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 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.

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 \div 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 $412$ , 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.

Operation Symbols for Division

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

Operation	Notation	Expression	Read as	Result
$Division$	$\div$ $\frac{a}{b}$ $b a$ $a / b$	$12 \div 4$ $\frac{12}{4}$ $412$ $12 / 4$	$Twelve divided by four$	$the quotient of 12 and 4$

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.

Translate from math notation to words.

ⓐ $64 \div 8$ ⓑ $\frac{42}{7}$ ⓒ $428$

Solution

ⓐ We read this as sixty-four divided by eight and the result is the quotient of sixty-four and eight.
ⓑ We read this as forty-two divided by seven and the result is the quotient of forty-two and seven.
ⓒ We read this as twenty-eight divided by four and the result is the quotient of twenty-eight and four.

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.

Model the division: $24 \div 8 .$

Solution

To find the quotient $24 \div 8,$ we want to know how many groups of $8$ are in $24 .$

Model the dividend. Start with $24$ counters.
An image of 24 counters placed randomly.

The divisor tell us the number of counters we want in each group. Form groups of $8$ counters.
An image of 24 counters, all contained in 3 bubbles, each bubble containing 8 counters.

Count the number of groups. There are $3$ groups.

$24 \div 8 = 3$

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 $3 \cdot 8 = 24 .$

Divide. Then check by multiplying. ⓐ $42 \div 6$ ⓑ $\frac{72}{9}$ ⓒ $763$

Solution

Illustrates division of 42 by 6, with calculation and verification steps.

ⓐ

$42 \div 6$

Divide 42 by 6. $7$

Check by multiplying.
$7 \cdot 6$

$42 ✓$
Demonstration of dividing 72 by 9, including the calculation and verification steps.

ⓑ

$\frac{72}{9}$

Divide 72 by 9. $8$

Check by multiplying.
$8 \cdot 9$

$72 ✓$
Step-by-step demonstration of dividing 63 by 7 using long division, including a multiplication check.

ⓒ

$763$

Divide 63 by 7. $9$

Check by multiplying.
$9 \cdot 7$

$63 ✓$

Illustrates division of 42 by 6, with calculation and verification steps.
ⓐ
	$42 \div 6$
Divide 42 by 6.	$7$
Check by multiplying. $7 \cdot 6$
$42 ✓$

Demonstration of dividing 72 by 9, including the calculation and verification steps.
ⓑ
	$\frac{72}{9}$
Divide 72 by 9.	$8$
Check by multiplying. $8 \cdot 9$
$72 ✓$

Step-by-step demonstration of dividing 63 by 7 using long division, including a multiplication check.
ⓒ
	$763$
Divide 63 by 7.	$9$
Check by multiplying. $9 \cdot 7$
$63 ✓$

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.

Division Properties of One

This table outlines basic division properties, presenting their descriptive rules and corresponding mathematical expressions.
Any number (except 0) divided by itself is one.	$a \div a = 1$
Any number divided by one is the same number.	$a \div 1 = a$

Divide. Then check by multiplying:

ⓐ $11 \div 11$
ⓑ $\frac{19}{1}$
ⓒ $17$

Solution

ⓐ

$11 \div 11$

A number divided by itself is 1. $1$

Check by multiplying.
$1 \cdot 11$

$11 ✓$
ⓑ

$\frac{19}{1}$

A number divided by 1 equals itself. $19$

Check by multiplying.
$19 \cdot 1$

$19 ✓$
ⓒ

$17$

A number divided by 1 equals itself. $7$

Check by multiplying.
$7 \cdot 1$

$7 ✓$

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.

Division Properties of Zero

Rules for division involving zero, showing verbal descriptions and corresponding mathematical expressions.
Zero divided by any number is 0.	$0 \div a = 0$
Dividing a number by zero is undefined.	$a \div 0$ undefined

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 $0 .$

Divide. Check by multiplying: ⓐ $0 \div 3$ ⓑ $10 / 0 .$

Solution

Demonstration and verification of the property that zero divided by any non-zero number equals zero.

