1.5 Divide Whole Numbers

Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the $12$ cookies in Figure 1.20 and want to package them in bags with $4$ cookies in each bag. How many bags would we need?

An image of three rows of four cookies to show twelve cookies. — **Figure 1.20**

You might put $4$ cookies in first bag, $4$ in the second bag, and so on until you run out of cookies. Doing it this way, you would fill $3$ bags.

An image of 3 bags of cookies, each bag containing 4 cookies.

In other words, starting with the $12$ cookies, you would take away, or subtract, $4$ cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.

Instead of subtracting $4$ repeatedly, we can write

$12 divided by 4$

We read this as twelve divided by four and the result is the quotient of $12$ and $4 .$ The quotient is $3$ because we can subtract $4$ from $12$ exactly $3$ times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is $12$ and the divisor is $4 .$

In the past you may have used the notation $4 12$ , but this division also can be written as $12 \div 4, 12 / 4, \frac{12}{4} .$ In each case the $12$ is the dividend and the $4$ is the divisor.

Note

Operation Symbols for Division

To represent and describe division, we can use symbols and words.

Operation	Notation	Expression	Read as	Result
$Division$	$\div$ $\frac{a}{b}$ $b a$ $a / b$	$12 divided by 4$ $\frac{12}{4}$ $4 12$ $12 / 4$	$Twelve divided by four$	$the quotient of 12 and 4$

Division is performed on two numbers at a time. When translating from math notation to English words, or English words to math notation, look for the words of and and to identify the numbers.

Model Division of Whole Numbers

As we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly.

Divide Whole Numbers

We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know $12 \div 4 = 3$ because $3 \cdot 4 = 12 .$ Knowing all the multiplication number facts is very important when doing division.

We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. In Example 2, we know $24 \div 8 = 3$ is correct because $3 \cdot 8 = 24 .$

Example 3

Divide. Then check by multiplying. ⓐ $42 divided by 6$ ⓑ $\frac{72}{9}$ ⓒ $7 63$

ⓐ
$42 divided by 6$
Divide 42 by 6. $7$
Check by multiplying.
$7 times 6$
$42 ✓$
ⓑ
$\frac{72}{9}$
Divide 72 by 9. $8$
Check by multiplying.
$8 times 9$
$72 ✓$
ⓒ
$7 63$
Divide 63 by 7. $9$
Check by multiplying.
$9 times 7$
$63 ✓$

What is the quotient when you divide a number by itself?

$\frac{15}{15} = 1 because 1 \cdot 15 = 15$

Dividing any number $(except 0)$ by itself produces a quotient of $1 .$ Also, any number divided by $1$ produces a quotient of the number. These two ideas are stated in the Division Properties of One.

Example 4

Divide. Then check by multiplying:

ⓐ $11 divided by 11$
ⓑ $\frac{19}{1}$
ⓒ $1 7$

ⓐ
$11 divided by 11$
A number divided by itself is 1. $1$
Check by multiplying.
$1 times 11$
$11 ✓$
ⓑ
$\frac{19}{1}$
A number divided by 1 equals itself. $19$
Check by multiplying.
$19 times 1$
$19 ✓$
ⓒ
$1 7$
A number divided by 1 equals itself. $7$
Check by multiplying.
$7 times 1$
$7 ✓$

Suppose we have $$0,$ and want to divide it among $3$ people. How much would each person get? Each person would get $$0.$ Zero divided by any number is $0 .$

Now suppose that we want to divide $$1 0$ by $0 .$ That means we would want to find a number that we multiply by $0$ to get $10 .$ This cannot happen because $0$ times any number is $0 .$ Division by zero is said to be undefined.

These two ideas make up the Division Properties of Zero.

Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away $0$ from $10 ?$ Because subtracting $0$ will never change the total, we will never get an answer. So we cannot divide a number by $0 .$

When the divisor or the dividend has more than one digit, it is usually easier to use the $4 12$ notation. This process is called long division. Let’s work through the process by dividing $78$ by $3 .$

Divide the first digit of the dividend, 7, by the divisor, 3.
The divisor 3 can go into 7 two times since $2 \times 3 = 6$ . Write the 2 above the 7 in the quotient.
Multiply the 2 in the quotient by 3 and write the product, 6, under the 7.
Subtract that product from the first digit in the dividend. Subtract $7 minus 6$ . Write the difference, 1, under the first digit in the dividend.
Bring down the next digit of the dividend. Bring down the 8.
Divide 18 by the divisor, 3. The divisor 3 goes into 18 six times.
Write 6 in the quotient above the 8.
Multiply the 6 in the quotient by the divisor and write the product, 18, under the dividend. Subtract 18 from 18.

