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Fractions - Adding Unlike Denominators - LCM

Least common multiples are very useful in finding the least common denominator when adding and subtracting fractions. Let’s look at one method to find least common multiples, as well as the purpose and meaning of them.

Purpose – To help in the addition and subtraction of fractions since it’s a requirement that
the denominators are the same
Meaning of Least Common Multiple (LCM) - multiples are the result of repeated addition or what you referred to as skip counting in Elementary school

For example - List the multiples of 2 and 3; to do so let’s skip count

2 >>> 2, 4, 6, 8, 12, 14, 16, 18, 24 . . .

3 >>> 3, 6, 9, 12, 15, 18, 21 . . .

What multiples are listed in both lists?
They are 6, 12, and 18. These are called the common multiples.
Which of the common multiples is the smallest? 6
Therefore, the least common multiple of 2 and 3 is 6.

Let’s apply what we just learned. Add ½ + 2/3. The denominators (the bottom numbers) are different. So, find a common denominator which is what was done above when the LCM was found. Thus, the common denominator for ½ and 2/3 is 6. This is also called the least common denominator.

Tip – write the problem vertically

1/2 = ? /6

+ 2/3 = ? /6
--------------------

Now, new numerators are needed.
Recall that multiples are the result of repeated addition, and a shortcut for repeated addition is multiplication. Look at the fraction ½, and ask “2 times what number equals the new denominator 6?” The answer is three. So, multiply the old numerator by 3 too. Thus, the new numerator is 3. In essence, you now have created an equivalent fraction for ½ which is 3/6. See below.

Next, look at the fraction 2/3 and ask “3 times what number equals the new denominator 6?” The answer is two. So, multiply the old numerator by 2. Thus, the new numerator is 4. In essence, you have created an equivalent fraction for 2/3 which is 4/6. See below.

Now, both fractions have the same denominator. Add.

1/2 = 3 /6

+ 2/3 = 4 /6
------------------------
7/6 = 1 1/6

The answer is 7/6 which is considered an improper fraction because the numerator is larger than the denominator. Therefore, it must be reduced to the lowest terms. Simply, divide 7 by 6. Six can go into 7 one time with 1 left over or remaining. Thus, the answer is 1 1/6. Note: the remainder became the numerator and the denominator remained as 6.


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