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Divide Whole Numbers

Module by: First Last. E-mail the author

Summary: By the end of this section, you will be able to:

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

Note: You are viewing an old style version of this document. The new style version is available here.

Note:

Before you get started, take this readiness quiz.

  1. 1) Multiply: 27·3.27·3.
    If you missed this problem, review (Reference).
  2. 2) Subtract: 4326.4326.
    If you missed this problem, review (Reference)
  3. 3) Multiply: 62(87).62(87).
    If you missed this problem, review (Reference).

Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the 1212 cookies in Figure 1 and want to package them in bags with 44 cookies in each bag. How many bags would we need?

Figure 1
An image of three rows of four cookies to show twelve cookies.

You might put 44 cookies in first bag, 44 in the second bag, and so on until you run out of cookies. Doing it this way, you would fill 33 bags.

An image of 3 bags of cookies, each bag containing 4 cookies.

In other words, starting with the 1212 cookies, you would take away, or subtract, 44 cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.

Instead of subtracting 44 repeatedly, we can write

12÷412÷4

We read this as twelve divided by four and the result is the quotient of 1212 and 4.4. The quotient is 33 because we can subtract 44 from 1212 exactly 33 times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is 1212 and the divisor is 4.4.

In the past you may have used the notation 412412, but this division also can be written as 12÷4,12/4,124.12÷4,12/4,124. In each case the 1212 is the dividend and the 44 is the divisor.

Note: Operation Symbols for Division:

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

Operation Notation Expression Read as Result
DivisionDivision ÷÷
abab
baba
a/ba/b
12÷412÷4
124124
412412
12/412/4
Twelve divided by fourTwelve divided by four the quotient of 12 and 4the 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

Problem 1

Translate from math notation to words.

64÷864÷8 427427 428428

Solution: 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.

Note:

Exercise 1

Translate from math notation to words:

84÷784÷7 186186 824824

Solution
  • 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

Note:

Exercise 2

Translate from math notation to words:

72÷972÷9 213213 654654

Solution
  • 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 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.

Note:

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

Example 2

Problem 1

Model the division: 24÷8.24÷8.

Solution: Solution

To find the quotient 24÷8,24÷8, we want to know how many groups of 88 are in 24.24.

Model the dividend. Start with 2424 counters.
An image of 24 counters placed randomly.

The divisor tell us the number of counters we want in each group. Form groups of 88 counters.
An image of 24 counters, all contained in 3 bubbles, each bubble containing 8 counters.

Count the number of groups. There are 33 groups.

24÷8=324÷8=3

Note:

Exercise 3

Model: 24÷6.24÷6.

Solution


No Alt Text

Note:

Exercise 4

Model: 42÷7.42÷7.

Solution


No Alt Text

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÷4=312÷4=3 because 3·4=12.3·4=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÷8=324÷8=3 is correct because 3·8=24.3·8=24.

Example 3

Problem 1

Divide. Then check by multiplying. 42÷642÷6 729729 763763

Solution: Solution
  •  
      42÷642÷6
    Divide 42 by 6. 77
    Check by multiplying.
    7·67·6
     
    4242  
  •  
      729729
    Divide 72 by 9. 88
    Check by multiplying.
    8·98·9
     
    7272  
  •  
      763763
    Divide 63 by 7. 99
    Check by multiplying.
    9·79·7
     
    6363  

Note:

Exercise 5

Divide. Then check by multiplying:

54÷654÷6 279279

Solution

9 3

Note:

Exercise 6

Divide. Then check by multiplying:

369369 840840

Solution

4 5

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

1515=1because1·15=151515=1because1·15=15

Dividing any number (except 0)(except 0) by itself produces a quotient of 1.1. Also, any number divided by 11 produces a quotient of the number. These two ideas are stated in the Division Properties of One.

