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Adding fractions using the Least Common Denominator

You have already learned how to add or subtract two fractions with the same
denominators: just add (or subtract) the numerators of the fractions :

However, to add fractions with different denominators, it's necessary to convert the
fractions to equivalent ones, which have the same denominator:

Ex: [converting to the denominator 6]
  [ multiplication of fractions ]
  [ common denominator 6 achieved]

In this example, finding the common denominator was pretty easy . The common
denominator is 6. We found it by multiplying the original denominators, 2 and 3.

Notice that 2 and 3 are both prime numbers.

What happens if the beginning denominators are not prime numbers?

We could find a common denominator as we did before: just multiply 6 and 15 together:

This involved a lot of work! Also, although it may not be obvious, the final answer is
not in lowest terms ; the fraction can be reduced:

We can save ourselves from working with larger numbers by finding the Least
Common Denominator
(LCD). The process for finding the LCD is outlined below; but
before we look at it, consider this: The "LCD process" is more complicated than simply
multiplying denominators together, and seems hard when you first learn it; but it's
worth the effort. Here are its advantages:

Systematic . Once you learn this method, it eliminates guess work.
• It al lows you to work with smaller numbers.
• Most importantly, the same method has to be used for problems that come up later
in Algebra.

Now we outline the LCD process. The process is based on factoring. It has three steps.

To find the LCD ( Least Common Denominator) of two or more fractions:


1. Factor each denominator into prime numbers.
Use exponential notation for the result. If a denominator is already prime, this
step is easy .

2. List all the different prime numbers that appeared in Step 1.

3. Raise each prime number to the largest exponent it receives in any of the
. Then multiply these together.

Here's an example: we will find the LCD of the following fractions (we won't bother to
carry out the addition of the fractions in this example).

Ex: Given and . What is the LCD of these two fractions?

Step 1. Factor 24 and 180, using exponential notation:

Step 2. List all the different prime numbers in Step 1:

2, 3, 5 are all the prime numbers that occurred.
(Notice that 2 and 3 occurred in both factorizations, and 5
occurred only in the second factorization.)

Step 3. Raise each prime number to its highest power, and multiply:

The first prime is the number 2. It's raised to the third power in the first
factorization, and it's raised to the second power in the second factorization. So its
highest power is 3. Write .

The second prime is 3. It's raised to the first power (so no exponent is written; 31
= 3) in the first factorization, and it's raised to the second power in the second
factorization. So its highest power is 2. Write .

The third prime is 5. It's not in the first factorization, and it's raised to the first
power in the second factorization . So its highest power is 1, which we don't
need to write. We write only 5.


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