Prealgebra 2e

Learning Objectives

By the end of this section, you will be able to:
• 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 $base-10base-10$ blocks?

If you missed this problem, review Example 1.2.

Be Prepared 1.2

Write the number three hundred forty-two thousand six using digits?
If you missed this problem, review Example 1.7.

A college student has a part-time job. Last week he worked $33$ hours on Monday and $44$ hours on Friday. To find the total number of hours he worked last week, he added $33$ and $4.4.$

The operation of addition combines numbers to get a sum. The notation we use to find the sum of $33$ and $44$ is:

$3+43+4$

We read this as three plus four and the result is the sum of three and four. The numbers $33$ and $44$ are called the addends. A math statement that includes numbers and operations is called an expression.

To describe addition, we can use symbols and words.

Operation Notation Expression Read as Result
Addition $++$ $3+43+4$ three plus four the sum of $33$ and $44$

Example 1.11

Translate from math notation to words:

1. $7+17+1$
2. $12+1412+14$
Try It 1.21

Translate from math notation to words:

1. $8+48+4$
2. $18+1118+11$
Try It 1.22

Translate from math notation to words:

1. $21+1621+16$
2. $100+200100+200$

Addition is really just counting. We will model addition with $base-10base-10$ blocks. Remember, a block represents $11$ and a rod represents $10.10.$ Let’s start by modeling the addition expression we just considered, $3+4.3+4.$

Each addend is less than $10,10,$ 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 $77$ blocks in all. We use an equal sign $(=)(=)$ to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that. $3+4=7.3+4=7.$

Example 1.12

Model the addition $2+6.2+6.$

Try It 1.23

Model: $3+6.3+6.$

Try It 1.24

Model: $5+1.5+1.$

When the result is $1010$ or more ones blocks, we will exchange the $1010$ blocks for one rod.

Example 1.13

Model the addition $5+8.5+8.$

Try It 1.25

Model the addition: $5+7.5+7.$

Try It 1.26

Model the addition: $6+8.6+8.$

Next we will model adding two digit numbers.

Example 1.14

Model the addition: $17+26.17+26.$

Try It 1.27

Model each addition: $15+27.15+27.$

Try It 1.28

Model each addition: $16+29.16+29.$

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.

The sum of any number $aa$ and $00$ is the number.

$a+0=a0+a=aa+0=a0+a=a$

Example 1.15

Find each sum:

1. $0+110+11$
2. $42+042+0$
Try It 1.29

Find each sum:

1. $0+190+19$
2. $39+039+0$
Try It 1.30

Find each sum:

1. $0+240+24$
2. $57+057+0$

Look at the pairs of sums.

 $2+3=52+3=5$ $3+2=53+2=5$ $4+7=114+7=11$ $7+4=117+4=11$ $8+9=178+9=17$ $9+8=179+8=17$

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.

Changing the order of the addends $aa$ and $bb$ does not change their sum.

$a+b=b+aa+b=b+a$

Example 1.16

1. $8+78+7$
2. $7+87+8$
Try It 1.31

Add: $9+79+7$ and $7+9.7+9.$

Try It 1.32

Add: $8+68+6$ and $6+8.6+8.$

Example 1.17

Add: $28+61.28+61.$

Try It 1.33

Add: $32+54.32+54.$

Try It 1.34

Add: $25+74.25+74.$

In the previous example, the sum of the ones and the sum of the tens were both less than $10.10.$ But what happens if the sum is $1010$ or more? Let’s use our $base-10base-10$ model to find out. Figure 1.10 shows the addition of $1717$ and $2626$ again.

Figure 1.10

When we add the ones, $7+6,7+6,$ we get $1313$ ones. Because we have more than $1010$ ones, we can exchange $1010$ of the ones for $11$ ten. Now we have $44$ tens and $33$ ones. Without using the model, we show this as a small red $11$ above the digits in the tens place.

When the sum in a place value column is greater than $9,9,$ we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, $1010$ ones for $11$ ten or $1010$ tens for $11$ hundred.

How To

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 $9,9,$ 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: $43+69.43+69.$

Try It 1.35

Add: $35+98.35+98.$

Try It 1.36

Add: $72+89.72+89.$

Example 1.19

Add: $324+586.324+586.$

Try It 1.37

Add: $456+376.456+376.$

Try It 1.38

Add: $269+578.269+578.$

Example 1.20

Add: $1,683+479.1,683+479.$

Try It 1.39

Add: $4,597+685.4,597+685.$

Try It 1.40

Add: $5,837+695.5,837+695.$

Example 1.21

Add: $21,357+861+8,596.21,357+861+8,596.$

Try It 1.41

Add: $46,195+397+6,281.46,195+397+6,281.$

Try It 1.42

Add: $53,762+196+7,458.53,762+196+7,458.$

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
sum
increased by
more than
total of
$11$ plus $22$
the sum of $33$ and $44$
$55$ increased by $66$
$88$ more than $77$
the total of $99$ and $55$
$66$ added to $44$
$1+21+2$
$3+43+4$
$5+65+6$
$7+87+8$
$9+59+5$
$4+64+6$
Table 1.2

Example 1.22

Translate and simplify: the sum of $1919$ and $23.23.$

Try It 1.43

Translate and simplify: the sum of $1717$ and $26.26.$

Try It 1.44

Translate and simplify: the sum of $2828$ and $14.14.$

Example 1.23

Translate and simplify: $2828$ increased by $31.31.$

Try It 1.45

Translate and simplify: $2929$ increased by $76.76.$

Try It 1.46

Translate and simplify: $3737$ increased by $69.69.$

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 $87,93,68,95,and8987,93,68,95,and89$ on the five tests of the semester. What is the total number of points he earned on the five tests?

