BMCC: Math Tutorials: Fractions:
Equivalent Fractions
Quick links to topics on this page:
Definition of Equivalent Fractions
Finding Equivalent Fractions
Writing Fractions in Simplest Form
Equivalent fractions are fractions that have different denominators, but the same overall value. By the "same overall value," I mean that the result is the same if the numerator is divided by the denominator. For instance, the fraction 
has a value of 0.75, because 3 divided by 4 equals 0.75:
The fraction 
also has a value of 0.75:
We say that and are equivalent fractions.
Suppose we have one fraction, and we want to determine an equivalent fraction with a particular denominator. Let's start with the fraction 
, and generate an equivalent fraction with a denominator of 18. The process used here is based on the multiplication property of one, which applies to fractions as well as to whole numbers.
Therefore, 
and are equivalent fractions.
The reason we were able to do this is that we didn't change the value of when we multiplied both the numerator and denominator by 3. In essence, we were multiplying the original fraction by 
:
Since is equal to 1, multiplying the original fraction by does not change its value. You can verify for yourself that both and have a value of 
. The two fractions have a different form, but the same value, so they are equivalent fractions.
Another example: write 
as an equivalent fraction with a denominator of 27.
Step 1: Divide the larger denominator by the smaller denominator. The larger denominator is 27, and the smaller denominator is 3. 27 divided by 3 gives a quotient of 9.
Step 2: Multiply both the numerator and the denominator of the original fraction by the quotient determined in step 1. We need to multiply both 2 and 3 by 9, like this:
Thus, 
and are equivalent fractions.
and a third example. . .
Write the number 5 as a fraction with a denominator of 20. To do this, you need to recall that any whole number n can be written as a fraction, as 
. That means the number 5 can be represented as 
. Now the problem looks like the same type we've been doing-- we need to represent as an equivalent fraction with a denominator of 20.
Step 1: Divide the larger denominator by the smaller denominator. The larger denominator is 20, and the smaller denominator is 1. 20 divided by 1 gives a quotient of 20.
Step 2: Multiply both the numerator and the denominator of the original fraction by the quotient determined in step 1. We need to multiply both 5 and 1 by 20, like this:
Thus, the number 5 is equivalent to 
.
If a fraction has no common factors in the numerator and denominator, the fraction is said to be in simplest form. If you have a set of equivalent fractions, no more than one of the set can be in simplest form. For instance, the fractions 
and 
are equivalent, but only is in simplest form. is not in simplest form because its numerator and denominator contain common factors.
What we mean by "common factors" in the numerator and denominator is whether there is any number that is a factor of both the numerator and denominator. To determine this, it is necessary to find the prime factorization for both numbers. If necessary, you can review how to determine the factors for a number, and how to determine the prime factorization of a number.
For the fraction , the prime factorization for 6 is 2 x 3, and the prime factorization for 10 is 2 x 5. The fraction can be represented with its prime factors as follows:
The number 2 is a prime factor for both the numerator and denominator in this example, so 2 is a common factor. Now we can take advantage of the multiplication property of one again, as we did above, because
Since 
is equal to 1, then 
. The fraction 
has no common factors in the numerator and denominator, so it cannot be reduced further. Thus,
expressed in simplest form is .
Another way to think of the process is simply to eliminate all common factors in the numerator and denominator-- for each occurrence of a common factor n in the numerator, you can eliminate that factor from the numerator and an equivalent occurrence of the factor from the denominator.
Example: express 
in simplest form.
The prime factorization for 8 is 2 x 2 x 2 , and the prime factorization for 58 is 2 x 29 , so can be represented as 
. 2 is a common factor, with three occurrences in the numerator, but only one occurrence in the denominator. Because it only appears once in the denominator, only one occurrence in the numerator can be correlated with the occurrence in the denominator, and thus only one occurrence in the numerator can be eliminated.
So expressed in simplest form is 
.
Example: express 
in simplest form.
You can verify for yourself as an exercise that the prime factorization for 30 is 2 x 3 x 5 , and the prime factorization for 42 is 2 x 3 x 7 . The original fraction expressed as prime factors looks like this:
We can see that both 2 and 3 are common factors, and that each has one occurrence in both the numerator and denominator. Correlating the common factors between the numerator and denominator allows us to eliminate those factors (because 
) like this:
Thus, expressed in simplest form is 
.
Next: Addition of Fractions
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