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Divide Whole Numbers

Use division notation

Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the $12$ cookies in 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 $412$, but this division also can be written as $12\div 4,\ 12\text{/}4,\frac{12}{4}.$ In each case the $12$ is the dividend and the $4$ is the divisor.

Example

Try it.

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.

Example

Try it.

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.

$24\div 8=3$

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 , we know $24\div 8=3$ is correct because $3\cdot 8=24.$

Example

Try it.

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

Solution

ⓐ
$42\div 6$
Divide 42 by 6. $7$
Check by multiplying.
$7\cdot 6$
$42✓$
ⓑ
$\frac{72}{9}$
Divide 72 by 9. $8$
Check by multiplying.
$8\cdot 9$
$72✓$
ⓒ
$763$
Divide 63 by 7. $9$
Check by multiplying.
$9\cdot 7$
$63✓$

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

\[\frac{15}{15}=1\ \text{because}\ 1\cdot 15=15\]

Dividing any number $\text{(except 0)}$ 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

Try it.

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 $\text{\$0},$ and want to divide it among $3$ people. How much would each person get? Each person would get $\text{\$0}.$ Zero divided by any number is $0.$

Now suppose that we want to divide $\text{\$1}0$ 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.

Example

Try it.

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

Solution

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

\[\text{So}\ 78\div 3=26.\]

Condensed — the full section is in OpenStax Prealgebra 2e.

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 .

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

Example

Try it.

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

Solution

The word quotient tells us to divide.

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

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

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

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.

Example

Try it.

Cecelia bought a $\text{160-ounce}$ box of oatmeal at the big box store. She wants to divide the $160$ ounces of oatmeal into $\text{8-ounce}$ 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.

Key Concepts

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

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.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Multiply: $27\cdot 3.$
If you missed this problem, review .

விடை தெரியப்படுத்து

$81$
Subtract: $43-26.$
If you missed this problem, review

விடை தெரியப்படுத்து

$17$
Multiply: $62(87).$
If you missed this problem, review .

விடை தெரியப்படுத்து

$5,394$
Translate from math notation to words.
ⓐ $64\div 8$ ⓑ $\frac{42}{7}$ ⓒ $428$
விடை தெரியப்படுத்து
- ⓐ 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.
Translate from math notation to words:
ⓐ $84\div 7$ ⓑ $\frac{18}{6}$ ⓒ $824$
விடை தெரியப்படுத்து
- ⓐ eighty-four divided by seven; the quotient of eighty-four and seven
- ⓑ eighteen divided by six; the quotient of eighteen and six.
- ⓒ twenty-four divided by eight; the quotient of twenty-four and eight
Translate from math notation to words:
ⓐ $72\div 9$ ⓑ $\frac{21}{3}$ ⓒ $654$
விடை தெரியப்படுத்து
- ⓐ seventy-two divided by nine; the quotient of seventy-two and nine
- ⓑ twenty-one divided by three; the quotient of twenty-one and three
- ⓒ fifty-four divided by six; the quotient of fifty-four and six
Model the division: $24\div 8.$

விடை தெரியப்படுத்து

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.
$24\div 8=3$
Model: $24\div 6.$

விடை தெரியப்படுத்து
Model: $42\div 7.$

விடை தெரியப்படுத்து
Divide. Then check by multiplying. ⓐ $42\div 6$ ⓑ $\frac{72}{9}$ ⓒ $763$
விடை தெரியப்படுத்து
- ⓐ
  $42\div 6$
  Divide 42 by 6. $7$
  Check by multiplying.
  $7\cdot 6$
  $42✓$
- ⓑ
  $\frac{72}{9}$
  Divide 72 by 9. $8$
  Check by multiplying.
  $8\cdot 9$
  $72✓$
- ⓒ
  $763$
  Divide 63 by 7. $9$
  Check by multiplying.
  $9\cdot 7$
  $63✓$
Divide. Then check by multiplying:
ⓐ $54\div 6$ ⓑ $\frac{27}{9}$

விடை தெரியப்படுத்து

ⓐ 9 ⓑ 3
Divide. Then check by multiplying:
ⓐ $\frac{36}{9}$ ⓑ $840$

விடை தெரியப்படுத்து

ⓐ 4 ⓑ 5
Divide. Then check by multiplying:
1. ⓐ $11\div 11$
2. ⓑ $\frac{19}{1}$
3. ⓒ $17$
விடை தெரியப்படுத்து
- ⓐ
  $11\div 11$
  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.
  $7\cdot 1$
  $7✓$
Divide. Then check by multiplying:
ⓐ $14\div 14$ ⓑ $\frac{27}{1}$
விடை தெரியப்படுத்து
1. ⓐ 1
2. ⓑ 27
Divide. Then check by multiplying:
ⓐ $\frac{16}{1}$ ⓑ $14$
விடை தெரியப்படுத்து
1. ⓐ 16
2. ⓑ 4
Divide. Check by multiplying: ⓐ $0\div 3$ ⓑ $10/0.$
விடை தெரியப்படுத்து
- ⓐ
  $0\div 3$
  Zero divided by any number is zero. $0$
  Check by multiplying.
  $0\cdot 3$
  $0✓$
- ⓑ
  $10/0$
  Division by zero is undefined. undefined
Divide. Then check by multiplying:
ⓐ $0\div 2$ ⓑ $17/0$

