Equivalent Fractions

Date: 12/19/96 at 19:07:34
From: Jose Velazquez
Subject: Equivalent fractions

Hi, 

My name is Nady and I'm in 6th grade.  I don't know how to do this, 
please help me!

These are equivalent fractions:

5/8 + 9/16 + 1/2 =

This confuses me too:

6/7 - 3/14 + 1/2 =

Thanks for your time and help.

Nady

Date: 01/25/97 at 13:27:06
From: Doctor Mason
Subject: Re: equivalent fractions

Dear Nady,

Equivalent fractions are fractions with the same value.  In other 
words, they are two names for the same number.  For example 5/8 and 
10/16 are equivalent fractions because they both describe the same 
amount, only in different ways.  You can tell this is true because 
when you reduce 10/16, you get the 5/8.

It seems from the problems that you submitted, you are asked to add 
and subtract fractional values.  In order to do this, you must first 
re-write each fractions with equivalent fractions that have the same 
denominator (bottom number) as all the other fractions.  We do this 
because a fraction represents a part of something, and in order to add 
or subtract them, these somethings or denominators must be same 
"size."  Just as a part of an apple is different from a part of an 
orange, "parts" of numbers are also different.

So to add 5/8 + 9/16 + 1/2, we must write all the fractions as 16ths.
We do this because 16 is the least common denominator (also known as 
the LCD) of 8, 16, and 2.  As you have probably already learned, the 
LCD of a set of numbers is the smallest number that all of the numbers 
in the set divide into evenly.  16 is the LCD of 8, 16, and 2 since 
16 is the smallest number which 8, 16, and 2 divide without remainder.  
So, we rewrite each of the fractions 5/8, 9/16, and 1/2 as equivalent 
fractions, each of which has 16 in the denominator.   

So, what is the equivalent fraction for 1/2 which has a denominator of 
16?  If we know what denominator we want, we ask ourselves, what 
number times the denominator we have gives the denominator we want:

    1       ?                          1     8     8
   ---  =  ---     Since 2 X 8 = 16,  --- X --- = ---
    2      16                          2     8    16

We multiply both numerator and denominator by 8 because 8/8 = 1, and 
we know when we multiply a number by "1" the original number does not 
change in value.  For this reason we know that 1/2 has the same value 
as 8/16 even though it doesn't look the same as 8/16.

We go through the same process to find that:

5/8 = 10/16    

Now we have that 5/8 + 9/16 + 1/2 = 10/16 + 9/16 + 8/16

Since the denominators are the same on the righthand side of the 
equation, we can go ahead and add, so we get:

10/16 + 9/16 + 8/16 = 27/16

So, 5/8 + 9/16 + 1/2 = 27/16, and we've found our answer!

Now that we have worked through the first of your two problems, I am 
confident that you can work through the second one, using the same 
ideas.  If you have more questions, please do write back!  Good luck!

-Doctors Mason and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/