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Fraction: Algebra without the Equations

Date: 8/23/96 at 23:36:49
From: Donna J Brewer
Subject: Solving Fraction Problems in a Different Way

Dear Dr. Math,

I am no wizard at math, but I discovered a cool way to solve certain
fraction problems. The only trouble is, I don't know why it works!  
It is bugging the heck out of me.

Here is an example.

2/3 of ? = 66

Now, I know that I have 3 equal groups, with a certain number in each
group.  But I don't know what that number is.

I found that if I divide the given sum (in this case 66) by the
number in the top of the fraction (in this case 2), I will ALWAYS 
come up with the number in each group (or a single portion of whatever 
the fraction is, one third in this instance).

Example: I divide 66 by the top number in the fraction (2) and get the
answer of 33. Since I now know how much comprises one portion, I 
simply multiply by the denominator to find out the unknown number.

In the above case, I multiply 33 by 3 and get 99.

My solved problem now reads 2/3 of 99 = 66

This solution ALWAYS works.

       3/4 of ? = 75
75 divided by 3 = 25

         25 x 4 = 100
     3/4 of 100 = 75.

Date: 8/26/96 at 13:25:34
From: Doctor Leigh
Subject: Re: Solving Fraction Problems in a Different Way

Thank you for your problem.  

Your solution works because you are doing the correct algebra without 
needing to write out the equations.  Let's take your sample problem.

3/4 of ? = 75

To do this algebraically you would set up an equation:

   3/4 * x = 75

From there you need to get x alone on one side of the equation.  
Normally you would just divide by 3/4, but you can also divide by 
three and multiply by four - which is the same thing as dividing 
by 3/4.  This is why your solution works. 
You are really just dividing by the fraction, but you have separated 
the division. 

Good work in taking something and changing it into what you 

-Doctor Leigh Ann,  The Math Forum
 Check out our web site!   
Associated Topics:
Middle School Algebra
Middle School Fractions

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