Adding and Subtracting Fractions

Date: 10/25/1999 at 08:45:58
From: Kevin Balandin
Subject: Fractions

I am in grade 7 and (can you believe it?) I can't add or subtract 
fractions! 

Please help me out.

Kevin

Date: 10/25/1999 at 15:44:27
From: Doctor Rick
Subject: Re: Fractions

Hi, Kevin.

Let's just go over some basics; if you have a problem with a 
particular part of this process, write again and we can work through 
it.

Start with money. It doesn't make much sense to add quarters and 
dimes, does it? If I have 2 quarters and 3 dimes, then I have 5 coins, 
but this doesn't tell you anything about how much money I have.

What you could do instead is to convert the quarters and dimes to 
cents:

            2 quarters + 3 dimes =
     2 * 25 cents + 3 * 10 cents = 
             50 cents + 30 cents = 
                        80 cents

Another way would be to convert the quarters and dimes into nickels. 
Each quarter becomes 5 nickels, and each dime becomes 2 nickels. So

        2 quarters + 3 dimes =
     2*5 nickels + 3*2 dimes =
      10 nickels + 6 nickels =
                  16 nickels

I can rewrite the problem as a sum of fractions, because a quarter is 
1/4 dollar, a dime is 1/10 dollar, and a nickel is 1/20 dollar.

      2     3    16
     --- + --- = --
      4    10    20

Before you can add two fractions, you need to rewrite one or both 
fractions so they have the same denominator. This is the same thing as 
converting different coins into the same kind of coin. You have 
learned about equivalent fractions, right? If you multiply the 
numerator and denominator of a fraction by the same number, you get 
another name for the same fraction - an equivalent fraction.

      3     3*2     6
     --- = ---- = ----
     10    10*2    20

If you want to add, say, 3/10 and 5/6, you need to find a pair of 
fractions that are equivalent to these but that have the same 
denominator. Let's write out some equivalent fractions:

      3/10 = 6/20 = 9/30 = 12/40 = 15/50 = ...

     5/6 = 10/12 = 15/18 = 20/24 = 25/30 = ...

You can see a pair of fractions that have the same denominator: 9/30 
and 25/30. Let's add them:

       3     5      9     25     34
     ---- + --- = ---- + ---- = ----
      10     6     30     30     30

Now we can simplify the answer, by reducing it to lowest terms. Both 
34 and 30 are divisible by 2, so a fraction equivalent to 34/30 is 
17/15 (dividing the numerator and denominator by 2). Finally, because 
17 is greater than 15, this is an improper fraction, so we can convert 
it into a mixed number:

     17   15    2      2
     -- = -- + -- = 1 --
     15   15   15     15


Subtraction works the same way. Here is an example:

      5     3    25    9   16    8
     --- - --- = -- - -- = -- = --
      6     10   30   30   30   15

You can find some further help with fractions in the links at the 
bottom of our Dr. Math FAQ on "Fractions, Decimals, Percentages":

  http://mathforum.org/dr.math/faq/faq.fractions.html   

In particular, you can find techniques for finding the smallest 
denominator that will work -- the Lowest Common Denominator (LCD), 
also called the Least Common Multiple (LCM). That's the trickiest part 
of adding fractions. You will find more help with this if you go to 
our Dr. Math Search page,

   http://mathforum.org/mathgrepform.html   

and search for   LCD LCM   selecting "At least one of your keywords".

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