Note: Division Properties of One:

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

Example 4

Problem 1

Divide. Then check by multiplying:

  1. 11÷1111÷11
  2. 191191
  3. 1717
Solution: Solution
  •  
      11÷1111÷11
    A number divided by itself is 1. 11
    Check by multiplying.
    1·111·11
     
    1111  
  •  
      191191
    A number divided by 1 equals itself. 1919
    Check by multiplying.
    19·119·1
     
    1919  
  •  
      1717
    A number divided by 1 equals itself. 77
    Check by multiplying.
    7·17·1
     
    77  

Note:

Exercise 7

Divide. Then check by multiplying:

14÷1414÷14 271271

Solution
  1. 1
  2. 27

Note:

Exercise 8

Divide. Then check by multiplying:

161161 1414

Solution
  1. 16
  2. 4

Suppose we have $0,$0, and want to divide it among 33 people. How much would each person get? Each person would get $0.$0. Zero divided by any number is 0.0.

Now suppose that we want to divide $10$10 by 0.0. That means we would want to find a number that we multiply by 00 to get 10.10. This cannot happen because 00 times any number is 0.0. Division by zero is said to be undefined.

These two ideas make up the Division Properties of Zero.

Note: Division Properties of Zero:

Zero divided by any number is 0. 0÷a=00÷a=0
Dividing a number by zero is undefined. a÷0a÷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 00 from 10?10? Because subtracting 00 will never change the total, we will never get an answer. So we cannot divide a number by 0.0.

Example 5

Problem 1

Divide. Check by multiplying: 0÷30÷3 10/0.10/0.

Solution: Solution
  •  
      0÷30÷3
    Zero divided by any number is zero. 00
    Check by multiplying.
    0·30·3
     
    00  
  •  
      10/010/0
    Division by zero is undefined. undefined

Note:

Exercise 9

Divide. Then check by multiplying:

0÷20÷2 17/017/0

Solution

0 undefined

Note:

Exercise 10

Divide. Then check by multiplying:

0÷60÷6 13/013/0

Solution

0 undefined

When the divisor or the dividend has more than one digit, it is usually easier to use the 412412 notation. This process is called long division. Let’s work through the process by dividing 7878 by 3.3.

Divide the first digit of the dividend, 7, by the divisor, 3.  
The divisor 3 can go into 7 two times since 2×3=62×3=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 2 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 7676. 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. CNX_BMath_Figure_01_05_043_img-06.png
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. 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.

So78÷3=26.So78÷3=26.

Check by multiplying the quotient times the divisor to get the dividend. Multiply 26×326×3 to make sure that product equals the dividend, 78.78.

216×3___78216×3___78
(4)

It does, so our answer is correct.

Note: Divide whole numbers.:

  1. Step 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. Step 2. Write the quotient above the dividend.
  3. Step 3. Multiply the quotient by the divisor and write the product under the dividend.
  4. Step 4. Subtract that product from the dividend.
  5. Step 5. Bring down the next digit of the dividend.
  6. Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
  7. Step 7. Check by multiplying the quotient times the divisor.

Example 6

Problem 1

Divide 2,596÷4.2,596÷4. Check by multiplying:

Solution: 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 25242524. 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÷4=6492,596÷4=649.  
Check by multiplying.
CNX_BMath_Figure_01_05_044_img-08.png
 

It equals the dividend, so our answer is correct.

Note:

Exercise 11

Divide. Then check by multiplying: 2,636÷42,636÷4

Solution

659

Note:

Exercise 12

Divide. Then check by multiplying: 2,716÷42,716÷4

Solution

679

Example 7

Problem 1

Divide 4,506÷6.4,506÷6. Check by multiplying:

Solution: 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.

Note:

Exercise 13

Divide. Then check by multiplying: 4,305÷5.4,305÷5.

Solution

861

Note:

Exercise 14

Divide. Then check by multiplying: 3,906÷6.3,906÷6.

Solution

651

Example 8

Problem 1

Divide 7,263÷9.7,263÷9. Check by multiplying.