ⓐ

$0 \div 3$

Zero divided by any number is zero. $0$

Check by multiplying.
$0 \cdot 3$

$0 ✓$
Illustration of division by zero being undefined, showing a mathematical expression and its resulting state.

ⓑ

$10 / 0$

Division by zero is undefined. undefined

Demonstration and verification of the property that zero divided by any non-zero number equals zero.
ⓐ
	$0 \div 3$
Zero divided by any number is zero.	$0$
Check by multiplying. $0 \cdot 3$
$0 ✓$

Illustration of division by zero being undefined, showing a mathematical expression and its resulting state.
ⓑ
	$10 / 0$
Division by zero is undefined.	undefined

When the divisor or the dividend has more than one digit, it is usually easier to use the $412$ 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 $7 - 6$ . 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, $78 .$

\begin{matrix} \overset{1}{2} 6 \\ \underset{___}{\times 3} \\ 78 ✓ \end{matrix}

It does, so our answer is correct.

Divide $2,596 \div 4 .$ Check by multiplying:

Solution

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 $25 - 24$ . 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.

Divide $4,506 \div 6 .$ Check by multiplying:

Solution

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.

Divide $7,263 \div 9 .$ Check by multiplying.

Solution

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 2.

An image of 28 cookies placed at random.

Try to put the cookies in groups of eight as in Figure 3.

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

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}

Divide $1,439 \div 4 .$ Check by multiplying.

Solution

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 $1,439 \div 4$ is $359$ with a remainder of $3 .$ Our answer is correct.

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

Solution

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. $1,462 \div 13$ is 112 with a remainder of 5.
Check by multiplying.

Our answer is correct.

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

Solution

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 $74,521 \div 241$ 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 19.

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 $12$	$12 \div 4$ $\frac{12}{4}$ $12 / 4$ $412$

Translate and simplify: the quotient of $51$ and $17 .$

Solution

The word quotient tells us to divide.

$\begin{matrix} the quotient of 51 and 17 \\ Translate. & 51 \div 17 \\ Divide. & 3 \end{matrix}$

We could just as correctly have translated the quotient of $51$ and $17$ using the notation

$1751 or \frac{51}{17} .$

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.

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?

Solution

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

Demonstrates the process of solving a division word problem: phrasing, math notation, simplification, and final answer.
Write a phrase.	160 ounces divided by 8 ounces
Translate to math notation.	$160 \div 8$
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
$Division$	$\div$ $\frac{a}{b}$ $b a$ $a / b$	$12 \div 4$ $\frac{12}{4}$ $412$ $12 / 4$	$Twelve divided by four$	$the quotient of 12 and 4$

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. $a \div 0$ 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.

$54 \div 9$

Solution

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

$\frac{56}{7}$

$\frac{32}{8}$

Solution

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

$642$

$48 \div 6$

Solution

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

$\frac{63}{9}$

$763$

Solution

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

$72 \div 8$

Model Division of Whole Numbers

In the following exercises, model the division.

$15 \div 5$

Solution

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

$10 \div 5$

$\frac{14}{7}$

Solution

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.