Try It 1.47

Mark is training for a bicycle race. Last week he rode $1818$ miles on Monday, $1515$ miles on Wednesday, $2626$ miles on Friday, $4949$ miles on Saturday, and $3232$ 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 $230,165,and325.230,165,and325.$ 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.

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.

Section 1.2 Exercises

Practice Makes Perfect

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

59.

$5+25+2$

60.

$6+36+3$

61.

$13+1813+18$

62.

$15+1615+16$

63.

$214+642214+642$

64.

$438+113438+113$

In the following exercises, model the addition.

65.

$2+42+4$

66.

$5+35+3$

67.

$8+48+4$

68.

$5+95+9$

69.

$14+7514+75$

70.

$15+6315+63$

71.

$16+2516+25$

72.

$14+2714+27$

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

73.
74.
75.
76.
77.
78.

79.
1. $0+130+13$
2. $13+013+0$
80.
1. $0+5,2800+5,280$
2. $5,280+05,280+0$
81.
1. $8+38+3$
2. $3+83+8$
82.
1. $7+57+5$
2. $5+75+7$
83.

$45+3345+33$

84.

$37+2237+22$

85.

$71+2871+28$

86.

$43+5343+53$

87.

$26+5926+59$

88.

$38+1738+17$

89.

$64+7864+78$

90.

$92+3992+39$

91.

$168+325168+325$

92.

$247+149247+149$

93.

$584+277584+277$

94.

$175+648175+648$

95.

$832+199832+199$

96.

$775+369775+369$

97.

$6,358+4926,358+492$

98.

$9,184+5789,184+578$

99.

$3,740+18,5933,740+18,593$

100.

$6,118+15,9906,118+15,990$

101.

$485,012+619,848485,012+619,848$

102.

$368,911+857,289368,911+857,289$

103.

$24,731+592+3,86824,731+592+3,868$

104.

$28,925+817+4,59328,925+817+4,593$

105.

$8,015+76,946+16,5708,015+76,946+16,570$

106.

$6,291+54,107+28,6356,291+54,107+28,635$

Translate Word Phrases to Math Notation

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

107.

the sum of $1313$ and $1818$

108.

the sum of $1212$ and $1919$

109.

the sum of $9090$ and $6565$

110.

the sum of $7070$ and $3838$

111.

$3333$ increased by $4949$

112.

$6868$ increased by $2525$

113.

$250250$ more than $599599$

114.

$115115$ more than $286286$

115.

the total of $628628$ and $7777$

116.

the total of $593593$ and $7979$

117.

$1,4821,482$ added to $915915$

118.

$2,7192,719$ added to $682682$

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 $1,100,1,100,$ the microwave cost $250,250,$ and the dishwasher cost $525.525.$ What was the total cost of these three appliances?

120.

Sports equipment Aiden bought a baseball bat, helmet, and glove. The bat cost $299,299,$ the helmet cost $35,35,$ and the glove cost $68.68.$ What was the total cost of Aiden’s sports equipment?

121.

Bike riding Ethan rode his bike $1414$ miles on Monday, $1919$ miles on Tuesday, $1212$ miles on Wednesday, $2525$ miles on Friday, and $6868$ miles on Saturday. What was the total number of miles Ethan rode?

122.

Business Chloe has a flower shop. Last week she made $1919$ floral arrangements on Monday, $1212$ on Tuesday, $2323$ on Wednesday, $2929$ on Thursday, and $4444$ on Friday. What was the total number of floral arrangements Chloe made?

123.

Apartment size Jackson lives in a $77$ room apartment. The number of square feet in each room is $238,120,156,196,100,132,238,120,156,196,100,132,$ and $225.225.$ What is the total number of square feet in all $77$ rooms?

124.

Weight Seven men rented a fishing boat. The weights of the men were $175,192,148,169,205,181,175,192,148,169,205,181,$ and $225225$ pounds. What was the total weight of the seven men?

125.

Salary Last year Natalie’s salary was $82,572.82,572.$ Two years ago, her salary was $79,316,79,316,$ and three years ago it was $75,298.75,298.$ 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 $292,540,505,875,292,540,505,875,$ and $423,699.423,699.$ 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 $16-ounce16-ounce$ drink for lunch. On the restaurant’s nutrition chart, she saw that each item had the following number of calories:

Grilled chicken salad – $320320$ calories
Ranch dressing – $170170$ calories
$16-ounce16-ounce$ drink – $150150$ 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 $12-oz12-oz$ chocolate shake for dinner. The restaurant’s nutrition chart lists the following calories for each item:

Grilled chicken sandwich – $420420$ calories
Small fries – $230230$ calories
$12-oz12-oz$ chocolate shake – $580580$ calories

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

137.

Test scores A students needs a total of $400400$ points on five tests to pass a course. The student scored $82,91,75,88,and70.82,91,75,88,and70.$ Did the student pass the course?

138.

Elevators The maximum weight capacity of an elevator is $11501150$ pounds. Six men are in the elevator. Their weights are $210,145,183,230,159,and164210,145,183,230,159,and164$ 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.

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?