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Prealgebra

1.5 Divide Whole Numbers

Prealgebra1.5 Divide Whole Numbers
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

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
Be Prepared 1.4

Before you get started, take this readiness quiz.

  1. Multiply: 27·3.27·3.
    If you missed this problem, review Example 1.44.
  2. Subtract: 4326.4326.
    If you missed this problem, review Example 1.32
  3. Multiply: 62(87).62(87).
    If you missed this problem, review Example 1.45.

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 1.13 and want to package them in bags with 44 cookies in each bag. How many bags would we need?

An image of three rows of four cookies to show twelve cookies.
Figure 1.13

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.

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 1.56

Translate from math notation to words.

64÷864÷8 427427 428428

Try It 1.111

Translate from math notation to words:

84÷784÷7 186186 824824

Try It 1.112

Translate from math notation to words:

72÷972÷9 213213 654654

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.

Manipulative Mathematics

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

Example 1.57

Model the division: 24÷8.24÷8.

Try It 1.113

Model: 24÷6.24÷6.

Try It 1.114

Model: 42÷7.42÷7.

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 1.57, we know 24÷8=324÷8=3 is correct because 3·8=24.3·8=24.

Example 1.58

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

Try It 1.115

Divide. Then check by multiplying:

54÷654÷6 279279

Try It 1.116

Divide. Then check by multiplying:

369369 840840

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.

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
Table 1.6

Example 1.59

Divide. Then check by multiplying:

  1. 11÷1111÷11
  2. 191191
  3. 1717
Try It 1.117

Divide. Then check by multiplying:

14÷1414÷14 271271

Try It 1.118

Divide. Then check by multiplying:

161161 1414

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.

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
Table 1.7

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 1.60

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

Try It 1.119

Divide. Then check by multiplying:

0÷20÷2 17/017/0

Try It 1.120

Divide. Then check by multiplying:

0÷60÷6 13/013/0

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

It does, so our answer is correct.

How To

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 1.61

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

Try It 1.121

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

Try It 1.122

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

Example 1.62

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

Try It 1.123

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

Try It 1.124

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

Example 1.63

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

Try It 1.125

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

Try It 1.126

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

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 1.14.

An image of 28 cookies placed at random.
Figure 1.14

Try to put the cookies in groups of eight as in Figure 1.15.

An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.
Figure 1.15

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 1.64

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

Try It 1.127

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

Try It 1.128

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

Example 1.65

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

Try It 1.129

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

Try It 1.130

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

Example 1.66

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

Try It 1.131

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

Try It 1.132

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

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 1.8.

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
Table 1.8

Example 1.67

Translate and simplify: the quotient of 5151 and 17.17.

Try It 1.133

Translate and simplify: the quotient of 9191 and 13.13.

Try It 1.134

Translate and simplify: the quotient of 5252 and 13.13.

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 1.68

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?

Try It 1.135

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?

Try It 1.136

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?

Section 1.5 Exercises

Practice Makes Perfect

Use Division Notation

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

343.

54÷954÷9

344.

567567

345.

328328

346.

642642

347.

48÷648÷6

348.

639639

349.

763763

350.

72÷872÷8

Model Division of Whole Numbers

In the following exercises, model the division.

351.

15÷515÷5

352.

10÷510÷5

353.

147147

354.

186186

355.

420420

356.

315315

357.

24÷624÷6

358.

16÷416÷4

Divide Whole Numbers

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

359.

18÷218÷2

360.

14÷214÷2

361.

273273

362.

303303

363.

428428

364.

436436

365.

455455

366.

355355

367.

72/872/8

368.

864864

369.

357357

370.

42÷742÷7

371.

15151515

372.

12121212

373.

43÷4343÷43

374.

37÷3737÷37

375.

231231

376.

291291

377.

19÷119÷1

378.

17÷117÷1

379.

0÷40÷4

380.

0÷80÷8

381.

5050

382.

9090

383.

260260

384.

320320

385.

120120

386.

160160

387.

72÷372÷3

388.

57÷357÷3

389.

968968

390.

786786

391.

54655465

392.

45284528

393.

924÷7924÷7

394.

861÷7861÷7

395.

5,22665,2266

396.

3,77683,7768

397.

431,324431,324

398.

546,855546,855

399.

7,209÷37,209÷3

400.

4,806÷34,806÷3

401.

5,406÷65,406÷6

402.

3,208÷43,208÷4

403.

42,81642,816

404.

63,62463,624

405.

91,881991,8819

406.

83,256883,2568

407.

2,470÷72,470÷7

408.

3,741÷73,741÷7

409.

855,305855,305

410.

951,492951,492

411.

431,1745431,1745


412.

297,2774297,2774

413.

130,016÷3130,016÷3

414.

105,609÷2105,609÷2

415.

155,735155,735

416.

4,933214,93321

417.

56,883÷6756,883÷67

418.

43,725/7543,725/75

419.

30,14431430,144314

420.

26,145÷41526,145÷415

421.

273542,195273542,195

422.

816,243÷462816,243÷462

Mixed Practice

In the following exercises, simplify.

423.

15(204)15(204)

424.

74·39174·391

425.

256184256184

426.

305262305262

427.

719+341719+341

428.

647+528647+528

429.

2587525875

430.

1104÷231104÷23

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

431.

the quotient of 4545 and 1515

432.

the quotient of 6464 and 1616

433.

the quotient of 288288 and 2424

434.

the quotient of 256256 and 3232

Divide Whole Numbers in Applications

In the following exercises, solve.

435.

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?

436.

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?

437.

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?

438.

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?

439.

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?

440.

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.

441.

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?

442.

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?

443.

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?

444.

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

445.

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?

446.

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?

447.

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?

448.

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

449.

Explain how you use the multiplication facts to help with division.

450.

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

451.

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?

452.

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?

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