We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.

$So 78 \div 3 = 26 .$

Check by multiplying the quotient times the divisor to get the dividend. Multiply $26 times 3$ to make sure that product equals the dividend, $78 .$

$2 1 6; times 3 ___; 78 ✓$

It does, so our answer is correct.

Example 6

Divide $2,596 divided by 4.$ Check by multiplying:

Let's rewrite the problem to set it up for long division.
Divide the first digit of the dividend, 2, by the divisor, 4.
Since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. The divisor 4 goes into 25 six times.
We write the 6 in the quotient above the 5.
Multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend.
Subtract that product from the first two digits in the dividend. Subtract $25 minus 24$ . Write the difference, 1, under the second digit in the dividend.
Now bring down the 9 and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19.
Bring down the 6 and repeat these steps. There are 9 fours in 36. Write the 9 over the 6. Multiply the 9 by 4 and subtract this product from 36.
So $2,596 \div 4 = 649$ .
Check by multiplying.

It equals the dividend, so our answer is correct.

Example 7

Divide $4,506 divided by 6.$ Check by multiplying:

Let's rewrite the problem to set it up for long division.
First we try to divide 6 into 4.
Since that won't work, we try 6 into 45. There are 7 sixes in 45. We write the 7 over the 5.
Multiply the 7 by 6 and subtract this product from 45.
Now bring down the 0 and repeat these steps. There are 5 sixes in 30. Write the 5 over the 0. Multiply the 5 by 6 and subtract this product from 30.
Now bring down the 6 and repeat these steps. There is 1 six in 6. Write the 1 over the 6. Multiply 1 by 6 and subtract this product from 6.
Check by multiplying.

It equals the dividend, so our answer is correct.

Example 8

Divide $7,263 divided by 9.$ Check by multiplying.

Let's rewrite the problem to set it up for long division.
First we try to divide 9 into 7.
Since that won't work, we try 9 into 72. There are 8 nines in 72. We write the 8 over the 2.
Multiply the 8 by 9 and subtract this product from 72.
Now bring down the 6 and repeat these steps. There are 0 nines in 6. Write the 0 over the 6. Multiply the 0 by 9 and subtract this product from 6.
Now bring down the 3 and repeat these steps. There are 7 nines in 63. Write the 7 over the 3. Multiply the 7 by 9 and subtract this product from 63.
Check by multiplying.

It equals the dividend, so our answer is correct.

So far all the division problems have worked out evenly. For example, if we had $24$ cookies and wanted to make bags of $8$ cookies, we would have $3$ bags. But what if there were $28$ cookies and we wanted to make bags of $8 ?$ Start with the $28$ cookies as shown in Figure 1.21.

An image of 28 cookies placed at random. — **Figure 1.21**

Try to put the cookies in groups of eight as in Figure 1.22.

An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles. — **Figure 1.22**

There are $3$ groups of eight cookies, and $4$ cookies left over. We call the $4$ cookies that are left over the remainder and show it by writing R4 next to the $3 .$ (The R stands for remainder.)

To check this division we multiply $3$ times $8$ to get $24,$ and then add the remainder of $4 .$

$\begin{matrix} 3 \\ \underset{___}{\times 8} \\ 24 \\ \underset{___}{+ 4} \\ 28 \end{matrix}$

Example 9

Divide $1,439 divided by 4.$ Check by multiplying.

Let's rewrite the problem to set it up for long division.
First we try to divide 4 into 1. Since that won't work, we try 4 into 14. There are 3 fours in 14. We write the 3 over the 4.
Multiply the 3 by 4 and subtract this product from 14.
Now bring down the 3 and repeat these steps. There are 5 fours in 23. Write the 5 over the 3. Multiply the 5 by 4 and subtract this product from 23.
Now bring down the 9 and repeat these steps. There are 9 fours in 39. Write the 9 over the 9. Multiply the 9 by 4 and subtract this product from 39. There are no more numbers to bring down, so we are done. The remainder is 3.
Check by multiplying.

