Beginning Algebra
Tutorial :
Fractions


Learning Objectives

 
After completing this tutorial, you should be able to:
  1. Know what the numerator and denominator of a fraction are.
  2. Find the prime factorization of a number.
  3. Simplify a fraction.
  4. Find the least common denominator of given fractions.
  5. Multiply, divide, add and subtract fractions.

 
 
 
 
Introduction

 
 
Do you ever feel like running and hiding when you see a fraction?  If so, you are not alone.  But don't fear help is here.  Hey that rhymes.  Anyway, in this tutorial we will be going over how to simplify, multiply, divide, add, and subtract fractions.  Sounds like we have our work cut out for us.  I think you are ready to tackle these fractions.

 
 
Tutorial

 
 
 
Fractions

   ,   where 

a = numerator

b = denominator


 
A numeric fraction is a quotient of two numbers.  The top number is called the numerator and the bottom number is referred to as the denominator.   The denominator cannot equal 0.


 

Prime Factorization

 
A prime number is a whole number that has two distinct factors, 1 and itself. 

Examples of prime numbers are 2, 3, 5, 7, 11, and 13.  The list can go on and on. 

Be careful, 1 is not a prime number because it only has one distinct factor which is 1. 

When you rewrite a number using prime factorization, you write that number as a product of prime numbers.

For example, the prime factorization of 12 would be 

12 = (2)(6) = (2)(2)(3). 

That last product is 12 and is made up of all prime numbers.


 
 
When is a Fraction Simplified?

 
Good question.  A fraction is simplified if the numerator and denominator do not have any common factors other than 1.  You can divide out common factors by  using the Fundamental Principle of Fractions, shown next.


 

Fundamental Principle of Fractions


 
In other words, if you divide out the same factor in both the numerator and the denominator, then you will end up with an equivalent expression.  An equivalent expression  is one that looks different, but has the same value.


 

Writing the Fraction in Lowest Terms
(or Simplifying the Fraction)

 
Step 1:  Write the numerator and denominator as a product of prime numbers.
 

Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.
 


 
 
 
Example 1:  Write the fraction in lowest terms. 

 
Step 1:  Write the numerator and denominator as a product of prime numbers.

 
*Rewrite 35 as a product of primes

 
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.

 
*Div. the common factor of 7 
out of both num. and den.

 
Note that even though the 7's divide out in the last step, there is still a 1 in the numerator.  7 is thought of as 7 times 1 (not 0). 

 
 
 
Example 2:  Write the fraction in lowest terms. 

 
Step 1:  Write the numerator and denominator as a product of prime numbers.

 
*Rewrite 90 as a product of primes
*Rewrite 50 as a product of primes

 
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.

 
*Div. the common factors of 2 and 5 
out of both num. and den.

 
 
 
 
Example 3:  Write the fraction in lowest terms. 

 
Step 1:  Write the numerator and denominator as a product of prime numbers.

 
3 and 5 are both prime numbers so the fraction is already written as a quotient of prime numbers

 
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.

 


 
There was no common factors to divide out.  The original fraction 3/5 was already written in lowest terms.

 
Multiplying Fractions


 
In other words, when multiplying fractions, multiply the numerators together to get the productís numerator and multiply the denominators together to get the productís denominator.

Make sure that you do reduce your answers, as shown above.  You may do this before you multiply or after.


 
 
Example 4:  Multiply.  Write the final answer in lowest terms.

 
*Write as prod. of num. over prod. of den.

*Div. the common factor of 5 
out of both num. and den.

 


 
 
 
Reciprocal


 
Two numbers are reciprocals of each other if their product is 1. 

In other words, you flip the number upside down.  The numerator becomes the denominator and vice versa. 

For example,  5 (which can be written as 5/1) and 1/5 are reciprocals.  3/4 and 4/3 are also reciprocals of each other.


 
 
Dividing Fractions


 
In other words, when dividing fractions, use the definition of division by rewriting it  as multiplication of the reciprocal and then proceed with the multiplication as explained above.

 
 
Example 5:  Divide.  Write the final answer in lowest terms.

 
*Rewrite as the mult. of the reciprocal
 

*Write as prod. of num. over prod. of den.

*Div. the common factor of 2 
out of both num. and den.

 


 

Adding or Subtracting Fractions 
with Common Denominators

   or 


 
Step 1: Combine the numerators together.
 

Step 2: Put the sum or difference found in step 1 over the common denominator.
 

Step 3: Reduce to lowest terms if necessary.
 