$\frac{18}{6}$

$420$

Solution

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

$315$

$24 \div 6$

Solution

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

$16 \div 4$

Divide Whole Numbers

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

$18 \div 2$

Solution

$14 \div 2$

$\frac{27}{3}$

Solution

$\frac{30}{3}$

$428$

Solution

$436$

$\frac{45}{5}$

Solution

$\frac{35}{5}$

$72 / 8$

Solution

$864$

$\frac{35}{7}$

Solution

$42 \div 7$

$1515$

Solution

$1212$

$43 \div 43$

Solution

$37 \div 37$

$\frac{23}{1}$

Solution

$\frac{29}{1}$

$19 \div 1$

Solution

$17 \div 1$

$0 \div 4$

Solution

$0 \div 8$

$\frac{5}{0}$

Solution

undefined

$\frac{9}{0}$

$\frac{26}{0}$

Solution

undefined

$\frac{32}{0}$

$120$

Solution

$160$

$72 \div 3$

Solution

$57 \div 3$

$\frac{96}{8}$

Solution

$\frac{78}{6}$

$5465$

Solution

$4528$

$924 \div 7$

Solution

132

$861 \div 7$

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

Solution

871

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

$4 31,324$

Solution

7,831

$5 46,855$

$7,209 \div 3$

Solution

2,403

$4,806 \div 3$

$5,406 \div 6$

Solution

901

$3,208 \div 4$

$4 2,816$

Solution

704

$6 3,624$

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

Solution

10,209

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

$2,470 \div 7$

Solution

352 R6

$3,741 \div 7$

$8 55,305$

Solution

6,913 R1

$9 51,492$

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

Solution

86,234 R4

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

$130,016 \div 3$

Solution

43,338 R2

$105,609 \div 2$

$15 5,735$

Solution

382 R5

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

$56,883 \div 67$

Solution

849

$43,725 / 75$

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

Solution

$26,145 \div 415$

$273 542,195$

Solution

1,986 R17

$816,243 \div 462$

Mixed Practice

In the following exercises, simplify.

$15 (204)$

Solution

3,060

$74 \cdot 391$

$256 - 184$

Solution

$305 - 262$

$719 + 341$

Solution

1,060

$647 + 528$

$25875$

Solution

$1104 \div 23$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

the quotient of $45$ and $15$

Solution

45 ÷ 15; 3

the quotient of $64$ and $16$

the quotient of $288$ and $24$

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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.

Solution

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.

ⓑ 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.

$0, 2, 99$

Solution

ⓐ 2, 99
ⓑ 0, 2, 99

$0, 3, 25$

$0, 4, 90$

Solution

ⓐ 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

Solution

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.

$472,981$

ⓐ $8$
ⓑ $4$
ⓒ $1$
ⓓ $7$
ⓔ $2$

Solution

ⓐ 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.

$5,280$

Solution

Five thousand, two hundred eighty

$204,614$

$5,012,582$

Solution

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

Solution

15,253

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

Solution

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.

$412$

Solution

410

$648$

$3,556$

Solution

3,560

$2,734$

In the following exercises, round to the nearest hundred.

$38,975$

Solution

39,000

$26,849$

$81,486$

Solution

81,500

$75,992$

Add Whole Numbers

Use Addition Notation

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

$4 + 3$

Solution

four plus three; the sum of four and three

$25 + 18$

$571 + 629$

Solution

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

$10,085 + 3,492$

Model Addition of Whole Numbers

In the following exercises, model the addition.

$6 + 7$

Solution

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

$38 + 14$

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.

Solution

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.

ⓐ $0 + 19$ ⓑ $19 + 0$

Solution

ⓐ 19
ⓑ 19

ⓐ $0 + 480$ ⓑ $480 + 0$

ⓐ $7 + 6$ ⓑ $6 + 7$

Solution

ⓐ 13
ⓑ 13

ⓐ $23 + 18$ ⓑ $18 + 23$

$44 + 35$

Solution

$63 + 29$

$96 + 58$

Solution

154

$375 + 591$

$7,281 + 12,546$

Solution

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 $12$

Solution

30 + 12; 42

$11$ increased by $8$

$25$ more than $39$

Solution

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?

Solution

$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.

Solution

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.