So $1,439 divided by 4$ is $359$ with a remainder of $3 .$ Our answer is correct.

Example 10

Divide and then check by multiplying: $1,461 divided by 13.$

Let's rewrite the problem to set it up for long division.	$13 1,461$
First we try to divide 13 into 1. Since that won't work, we try 13 into 14. There is 1 thirteen in 14. We write the 1 over the 4.
Multiply the 1 by 13 and subtract this product from 14.
Now bring down the 6 and repeat these steps. There is 1 thirteen in 16. Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16.
Now bring down the 1 and repeat these steps. There are 2 thirteens in 31. Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. There are no more numbers to bring down, so we are done. The remainder is 5. $1,462 divided by 13$ is 112 with a remainder of 5.
Check by multiplying.

Our answer is correct.

Example 11

Divide and check by multiplying: $74,521 divided by 241.$

Let's rewrite the problem to set it up for long division.	$241 74,521$
First we try to divide 241 into 7. Since that won’t work, we try 241 into 74. That still won’t work, so we try 241 into 745. Since 2 divides into 7 three times, we try 3. Since $3 \times 241 = 723$ , we write the 3 over the 5 in 745. Note that 4 would be too large because $4 \times 241 = 964$ , which is greater than 745.
Multiply the 3 by 241 and subtract this product from 745.
Now bring down the 2 and repeat these steps. 241 does not divide into 222. We write a 0 over the 2 as a placeholder and then continue.
Now bring down the 1 and repeat these steps. Try 9. Since $9 \times 241 = 2,169$ , we write the 9 over the 1. Multiply the 9 by 241 and subtract this product from 2,221.
There are no more numbers to bring down, so we are finished. The remainder is 52. So $74,521 divided by 241$ is 309 with a remainder of 52.
Check by multiplying.

Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them.

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation for division into words. Now we’ll translate word phrases into math notation. Some of the words that indicate division are given in Table 1.8.

**Table 1.8**
Operation	Word Phrase	Example	Expression
Division	divided by quotient of divided into	$12$ divided by $4$ the quotient of $12$ and $4$ $4$ divided into $12$	$12 divided by 4$ $\frac{12}{4}$ $12 / 4$ $4 12$

Divide Whole Numbers in Applications

We will use the same strategy we used in previous sections to solve applications. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify it to get the answer. Finally, we write a sentence to answer the question.

Example 13

Cecelia bought a $160-ounce$ box of oatmeal at the big box store. She wants to divide the $160$ ounces of oatmeal into $8-ounce$ servings. She will put each serving into a plastic bag so she can take one bag to work each day. How many servings will she get from the big box?

We are asked to find the how many servings she will get from the big box.

Write a phrase.	160 ounces divided by 8 ounces
Translate to math notation.	$160 divided by 8$
Simplify by dividing.	$20$
Write a sentence to answer the question.	Cecelia will get 20 servings from the big box.

Key Concepts

Operation	Notation	Expression	Read as	Result
$Division$	$\div$ $\frac{a}{b}$ $b a$ $a / b$	$12 divided by 4$ $\frac{12}{4}$ $4 12$ $12 / 4$	$Twelve divided by four$	$the quotient of 12 and 4$

Division Properties of One
- Any number (except 0) divided by itself is one. $a \div a = 1$
- Any number divided by one is the same number. $a \div 1 = a$
Division Properties of Zero
- Zero divided by any number is 0. $0 \div a = 0$
- Dividing a number by zero is undefined. $a divided by 0$ undefined
Divide whole numbers.
1. Divide the first digit of the dividend by the divisor.
  If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
2. Write the quotient above the dividend.
3. Multiply the quotient by the divisor and write the product under the dividend.
4. Subtract that product from the dividend.
5. Bring down the next digit of the dividend.
6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
7. Check by multiplying the quotient times the divisor.

Section Exercises

Practice Makes Perfect

Use Division Notation

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

$54 divided by 9$

fifty-four divided by nine; the quotient of fifty-four and nine

$\frac{56}{7}$

$\frac{32}{8}$

thirty-two divided by eight; the quotient of thirty-two and eight

$6 42$

$48 divided by 6$

forty-eight divided by six; the quotient of forty-eight and six

$\frac{63}{9}$

$7 63$

sixty-three divided by seven; the quotient of sixty-three and seven

$72 divided by 8$

Model Division of Whole Numbers

In the following exercises, model the division.