Solution: 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 are 7 nines in 63. Write the 7 over the 3.
Multiply the 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.

Note:

Exercise 15

Divide. Then check by multiplying: 4,928÷7.4,928÷7.

Solution

704

Note:

Exercise 16

Divide. Then check by multiplying: 5,663÷7.5,663÷7.

Solution

809

So far all the division problems have worked out evenly. For example, if we had 2424 cookies and wanted to make bags of 88 cookies, we would have 33 bags. But what if there were 2828 cookies and we wanted to make bags of 8?8? Start with the 2828 cookies as shown in Figure 2.

Figure 2
An image of 28 cookies placed at random.

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

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

There are 33 groups of eight cookies, and 44 cookies left over. We call the 44 cookies that are left over the remainder and show it by writing R4 next to the 3.3. (The R stands for remainder.)

To check this division we multiply 33 times 88 to get 24,24, and then add the remainder of 4.4.

3×8___24+4___283×8___24+4___28

Example 9

Problem 1

Divide 1,439÷4.1,439÷4. Check by multiplying.

Solution: 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. 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.
CNX_BMath_Figure_01_05_047_img-05.png
Check by multiplying.
CNX_BMath_Figure_01_05_047_img-06.png
 

So 1,439÷41,439÷4 is 359359 with a remainder of 3.3. Our answer is correct.

Note:

Exercise 17

Divide. Then check by multiplying: 3,812÷8.3,812÷8.

Solution

476 with a remainder of 4

Note:

Exercise 18

Divide. Then check by multiplying: 4,319÷8.4,319÷8.

Solution

539 with a remainder of 7

Example 10

Problem 1

Divide and then check by multiplying: 1,461÷13.1,461÷13.

Solution: Solution
Let's rewrite the problem to set it up for long division. 131,461131,461
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.
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÷131,462÷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.

Note:

Exercise 19

Divide. Then check by multiplying: 1,493÷13.1,493÷13.

Solution

114 R11

Note:

Exercise 20

Divide. Then check by multiplying: 1,461÷12.1,461÷12.

Solution

121 R9

Example 11

Problem 1

Divide and check by multiplying: 74,521÷241.74,521÷241.

Solution: Solution
Let's rewrite the problem to set it up for long division. 24174,52124174,521
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 3×241=7233×241=723, we write the 3 over the 5 in 745.
Note that 4 would be too large because 4×241=9644×241=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 9×241=2,1699×241=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-04.png
There are no more numbers to bring down, so we are finished. The remainder is 52. So 74,521÷24174,521÷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.

Note:

Exercise 21

Divide. Then check by multiplying: 78,641÷256.78,641÷256.

Solution

307 R49

Note:

Exercise 22

Divide. Then check by multiplying: 76,461÷248.76,461÷248.

Solution

308 R77

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.

Table 19
Operation Word Phrase Example Expression
Division divided by
quotient of
divided into
1212 divided by 44
the quotient of 1212 and 44
44 divided into 1212
12÷412÷4
124124
12/412/4
412412

Example 12

Problem 1

Translate and simplify: the quotient of 5151 and 17.17.

Solution: Solution

The word quotient tells us to divide.

the quotient of 51 and 17Translate.51÷17Divide.3the quotient of 51 and 17Translate.51÷17Divide.3

We could just as correctly have translated the quotient of 5151 and 1717 using the notation

1751or5117.1751or5117.

Note:

Exercise 23

Translate and simplify: the quotient of 9191 and 13.13.

Solution

91 ÷ 13; 7

Note:

Exercise 24

Translate and simplify: the quotient of 5252 and 13.13.

Solution

52 ÷ 13; 4

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

Problem 1

Cecelia bought a 160-ounce160-ounce box of oatmeal at the big box store. She wants to divide the 160160 ounces of oatmeal into 8-ounce8-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: 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÷8160÷8
Simplify by dividing. 2020
Write a sentence to answer the question.Cecelia will get 20 servings from the big box.

