# Whole Numbers

## Divide Whole Numbers


### Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the  cookies in  and want to package them in bags with  cookies in each bag. How many bags would we need?
You might put  cookies in first bag,  in the second bag, and so on until you run out of cookies. Doing it this way, you would fill  bags.
In other words, starting with the  cookies, you would take away, or subtract,  cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.
Instead of subtracting  repeatedly, we can write
We read this as twelve divided by four and the result is the quotient of  and  The quotient is  because we can subtract  from  exactly  times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is  and the divisor is 
In the past you may have used the notation , but this division also can be written as  In each case the  is the dividend and the  is the divisor.

### 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  because  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 , we know  is correct because 
What is the quotient when you divide a number by itself?
Dividing any number  by itself produces a quotient of  Also, any number divided by  produces a quotient of the number. These two ideas are stated in the Division Properties of One.
Suppose we have  and want to divide it among  people. How much would each person get? Each person would get  Zero divided by any number is 
Now suppose that we want to divide  by  That means we would want to find a number that we multiply by  to get  This cannot happen because  times any number is  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  from  Because subtracting  will never change the total, we will never get an answer. So we cannot divide a number by 
When the divisor or the dividend has more than one digit, it is usually easier to use the  notation. This process is called long division. Let’s work through the process by dividing  by 
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.
Check by multiplying the quotient times the divisor to get the dividend. Multiply  to make sure that product equals the dividend, 
It does, so our answer is correct.
So far all the division problems have worked out evenly. For example, if we had  cookies and wanted to make bags of  cookies, we would have  bags. But what if there were  cookies and we wanted to make bags of  Start with the  cookies as shown in .
Try to put the cookies in groups of eight as in .
There are  groups of eight cookies, and  cookies left over. We call the  cookies that are left over the remainder and show it by writing R4 next to the  (The R stands for remainder.)
To check this division we multiply  times  to get  and then add the remainder of 

### 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 .

### 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.

### Key Concepts

1. Division Properties of One

2. Division Properties of Zero

3. Divide whole numbers.


### Section Exercises


### Practice Makes Perfect

Use Division Notation 
In the following exercises, translate from math notation to words.
Model Division of Whole Numbers 
In the following exercises, model the division.
Divide Whole Numbers 
In the following exercises, divide. Then check by multiplying.
Mixed Practice 
In the following exercises, simplify.
Translate Word Phrases to Algebraic Expressions 
In the following exercises, translate and simplify.
Divide Whole Numbers in Applications 
In the following exercises, solve.
Mixed Practice 
In the following exercises, solve.

### Writing Exercises


### Everyday Math


### Self Check

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

ⓑ 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.
Model Whole Numbers 
In the following exercises, model each number using  blocks and then show its value using place value notation.
Identify the Place Value of a Digit 
In the following exercises, find the place value of the given digits.
Use Place Value to Name Whole Numbers 
In the following exercises, name each number in words.
Use Place Value to Write Whole Numbers 
In the following exercises, write as a whole number using digits.
Round Whole Numbers 
In the following exercises, round to the nearest ten.
In the following exercises, round to the nearest hundred.

### Add Whole Numbers

Use Addition Notation 
In the following exercises, translate the following from math notation to words.
Model Addition of Whole Numbers 
In the following exercises, model the addition.
Add Whole Numbers 
In the following exercises, fill in the missing values in each chart.
In the following exercises, add.
Translate Word Phrases to Math Notation 
In the following exercises, translate each phrase into math notation and then simplify.
Add Whole Numbers in Applications 
In the following exercises, solve.
In the following exercises, find the perimeter of each figure.

### Subtract Whole Numbers

Use Subtraction Notation 
In the following exercises, translate the following from math notation to words.
Model Subtraction of Whole Numbers 
In the following exercises, model the subtraction.
Subtract Whole Numbers 
In the following exercises, subtract and then check by adding.
Translate Word Phrases to Math Notation 
In the following exercises, translate and simplify.
Subtract Whole Numbers in Applications 
In the following exercises, solve.

### Multiply Whole Numbers

Use Multiplication Notation 
In the following exercises, translate from math notation to words.
Model Multiplication of Whole Numbers 
In the following exercises, model the multiplication.
Multiply Whole Numbers 
In the following exercises, fill in the missing values in each chart.
In the following exercises, multiply.
Translate Word Phrases to Math Notation 
In the following exercises, translate and simplify.
Multiply Whole Numbers in Applications 
In the following exercises, solve.

### Divide Whole Numbers

Use Division Notation 
Translate from math notation to words.
Model Division of Whole Numbers 
In the following exercises, model.
Divide Whole Numbers 
In the following exercises, divide. Then check by multiplying.
Translate Word Phrases to Math Notation 
In the following exercises, translate and simplify.
Divide Whole Numbers in Applications 
In the following exercises, solve.

### Chapter Practice Test

Simplify.
Translate each phrase to math notation and then simplify.
In the following exercises, solve.