$15 divided by 5$

Three distinct clusters of grey circles, each enclosed by a blob-like outline, illustrating concepts of grouping or classification.

$10 divided by 5$

$\frac{14}{7}$

Two irregular shapes enclose groups of grey circles, depicting distinct clusters or collections of items. This image could represent separated data points or cellular groupings.

$\frac{18}{6}$

$4 20$

An illustration showing multiple clusters of gray circles, with each group enclosed by an irregular outline, representing data grouping or segmentation.

$3 15$

$24 divided by 6$

An illustration showing clusters of gray circles, with each group enclosed by an irregular outline, representing data grouping or segmentation.

$16 divided by 4$

Divide Whole Numbers

In the following exercises, divide. Then check by multiplying.

$18 divided by 2$

$14 divided by 2$

$\frac{27}{3}$

$\frac{30}{3}$

$4 28$

$4 36$

$\frac{45}{5}$

$\frac{35}{5}$

$72 / 8$

$8 64$

$\frac{35}{7}$

$42 divided by 7$

$15 15$

$12 12$

$43 divided by 43$

$37 divided by 37$

$\frac{23}{1}$

$\frac{29}{1}$

$19 divided by 1$

$17 divided by 1$

$0 divided by 4$

$0 divided by 8$

$\frac{5}{0}$

undefined

$\frac{9}{0}$

$\frac{26}{0}$

undefined

$\frac{32}{0}$

$12 0$

$16 0$

$72 divided by 3$

$57 divided by 3$

$\frac{96}{8}$

$\frac{78}{6}$

$5 465$

$4 528$

$924 divided by 7$

132

$861 divided by 7$

$\frac{5,226}{6}$

871

$\frac{3,776}{8}$

$4 31,324$

7,831

$5 46,855$

$7,209 divided by 3$

2,403

$4,806 divided by 3$

$5,406 divided by 6$

901

$3,208 divided by 4$

$4 2,816$

704

$6 3,624$

$\frac{91,881}{9}$

10,209

$\frac{83,256}{8}$

$2,470 divided by 7$

352 R6

$3,741 divided by 7$

$8 55,305$

6,913 R1

$9 51,492$

$\frac{431,174}{5}$

86,234 R4

$\frac{297,277}{4}$

$130,016 divided by 3$

43,338 R2

$105,609 divided by 2$

$15 5,735$

382 R5

$\frac{4,933}{21}$

$56,883 divided by 67$

849

$43,725 / 75$

$\frac{30,144}{314}$

$26,145 divided by 415$

$273 542,195$

1,986 R17

$816,243 divided by 462$

Mixed Practice

In the following exercises, simplify.

$15 open parenthesis 204 close parenthesis$

3,060

$74 times 391$

$256 minus 184$

$305 minus 262$

$719 plus 341$

1,060

$647 plus 528$

$25 875$

$1104 divided by 23$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

the quotient of $45$ and $15$

45 ÷ 15; 3

the quotient of $64$ and $16$

the quotient of $288$ and $24$

288 ÷ 24; 12

the quotient of $256$ and $32$

Divide Whole Numbers in Applications

In the following exercises, solve.

Trail mix Ric bought $64$ ounces of trail mix. He wants to divide it into small bags, with $2$ ounces of trail mix in each bag. How many bags can Ric fill?

Ric can fill 32 bags.

Crackers Evie bought a $42$ ounce box of crackers. She wants to divide it into bags with $3$ ounces of crackers in each bag. How many bags can Evie fill?

Astronomy class There are $125$ students in an astronomy class. The professor assigns them into groups of $5 .$ How many groups of students are there?

There are 25 groups.

Flower shop Melissa’s flower shop got a shipment of $152$ roses. She wants to make bouquets of $8$ roses each. How many bouquets can Melissa make?

Baking One roll of plastic wrap is $48$ feet long. Marta uses $3$ feet of plastic wrap to wrap each cake she bakes. How many cakes can she wrap from one roll?

Marta can wrap 16 cakes from 1 roll.

Dental floss One package of dental floss is $54$ feet long. Brian uses $2$ feet of dental floss every day. How many days will one package of dental floss last Brian?

Mixed Practice

In the following exercises, solve.

Miles per gallon Susana’s hybrid car gets $45$ miles per gallon. Her son’s truck gets $17$ miles per gallon. What is the difference in miles per gallon between Susana’s car and her son’s truck?