Note:

Exercise 25

Marcus is setting out animal crackers for snacks at the preschool. He wants to put 99 crackers in each cup. One box of animal crackers contains 135135 crackers. How many cups can he fill from one box of crackers?

Solution

Marcus can fill 15 cups.

Note:

Exercise 26

Andrea is making bows for the girls in her dance class to wear at the recital. Each bow takes 44 feet of ribbon, and 3636 feet of ribbon are on one spool. How many bows can Andrea make from one spool of ribbon?

Solution

Andrea can make 9 bows.

Key Concepts

Operation Notation Expression Read as Result
DivisionDivision ÷÷
abab
baba
a/ba/b
12÷412÷4
124124
412412
12/412/4
Twelve divided by fourTwelve divided by four the quotient of 12 and 4the quotient of 12 and 4
  • Division Properties of One
    • Any number (except 0) divided by itself is one. a÷a=1a÷a=1
    • Any number divided by one is the same number. a÷1=aa÷1=a
  • Division Properties of Zero
    • Zero divided by any number is 0. 0÷a=00÷a=0
    • Dividing a number by zero is undefined. a÷0a÷0 undefined
  • Divide whole numbers.
    1. Step 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. Step 2. Write the quotient above the dividend.
    3. Step 3. Multiply the quotient by the divisor and write the product under the dividend.
    4. Step 4. Subtract that product from the dividend.
    5. Step 5. Bring down the next digit of the dividend.
    6. Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
    7. Step 7. Check by multiplying the quotient times the divisor.

Practice Makes Perfect

Use Division Notation

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

Exercise 27

54÷954÷9

Solution

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

Exercise 28

567567

Exercise 29

328328

Solution

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

Exercise 30

642642

Exercise 31

48÷648÷6

Solution

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

Exercise 32

639639

Exercise 33

763763

Solution

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

Exercise 34

72÷872÷8

Model Division of Whole Numbers

In the following exercises, model the division.

