Fraction Arithmetic
I hope everyone had a good holiday season. I certainly enjoyed mine. Over this season, I had a chance to speak with some youth and their parents. Funny that whenever we discuss that I have a PhD in applied mathematics, the topics of the children struggling in mathematics and the possibility of tutoring their children always seem to come up. I have no problem with tutoring and I actively participate in such sessions in my spare time. However I will say that it is sometimes a difficult task to do this job over such a short time period. Needless to say, I felt bad that I couldn’t have been of more assistance.
So, this being the holiday season and all, I decided to make somewhat of a new years resolution to focus this site more towards some of the things that the youth struggle with to hopefully be able to answer some of their questions.
With that being said, the first area that I decided to look at was fractions. This is one of the first areas where the youth begin to dislike mathematics. I feel like regardless of how much teachers and professors speak of the importance of understanding these processes, many students simply never grasp the procedures involved, partially because they never get used to the rules associated with these matters.
In this first script on fractions, I’ve focused on four types of problems corresponding to the four basic operations of arithmetic: Addition, Subtraction, Multiplication and Division.
To add two fractions of the form
We use the formula

+ 

= 
num_{1}*den_{2} + num_{2}*den_{1} 
den_{1}*den_{2} 

Lets take a moment to consider where this formula comes from. In order to be able to add fractions we first need to obtain a common denominator for the two fractions. One way that always works to obtain a common denominator is to multiply the denominators of the two fractions. So in the formula above, the denominator on the right hand side of the equals sign is the product of the two denominators on the left hand side. Once we have a common denominator, we need to rewrite each of the two fractions in terms of this common denominator.

+ 

= 
num_{1}*den_{2} 
den_{1}*den_{2} 

+ 
num_{2}*den_{1} 
den_{1}*den_{2} 

The formula for subtracting fractions is similar, with the notable difference of a subtraction in the place of addition.

 

= 
num_{1}*den_{2} – num_{2}*den_{1} 
den_{1}*den_{2} 

To multiply two fractions (also known as taking the product of two fractions, the resulting numerator is the product of the two initial numerators, and likewise the resulting denominator is the product of the two initial denominators.

* 

= 
num_{1}*num_{2} 
den_{1}*den_{2} 



Finally, remembering that division is the inverse of multiplication, we can derive the formula to divide two fractions by multiplying by the inverse of the fractions:

÷ 

= 

* 

= 
num_{1}*den_{2} 
den_{1}*num_{2} 



The next step in each of these operations is to reduce the fraction to lowest terms. One way of doing this is by considering Euclid’s GCD algorithm which is available here.
The script is available to practice your work on fractions at
http://www.learninglover.com/examples.php?id=31