The difference is 28 miles per gallon.

Distance Mayra lives $53$ miles from her mother’s house and $71$ miles from her mother-in-law’s house. How much farther is Mayra from her mother-in-law’s house than from her mother’s house?

Field trip The $45$ students in a Geology class will go on a field trip, using the college’s vans. Each van can hold $9$ students. How many vans will they need for the field trip?

They will need 5 vans for the field trip

Potting soil Aki bought a $128$ ounce bag of potting soil. How many $4$ ounce pots can he fill from the bag?

Hiking Bill hiked $8$ miles on the first day of his backpacking trip, $14$ miles the second day, $11$ miles the third day, and $17$ miles the fourth day. What is the total number of miles Bill hiked?

Bill hiked 50 miles

Reading Last night Emily read $6$ pages in her Business textbook, $26$ pages in her History text, $15$ pages in her Psychology text, and $9$ pages in her math text. What is the total number of pages Emily read?

Patients LaVonne treats $12$ patients each day in her dental office. Last week she worked $4$ days. How many patients did she treat last week?

LaVonne treated 48 patients last week.

Scouts There are $14$ boys in Dave’s scout troop. At summer camp, each boy earned $5$ merit badges. What was the total number of merit badges earned by Dave’s scout troop at summer camp?

Writing Exercises

Contact lenses Jenna puts in a new pair of contact lenses every $14$ days. How many pairs of contact lenses does she need for $365$ days?

Jenna uses 26 pairs of contact lenses, but there is 1 day left over, so she needs 27 pairs for 365 days.

Cat food One bag of cat food feeds Lara’s cat for $25$ days. How many bags of cat food does Lara need for $365$ days?

Everyday Math

Explain how you use the multiplication facts to help with division.

Answers may vary. Using multiplication facts can help you check your answers once you’ve finished division.

Oswaldo divided $300$ by $8$ and said his answer was $37$ with a remainder of $4 .$ How can you check to make sure he is correct?

Self Check

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

A self-assessment checklist for division and algebraic expressions, allowing students to rate their understanding as confident, needing some help, or not understanding. — **Figure 1.23**

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter Review Exercises

Introduction to Whole Numbers

Identify Counting Numbers and Whole Numbers

In the following exercises, determine which of the following are (a) counting numbers (b) whole numbers.

$0, 2, 99$

ⓐ 2, 99
ⓑ 0, 2, 99

$0, 3, 25$

$0, 4, 90$

ⓐ 4, 90
ⓑ 0, 4, 90

$0, 1, 75$

Model Whole Numbers

In the following exercises, model each number using $base-10$ blocks and then show its value using place value notation.

258

The image shows sets of hundred blocks, sets of blocks of 10, and single blocks. Below, the types of blocks and their total values are organized in a table with a total block values.

104

Identify the Place Value of a Digit

In the following exercises, find the place value of the given digits.

$472,981$

ⓐ $8$
ⓑ $4$
ⓒ $1$
ⓓ $7$
ⓔ $2$

ⓐ tens
ⓑ hundred thousands
ⓒ ones
ⓓ ten thousands
ⓔ thousands

$12,403,295$

ⓐ $4$
ⓑ $0$
ⓒ $1$
ⓓ $9$
ⓔ $3$

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

$5,280$

Five thousand, two hundred eighty

$204,614$

$5,012,582$

Five million, twelve thousand, five hundred eighty-two

$31,640,976$

Use Place Value to Write Whole Numbers

In the following exercises, write as a whole number using digits.

six hundred two

fifteen thousand, two hundred fifty-three

15,253

three hundred forty million, nine hundred twelve thousand, sixty-one

340,912,061

two billion, four hundred ninety-two million, seven hundred eleven thousand, two

Round Whole Numbers

In the following exercises, round to the nearest ten.

$412$

410

$648$

$3,556$

3,560

$2,734$

In the following exercises, round to the nearest hundred.

$38,975$

39,000

$26,849$

$81,486$

81,500

$75,992$

Add Whole Numbers

Use Addition Notation

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

$4 plus 3$

four plus three; the sum of four and three

$25 plus 18$

$571 plus 629$

five hundred seventy-one plus six hundred twenty-nine; the sum of five hundred seventy-one and six hundred twenty-nine

$10,085 plus 3,492$

Model Addition of Whole Numbers

In the following exercises, model the addition.