Exercise 35

Exercise 36

10÷510÷5

Exercise 37

Exercise 38

186186

Exercise 39

Exercise 40

315315

Exercise 41

Exercise 42

16÷416÷4

Divide Whole Numbers

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

Exercise 43

Exercise 44

14÷214÷2

Exercise 45

Exercise 46

303303

Exercise 47

Exercise 48

436436

Exercise 49

Exercise 50

355355

Exercise 51

Exercise 52

864864

Exercise 53

Exercise 54

42÷742÷7

Exercise 55

Exercise 56

12121212

Exercise 57

Exercise 58

37÷3737÷37

Exercise 59

Exercise 60

291291

Exercise 61

Exercise 62

17÷117÷1

Exercise 63

Exercise 64

0÷80÷8

Exercise 65

Exercise 66

9090

Exercise 67

260260

Solution

undefined

Exercise 68

320320

Exercise 69

Exercise 70

160160

Exercise 71

Exercise 72

57÷357÷3

Exercise 73

Exercise 74

786786

Exercise 75

Exercise 76

45284528

Exercise 77

924÷7924÷7

Solution

132

Exercise 78

861÷7861÷7

Exercise 79

5,22665,2266

Solution

871

Exercise 80

3,77683,7768

Exercise 81

431,324431,324

Solution

7,831

Exercise 82

546,855546,855

Exercise 83

7,209÷37,209÷3

Solution

2,403

Exercise 84

4,806÷34,806÷3

Exercise 85

5,406÷65,406÷6

Solution

901

Exercise 86

3,208÷43,208÷4

Exercise 87

42,81642,816

Solution

704

Exercise 88

63,62463,624

Exercise 89

91,881991,8819

Solution

10,209

Exercise 90

83,256883,2568

Exercise 91

2,470÷72,470÷7

Solution

352 R6

Exercise 92

3,741÷73,741÷7

Exercise 93

855,305855,305

Solution

6,913 R1

Exercise 94

951,492951,492

Exercise 95

431,1745431,1745


Solution

86,234 R4

Exercise 96

297,2774297,2774

Exercise 97

130,016÷3130,016÷3

Solution

43,338 R2

Exercise 98

105,609÷2105,609÷2

Exercise 99

155,735155,735

Solution

382 R5

Exercise 100

4,933214,93321

Exercise 101

56,883÷6756,883÷67

Solution

849

Exercise 102

43,725/7543,725/75

Exercise 103

30,14431430,144314

Solution

96

Exercise 104

26,145÷41526,145÷415

Exercise 105

273542,195273542,195

Solution

1,986 R17

Exercise 106

816,243÷462816,243÷462

Mixed Practice

In the following exercises, simplify.

Exercise 107

15(204)15(204)

Solution

3,060

Exercise 108

74·39174·391

Exercise 109

256184256184

Solution

72

Exercise 110

305262305262

Exercise 111

719+341719+341

Solution

1,060

Exercise 112

647+528647+528

Exercise 113

Exercise 114

1104÷231104÷23

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

Exercise 115

the quotient of 4545 and 1515

Solution

45 ÷ 15; 3

Exercise 116

the quotient of 6464 and 1616

Exercise 117

the quotient of 288288 and 2424

Solution

288 ÷ 24; 12

Exercise 118

the quotient of 256256 and 3232

Divide Whole Numbers in Applications

In the following exercises, solve.

Exercise 119

Trail mix Ric bought 6464 ounces of trail mix. He wants to divide it into small bags, with 22 ounces of trail mix in each bag. How many bags can Ric fill?

Solution

Ric can fill 32 bags.

Exercise 120

Crackers Evie bought a 4242 ounce box of crackers. She wants to divide it into bags with 33 ounces of crackers in each bag. How many bags can Evie fill?

Exercise 121

Astronomy class There are 125125 students in an astronomy class. The professor assigns them into groups of 5.5. How many groups of students are there?

Solution

There are 25 groups.

Exercise 122

Flower shop Melissa’s flower shop got a shipment of 152152 roses. She wants to make bouquets of 88 roses each. How many bouquets can Melissa make?

Exercise 123

Baking One roll of plastic wrap is 4848 feet long. Marta uses 33 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.

Exercise 124

Dental floss One package of dental floss is 5454 feet long. Brian uses 22 feet of dental floss every day. How many days will one package of dental floss last Brian?

Mixed Practice

In the following exercises, solve.

Exercise 125

Miles per gallon Susana’s hybrid car gets 4545 miles per gallon. Her son’s truck gets 1717 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.

Exercise 126

Distance Mayra lives 5353 miles from her mother’s house and 7171 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?

Exercise 127

Field trip The 4545 students in a Geology class will go on a field trip, using the college’s vans. Each van can hold 99 students. How many vans will they need for the field trip?

Solution

They will need 5 vans for the field trip

Exercise 128

Potting soil Aki bought a 128128 ounce bag of potting soil. How many 44 ounce pots can he fill from the bag?

Exercise 129

Hiking Bill hiked 88 miles on the first day of his backpacking trip, 1414 miles the second day, 1111 miles the third day, and 1717 miles the fourth day. What is the total number of miles Bill hiked?

Solution

Bill hiked 50 miles

Exercise 130

Reading Last night Emily read 66 pages in her Business textbook, 2626 pages in her History text, 1515 pages in her Psychology text, and 99 pages in her math text. What is the total number of pages Emily read?

Exercise 131

Patients LaVonne treats 1212 patients each day in her dental office. Last week she worked 44 days. How many patients did she treat last week?

Solution

LaVonne treated 48 patients last week.

Exercise 132

Scouts There are 1414 boys in Dave’s scout troop. At summer camp, each boy earned 55 merit badges. What was the total number of merit badges earned by Dave’s scout troop at summer camp?

Writing Exercises

Exercise 133

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.

