Elementary Algebra

Elementary Algebra1.6 Add and Subtract Fractions

### 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.

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

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. 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

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