Recall that the bottom number in a fraction is the denominator, where "denominator" refers to the name (that is, to the type) of the fraction. Halves are different from thirds, which are different from fourths, etc. The top number is the numerator, where "numerator" refers to the count. For example, the fraction tells you that you're working with a thing called "sevenths", and that you have five of them.
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You've heard the expression, "You can't add apples and oranges". In the context of fractions, this means that we cannot add different types of fractions. To add two fractions, they must have the same (that is, a shared or "common") denominator. We cannot combine and , because (for the purposes of this example) halves are apples and thirds are oranges.
But it's perfectly okay to add apples to apples. So let's start with adding fractions that are of the same type (that is, that share the same denominator).
To add two fractions that have the same denominator, follow these steps:
Adding two fractions that have the same denominator is really just that easy: add across the top, and simplify if you can.
These two fractions are of the same type: they're both fifths. The first fraction has three of them; the second fraction has one of them. That means that, in total, we have four of them. In other words:
There are no factors shared by 4 and 5, so this fraction cannot be reduced. Then my answer is:
Subtracting same-denominator fractions works the same way: you simply subtract across the top and, if possible, simplify your answer.
These two fractions are of the same type: they're both eighths. I've got seven of them, from which I'm subtracting five of them. This should leave me with two of them. Of course, each one-eighth is half of a one-fourth, so two of the eighths would be a single fourth.
Note: This isn't important, but fractions with a numerator of 1 are called "unit" fractions, because the number 1 is the basic unit in arithmetic. The ancient Egyptians only worked with unit fractions.
Believe it or not, many otherwise-advanced ancient civilizations (including the ancient Egyptians) never figured out the concept of common denominators. (Europeans were quite late to the party when it came to fractions. Egyptian, and later Roman, notation was poor, but they at least understood some of the basic concepts.) So don't feel bad if you have some trouble with the computations.
To add fractions which have different (or "unlike") denominators, follow these steps:
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Mathematically, as long as you simplify your final fraction after adding the numerators, then there is no difference between using the Least Common Denominator (the lcm) and using the product of the original denominators. Howevwer, there can be a difference in practice, which may influence your preference.
The lcm will always be the smallest ("least") shared ("common") denominator, but you may feel like taking the time to find the lcm is a waste; you'd rather just simplify at the end. On the other hand, using the lcm will make sure that you have as little simplifying (that is, reducing) at the end of your work. But you'd need to check for any possible simplification anyway, so maybe it's a toss-up...?
You should be aware, though, that some instructors absolutely insist on you always finding and using the lcm, so you could lose points if you don't. (In my classes, I really don't care, as long as your steps are valid and your answer is correct. But that's just me.) Many times, the lcm will turn out to be the product of the denominators, anyway. Use your best judgment.
A useful form of 1 is a fraction formed by a number divided by itself. Multiplying a fraction by 1 doesn't change its value, but multiplying a fraction by a form of 1 will change its denominator (and its numerator), so that you can convert a given type of fraction (thirds, fifths, eighths, whatever) into a type that better suits the given context. We multiply fractions by useful forms of 1 to convert fractions to common denominators.
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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. The prime-factors table makes it easy to find the lcm:
4: 2*2 5: 5 ---:----------- LCM: 2*2*5 = 20
To be able to add these fractions, I need 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, the 4 needs to become a 20, so I will multiply by :
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 fraction's value is stated.
In the case of the second fraction, the 5 needs to become a 20, so I will multiply by :
The fourths and fifths are now both twentieths; I'm finally in an all-apples situation. Now I can finally 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
In this case, the lcm also turned out to be the product of the two denominators.
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 polynomial fractions, which are called "rational expressions".
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 neat in their 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. So my answer is:
Subtracting fractions works the same way, other than that you'll be subtracting the numerators (and then simplifying, if possible) instead of adding them.
First, I'll find the lcm of the two denominators:
25: 5*5 35: 5 *7 ---:------------ LCM: 5*5*7 = 175
To convert these fractions 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; the denominator, 175, factors as 5×5×7. There are no common factors, so there's nothing I can cancel; the fraction can't be further simplified.
Then my final answer is
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