Exercise 134

Oswaldo divided 300300 by 88 and said his answer was 3737 with a remainder of 4.4. How can you check to make sure he is correct?

Everyday Math

Exercise 135

Contact lenses Jenna puts in a new pair of contact lenses every 1414 days. How many pairs of contact lenses does she need for 365365 days?

Solution

Jenna uses 26 pairs of contact lenses, but there is 1 day left over, so she needs 27 pairs for 365 days.

Exercise 136

Cat food One bag of cat food feeds Lara’s cat for 2525 days. How many bags of cat food does Lara need for 365365 days?

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.

Exercise 137

0,2,990,2,99

Solution

  1. 2, 99
  2. 0, 2, 99

Exercise 138

0,3,250,3,25

Exercise 139

0,4,900,4,90

Solution
  1. 4, 90
  2. 0, 4, 90

Exercise 140

0,1,750,1,75

Model Whole Numbers

In the following exercises, model each number using base-10base-10 blocks and then show its value using place value notation.

Exercise 141

Exercise 142

104

Identify the Place Value of a Digit

In the following exercises, find the place value of the given digits.

Exercise 143

472,981472,981

  1. 88
  2. 44
  3. 11
  4. 77
  5. 22
Solution
  1. tens
  2. hundred thousands
  3. ones
  4. thousands
  5. ten thousands

Exercise 144

12,403,29512,403,295

  1. 44
  2. 00
  3. 11
  4. 99
  5. 33

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

Exercise 145

5,2805,280

Solution

Five thousand two hundred eighty

Exercise 146

204,614204,614

Exercise 147

5,012,5825,012,582

Solution

Five million twelve thousand five hundred eighty-two

Exercise 148

31,640,97631,640,976

Use Place Value to Write Whole Numbers

In the following exercises, write as a whole number using digits.

Exercise 149

six hundred two

Exercise 150

fifteen thousand, two hundred fifty-three

Solution

15,253

Exercise 151

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

Solution

340,912,061

Exercise 152

two billion, four hundred ninety-two million, seven hundred eleven thousand, two

Round Whole Numbers

In the following exercises, round to the nearest ten.

Exercise 153

Exercise 154

648648

Exercise 155

3,5563,556

Solution

3,560

Exercise 156

2,7342,734

In the following exercises, round to the nearest hundred.

Exercise 157

38,97538,975

Solution

39,000

Exercise 158

26,84926,849

Exercise 159

81,48681,486

Solution

81,500

Exercise 160

75,99275,992

Add Whole Numbers

Use Addition Notation

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

Exercise 161

4+34+3

Solution

four plus three; the sum of four and three

Exercise 162

25+1825+18

Exercise 163

571+629571+629

Solution

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

Exercise 164

10,085+3,49210,085+3,492

Model Addition of Whole Numbers

In the following exercises, model the addition.

Exercise 165

Exercise 166

38+1438+14

Add Whole Numbers

In the following exercises, fill in the missing values in each chart.

Exercise 168

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.

Exercise 169

0+190+19 19+019+0

Solution

  1. 19
  2. 19

Exercise 170

0+4800+480 480+0480+0

Exercise 171

7+67+6 6+76+7

Solution

  1. 13
  2. 13

Exercise 172

23+1823+18 18+2318+23

Exercise 173

Exercise 174

63+2963+29

Exercise 175

Exercise 176

375+591375+591

Exercise 177

7,281+12,5467,281+12,546

Solution

19,827

Exercise 178

5,280+16,324+9,7315,280+16,324+9,731

Translate Word Phrases to Math Notation

In the following exercises, translate each phrase into math notation and then simplify.

Exercise 179

the sum of 3030 and 1212

Solution

30 + 12; 42

Exercise 180

1111 increased by 88

Exercise 181

2525 more than 3939

Solution

39 + 25; 64

Exercise 182

total of 1515 and 5050

Add Whole Numbers in Applications

In the following exercises, solve.

