Divide Whole Numbers

1.5 Divide Whole Numbers

The topics covered in this section are:

  1. Use division notation
  2. Model division of whole numbers
  3. Divide whole numbers
  4. Translate word phrases to math notation
  5. Divide whole numbers in applications

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?

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 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.

OperationNotationExpressionRead asResult
Division$\div$
$\frac{a}{b}$
b)a
$a/b$
$12 \div 4$
$\frac{12}{4}$
4)12
$12/4$
Twelve divided by fourthe 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.

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.

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.

$24 \div 8 = 3$

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

1.$42 \div 6$
Divide $42$ by $6$.$7$
Check by multiplying.
$7 \cdot 6$
$42$✓
2.$\frac{72}{9}$
Divide $72$ by $9$.$8$
Check by multiplying.
$8 \cdot 9$
$72$✓
3.7)63
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

Any number (except $0$) divided by itself is one.$a \div a = 1$
Any number divided by one is the same number.$a \div 1 = a$

Example 4

Divide. Then check by multiplying:

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

1.$11 \div 11$
A number divided by itself is $1$.$1$
Check by multiplying.
$1 \cdot 11$
$11$✓
2.$\frac{19}{1}$
A number divided by $1$ equals itself.$19$
Check by multiplying.
$19 \cdot 1$
$19$✓
3.1)7
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 $\$10$ 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

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 $0$.

Example 5

Divide. Then check by multiplying:

  • $0 \div 3$
  • $10/0$
Solution

1.$0 \div 3$
Zero divided by any number is zero.$0$
Check by multiplying.
$0 \cdot 3$
$0$✓
2.$10/0$
Division by zero is undefined.undefined

When the divisor or the dividend has more than one digit, it is usually easier to use the 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.CNX_BMath_Figure_01_05_043_img-02.png
Multiply the $2$ in the quotient by $3$ and write the product, $6$, under the $7$.CNX_BMath_Figure_01_05_043_img-03.png
Subtract that product from the first digit in the dividend. Subtract $7-6$. Write the difference, $1$, under the first digit in the dividend.CNX_BMath_Figure_01_05_043_img-04.png
Bring down the next digit of the dividend. Bring down the $8$.CNX_BMath_Figure_01_05_043_img-05.png
Divide $18$ by the divisor, $3$. The divisor $3$ goes into $18$ six times.
Write $6$ in the quotient above the $8$.
CNX_BMath_Figure_01_05_043_img-06.png
Multiply the $6$ in the quotient by the divisor and write the product, $18$, under the dividend. Subtract $18$ from the $18$.CNX_BMath_Figure_01_05_043_img-07.png

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

Let’s rewrite the problem to set it up for long division.CNX_BMath_Figure_01_05_044_img-01.png
Divide the first digit of the dividend, $2$, by the divisor, $4$.CNX_BMath_Figure_01_05_044_img-02.png
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 $5$.CNX_BMath_Figure_01_05_044_img-03.png
Multiply the $6$ in the quotient by the divisor $4$ and write the product, $24$, under the first two digits in the dividend.CNX_BMath_Figure_01_05_044_img-04.png
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.CNX_BMath_Figure_01_05_044_img-05.png
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 $19$.CNX_BMath_Figure_01_05_044_img-06.png
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 $36$.CNX_BMath_Figure_01_05_044_img-07.png
so $2,596 \div 4 = 649$
Check by multiplying.CNX_BMath_Figure_01_05_044_img-08.png

It equals the dividend, so our answer is correct.

Example 7

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

Solution

Let’s rewrite the problem to set it up for long division.CNX_BMath_Figure_01_05_045_img-01.png
First we try to divide $6$ into $4$.CNX_BMath_Figure_01_05_045_img-02.png
Since that won’t work, we try $6$ into $45$.
There are $7$ sixes in $45$. We write the $7$ over the $5$.
CNX_BMath_Figure_01_05_045_img-03.png
Multiply the $7$ by $6$ and subtract this product from $45$.CNX_BMath_Figure_01_05_045_img-04.png
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 $30$.
CNX_BMath_Figure_01_05_045_img-05.png
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 $6$.
CNX_BMath_Figure_01_05_045_img-06.png
Check by multiplying.CNX_BMath_Figure_01_05_045_img-07.png

It equals the dividend, so our answer is correct.

Example 8

Divide $7,263 \div 9$. Then check by multiplying:

Solution

Let’s rewrite the problem to set it up for long division.CNX_BMath_Figure_01_05_046_img-01.png
First we try to divide $9$ into $7$.CNX_BMath_Figure_01_05_046_img-02.png
Since that won’t work, we try $9$ into $72$.
There are $8$ nines in $72$. We write the $8$ over the $2$.
CNX_BMath_Figure_01_05_046_img-03.png
Multiply the $8$ by $9$ and subtract this product from $72$.CNX_BMath_Figure_01_05_046_img-04.png
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 $6$.
CNX_BMath_Figure_01_05_046_img-05.png
Now bring down the $3$ and repeat these steps. There is $7$ nines in $63$.
Write the $7$ over the $3$. Multiply $7$ by $9$ and subtract this product from $63$.
CNX_BMath_Figure_01_05_046_img-06.png
Check by multiplying.CNX_BMath_Figure_01_05_046_img-07.png

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.

An image of 28 cookies placed at random.

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

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 $4$.

Example 9

Divide $1,439 \div 4$. Then check by multiplying:

Solution

Let’s rewrite the problem to set it up for long division.CNX_BMath_Figure_01_05_047_img-01.png
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 $4$.
CNX_BMath_Figure_01_05_047_img-02.png
Multiply the $3$ by $4$ and subtract this product from $14$.CNX_BMath_Figure_01_05_047_img-03.png
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 $23$.
CNX_BMath_Figure_01_05_047_img-04.png
Now bring down the $9$ and repeat these steps. 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 $3$.
CNX_BMath_Figure_01_05_047_img-05.png
Check by multiplying.CNX_BMath_Figure_01_05_047_img-06.png

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

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 are $1$ thirteen in $14$. We write the $1$ over the $4$.
CNX_BMath_Figure_01_05_048_img-02.png
Multiply the $1$ by $13$ and subtract this product from $14$.CNX_BMath_Figure_01_05_048_img-03.png
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 $16$.
CNX_BMath_Figure_01_05_048_img-04.png
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 $5$.
CNX_BMath_Figure_01_05_048_img-05.png
Check by multiplying.CNX_BMath_Figure_01_05_048_img-06.png

Our answer is correct.

Example 11

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 $745$. 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 $745$.CNX_BMath_Figure_01_05_049_img-02.png
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.
CNX_BMath_Figure_01_05_049_img-03.png
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 $2,221$.CNX_BMath_Figure_01_05_049_img-03.png
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.CNX_BMath_Figure_01_05_049_img-05.png

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.

OperationNotationExpressionRead asResult
Division$\div$
$\frac{a}{b}$
b)a
$a/b$
$12 \div 4$
$\frac{12}{4}$
4)12
$12/4$
Twelve divided by fourthe quotient of $12$ and $4$

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.

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.
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