$6 plus 7$

A row of light gray vertical panels with subtle patterns, reflecting on a darker surface below, creating an abstract and minimalist composition.

$38 plus 14$

Add Whole Numbers

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

A table with 11 rows down and 11 rows across. The first row and first column are headers and include the numbers 0 through 9 both across and down, with a plus sign in the first cell. The numbers across in the second row down appear as follows: 0,0, 1, null, 3, 4, null, 6, 7, null, 9. The numbers across in the third row down appear as follows: 1, 1, 2, 3, 4, null, null, 7, 8, 9, null. The numbers in the fourth row down appear across as follows: 2, null, 3,4,5,6,7,8, null, 10, 11. The numbers across in the fifth row down appear as follows: 3, 3, null, 5, null, 7, 8, null, 10, null 12. The numbers across in the sixth row down appear as follows: 4, 4, 5, null, null, 8, 9,null, null, 12, null. The numbers across in the seventh row down appear as follows: 5, 5, null, 7, 8, null null, 11, null, 13, null. The numbers across in the eighth row down appear as follows:6, 6, 7, 8, null, 10, null, null, 13, null, 15. The numbers across in the ninth row down appear as follows: null, null, 9, null, null, 12, 13, null, 15, 16. The numbers across in the tenth row down appear as follows: 8,8,9,null, 11, null, null, 14, null, eleventh row down appear as follows: 9, 9, 10, 11, null, 13, 14, null, null, 17, null.

A clear and simple addition table showing the sums of numbers from 0 to 9. It's a fundamental tool for understanding basic arithmetic and number relationships in a grid format.

This table is 5 rows and 8 columns. The top row is a header row and includes the numbers 3 through 9, one number to each cell. The rows down include 6, 7, 8, and 9. There is a plus sign in the first cell. All cells are null.

In the following exercises, add.

ⓐ $0 plus 19$ ⓑ $19 plus 0$

ⓐ 19
ⓑ 19

ⓐ $0 plus 480$ ⓑ $480 plus 0$

ⓐ $7 plus 6$ ⓑ $6 plus 7$

ⓐ 13
ⓑ 13

ⓐ $23 plus 18$ ⓑ $18 plus 23$

$44 plus 35$

$63 plus 29$

$96 plus 58$

154

$375 plus 591$

$7,281 plus 12,546$

19,827

$5,280 + 16,324 + 9,731$

Translate Word Phrases to Math Notation

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

the sum of $30$ and $12$

30 + 12; 42

$11$ increased by $8$

$25$ more than $39$

39 + 25; 64

total of $15$ and $50$

Add Whole Numbers in Applications

In the following exercises, solve.

Shopping for an interview Nathan bought a new shirt, tie, and slacks to wear to a job interview. The shirt cost $$24,$ the tie cost $$14,$ and the slacks cost $$38.$ What was Nathan’s total cost?

$76

Running Jackson ran $4$ miles on Monday, $12$ miles on Tuesday, $1$ mile on Wednesday, $8$ miles on Thursday, and $5$ miles on Friday. What was the total number of miles Jackson ran?

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

An image of a rectangle that is 8 feet tall and 15 feet wide.

46 feet

An image of a right triangle that has a base of 12 centimeters, height of 5 centimeters, and diagonal hypotenuse of 13 centimeters.

Subtract Whole Numbers

Use Subtraction Notation

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

$14 minus 5$

fourteen minus five; the difference of fourteen and five

$40 minus 15$

$351 minus 249$

three hundred fifty-one minus two hundred forty-nine; the difference between three hundred fifty-one and two hundred forty-nine

$5,724 minus 2,918$

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

$18 minus 4$

The image uses blocks to demonstrate the subtraction problem 18 – 4.

$41 minus 29$

Subtract Whole Numbers

In the following exercises, subtract and then check by adding.

$8 minus 5$

$12 minus 7$

$23 minus 9$

$46 minus 21$

$82 minus 59$

$110 minus 87$

$539 minus 217$

322

$415 minus 296$

$1,020 minus 640$

380

$8,355 minus 3,947$

$1 0,000 minus 15$

9,985

$54,925 minus 35,647$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

the difference of nineteen and thirteen

19 − 13; 6

subtract sixty-five from one hundred

seventy-four decreased by eight

74 − 8; 66

twenty-three less than forty-one

Subtract Whole Numbers in Applications

In the following exercises, solve.

