To add fractions, you have to have "common" (shared) denominators. As the proverb says, you can only add apples to apples, not apples to oranges. In the context of adding fractions, you can't combine, say, 1/4 and 2/5. Because they are fractions of different types (one being fourths and the other being fifths), they are "apples and oranges". To add them, you first have to convert to an all-apples denominator; in this case, you'd convert to twentieths, getting 5/20 and 8/20 as your all-apples fractions.
Believe it or not, many otherwise-advanced ancient civilizations (such as the ancient Egyptians) never figured out the common-denominator concept. So don't feel bad if you have some trouble with the computations!
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The basic idea with converting to common denominators is to multiply fractions by useful forms of 1. What does this mean? Take a look:
Before I can add these fractions, I have to find their common denominator. The lowest (smallest) common denominator is just the Least Common Multiple (LCM) of the two denominators, 4 and 5. The prime factorizations and LCM of the denominators 4 and 5 are:
4: 2*2 5: 5 ---:----------- LCM: 2*2*5 = 20
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In other words, I have to convert the fourths and fifths into twentieths. I'll do this by multiplying by a useful form of 1. In the case of the first fraction, ^{1}/_{4}, the 4 needs to become a 20, so I need to multiply the 4 by 5. To keep the fraction equal to its original value, I'll have to multiply the top by 5, too. In other words, I'll multiply the fraction by ^{5}/_{5}, which is just a useful form of the number 1:
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Because I multiplied by (a useful form of) 1, I haven't changed the actual value of the fraction. All I've changed is how the value is stated.
In the case of the second fraction, ^{2}/_{5}, the 5 needs to become a 20, so I have to multiply the 5 by 4. To keep the fraction equal to the same value, I also have to multiply the top by 4, too. In other words, I'll multiply by ^{4}/_{4}, which is just a useful form of 1:
The fourths and fifths are now both twentieths; I'm finally in an all-apples situation. Only now can I actually add the fractions. To add these "apples", I add the numerators:
The numerator, 13, is prime, and it isn't a factor of 20, so there's no cancellation that I can do.
My simplified final answer is .
By the way, your calculator may be able to do all of this for you; check your manual. But make sure you at least understand the basic idea, because you'll need this process later in algebra, when you get to fractions with polynomials, called "rational expressions".
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First, I'll find the LCM of the two denominators:
15: 3*5 5: 5 ---:--------- LCM: 3*5 = 15
Since 5 is a factor of 15, then the LCM is 15; in particular, one of the fractions is already in LCM form. I'll convert the other fraction to this common denominator, add, and, if possible, simplify:
There are no common factors, so nothing simplifies.
My final answer is .
First I'll find the LCM of the two denominators:
8: 2*2*2 6: 2 *3 ---:------------- LCM: 2*2*2*3 = 24
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Notice that 8 and 6 both have 2 as a factor. The point of lining the factors up nice and neatly in columns, as I've done above, is to help avoid over-duplication of factors when finding the LCM. Be careful: there are only three 2's in the LCM, not four.
To convert the first fraction to a denominator of 24, I'll multiply, top and bottom, by 3. To convert the second fraction's denominator, I'll multiply, top and bottom, by 4.
The instructions don't say to express the answer in mixed-number form, so I'll leave it as an improper fraction. There are no common factors between the numerator and denominator, so I can't simplify any further.
First, I'll find the LCM of the three denominators:
7: 7 52: 2*2 *13 4: 2*2 ---:--------------- LCM: 2*2*7*13 = 364
Now I'll convert the three fractions to the common denominator, add, and then see if I can simplify.
Because 4 was a common factor of 1072 and 364, I was able to cancel this out and simplify to get my final answer:
First, I'll find the LCM of the two denominators:
25: 5*5 35: 5 *7 ---:------------ LCM: 5*5*7 = 175
To convert to the LCM, I'll multiply the first fraction, top and bottom, by 7, and the second fraction, top and bottom, by 5.
The numerator, 106, factors as 2×53, and 53 is prime, so there's nothing I can cancel; the fraction can't be further simplified.
Then my final answer is .
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