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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

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Kendra Reply
Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee 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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