Exercise 183

Shopping for an interview Nathan bought a new shirt, tie, and slacks to wear to a job interview. The shirt cost $24,$24, the tie cost $14,$14, and the slacks cost $38.$38. What was Nathan’s total cost?

Solution

$76

Exercise 184

Running Jackson ran 44 miles on Monday, 1212 miles on Tuesday, 11 mile on Wednesday, 88 miles on Thursday, and 55 miles on Friday. What was the total number of miles Jackson ran?

In the following exercises, find the perimeter of each figure.

Exercise 185

Exercise 186

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.

Exercise 187

145145

Solution

fourteen minus five; the difference of fourteen and five

Exercise 188

40154015

Exercise 189

351249351249

Solution

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

Exercise 190

5,7242,9185,7242,918

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

Exercise 191

Exercise 192

41294129

Subtract Whole Numbers

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

Exercise 193

Exercise 194

127127

Exercise 195

Exercise 196

46214621

Exercise 197

82598259

Solution

23

Exercise 198

1108711087

Exercise 199

539217539217

Solution

322

Exercise 200

415296415296

Exercise 201

1,0206401,020640

Solution

380

Exercise 202

8,3553,9478,3553,947

Exercise 203

10,0001510,00015

Solution

9,985

Exercise 204

54,92535,64754,92535,647

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

Exercise 205

the difference of nineteen and thirteen

Solution

19 − 13; 6

Exercise 206

subtract sixty-five from one hundred

Exercise 207

seventy-four decreased by eight

Solution

74 − 8; 66

Exercise 208

twenty-three less than forty-one

Subtract Whole Numbers in Applications

In the following exercises, solve.

Exercise 209

Temperature The high temperature in Peoria one day was 8686 degrees Fahrenheit and the low temperature was 2828 degrees Fahrenheit. What was the difference between the high and low temperatures?

Solution

58 degrees Fahrenheit

Exercise 210

Savings Lynn wants to go on a cruise that costs $2,485.$2,485. She has $948$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.

Exercise 211

8×58×5

Solution

eight times five the product of eight and five

Exercise 212

6·146·14

Exercise 213

(10)(95)(10)(95)

Solution

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

Exercise 214

54(72)54(72)

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

Exercise 215

Exercise 216

3×83×8

Multiply Whole Numbers

In the following exercises, fill in the missing values in each chart.

Exercise 218

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.

Exercise 219

Exercise 220

(256)0(256)0

Exercise 221

Exercise 222

(4,789)1(4,789)1

Exercise 223

7·47·4 4·74·7

Solution
  1. 28
  2. 28

Exercise 224

(25)(6)(25)(6)

Exercise 225

9,261×39,261×3

Solution

27,783

Exercise 226

48·7648·76

Exercise 227

64·1064·10

Solution

640

Exercise 228

1,000(22)1,000(22)

Exercise 229

162×493162×493

Solution

79,866

Exercise 230

(601)(943)(601)(943)

Exercise 231

3,624×5173,624×517

Solution

1,873,608

Exercise 232

10,538·2210,538·22

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

Exercise 233

the product of 1515 and 2828

Solution

15(28); 420

Exercise 234

ninety-four times thirty-three

Exercise 235

twice 575575

Solution

2(575); 1,150

Exercise 236

ten times two hundred sixty-four

Multiply Whole Numbers in Applications

In the following exercises, solve.

Exercise 237

Gardening Geniece bought 88 packs of marigolds to plant in her yard. Each pack has 66 flowers. How many marigolds did Geniece buy?

Solution

48 marigolds

Exercise 238

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 44 cups of rice, how many cups of water does she need?

Exercise 239

Multiplex There are twelve theaters at the multiplex and each theater has 150150 seats. What is the total number of seats at the multiplex?

Solution

1,800 seats

Exercise 240

Roofing Lewis needs to put new shingles on his roof. The roof is a rectangle, 3030 feet by 2424 feet. What is the area of the roof?