Temperature The high temperature in Peoria one day was $86$ degrees Fahrenheit and the low temperature was $28$ degrees Fahrenheit. What was the difference between the high and low temperatures?

58 degrees Fahrenheit

Savings Lynn wants to go on a cruise that costs $$2,485.$ She has $$948$ in her vacation savings account. How much more does she need to save in order to pay for the cruise?

Multiply Whole Numbers

Use Multiplication Notation

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

$8 times 5$

eight times five; the product of eight and five

$6 times 14$

$open parenthesis 10 close parenthesis open parenthesis 95 close parenthesis$

ten times ninety-five; the product of ten and ninety-five

$54 open parenthesis 72 close parenthesis$

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

$2 times 4$

The image uses circles to demonstrate the multiplication problem of 2 × 4.

$3 times 8$

Multiply Whole Numbers

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

A table with 10 rows down and 10 rows across. The first row and first column are headers and include the numbers 0 through 9 both across and down, with a plus sign in the first cell. The numbers across in the second row down appear as follows: 0, 0, 0, 0, 0, 0, null, 0, null, 0,0. The numbers across in the third row down appear as follows: 1, 0, 1, 2, null, 4, 5, 6, 7, null, 9. The numbers across in the fourth row down appear as follows: 2, 0, null, 4, null, 8, 10, null, 14, 16, null. The numbers across in the fifth row down appear as follows: 3, null, 3, null, 9, null, null, 18, null, 24, null. The numbers across in the sixth row down appear as follows: 4, 0, 4, 0, 12, null, null, 24, null, null, 36. The numbers across in the seventh row down appear as follows:5, 0, 5, 10, null, 20, null, 30, 35, 40, 45. The numbers across in the eighth row down appear as follows: 6, null, null, 12, 18, null, null, 36, 42, null, 54. The numbers across in the ninth row down appear as follows: 7, 0, 7, null, 21, null, 35, null, null, 56, 63. The numbers in the tenth row down appear as follows: 8, 0, 8, 16, null, 32, null, 48, null, 64, null. The numbers in the eleventh row down appear across as follows: 9, null, null, 18, 27, 36, null, null, 63, 72, null.

A multiplication table from 0x0 to 9x9, displaying the product of each row and column intersection. The table is shaded with alternating light and dark grey rows for readability. Perfect for learning basic math.

An image of a table with 8 columns and 5 rows. The cells in the first row and first column are shaded darker than the other cells. The first row has the values “x; 3; 4; 5; 6; 7; 8; 9”. The first column has the values “x; 6; 7; 8; 9”. All other cells are null.

In the following exercises, multiply.

$0 times 14$

$open parenthesis 256 close parenthesis 0$

$1 times 99$

$open parenthesis 4,789 close parenthesis 1$

ⓐ $7 times 4$ ⓑ $4 times 7$

ⓐ 28
ⓑ 28

$open parenthesis 25 close parenthesis open parenthesis 6 close parenthesis$

$9,261 times 3$

27,783

$48 times 76$

$64 times 10$

640

$1,000 open parenthesis 22 close parenthesis$

$162 times 493$

79,866

$open parenthesis 601 close parenthesis open parenthesis 943 close parenthesis$

$3,624 times 517$

1,873,608

$10,538 times 22$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

the product of $15$ and $28$

15(28); 420

ninety-four times thirty-three

twice $575$

2(575); 1,150

ten times two hundred sixty-four

Multiply Whole Numbers in Applications

In the following exercises, solve.

Gardening Geniece bought $8$ packs of marigolds to plant in her yard. Each pack has $6$ flowers. How many marigolds did Geniece buy?

48 marigolds

Cooking Ratika is making rice for a dinner party. The number of cups of water is twice the number of cups of rice. If Ratika plans to use $4$ cups of rice, how many cups of water does she need?

Multiplex There are twelve theaters at the multiplex and each theater has $150$ seats. What is the total number of seats at the multiplex?

1,800 seats

Roofing Lewis needs to put new shingles on his roof. The roof is a rectangle, $30$ feet by $24$ feet. What is the area of the roof?

Divide Whole Numbers

Use Division Notation

Translate from math notation to words.

