When adding (or subtracting)
fractions with the same denominators, add (or
subtract) their numerators.
Practice
Solutions
When the denominators are not the
same, you have to rewrite the fractions so that they do have the same denominator. There are two common
methods of doing this. The
first is the easiest. The second takes more effort but can result in smaller quantities and less
reducing. (When the denominators have no common divisors, these two methods are the same).
The easiest way to get a common
denominator is to multiply the first fraction by the second denominator over itself and the second
fraction by the first
denominator over itself.
Practice
Solutions
Our goal is to add/subtract two
fractions with the same denominator. In the
previous examples and practice problems, we found a common denominator.
Now we will find the least common denominator (LCD). For example in
we could compute.
But we
really only need to rewrite 1/3:
While 18
is a common denominator in the above example, 6 is the smallest common
denominator. When denominators get more complicated, either by being large or
having variables in them, you will find it easier to use the LCD to add or
subtract fractions. The solution might require less reducing, too.
In the
following practice problems one of the denominators will be the LCD; you only
need to rewrite the other.
Practice
Solutions
There
are a couple of ways of finding the LCD. Take for example
. We could list the multiples of 12 and 14—the first number
that appears on each list will be the LCD:
12, 24, 36, 48,
60, 72, 84 and 14, 28, 42, 56, 70, 84.
Because
84 is the first number on each list, 84 is the LCD for 1/12
and 9/14 . This method works fine as long as your lists
are not too long. But what if your denominators are 6 and 291? The LCD for
these denominators (which is 582) occurs 97th on the list of multiples of 6.
We can
use the prime factors of the denominators to find the LCD more efficiently. The
LCD will consist of every prime factor in each denominator (at its most
frequent occurrence). To find the LCD for 1/12 and 9/14
factor 12 and 14 into their prime factorizations: 12 = 2 · 2 · 3 and 14 = 2 · 7. There are
two 2s and one 3 in the prime factorization of 12, so the LCD will have two 2s and
one 3. There is one 2 in the prime factorization of 14, but this 2 is covered
by the 2s from 12. There is one 7 in the prime factorization of 14, so the LCD
will also have a 7 as a factor. Once you have computed the LCD, divide the LCD
by each denominator. Multiply each fraction by this number over itself.
LCD = 2 · 2 · 3 · 7 = 84
Practice
Solutions
Sumber
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