Divide Whole Numbers

Use Division Notation

Translate from math notation to words.

Exercise 241

54÷954÷9

Solution

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

Exercise 242

42/742/7

Exercise 243

728728

Solution

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

Exercise 244

648648

Model Division of Whole Numbers

In the following exercises, model.

Exercise 245

Exercise 246

312312

Divide Whole Numbers

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

Exercise 247

Exercise 248

328328

Exercise 249

Exercise 250

26262626

Exercise 251

Exercise 252

0÷520÷52

Exercise 253

100÷0100÷0

Solution

undefined

Exercise 254

35553555

Exercise 255

3828÷63828÷6

Solution

638

Exercise 256

311,519311,519

Exercise 257

750525750525

Solution

300 R5

Exercise 258

5,166÷425,166÷42

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

Exercise 259

the quotient of 6464 and 1616

Solution

64 ÷ 16; 4

Exercise 260

the quotient of 572572 and 5252

Divide Whole Numbers in Applications

In the following exercises, solve.

Exercise 261

Ribbon One spool of ribbon is 2727 feet. Lizbeth uses 33 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

Exercise 262

Juice One carton of fruit juice is 128128 ounces. How many 44 ounce cups can Shayla fill from one carton of juice?

Chapter Practice Test

Exercise 263

Determine which of the following numbers are

  1. counting numbers
  2. whole numbers.

0,4,870,4,87

Solution

  1. 4, 87
  2. 0, 4, 8

Exercise 264

Find the place value of the given digits in the number 549,362.549,362.

  1. 99
  2. 66
  3. 22
  4. 55

Exercise 265

Write each number as a whole number using digits.

  1. six hundred thirteen
  2. fifty-five thousand two hundred eight

Solution

  1. 613
  2. 55,208

Exercise 266

Round 25,84925,849 to the nearest hundred.

Simplify.

Exercise 267

Exercise 268

65426542

Exercise 269

Exercise 270

1,000×81,000×8

Exercise 271

90589058

Solution

32

Exercise 272

73+8973+89

Exercise 273

(0)(12,675)(0)(12,675)

Solution

0

Exercise 274

634+255634+255

Exercise 275

Exercise 276

81288128

Exercise 277

1457914579

Solution

66

Exercise 278

299+836299+836

Exercise 279

7·4757·475

Solution

3,325

Exercise 280

8,528+7048,528+704

Exercise 281

35(14)35(14)

Solution

490

Exercise 282

260260

Exercise 283

733291733291

Solution

442

Exercise 284

4,9161,5384,9161,538

Exercise 285

495÷45495÷45

Solution

11

Exercise 286

52×98352×983

Translate each phrase to math notation and then simplify.

Exercise 287

The sum of 1616 and 5858

Solution

16 + 58; 74

Exercise 288

The product of 99 and 1515

Exercise 289

The difference of 3232 and 1818

Solution

32 − 18; 14

Exercise 290

The quotient of 6363 and 2121

Exercise 291

Twice 524524

Solution

2(524); 1,048

Exercise 292

2929 more than 3232

Exercise 293

5050 less than 300300

Solution

300 − 50; 250

In the following exercises, solve.

Exercise 294

LaVelle buys a jumbo bag of 8484 candies to make favor bags for her son’s party. If she wants to make 1212 bags, how many candies should she put in each bag?

Exercise 295

Last month, Stan’s take-home pay was $3,816$3,816 and his expenses were $3,472.$3,472. How much of his take-home pay did Stan have left after he paid his expenses?

Solution

Stan had $344 left.

Exercise 296

Each class at Greenville School has 2222 children enrolled. The school has 2424 classes. How many children are enrolled at Greenville School?

Exercise 297

Clayton walked 1212 blocks to his mother’s house, 66 blocks to the gym, and 99 blocks to the grocery store before walking the last 33 blocks home. What was the total number of blocks that Clayton walked?

Solution

Clayton walked 30 blocks.

Glossary

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.

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