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Elementary Algebra

1.6 Add and Subtract Fractions

Elementary Algebra1.6 Add and Subtract Fractions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index

Learning Objectives

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

  • Add or subtract fractions with a common denominator
  • Add or subtract fractions with different denominators
  • Use the order of operations to simplify complex fractions
  • Evaluate variable expressions with fractions
Be Prepared 1.6

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions.

Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Fraction Addition and Subtraction

If a,b,andca,b,andc are numbers where c≠0,c≠0, then

ac+bc=a+bcandac−bc=a−bcac+bc=a+bcandac−bc=a−bc

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

Manipulative Mathematics

Doing the Manipulative Mathematics activities “Model Fraction Addition” and “Model Fraction Subtraction” will help you develop a better understanding of adding and subtracting fractions.

Example 1.77

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

Try It 1.153

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

Try It 1.154

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

Example 1.78

Find the difference: −2324−1324.−2324−1324.

Try It 1.155

Find the difference: −1928−728.−1928−728.

Try It 1.156

Find the difference: −2732−132.−2732−132.

Example 1.79

Simplify: −10x−4x.−10x−4x.

Try It 1.157

Find the difference: −9x−7x.−9x−7x.

Try It 1.158

Find the difference: −17a−5a.−17a−5a.

Now we will do an example that has both addition and subtraction.

Example 1.80

Simplify: 38+(−58)−18.38+(−58)−18.

Try It 1.159

Simplify: 25+(−49)−79.25+(−49)−79.

Try It 1.160

Simplify: 59+(−49)−79.59+(−49)−79.

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Finding the Least Common Denominator” will help you develop a better understanding of the LCD.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

Example 1.81

How to Add or Subtract Fractions

Add: 712+518.712+518.

Try It 1.161

Add: 712+1115.712+1115.

Try It 1.162

Add: 1315+1720.1315+1720.

How To

Add or Subtract Fractions.

  1. Step 1. Do they have a common denominator?
    • Yes—go to step 2.
    • No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
  2. Step 2. Add or subtract the fractions.
  3. Step 3. Simplify, if possible.

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked “missing factors.” The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).

In Example 1.81, the LCD, 36, has two factors of 2 and two factors of 3.3.

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.

We will apply this method as we subtract the fractions in Example 1.82.

Example 1.82

Subtract: 715−1924.715−1924.

Try It 1.163

Subtract: 1324−1732.1324−1732.

Try It 1.164

Subtract: 2132−928.2132−928.

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

Example 1.83

Add: 35+x8.35+x8.

Try It 1.165

Add: y6+79.y6+79.

Try It 1.166

Add: x6+715.x6+715.

We now have all four operations for fractions. Table 1.26 summarizes fraction operations.

Fraction Multiplication Fraction Division
ab·cd=acbdab·cd=acbd

Multiply the numerators and multiply the denominators		 ab÷cd=ab·dcab÷cd=ab·dc

Multiply the first fraction by the reciprocal of the second.
Fraction Addition Fraction Subtraction
ac+bc=a+bcac+bc=a+bc

Add the numerators and place the sum over the common denominator.		 ac−bc=a−bcac−bc=a−bc

Subtract the numerators and place the difference over the common denominator.
To multiply or divide fractions, an LCD is NOT needed.
 To add or subtract fractions, an LCD is needed.
Table 1.26

Example 1.84

Simplify: ⓐ 5x6−3105x6−310 ⓑ 5x6·310.5x6·310.

Try It 1.167

Simplify: ⓐ 3a4−893a4−89 ⓑ 3a4·89.3a4·89.

Try It 1.168

Simplify: ⓐ 4k5−164k5−16 ⓑ 4k5·16.4k5·16.

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction 34583458 by dividing 3434 by 58.58.

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

Example 1.85

How to Simplify Complex Fractions

Simplify: (12)24+32.(12)24+32.

Try It 1.169

Simplify: (13)223+2.(13)223+2.

Try It 1.170

Simplify: 1+42(14)2.1+42(14)2.

How To

Simplify Complex Fractions.

  1. Step 1. Simplify the numerator.
  2. Step 2. Simplify the denominator.
  3. Step 3. Divide the numerator by the denominator. Simplify if possible.

Example 1.86

Simplify: 12+2334−16.12+2334−16.

Try It 1.171

Simplify: 13+1234−13.13+1234−13.

Try It 1.172

Simplify: 23−1214+13.23−1214+13.

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Example 1.87

Evaluate x+13x+13 when ⓐ x=−13x=−13 ⓑ x=−34.x=−34.

Try It 1.173

Evaluate x+34x+34 when ⓐ x=−74x=−74 ⓑ x=−54.x=−54.

Try It 1.174

Evaluate y+12y+12 when ⓐ y=23y=23 ⓑ y=−34.y=−34.

Example 1.88

Evaluate −56−y−56−y when y=−23.y=−23.

Try It 1.175

Evaluate −12−y−12−y when y=−14.y=−14.

Try It 1.176

Evaluate −38−y−38−y when x=−52.x=−52.

Example 1.89

Evaluate 2x2y2x2y when x=14x=14 and y=−23.y=−23.

Try It 1.177

Evaluate 3ab23ab2 when a=−23a=−23 and b=−12.b=−12.

Try It 1.178

Evaluate 4c3d4c3d when c=−12c=−12 and d=−43.d=−43.

The next example will have only variables, no constants.

Example 1.90

Evaluate p+qrp+qr when p=−4,q=−2,andr=8.p=−4,q=−2,andr=8.

