Objectives
After reviewing this unit, you will be able to:
 Multiply fractions.
 Divide fractions.
 Add fractions.
 Subtract fractions.
Multiplying
Fractions
Multiplying two fractions is the easiest of any of the operations.
Rule for Multiplication of Fractions
When multiplying fractions, you simply multiply the numerators
together and then multiply the denominators together. Simplify
the result.

This works whether the denominators are the same or not.
 So, if you wish to multiply the fractions 3/2 and 4/3 together
you get 12/6.


 As with any solution, you should report the answer in simplified
form. The fraction 12/6 can be simplified to 2.


You should recall that any number divided by itself is
1, so 6/6=1. In other words, if you find the same number on both
the top and the bottom of a fraction, you can cancel it out.
Example
What do you get when you multiply 1/2 and 3/7?
The result of multiplying these two fractions is 3/14.
The fraction 3/14 cannot be simplified any further; it is
in its simplest form.
Dividing Fractions
Dividing one fraction by another is almost as easy as multiplying
two fractions. It even involves multiplying fractions! First,
let's look at how division of two fractions may be represented.
If we wish to divide 3/5 by 2/3, we could write that as:
or
Rule for
Division of Fractions
When you divide two fractions, you take the reciprocal of the
second fraction, or bottom fraction, and multiply. (Taking the
reciprocal of a fraction means to flip it over.)

As with multiplication, this works whether the denominators
are the same or not.
So, if you wish to divide the fraction 3/2 by 4/3, you get the
result shown at the right. As with any solution, you should report
the answer in its simplified form. In this case, 9/8 is in its
simplest form. 

Example
What do you get when you divide 12/17 by 6/7?
The answer is 14/17.
 We take the reciprocal of the second fraction and multiplying
it by the first. We get 82/102, which, however, is not in its
simplified form.


 One easy way to simplify this fraction is go back to the
step before the numerator and denominator were multiplied.


 To reduce a fraction to its simplest form, we need to find
the prime factors of both the numerator and denominator (This
was shown in the unit on Equivalent
Fractions). When we do this for the numerator and denominator
we find we can cancel out a 2 and a 3 from the top and bottom.
This gives us the result 14/17.


Adding and Subtracting Fractions
When adding and subtracting fractions, the fractions being added
or subtracted must have the same denominator. When denominators
are different, you will need to convert each fraction into an
equivalent fraction by finding the least common denominator (LCD)
for the fractions. The two new fractions should have the same
denominator, making them easy to add or subtract. (Determining
the LCD of a set of fractions was reviewed in the unit Comparing
Fractions.)
Rule for
Addition of Fractions
When adding fractions, you must make sure that the fractions
being added have the same denominator. If they do not, find the
LCD for the fractions and put each in its equivalent form. Then,
simply add the numerators of the fractions.
Rule for
Subtraction of Fractions
When subtracting fractions, you must make sure that the fractions
being subtracted have the same denominator. If they do not, find
the LCD for the fractions and put each in its equivalent form.
Then, simply subtract the numerators of the fractions.

This rule can be broken down into several steps:
 Determine whether the fractions have
the same denominator.
If the denominators are the same, move to step 4.
 If the denominators are different,
find the LCD for the fractions being added.
(This process is explained in detail
in the previous
unit.)
 Find the equivalent fractions with
the LCD in the denominator.
 Add or subtract the numerators of the
fractions.
 Simplify the resulting fraction.
If we have the fractions 1/6 and 2/6, and wish to add them,
we follow our steps:
 Determine whether the fractions have
the same denominator. If the denominators are the same, move
to step 4.

The fractions 1/6 and 2/6 have the same
denominator, so we can move to step 4. 
 Add the numerators of the fractions.


 Simplify the resulting fraction.
This fraction can be simplified to 1/2.


Visually, this would look like:
Now let's try an example.
Example of
Adding Fractions
What is the sum of 3/4 and 1/3?
The answer is 13/12.
Following the steps:
 Determine whether the fractions have
the same denominator.
If the denominators are the same, move to step 4.
First, you should notice that the two fractions do not
have the same denominator. This means we need to find the LCD
for the two fractions.
 Find the LCD for the fractions being
added.
 Write the prime factors for
the denominator of each fraction.
 The prime factors of 4 are: 2 and 2.
 The prime factor of 3 is: 3
 Note all prime factors that occur.
For each prime factor that occurs, determine in which denominator
it occurs the most. Write down the prime factor the number of
times it occurs in that one denominator.
The prime factors that occur are 2, 2, and 3.
 Calculate the LCD of your fractions.
To do this, multiply the factors selected in step 2b.
2 x 2 x 3= 12,
12 is our LCD.
 Find the equivalent fractions that
have the LCD in the denominator.
Let's start with 3/4. The prime factor missing from this denominator
is a 3. So, 3 is the multiplier for 3/4. 

For the fraction 1/3, the prime factors that are missing are
2 and 2. Since 2 x 2= 4, 4 is the multiplier for the fraction
1/3. 

 Add the numerators of the fractions.
Now that we have found the fractions that are equivalent to the
ones we are adding, and these have the same denominator, we can
add the fractions together. 

We can see that the fraction we are adding are 9/12 and
4/12, which equals 13/12.
 Simplify the resulting fraction.
The answer of 13/12 is in its simplest form.
The steps for subtracting fractions are the same as for addition.
The only difference is substituting subtraction for addition.
If we wish to subtract 1/8 from 4/8, we can follow the steps outlined
above.
 Determine whether the fractions have
the same denominator. If the denominators are the same, move
to step 4.

They are the same, so we can skip to step 4. 
 Subtract the numerators of the fractions.


 Simplify the resulting fraction.

This fraction, 3/8, is in its simplest form. 
As you can see, addition and subtraction of fractions is similar
to adding and subtracting whole numbers. The important point is
to be sure the fractions being added or subtracted have the same
denominator.
You should now know how to add, subtract, multiply and divide
fractions. Try the practice below to be sure you understand how
to perform these operations on fractions.
