Two fractions are said to be equivalent if they have the
same value . Naturally, one approach we
could use to de termine if two fractions are equivalent is to convert each
fraction to a decimal. For
example, since 
, the fractions 3/5 and 15/25
are equivalent, and we could
write 
. Alternatively, consider the following
forms of the number 1:
Clearly 2 parts out of 2 is equal to 1, as is 100 parts out of 100, or n parts
out of n. Now consider
the following property (the Fundamental Property of Fractions):
If a, b, and c are non zero :
This statement is saying if both the numerator and denominator of a fraction
have the same factor
(called a common factor ), then that factor can be eliminated resulting in an
equivalent fraction.
It is true because 
, so we are multiplying
the fraction 
(a form of 1) to
result in the fraction 
. Recall that
multiplying a number by 1 does not change its value (the
Identity Property of Multiplication). Using our fractions 3/5 and 15/25 , note
that:
Thus 
and 
are equivalent fractions, or 
.
Example 1
Determine whether the two fractions are equivalent by using the Fundamental
Property of Fractions.
Solution
a. For the two fractions to be equivalent, there must be a form of 1 (or a
common factor) which can be multiplied by one fraction to create the other.
Note that:
Since 5/5 is a form of 1, the two fractions are equivalent.
b. We must find a form of 1 (or common factor) which can
be multiplied by
one fraction to create the other. Note that:
Since 12/12 is a form of 1, the two fractions are equivalent.
c. We must find a form of 1 (or common factor) which can
be multiplied by
one fraction to create the other. Note that:
Since 7/8 is not a form of 1, the two fractions are not equivalent. An
alternate way to verify this is to convert each fraction to decimal:
Note that these two decimal forms are not the same.
d. We must find a form of 1 (or common factor) which can
be multiplied by
one fraction to create the other. Note that:
Since xy/xy is a form of 1, the two fractions are equivalent.
Given a fraction, could you find other fractions which are
equivalent to it. For example, given
the fraction 3/7 , what would be some other fractions equivalent to it? We could
multiply by
different forms of 1:
Note that we could list as many equivalent fractions as we can list forms of 1,
which is infinite.
Example 2
For each fraction, list three equivalent fractions. Use variables in at
least one of
your fractions.
Solution
a. Using the fractions 
, and ab/ab as forms
of 1 (yours will probably be
different):
Three equivalent fractions are
b. Using the fractions 
as forms of 1 (yours will probably be
different):
Three equivalent fractions are 
c. Using the fractions 
as forms of 1 (again
yours will
probably be different):
Three equivalent fractions are 
. Note how we
multiplied the numbers, and how the exponent was used to represent x • x in
the third fraction.
In addition and subtraction of fractions, it will be
necessary to convert a fraction to a specified
denominator. For example, given the fraction 5/6, how could this fraction be
converted to one
with a denominator of 72? That is, what numerator x would result in
being equivalent?
Since 6 •12 = 72 (we can find 12 by dividing 6 into 72), the form of 1 to use is
12/12. Thus:
The missing numerator is x = 60 . This idea is often referred to as building
fractions.
Example 3 Find the variable such that the two given
fractions are equivalent.
Solution a. Since 70 ÷ 14 = 5 , the form of 1 to use is 5/5. Therefore:
The missing numerator is x = 45.
b. Since 33 ÷ 3 = 11, the form of 1 to use is 11/11.
Therefore:
The missing denominator is y = 44.
c. Since 120 ÷ 15 = 8 , the form of 1 to use is 8/8 .
Instead of multiplying the
first fraction by 8/8, we can alternatively divide the second fraction:
The missing numerator is a = 12. Note how we used the idea that division is
the inverse of multiplication to do this problem.
d. Since 200 ÷ 5 = 40 , the form of 1 to use is 40/40.
Again, we do this problem
“backwards” by dividing the second fraction:
The missing denominator is b = 8.
This last example leads to the idea of simplifying (or reducing) fractions. That
is, given a
fraction such as 32/40, can we apply the Fundamental Property of Fractions to
reduce the numbers
to a “simpler” form? Using the form of 1 as 8/8, we can write:
We say that 32/40 reduces to 4/5. Note that 4/5 does not reduce further, since
there is no other form
of 1 we can use in the Fundamental Property of Fractions. But where did 8/8 come
from? Recall
from Chapter 1 that the greatest common factor (GCF) of 32 and 40 is the largest
number that
will divide into both 32 and 40, which is precisely the number 8. In other
words, using the GCF
of the numerator and denominator as the common factor will always result in the
form of 1 to
use. In the past, you may have learned to reduce fractions by dividing the
numerator and
denominator by the same number (this is the same as our form of 1). The big
problem, however,
is knowing when to stop.
For example, we can attempt to reduce 32/40 as:
However, the result can be reduced further. Thus the GCF becomes the quickest
(and safest, in
terms of errors) approach to simplify fractions.
Example 4 Use the greatest common factor to
simplify each fraction.
Solution
a. The GCF of 56 and 80 is 8, so the form of 1 to use is 8/8. Therefore:
b. The GCF of 25 and 150 is 25, so the form of 1 to use is
25/25. Therefore:
Note how our “invisible” factor of 1 is used in this fraction.
c. The GCF of 48 and 132 is 12, so the form of 1 to use is
12/12. Therefore:
d. The GCF of 5xy and 10x is 5x , so the form of 1 to use
is 5x/5x. Therefore:
This illustrates how we can simplify fractions with symbols also.
Thus far, we have found the GCF by guessing at it, but
recall our alternate approach using
primes, which works particularly well for larger numbers. For example, to reduce
the fraction
, it would be difficult to guess at the GCF
of 168 and 180. We first factor each number into
primes:
168 = 8 • 21 = (2 • 4) • (3 • 7) = (2 • 2 • 2) • (3 • 7) = 2 • 2 • 2 • 3 • 7
180 = 10 •18 = (2 • 5) • (3 • 6) = (2 • 5) • (3 • 2 • 3) = 2 • 2 • 3 • 3 • 5
Instead of finding the GCF, we will use the primes in our fraction, remembering
that common
factors of the numerator and denominator will cancel:
 |
prime factorizations |
 |
cancelling common factors |
 |
writing the remaining factors |
 |
multiplying |
For fractions with larger numbers, this is usually the
most efficient, and more importantly the
most accurate, approach.
