Beginning
Algebra Tutorial : Fractions
Learning
Objectives 
After completing this tutorial, you should be able to:
 Know what the numerator and denominator of a fraction are.
 Find the prime factorization of a number.
 Simplify a fraction.
 Find the least common denominator of given fractions.
 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. 
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) 
Example 1: Write the
fraction in lowest terms. 

*Rewrite 35 as a product of
primes 

*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. 

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

*Div. the common factors of 2 and
5 out of both num. and
den. 
Example 3: Write the
fraction in lowest terms. 
3 and 5 are both prime numbers so the fraction is already
written as a quotient of prime
numbers 
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 product’s numerator and multiply the
denominators together to get the product’s 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. 
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 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 
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 
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. 

*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. 

*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.

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. 
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. 

*Write the sum and difference over the
common den.
*Div. the common factor of 3 out of both
num. and den.