விடை தெரியப்படுத்து

ⓐ 0 ⓑ undefined
Divide. Then check by multiplying:
ⓐ $0\div 6$ ⓑ $13/0$

விடை தெரியப்படுத்து

ⓐ 0 ⓑ undefined

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

விடை தெரியப்படுத்து

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. Then check by multiplying: $2,636\div 4$

விடை தெரியப்படுத்து

659
Divide. Then check by multiplying: $2,716\div 4$

விடை தெரியப்படுத்து

679

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

விடை தெரியப்படுத்து

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. Then check by multiplying: $4,305\div 5.$

விடை தெரியப்படுத்து

861
Divide. Then check by multiplying: $3,906\div 6.$

விடை தெரியப்படுத்து

651

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

விடை தெரியப்படுத்து

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.

Divide. Then check by multiplying: $4,928\div 7.$

விடை தெரியப்படுத்து

704
Divide. Then check by multiplying: $5,663\div 7.$

விடை தெரியப்படுத்து

809

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

விடை தெரியப்படுத்து

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. Then check by multiplying: $3,812\div 8.$

விடை தெரியப்படுத்து

476 with a remainder of 4
Divide. Then check by multiplying: $4,319\div 8.$

விடை தெரியப்படுத்து

539 with a remainder of 7

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

விடை தெரியப்படுத்து

Let's rewrite the problem to set it up for long division.	$131,461$
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. Then check by multiplying: $1,493\div 13.$

விடை தெரியப்படுத்து

114 R11
Divide. Then check by multiplying: $1,461\div 12.$

விடை தெரியப்படுத்து

121 R9

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

விடை தெரியப்படுத்து

Let's rewrite the problem to set it up for long division.	$24174,521$
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.

Divide. Then check by multiplying: $78,641\div 256.$

விடை தெரியப்படுத்து

307 R49
Divide. Then check by multiplying: $76,461\div 248.$

விடை தெரியப்படுத்து

308 R77
Translate and simplify: the quotient of $51$ and $17.$

விடை தெரியப்படுத்து

The word quotient tells us to divide.
$\begin{array}{llll} & & & \text{the quotient of 51 and 17} \\ \text{Translate.} & & & 51\div 17 \\ \text{Divide.} & & & 3\end{array}$
We could just as correctly have translated the quotient of $51$ and $17$ using the notation
$1751\ \text{or}\ \frac{51}{17}.$
Translate and simplify: the quotient of $91$ and $13.$

விடை தெரியப்படுத்து

91 ÷ 13; 7
Translate and simplify: the quotient of $52$ and $13.$

விடை தெரியப்படுத்து

52 ÷ 13; 4
Cecelia bought a $\text{160-ounce}$ box of oatmeal at the big box store. She wants to divide the $160$ ounces of oatmeal into $\text{8-ounce}$ 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?

விடை தெரியப்படுத்து

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.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Divide Whole Numbers

Use division notation
Model division of whole numbers
Divide whole numbers
Translate word phrases to math notation
Divide whole numbers in applications

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

உங்களை முயற்சிக்கவும்

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

மேலும் Arithmetic

Order of operations Fractions Percentages Powers and roots

ⓐ
	\(42\div 6\)
Divide 42 by 6.	\(7\)
Check by multiplying. \(7\cdot 6\)
\(42✓\)

ⓑ
	\(\frac{72}{9}\)
Divide 72 by 9.	\(8\)
Check by multiplying. \(8\cdot 9\)
\(72✓\)

ⓒ
	\(763\)
Divide 63 by 7.	\(9\)
Check by multiplying. \(9\cdot 7\)
\(63✓\)

ⓐ
	\(11\div 11\)
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. \(19\cdot 1\)
\(19✓\)

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

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

Zero divided by any number is 0.	\(0\div a=0\)
Dividing a number by zero is undefined.	\(a\div 0\) undefined

ⓐ
	\(0\div 3\)
Zero divided by any number is zero.	\(0\)
Check by multiplying. \(0\cdot 3\)
\(0✓\)

Let's rewrite the problem to set it up for long division.	\(131,461\)
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.

Let's rewrite the problem to set it up for long division.	\(24174,521\)
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.

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

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