Adding and Subtracting Fractions Help
Adding and Subtracting Fractions - Same and Different Denominators
When adding (or subtracting) fractions with the same denominators, add (or subtract) their numerators.
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.)
Finding A Common Denominator
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.
In the first denominator is 2 and the second denominator is 7. Multiply by and multiply by .
Find practice problems and solutions at Adding and Subtracting Fractions Practice Problems - Set 1 and Set 2.
Finding the Least Common Denominator (LCD)
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 :
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.
Find practice problems and solutions at Adding and Subtracting Fractions Practice Problems - Set 3.
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 and . 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 and 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.
Find practice problems and solutions at Adding and Subtracting Fractions Practice Problems - Set 4.
Adding More than Two Fractions
Finding the LCD for three or more fractions is pretty much the same as finding the LCD for two fractions. Factor each denominator into its prime factorization and list the primes that appear in each. Divide the LCD by each denominator. Multiply each fraction by this number over itself.
Prime factorization of the denominators:
5 = 5 15 = 3 · 5 20 = 2 · 2 · 5
The LCD = 2 · 2 · 3 · 5 = 60
Prime factorization of the denominators:
10 = 2 · 5 12 = 2 · 2 · 3 18 = 2 · 3 · 3
LCD = 2 · 2 · 3 · 3 · 5 = 180
Find practice problems and solutions at
Practice problems for this concept can be found at: Algebra Fractions Practice Test.
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