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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

A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Mikaela Reply
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?
Sam Reply
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Mckenzie Reply
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?
Jake Reply
enrique borrowed $23,500 to buy a car he pays his uncle 2% interest on the $4,500 he borrowed from him and he pays the bank 11.5% interest on the rest. what average interest rate does he pay on the total $23,500
Nakiya Reply
13.5
Pervaiz
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Bridget Reply
The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used. Find the payment for a month when Randy used 15 units of water.
Bridget
help me understand graphs
Marlene Reply
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
what's the formula given? f(x)=?
Melissa
It shows a graph that I wish I could send photo of to you on here
Marlene
Which problem specifically?
Melissa
which problem?
Melissa
I don't know any to be honest. But whatever you can help me with for I can practice will help
Marlene
I got it. sorry, was out and about. I'll look at it now.
Melissa
Thank you. I appreciate it because my teacher assumes I know this. My teacher before him never went over this and several other things.
Marlene
I just responded.
Melissa
Thank you
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
WENDY Reply
State the question clearly please
Rich
write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
0.81818
Rich
5/100 = .05 but Rich is right that 9/11 = .81818
Melissa
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
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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