 

Why do we have to have a common denominator when we add or subtract fractions?????
Another good question.  The denominator indicates what type of fraction that you have and the numerator is counting up how many of that type you have.  You can only directly combine fractions that are of the same type (have the same denominator).  For example if 2 was my denominator, I would be counting up how many halves I had, if 3 was my denominator, I would be counting up how many thirds I had.  But, I would not be able to add a fraction with a denominator of 2 directly with a fraction that had a denominator of 3 because they are not the same type of fraction.  I would have to find a common denominator first before I could combine, which we will cover after this example.


 
Example 6:  Add.  Write the final answer in lowest terms.

 
 
Step 1: Combine the numerators together.
AND
Step 2: Put the sum or difference found in step 1 over the common denominator.

 

*Write the sum over the common den.

 
Step 3: Reduce to lowest terms if necessary.

 
Since 5 and 7 are prime numbers that have no factors in common, 5/7 is already in lowest terms.


 

Least Common Denominator (LCD)

 
The LCD is the smallest number divisible by all the denominators.


 

Equivalent Fractions

 
Equivalent fractions are fractions that look different but have the same value. 

You can achieve this by multiplying the top and bottom by the same number.  This is like taking it times 1.  You can write 1 as any non zero number over itself.  For example 5/5 or 7/7.  1 is the identity number for multiplication.  In other words, when you multiply a number by 1, it keeps its identity or stays the same value. 


 
 
 
Example 7:  Write the fraction as an equivalent fraction with the given denominator.   with the denominator of 20.

 
 
*What number times 5  will result in 20?
 

*Multiply num. and den. by 4.

 


 
In this case, we do not want to reduce it to lowest terms because the problem asks us to write it with a denominator of 20, which is what we have.


 

Rewriting Mixed Numbers as
Improper Fractions


 
In some problems you may start off with a mixed number and need to rewrite it as an improper fraction.  You can do this by multiplying the denominator times the whole number and then add it to the numerator.  Then, place this number over the existing denominator.

An improper fraction is a fraction in which the numerator is larger than the denominator.


 
 
Example 8:  Rewrite the mixed fraction as an improper fraction.

 
*Mixed number

*Mult. den.  4 times whole number 7 
and add it to num.  3.
 
 

*Improper fraction
 


 
 
 
Adding or Subtracting Fractions 
Without Common Denominators 

 
As mentioned above, you need to have common denominators before you can add or subtract fractions together.

Step 1: Find the Least Common Denominator (LCD) for all denominators. 
 

Step 2: Rewrite fractions into equivalent fractions with the common denominator.
 

Step 3: Add and subtract the fractions with common denominators as described above.


 
 
Example 9:  Add.  Write the final answer in lowest terms.

 
Rewriting the numbers as fractions we get:

 

*Rewrite whole number 7 as 7/1
*Rewrite mixed number 2 3/4 as 11/4

 
Step 1: Find the Least Common Denominator (LCD) for all denominators. 

 
The first fraction has a denominator of 1 and the second fraction has a denominator of 4.  What is the smallest number that is divisible by both 1 and 4.  If you said 4, you are correct? 

Therefore, the LCD is 4.


 
Step 2: Rewrite fractions into equivalent fractions with the common denominator.

 
*What number times 1  will result in 4?

*Multiply num. and den. by 4.
 


 
The fraction 11/4 already has a denominator of 4, so we do not have to rewrite it.

 
Step 3: Add and subtract the fractions with common denominators as described above.

 

*Write the sum over the common den.
 

 


 
Note that this fraction is in simplest form.  There are no common factors that we can divide out of the numerator and denominator

 
 
 
 
Example 10:  Add and subtract.  Write the final answer in lowest terms.

 
Step 1: Find the Least Common Denominator (LCD) for all denominators. 

 
The first fraction has a denominator of 3, the second has a denominator of 5, and the third has a denominator of 15.  What is the smallest number that is divisible by 3, 5, and 15?  If you said 15, you are correct? 

Therefore, the LCD is 15.


 
Step 2: Rewrite fractions into equivalent fractions with the common denominator.

 
Writing an equivalent fraction of 2/3 with the LCD of 15 we get:

 
*What number times 3  will result in 15?

*Multiply num. and den. by 5.
 


 
Writing an equivalent fraction of 4/5 with the LCD of 15 we get:

 
*What number times 5  will result in 15?

*Multiply num. and den. by 3.
 


 
The fraction 1/15 already has a denominator of 15, so we do not have to rewrite it.

 
Step 3: Add and subtract the fractions with common denominators as described above.

 
*Write the sum and difference over the common den.
 
 
 

*Div. the common factor of 3 out of both num. and den.