Factoring Fractions

Date: 05/31/2001 at 17:44:48
From: Genise Gorman
Subject: Adding fractions with unlike denominators

I am having a major problem trying to figure out how to add fractions 
that have different denominators. I'm factoring, but once I factor I 
get completely lost. Can you help me, please, or at least give me a 
couple of ways of figuring this out? Thank you.

Date: 06/01/2001 at 12:34:23
From: Doctor Rick
Subject: Re: Adding fractions with unlike denominators

Hi, Genise.

You get lost while going through the Fraction Woods? Maybe a map will 
help. There are landmarks along the way:

1. Find a common denominator, preferably the least common denominator.

2. Replace each fraction (if needed) by an equivalent fraction with 
   the common denominator.

3. Add the numerators of these equivalent fractions, and use the 
   common denominator when you write the sum.

4. Reduce the sum fraction to lowest terms ("cancel" any common 
   factors in the numerator and denominator).

It sounds like you're getting lost on your way to the first landmark. 
For the time being, you can make this step easier by NOT trying to 
find the LEAST common denominator. The product of the two denominators 
is a perfectly acceptable denominator. You'll have to go farther in 
the last leg of the trip - the numbers will be bigger, and you're more 
likely to find common factors. But you'll be close to the edge of the 
woods by then, so you may be less likely to get lost.

Here's an example, using the least common denominator first, then just 
using the product of the denominators.

   7     5
  --- + --- = ?
   20    36

Step 1: Find the least common denominator of 20 and 36. Their prime 
factorizations are 

  20 = 2*2    *5
  36 = 2*2*3*3
       -----------
 LCD = 2*2*3*3*5 = 180

Where the two factorizations have a factor in common, I wrote one 
below the other. The numbers in a column are always the same. Then to 
find the LCD, I took one number from each column, and multiplied them 
together.

Step 2: Find equivalent fractions with denominators of 180.

To convert 7/20 to a fraction with a denominator of 180, we have to 
multiply the denominator (20) by 9. To keep the fraction equivalent, 
we have to multiply the numerator by the same number (9).

   7     7*9    63
  --- = ---- = ---
   20   20*9   180

To convert 5/36 to 180ths, we have to multiply the numerator and 
denominator by 5:

   5     5*5    25
  --- = ---- = ---
   36   36*5   180

Step 3: Add the numerators and use the common denominator.

   63    25    88
  --- + --- = ---
  180   180   180

Step 4: Reduce to lowest terms. To do this, we factor the numbers:

   88 = 2*2*2      *11
  180 = 2*2  *3*3*5

The common factor is 2*2 = 4. Divide each number by 4 (or keep the 
factors in each number that aren't in the other number) to get

   88/4   22
  ----- = --
  180/4   45

Now I'll do the same problem with the simpler first step.

Step 1: Find a common denominator by multiplying the two denominators.

  20 * 36 = 720. (That's bigger than the first way, but it's okay.)

Step 2: Find equivalent fractions with this common denominator. To 
convert 7/20 to 720ths, we need to multiply the numerator and 
denominator by 36.

   7     7*36   252
  --- = ----- = ---
   20   20*36   720

To convert 5/36 to 720ths, we have to multiply the numerator and 
denominator by 20.

   5     5*20   100
  --- = ----- = ---
   36   36*20   720

Step 3: Add the numerators and use the common denominator.

  252   100   352
  --- + --- = ---
  720   720   720

Step 4: Reduce to lowest terms. To do this, factor each number:

  352 = 2*2*2*2*2      *11

  720 = 2*2*2*2  *3*3*5

The common factor is 16. Divide 352 by 16 (or keep the factors that 
aren't in the factorization of 720) to get 22. Divide 720 by 16 (or 
keep the factors that aren't in the factorization of 352) to get 45.

The answer is 22/45, the same answer I got the first time.

Perhaps my example of how to find the LCD will help keep you from 
getting lost. If so, do it the first way. If you still tend to get 
lost, try the second way.

If you need more ideas, our Dr. Math Archives contain lots of answers 
to questions about adding fractions. Just go to our Search Dr. Math 
page:

  http://mathforum.org/mathgrepform.html   

type in the words:

  adding fractions

select "that exact phrase," and click the Search button. Maybe one of 
these answers will help you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/