$14 - 5$

Solution

fourteen minus five; the difference of fourteen and five

$40 - 15$

$351 - 249$

Solution

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

$5,724 - 2,918$

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

$18 - 4$

Solution

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

$41 - 29$

Subtract Whole Numbers

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

$8 - 5$

Solution

$12 - 7$

$23 - 9$

Solution

$46 - 21$

$82 - 59$

Solution

$110 - 87$

$539 - 217$

Solution

322

$415 - 296$

$1,020 - 640$

Solution

380

$8,355 - 3,947$

$1 0,000 - 15$

Solution

9,985

$54,925 - 35,647$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

the difference of nineteen and thirteen

Solution

19 − 13; 6

subtract sixty-five from one hundred

seventy-four decreased by eight

Solution

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?

Solution

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.

$8 \times 5$

Solution

eight times five; the product of eight and five

$6 \cdot 14$

$(10) (95)$

Solution

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

$54 (72)$

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

$2 \times 4$

Solution

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.

Solution

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.

$0 \cdot 14$

Solution

$(256) 0$

$1 \cdot 99$

Solution

$(4,789) 1$

ⓐ $7 \cdot 4$ ⓑ $4 \cdot 7$

Solution

ⓐ 28
ⓑ 28

$(25) (6)$

$9,261 \times 3$

Solution

27,783

$48 \cdot 76$

$64 \cdot 10$

Solution

640

$1,000 (22)$

$162 \times 493$

Solution

79,866

$(601) (943)$

$3,624 \times 517$

Solution

1,873,608

$10,538 \cdot 22$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

the product of $15$ and $28$

Solution

15(28); 420

ninety-four times thirty-three

twice $575$

Solution

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?

Solution

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?

Solution

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.

$54 \div 9$

Solution

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

$42 / 7$

$\frac{72}{8}$

Solution

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

$648$

Model Division of Whole Numbers

In the following exercises, model.

$8 \div 2$

Solution

The image shows clusters of grey circles to demonstrate 8 divided by 2.

$312$

Divide Whole Numbers

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

$14 \div 2$

Solution

$\frac{32}{8}$

$52 \div 4$

Solution

$2626$

$\frac{97}{1}$

Solution

$0 \div 52$

$100 \div 0$

Solution

undefined

$\frac{355}{5}$

$3828 \div 6$

Solution

638

$31 1,519$

$\frac{7505}{25}$

Solution

300 R5

$5,166 \div 42$

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

the quotient of $64$ and $16$

Solution

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?

Solution

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.

$0, 4, 87$

Solution

ⓐ 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

Solution

ⓐ 613
ⓑ 55,208

Round $25,849$ to the nearest hundred.

Simplify.

$45 + 23$

Solution

$65 - 42$

$85 \div 5$

Solution

$1,000 \times 8$

$90 - 58$

Solution

$73 + 89$

$(0) (12,675)$

Solution

$634 + 255$

$\frac{0}{9}$

Solution

$8128$

$145 - 79$

Solution

$299 + 836$

$7 \cdot 475$

Solution

3,325

$8,528 + 704$

$35 (14)$

Solution

490

$\frac{26}{0}$

$733 - 291$

Solution

442

$4,916 - 1,538$

$495 \div 45$

Solution

$52 \times 983$

Translate each phrase to math notation and then simplify.

The sum of $16$ and $58$

Solution

16 + 58; 74

The product of $9$ and $15$

The difference of $32$ and $18$

Solution

32 − 18; 14

The quotient of $63$ and $21$

Twice $524$

Solution

2(524); 1,048

$29$ more than $32$

$50$ less than $300$

Solution

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?

Solution

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?

Solution

Clayton walked 30 blocks.

dividend

When dividing two numbers, the dividend is the number being divided.

divisor

When dividing two numbers, the divisor is the number dividing the dividend.

quotient

The quotient is the result of dividing two numbers.

ⓐ
	$11 \div 11$
A number divided by itself is 1.	$1$
Check by multiplying. $1 \cdot 11$
$11 ✓$

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

ⓒ
	$17$
A number divided by 1 equals itself.	$7$
Check by multiplying. $7 \cdot 1$
$7 ✓$