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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

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
rectangular field solutions
Navin Reply
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
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
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence 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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