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Simplify: 3 a 4 8 9 3 a 4 · 8 9 .

27 a 32 36 2 a 3

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Simplify: 4 k 5 1 6 4 k 5 · 1 6 .

24 k 5 30 2 k 15

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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 3 4 5 8 by dividing 3 4 by 5 8 .

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.

How to simplify complex fractions

Simplify: ( 1 2 ) 2 4 + 3 2 .

Solution

In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have “Step 1. Simplify the numerator. Remember one half squared means one half times one half.” To the right of this, we have the quantity (1/2) squared all over the quantity (4 plus 3 squared). Then, we have 1/4 over the quantity (4 plus 3 squared). The next line’s direction reads “Step 2. Simplify the denominator.” To the right of this, we have 1/4 over the quantity (4 plus 9), under which we have 1/4 over 13. The final step is “Step 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.” To the right we have 1/4 divided by 13. Then we have 1/4 times 1/13, which equals 1/52.
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Simplify: ( 1 3 ) 2 2 3 + 2 .

1 90

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Simplify: 1 + 4 2 ( 1 4 ) 2 .

272

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Simplify complex fractions.

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

Simplify: 1 2 + 2 3 3 4 1 6 .

Solution

It may help to put parentheses around the numerator and the denominator.

( 1 2 + 2 3 ) ( 3 4 1 6 ) Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12). ( 3 6 + 4 6 ) ( 9 12 2 12 ) Simplify. ( 7 6 ) ( 7 12 ) Divide the numerator by the denominator. 7 6 ÷ 7 12 Simplify. 7 6 · 12 7 Divide out common factors. 7 · 6 · 2 6 · 7 Simplify. 2

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Simplify: 1 3 + 1 2 3 4 1 3 .

2

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Simplify: 2 3 1 2 1 4 + 1 3 .

2 7

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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.

Evaluate x + 1 3 when x = 1 3 x = 3 4 .

  1. To evaluate x + 1 3 when x = 1 3 , substitute 1 3 for x in the expression.
    .
    . .
    Simplify. 0


  2. To evaluate x + 1 3 when x = 3 4 , we substitute 3 4 for x in the expression.
    .
    . .
    Rewrite as equivalent fractions with the LCD, 12. .
    Simplify. .
    Add. 5 12
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Evaluate x + 3 4 when x = 7 4 x = 5 4 .

−1 1 2

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Evaluate y + 1 2 when y = 2 3 y = 3 4 .

7 6 1 12

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Evaluate 5 6 y when y = 2 3 .

Solution

.
. .
Rewrite as equivalent fractions with the LCD, 6.          .
Subtract. .
Simplify. 1 6
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Evaluate 1 2 y when y = 1 4 .

1 4

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Evaluate 3 8 y when x = 5 2 .

17 8

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Evaluate 2 x 2 y when x = 1 4 and y = 2 3 .

Solution

Substitute the values into the expression.

2 x 2 y
. .
Simplify exponents first. 2 ( 1 16 ) ( 2 3 )
Multiply. Divide out the common factors. Notice we write 16 as 2 2 4 to make it easy to remove common factors. 2 1 2 2 2 4 3
Simplify. 1 12

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Evaluate 3 a b 2 when a = 2 3 and b = 1 2 .

1 2

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Evaluate 4 c 3 d when c = 1 2 and d = 4 3 .

2 3

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The next example will have only variables, no constants.

Evaluate p + q r when p = −4 , q = −2 , and r = 8 .

Solution

To evaluate p + q r when p = −4 , q = −2 , and r = 8 , we substitute the values into the expression.

p + q r
. .
Add in the numerator first. −6 8
Simplify. 3 4

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Evaluate a + b c when a = −8 , b = −7 , and c = 6 .

5 2

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Evaluate x + y z when x = 9 , y = −18 , and z = −6 .

3 2

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Key concepts

  • Fraction Addition and Subtraction: If a , b , and c are numbers where c 0 , then
    a c + b c = a + b c and a c b c = a b c .
    To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
  • Strategy for Adding or Subtracting Fractions
    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. Add or subtract the fractions.
    3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
  • Simplify Complex Fractions
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Divide the numerator by the denominator. Simplify if possible.

Practice makes perfect

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

3 16 + ( 7 16 )

5 8

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6 13 + ( 10 13 ) + ( 12 13 )

16 13

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5 12 + ( 7 12 ) + ( 11 12 )

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In the following exercises, subtract.

5 y 8 7 8

5 y 7 8

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23 u 15 u

38 u

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3 5 ( 4 5 )

1 5

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7 9 ( 5 9 )

2 9

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8 11 ( 5 11 )

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Mixed Practice

In the following exercises, simplify.

5 18 · 9 10

1 4

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7 24 + 2 24

5 24

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Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

11 30 + 27 40

37 120

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13 30 + 25 42

17 105

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39 56 22 35

53 40

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2 3 ( 3 4 )

1 12

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y 4 3 5

4 y 12 20

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Mixed Practice

In the following exercises, simplify.

2 3 + 1 6 2 3 ÷ 1 6

5 6 4

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2 5 1 8 2 5 · 1 8

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5 n 6 ÷ 8 15 5 n 6 8 15

25 n 16 25 n 16 30

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3 a 8 ÷ 7 12 3 a 8 7 12

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3 8 ÷ ( 3 10 )

5 4

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7 15 y 4

−28 15 y 60

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Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

( 3 5 ) 2 ( 3 7 ) 2

49 25

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7 8 2 3 1 2 + 3 8

5 21

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12 ( 9 20 4 15 )

11 5

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( 5 9 + 1 6 ) ÷ ( 2 3 1 2 )

13 3

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( 3 4 + 1 6 ) ÷ ( 5 8 1 3 )

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Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

x + ( 5 6 ) when
x = 1 3
x = 1 6

1 2 −1

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x + ( 11 12 ) when
x = 11 12
x = 3 4

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x 2 5 when
x = 3 5
x = 3 5

1 5 −1

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x 1 3 when
x = 2 3
x = 2 3

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7 10 w when
w = 1 2
w = 1 2

1 5 6 5

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5 12 w when
w = 1 4
w = 1 4

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2 x 2 y 3 when x = 2 3 and y = 1 2

1 9

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8 u 2 v 3 when u = 3 4 and v = 1 2

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a + b a b when a = −3 , b = 8

5 11

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r s r + s when r = 10 , s = −5

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Everyday math

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs 1 2 yard of print fabric and 3 8 yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

7 8 yard

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Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs 1 2 cup of sugar for the chocolate chip cookies and 1 4 of sugar for the oatmeal cookies. How much sugar does she need altogether?

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Writing exercises

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

Answers may vary

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How do you find the LCD of 2 fractions?

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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?

Questions & Answers

4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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