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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 [link] , the LCD, 36, has two factors of 2 and two factors of 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 [link] .

Subtract: 7 15 19 24 .

Solution

Do the fractions have a common denominator? No, so we need to find the LCD.

Find the LCD. .
Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
Rewrite as equivalent fractions with the LCD. .
Simplify. .
Subtract. 39 120
Check to see if the answer can be simplified. 13 3 40 3
Both 39 and 120 have a factor of 3.
Simplify. 13 40

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

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Subtract: 13 24 17 32 .

1 96

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Subtract: 21 32 9 28 .

75 224

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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.

Add: 3 5 + x 8 .

Solution

The fractions have different denominators.

.
Find the LCD. .
Rewrite as equivalent fractions with the LCD. .
Simplify. .
Add. .

Remember, we can only add like terms: 24 and 5 x are not like terms.

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Add: y 6 + 7 9 .

9 y + 42 54

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Add: x 6 + 7 15 .

15 x + 42 135

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We now have all four operations for fractions. [link] summarizes fraction operations .

Fraction Multiplication Fraction Division
a b · c d = a c b d

Multiply the numerators and multiply the denominators
a b ÷ c d = a b · d c

Multiply the first fraction by the reciprocal of the second.
Fraction Addition Fraction Subtraction
a c + b c = a + b c

Add the numerators and place the sum over the common denominator.
a c b c = a b c

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.

Simplify: 5 x 6 3 10 5 x 6 · 3 10 .

Solution

First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

  1. What is the operation? The operation is subtraction.
    Do the fractions have a common denominator? No. 5 x 6 3 10 Rewrite each fraction as an equivalent fraction with the LCD. 5 x · 5 6 · 5 3 · 3 10 · 3 25 x 30 9 30 Subtract the numerators and place the difference over the common denominators. 25 x 9 30 Simplify, if possible There are no common factors. The fraction is simplified.
  2. What is the operation? Multiplication.
    5 x 6 · 3 10 To multiply fractions, multiply the numerators and multiply the denominators. 5 x · 3 6 · 10 Rewrite, showing common factors. Remove common factors. 5 x · 3 2 · 3 · 2 · 5 Simplify. x 4

Notice we needed an LCD to add 5 x 6 3 10 , but not to multiply 5 x 6 · 3 10 .

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Questions & Answers

a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
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usman
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Peggie
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Frank
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Ramil
copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
Sandra Reply
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Penny Reply
2
Letha
4
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4
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4
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C+4=8 -4 -4 C =4
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Letha
4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Josh Reply
Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
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patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
Michelle Reply
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
Amber Reply
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bismark
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Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone, it would take her 120 days. How long would it take Deb to write the book alone?
Hailey Reply
Tell Deb to write the book alone and report back on how long it took her. Deb could get bored and stop, she could get sick next week and prolong the time, she could die next month. This question is impossible to answer.
vV
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tyler
is there a thumbs up mathematical symbol in algebra? 👍²= vV 90days x 120days =10800 120days - 90days = 30 10800days/30days= 360days? (ad) derall=1440 mins (90x ad )+(120 x ad )/(120 x ad ))-((90d x ad)= you go glen coco! I'll see my self out
Nick
Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
genevieve Reply
6.25
Ciid
6.25
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Twila
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
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Ciid
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj
tyler
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tyler
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tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
help
Twila
draw a diagram first
John
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 many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
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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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