How have you used models to help you learn the addition facts?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?