Try It 1.179

Evaluate a+bca+bc when a=−8,b=−7,andc=6.a=−8,b=−7,andc=6.

Try It 1.180

Evaluate x+yzx+yz when x=9,y=−18,andz=−6.x=9,y=−18,andz=−6.

Section 1.6 Exercises

Practice Makes Perfect

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

425.

613+513613+513

426.

415+715415+715

427.

x4+34x4+34

428.

8q+6q8q+6q

429.

−316+(−716)−316+(−716)

430.

−516+(−916)−516+(−916)

431.

−817+1517−817+1517

432.

−919+1719−919+1719

433.

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

434.

512+(−712)+(−1112)512+(−712)+(−1112)

In the following exercises, subtract.

435.

1115−7151115−715

436.

913−413913−413

437.

1112−5121112−512

438.

712−512712−512

439.

1921−4211921−421

440.

1721−8211721−821

441.

5y8−785y8−78

442.

11z13−81311z13−813

443.

−23u−15u−23u−15u

444.

−29v−26v−29v−26v

445.

−35−(−45)−35−(−45)

446.

−37−(−57)−37−(−57)

447.

−79−(−59)−79−(−59)

448.

−811−(−511)−811−(−511)

Mixed Practice

In the following exercises, simplify.

449.

−518·910−518·910

450.

−314·712−314·712

451.

n5−45n5−45

452.

611−s11611−s11

453.

−724+224−724+224

454.

−518+118−518+118

455.

815÷125815÷125

456.

712÷928712÷928

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

457.

12+1712+17

458.

13+1813+18

459.

13−(−19)13−(−19)

460.

14−(−18)14−(−18)

461.

712+58712+58

462.

512+38512+38

463.

712−916712−916

464.

716−512716−512

465.

23−3823−38

466.

56−3456−34

467.

−1130+2740−1130+2740

468.

−920+1730−920+1730

469.

−1330+2542−1330+2542

470.

−2330+548−2330+548

471.

−3956−2235−3956−2235

472.

−3349−1835−3349−1835

473.

−23−(−34)−23−(−34)

474.

−34−(−45)−34−(−45)

475.

1+781+78

476.

1−3101−310

477.

x3+14x3+14

478.

y2+23y2+23

479.

y4−35y4−35

480.

x5−14x5−14

Mixed Practice

In the following exercises, simplify.

481.

ⓐ 23+1623+16 ⓑ 23÷1623÷16

482.

ⓐ −25−18−25−18 ⓑ −25·18−25·18

483.

ⓐ 5n6÷8155n6÷815 ⓑ 5n6−8155n6−815

484.

ⓐ 3a8÷7123a8÷712 ⓑ 3a8−7123a8−712

485.

−38÷(−310)−38÷(−310)

486.

−512÷(−59)−512÷(−59)

487.

−38+512−38+512

488.

−18+712−18+712

489.

56−1956−19

490.

59−1659−16

491.

−715−y4−715−y4

492.

−38−x11−38−x11

493.

1112a·9a161112a·9a16

494.

10y13·815y10y13·815y

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

495.

23+42(23)223+42(23)2

496.

33−32(34)233−32(34)2

497.

(35)2(37)2(35)2(37)2

498.

(34)2(58)2(34)2(58)2

499.

213+15213+15

500.

514+13514+13

501.

78−2312+3878−2312+38

502.

34−3514+2534−3514+25

503.

12+23·51212+23·512

504.

13+25·3413+25·34

505.

1−35÷1101−35÷110

506.

1−56÷1121−56÷112

507.

23+16+3423+16+34

508.

23+14+3523+14+35

509.

38−16+3438−16+34

510.

25+58−3425+58−34

511.

12(920−415)12(920−415)

512.

8(1516−56)8(1516−56)

513.

58+16192458+161924

514.

16+310143016+3101430

515.

(59+16)÷(23−12)(59+16)÷(23−12)

516.

(34+16)÷(58−13)(34+16)÷(58−13)

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

517.

x+(−56)x+(−56) when
ⓐ x=13x=13
ⓑ x=−16x=−16

518.

x+(−1112)x+(−1112) when
ⓐ x=1112x=1112
ⓑ x=34x=34

519.

x−25x−25 when
ⓐ x=35x=35
ⓑ x=−35x=−35

520.

x−13x−13 when
ⓐ x=23x=23
ⓑ x=−23x=−23

521.

710−w710−w when
ⓐ w=12w=12
ⓑ w=−12w=−12

522.

512−w512−w when
ⓐ w=14w=14
ⓑ w=−14w=−14

523.

2x2y32x2y3 when x=−23x=−23 and y=−12y=−12

524.

8u2v38u2v3 when u=−34u=−34 and v=−12v=−12

525.

a+ba−ba+ba−b when a=−3,b=8a=−3,b=8

526.

r−sr+sr−sr+s when r=10,s=−5r=10,s=−5

Everyday Math

527.

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs 1212 yard of print fabric and 3838 yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

528.

Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs 1212 cup of sugar for the chocolate chip cookies and 1414 of sugar for the oatmeal cookies. How much sugar does she need altogether?

Writing Exercises

529.

Why do you need a common denominator to add or subtract fractions? Explain.

530.

How do you find the LCD of 2 fractions?

Self Check

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

This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “add and subtract fractions with different denominators,” “identify and use fraction operations,” “use the order of operations to simplify complex fractions,” and “evaluate variable expressions with fractions.” The rest of the cells are blank.

ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

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