Solving Two Equations with Mixed Fractions

Date: 05/13/98 at 19:15:53
From: gogeta
Subject: pre algebra I really don't know what it is called so all I 
can say is that

  2         1     1
2 - (x) - 1 - = 4 - 
  3         4     3

I have tried all things to get this, but I can't get it.

Date: 05/18/98 at 13:00:36
From: Doctor Jeremiah
Subject: Re: pre algebra I really don't know what it is called so all 
I can say is that

Hi Gogeta:

I can see a couple of things that need to be done in order to solve 
this.

1) Make improper fractions out of the mixed fractions. This will make
   the problem much easier to do.

   For example:  
 
        2
      2 -
        3 
      
   is a mixed fraction because it has a whole number (the "2") and a 
   fraction (the "2/3").

   You need to find out how many thirds 2 2/3 really is. One whole
   would be three thirds (3/3), and two would be twice that many
   (6/3).

   So 2 2/3 would be:
   
      2 2/3 = 2 + 2/3 = (6/3) + (2/3) = 8/3

2) Make the bottom of all the fractions the same number.

   See how the problem adds thirds to fourths? That makes the problem 
   impossible. We need to change it so that we are adding the same 
   kind of fractions.

   So we need to decide what the number on the bottom should be. It 
   needs to be the smallest number that is big enough to be a multiple 
   of all the fraction's current bottom numbers. This new number that 
   will be on the bottom is called the "lowest common denominator."

   The smallest number that is a multiple of 3 and 4 both is 12.
   We fix the equation by multiplying both the top and the bottom
   of every term in the equation by whatever value will make the 
   bottom of the fraction equal to 12.

   For 8/3, we need to multiply the top and bottom by 4. If we do that 
   for every term in the equation, then all the fractions will have
   12 on the bottom.

      8   4   32
      - * - = --
      3   4   12

3) Simplify the equation. Now that everything on both sides has a 12 
   on the bottom, we can cancel all the 12s on the bottom by 
   multiplying both sides by 12 to get rid of them all. This leaves us 
   with a very simple equation to solve.

Here is another example to help you see what is going on:

     1             1
   4 - (x) - 2 = 3 -
     5             3

1) Make improper fractions out of the mixed fractions:

        1       1   4*5   1   20   1   21
      4 - = 4 + - = --- + - = -- + - = --
        5       5    5    5    5   5    5

   In the same way:

         1   10
       3 - = --
         3    3

   This gives us the equation:

       21            10
       -- (x) - 2 =  --
        5             3

2) Create a lowest common denominator:

   Remember, you have to do this even for terms that are not
   fractions, so that you can simplify later.

      21   3           15   10   5
      -- * - (x) - 2 * -- = -- * -
       5   3           15    3   5

      63       30   50
      -- (x) - -- = --
      15       15   15

3) Simplify: (cancel all the 15s on the bottom by multiplying
   both sides by 15)

           / 63       30 \    50 
      15 * | -- (x) - -- | =  -- * 15
           \ 15       15 /    15

           63            30   50 
      15 * -- (x) - 15 * -- = -- * 15
           15            15   15

      63(x) - 30 = 50

Now it is easy to solve:

      63(x) = 80

      x = 80/63

Hope that helps some. If I can help some more, please write back.

-Doctor Jeremiah, The Math Forum
Check out our web site! http://mathforum.org/dr.math/