Beginning Algebra
Tutorial 3:
Fractions
 Learning Objectives
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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.
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Introduction
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| 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.
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When is a Fraction Simplified?
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| 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
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| 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.
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|
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.
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Dividing Fractions
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| 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.
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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)
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| 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
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| 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.
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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
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Adding or Subtracting
Fractions
Without Common Denominators
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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.
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| 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.
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| 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.
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| 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.
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Practice Problems
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| These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer.
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 Practice
Problem 1a:
Write the number as a product of
primes.
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Practice
Problems 2a - 2b:
Write the fraction in lowest terms.
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Practice
Problems 3a - 3e:
Perform the following
operations. Write answers
in the lowest terms.
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Need Extra Help on These Topics?
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All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on June 21, 2003 by Kim Seward. |