$54 divided by 9$

fifty-four divided by nine; the quotient of fifty-four and nine

$42 / 7$

$\frac{72}{8}$

seventy-two divided by eight; the quotient of seventy-two and eight

$6 48$

Model Division of Whole Numbers

In the following exercises, model.

$8 divided by 2$

$3 12$

Divide Whole Numbers

In the following exercises, divide. Then check by multiplying.

$14 divided by 2$

$\frac{32}{8}$

$52 divided by 4$

$26 26$

$\frac{97}{1}$

$0 divided by 52$

$100 divided by 0$

undefined

$\frac{355}{5}$

$3828 divided by 6$

638

$31 1,519$

$\frac{7505}{25}$

300 R5

$5,166 divided by 42$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

the quotient of $64$ and $16$

64 ÷ 16; 4

the quotient of $572$ and $52$

Divide Whole Numbers in Applications

In the following exercises, solve.

Ribbon One spool of ribbon is $27$ feet. Lizbeth uses $3$ feet of ribbon for each gift basket that she wraps. How many gift baskets can Lizbeth wrap from one spool of ribbon?

9 baskets

Juice One carton of fruit juice is $128$ ounces. How many $4$ ounce cups can Shayla fill from one carton of juice?

Chapter Practice Test

Determine which of the following numbers are

ⓐ counting numbers
ⓑ whole numbers.

$0, 4, 87$

ⓐ 4, 87
ⓑ 0, 4, 87

Find the place value of the given digits in the number $549,362.$

ⓐ $9$
ⓑ $6$
ⓒ $2$
ⓓ $5$

Write each number as a whole number using digits.

ⓐ six hundred thirteen
ⓑ fifty-five thousand two hundred eight

ⓐ 613
ⓑ 55,208

Round $25,849$ to the nearest hundred.

Simplify.

$45 plus 23$

$65 minus 42$

$85 divided by 5$

$1,000 times 8$

$90 minus 58$

$73 plus 89$

$open parenthesis 0 close parenthesis open parenthesis 12,675 close parenthesis$

$634 plus 255$

$\frac{0}{9}$

$8 128$

$145 minus 79$

$299 plus 836$

$7 times 475$

3,325

$8,528 plus 704$

$35 open parenthesis 14 close parenthesis$

490

$\frac{26}{0}$

$733 minus 291$

442

$4,916 minus 1,538$

$495 divided by 45$

$52 times 983$

Translate each phrase to math notation and then simplify.

The sum of $16$ and $58$

16 + 58; 74

The product of $9$ and $15$

The difference of $32$ and $18$

32 − 18; 14

The quotient of $63$ and $21$

Twice $524$

2(524); 1,048

$29$ more than $32$

$50$ less than $300$

300 − 50; 250

In the following exercises, solve.

LaVelle buys a jumbo bag of $84$ candies to make favor bags for her son’s party. If she wants to make $12$ bags, how many candies should she put in each bag?

Last month, Stan’s take-home pay was $$3,816$ and his expenses were $$3,472.$ How much of his take-home pay did Stan have left after he paid his expenses?

Stan had $344 left.

Each class at Greenville School has $22$ children enrolled. The school has $24$ classes. How many children are enrolled at Greenville School?

Clayton walked $12$ blocks to his mother’s house, $6$ blocks to the gym, and $9$ blocks to the grocery store before walking the last $3$ blocks home. What was the total number of blocks that Clayton walked?

Clayton walked 30 blocks.

ⓐ
	$0 divided by 3$
Zero divided by any number is zero.	$0$
Check by multiplying. $0 times 3$
$0 ✓$

ⓐ
	$42 divided by 6$
Divide 42 by 6.	$7$
Check by multiplying. $7 times 6$
$42 ✓$

ⓑ
	$\frac{72}{9}$
Divide 72 by 9.	$8$
Check by multiplying. $8 times 9$
$72 ✓$

ⓒ
	$7 63$
Divide 63 by 7.	$9$
Check by multiplying. $9 times 7$
$63 ✓$

ⓐ
	$11 divided by 11$
A number divided by itself is 1.	$1$
Check by multiplying. $1 times 11$
$11 ✓$

ⓑ
	$\frac{19}{1}$
A number divided by 1 equals itself.	$19$
Check by multiplying. $19 times 1$
$19 ✓$

ⓒ
	$1 7$
A number divided by 1 equals itself.	$7$
Check by multiplying. $7 times 1$
$7 ✓$