# 1.5 Divide Whole Numbers

The topics covered in this section are:

## 1.5.1 Use Division Notation

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

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.

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 4)12, but this division can also 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.

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.

#### Example 1

Translate from math notation to words:

• $64 \div 8$
• $\frac{42}{7}$
• 4)28
Solution
1. We read this as sixty-four divided by eight and the result is the quotient of sixty-four and eight.
2. We read this as forty-two divided by seven and the result is the quotient of forty-two and seven.
3. We read this as twenty-eight divided by four and the result is the quotient of twenty-eight and four.

## 1.5.2 Model Division of Whole Numbers

### MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity Model Division of Whole Numbers will help you develop a better understanding of dividing whole numbers.

#### Example 2

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.

The divisor tell us the number of counters we want in each group. Form groups of $8$ counters.

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

$24 \div 8 = 3$

## 1.5.3 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 $3 \cdot 8 =24$.

#### Example 3

Divide. Then check by multiplying:

• $42 \div 6$
• $\frac{72}{9}$
• 7)63
Solution

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

#### Example 4

Divide. Then check by multiplying:

• $11 \div 11$
• $\frac{19}{1}$
• 1)7
Solution

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 $\$10$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 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$0$. #### Example 5 Divide. Then check by multiplying: •$0 \div 3$•$10/0$Solution 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$3$. 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,$78$. It does, so our answer is correct. ### How To: 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. #### Example 6 Divide$2,596 \div 4$. Then check by multiplying: Solution It equals the dividend, so our answer is correct. #### Example 7 Divide$4,506 \div 6$. Then check by multiplying: Solution It equals the dividend, so our answer is correct. #### Example 8 Divide$7,263 \div 9$. Then check by multiplying: Solution 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 the Figure below. Try to put the cookies in groups of eight as in the Figure below. 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$4$. #### Example 9 Divide$1,439 \div 4$. Then check by multiplying: Solution So$1,439 \div 4$is$359$with a remainder of$3$. Our answer is correct. #### Example 10 Divide$1,461 \div 13$. Then check by multiplying: Solution Our answer is correct. #### Example 11 Divide and check by multiplying:$74,521 \div 241$. Solution 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. ## 1.5.4 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 the table below. #### Example 12 Translate and simplify: the quotient of$51$and$17$. Solution The word quotient tells us to divide. the quotient of$51$and$17$Translate.$51 \div 17$Divide.$3$We could just as correctly have translated the quotient of$51$and$17$using the notation 4)12 or$\frac{51}{17}$. ## 1.5.5 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?

Solution

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