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Prealgebra

4.4 Add and Subtract Fractions with Common Denominators

Prealgebra4.4 Add and Subtract Fractions with Common Denominators
  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:
  • Model fraction addition
  • Add fractions with a common denominator
  • Model fraction subtraction
  • Subtract fractions with a common denominator
Be Prepared 4.4

Before you get started, take this readiness quiz.

  1. Simplify: 2x+9+3x4.2x+9+3x4.
    If you missed this problem, review Example 2.22.
  2. Draw a model of the fraction 34.34.
    If you missed this problem, review Example 4.2.
  3. Simplify: 3+26.3+26.
    If you missed this problem, review Example 4.48.

Model Fraction Addition

How many quarters are pictured? One quarter plus 22 quarters equals 33 quarters.

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

142434one quarter+two quarters=three quarters142434one quarter+two quarters=three quarters

Let’s use fraction circles to model the same example, 14+24.14+24.

Start with one 1414 piece. . .
Add two more 1414pieces. . .
The result is 3434. . .

So again, we see that

14+24=3414+24=34

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Model Fraction Addition" will help you develop a better understanding of adding fractions

Example 4.52

Use a model to find the sum 38+28.38+28.

Try It 4.103

Use a model to find each sum. Show a diagram to illustrate your model.

18+4818+48

Try It 4.104

Use a model to find each sum. Show a diagram to illustrate your model.

16+4616+46

Add Fractions with a Common Denominator

Example 4.52 shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

Fraction Addition

If a,b,andca,b,andc are numbers where c0,c0, then

ac+bc=a+bcac+bc=a+bc

To add fractions with a common denominators, add the numerators and place the sum over the common denominator.

Example 4.53

Find the sum: 35+15.35+15.

Try It 4.105

Find each sum: 36+26.36+26.

Try It 4.106

Find each sum: 310+710.310+710.

Example 4.54

Find the sum: x3+23.x3+23.

Try It 4.107

Find the sum: x4+34.x4+34.

Try It 4.108

Find the sum: y8+58.y8+58.

Example 4.55

Find the sum: 9d+3d.9d+3d.

Try It 4.109

Find the sum: 7d+8d.7d+8d.

Try It 4.110

Find the sum: 6m+9m.6m+9m.

Example 4.56

Find the sum: 2n11+5n11.2n11+5n11.

Try It 4.111

Find the sum: 3p8+6p8.3p8+6p8.

Try It 4.112

Find the sum: 2q5+7q5.2q5+7q5.

Example 4.57

Find the sum: 312+(512).312+(512).

Try It 4.113

Find each sum: 415+(615).415+(615).

Try It 4.114

Find each sum: 521+(921).521+(921).

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 1212 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 712712 of the pizza) left in the box. If Leonardo eats 22 of these remaining pieces (or 212212 of the pizza), how much is left? There would be 55 pieces left (or 512512 of the pizza).

712212=512712212=512

Let’s use fraction circles to model the same example, 712212.712212.

Start with seven 112112 pieces. Take away two 112112 pieces. How many twelfths are left?

The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange.

Again, we have five twelfths, 512.512.

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Model Fraction Subtraction" will help you develop a better understanding of subtracting fractions.

Example 4.58

Use fraction circles to find the difference: 4515.4515.

Try It 4.115

Use a model to find each difference. Show a diagram to illustrate your model.

78487848

Try It 4.116

Use a model to find each difference. Show a diagram to illustrate your model.

56465646

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Fraction Subtraction

If a,b,andca,b,andc are numbers where c0,c0, then

acbc=abcacbc=abc

To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Example 4.59

Find the difference: 23241424.23241424.

Try It 4.117

Find the difference: 1928728.1928728.

Try It 4.118

Find the difference: 27321132.27321132.

Example 4.60

Find the difference: y616.y616.

Try It 4.119

Find the difference: x727.x727.

Try It 4.120

Find the difference: y141314.y141314.

Example 4.61

Find the difference: 10x4x.10x4x.

Try It 4.121

Find the difference: 9x7x.9x7x.

Try It 4.122

Find the difference: 17a5a.17a5a.

Now lets do an example that involves both addition and subtraction.

Example 4.62

Simplify: 38+(58)18.38+(58)18.

Try It 4.123

Simplify: 25+(45)35.25+(45)35.

Try It 4.124

Simplify: 59+(49)79.59+(49)79.

Section 4.4 Exercises

Practice Makes Perfect

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

254.

25+1525+15

255.

310+410310+410

256.

16+3616+36

257.

38+3838+38

Add Fractions with a Common Denominator

In the following exercises, find each sum.

258.

49+1949+19

259.

29+5929+59

260.

613+713613+713

261.

915+715915+715

262.

x4+34x4+34

263.

y3+23y3+23

264.

7p+9p7p+9p

265.

8q+6q8q+6q

266.

8b9+3b98b9+3b9

267.

5a7+4a75a7+4a7

268.

−12y8+3y8−12y8+3y8

269.

−11x5+7x5−11x5+7x5

270.

18+(38)18+(38)

271.

18+(58)18+(58)

272.

316+(716)316+(716)

273.

516+(916)516+(916)

274.

817+1517817+1517

275.

919+1719919+1719

276.

613+(1013)+(1213)613+(1013)+(1213)

277.

512+(712)+(1112)512+(712)+(1112)

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

278.

58285828

279.

56265626

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

280.

45154515

281.

45354535

282.

11157151115715

283.

913413913413

284.

11125121112512

285.

712512712512

286.

42119214211921

287.

8916989169

288.

y17917y17917

289.

x19819x19819

290.

5y8785y878

291.

11z1381311z13813

292.

8d3d8d3d

293.

7c7c7c7c

294.

23u15u23u15u

295.

29v26v29v26v

296.

6c75c76c75c7

297.

12d119d1112d119d11

298.

−4r135r13−4r135r13

299.

−7s37s3−7s37s3

300.

35(45)35(45)

301.

37(57)37(57)

302.

79(59)79(59)

303.

811(511)811(511)

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

304.

518·910518·910

305.

314·712314·712

306.

n545n545

307.

611s11611s11

308.

724+224724+224

309.

518+118518+118

310.

815÷125815÷125

311.

712÷928712÷928

Everyday Math

312.

Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has 610610 of a pound of nuts and 310310 of a pound of raisins. How much trail mix can he make?

313.

Baking Janet needs 5858 of a cup of flour for a recipe she is making. She only has 3838 of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?

Writing Exercises

314.

Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know.
Bits in case: 116116, 1818, ___, ___, 516516, 3838, ___, 1212, 916916, 5858.
Bits that fell out: 716716, 316316, 1414.

315.

After a party, Lupe has 512512 of a cheese pizza, 412412 of a pepperoni pizza, and 412412 of a veggie pizza left. Will all the slices fit into 11 pizza box? Explain your reasoning.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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