BMCC: Math Tutorials: Fractions:
Division of Fractions
Quick links to topics on this page:
Defining Some Terms
Division of Two Fractions
Division of Fractions, Whole Numbers, and Mixed Numbers
Before discussing the process of dividing fractions, there are some terms that should be introduced. First, the act of switching the numerator and denominator of a fraction is called inverting the fraction. In a general case, for a fraction 
, inverting the fraction gives another fraction: 
. Here are some examples of what happens when fractions are inverted:
becomes 
becomes 
Note that this last example is based on the fact that 6 can also be written as 
, which when inverted becomes .
There are a few things to notice about the process of inversion. Unless a fraction is equal to 1, the numerator and denominator will have different values-- one of them will be larger than the other. After inversion, the other value will be larger. That is, if you start with a fraction in which the denominator is larger than the numerator, after inversion the numerator will be larger than the denominator. Similarly, if the numerator is larger than the denominator in the original fraction, in the inverted fraction the denominator will be larger than the numerator.
The upshot of this is that if a proper fraction (value less than 1) is inverted, the resulting fraction will be an improper fraction (value greater than 1). Likewise, if the original fraction is an improper fraction, the inverted fraction will be a proper fraction.
Here's a summary of how the original and inverted fractions compare when the original fraction is a proper fraction:
| ORIGINAL | INVERTED |
| proper fraction | improper fraction |
| value < 1 | value > 1 |
| numerator < denominator | numerator > denominator |
And the analogous summary for when the original fraction is an improper fraction:
| ORIGINAL | INVERTED |
| improper fraction | proper fraction |
| value > 1 | value < 1 |
| numerator > denominator | numerator < denominator |
Remember that all whole numbers are really improper fractions, because any whole number n can also be written as 
. Thus, the inversion of a whole number, 
, will always be a proper fraction with a value less than 1, unless the original whole number n was equal to 1 . When 1 (which can also be written as 
, or 
, or 
, etc.) is inverted, you still end up with a value of 1 .
The second term you should become familiar with is reciprocal. The reciprocal is simply what we have been referring to as the "inverted fraction." Starting with any original fraction , the reciprocal is the fraction . For instance, the reciprocal of 
is 
.
Finally, just a quick reminder of the terms used in division problems. Consider the general division problem 
, which can also be represented as 
. This would be read outloud as "a divided by b equals c." The number being divided into, a in this example, is called the dividend. The number the dividend is being divided by, b in this example, is called the divisor. The resulting answer, c in this example, is called the quotient.
Demonstrating these terms with the words themselves gives:
For example, in the problem 
, the dividend is 6 , the divisor is 3 , and the quotient is 2 .
Remember that, unlike in multiplication, division is order dependent. 6 divided by 3 is not the same as 3 divided by 6 , so make sure you keep track of the divisor and dividend when doing division problems.
OK, on to division of fractions.
Division problems involving two fractions are solved by rewriting the problems as related multiplication problems. Dividing one fraction by another is equivalent to multiplying the original dividend by the reciprocal of the original divisor. Shown in general terms,
The multiplication of 
is performed in the normal way (if necessary, you can review how to multiply two fractions).
Example: Solve 
.
We first rewrite this as a multiplication problem in which the original dividend is multiplied by the reciprocal of the original divisor. The divisor is 
and its reciprocal is 
, so the original division problem now becomes a multiplication problem: 
. This is solved the usual way, as
The fraction 
is not in simplest form-- it can be reduced to 
.
Another example: Solve 
.
The reciprocal of 
is 
, so the division problem becomes the multiplication problem 
, which is solved as
is an improper fraction and can be rewritten as the mixed number 
. Since 
is already in simplest form, the final mixed number does not have to be further simplified.
Division of whole numbers and/or mixed numbers is essentially identical to division of fractions, with the exception that any whole numbers and mixed numbers must first be rewritten as improper fractions.
Example: Divide 
.
First we need to rewrite the number 6 as an improper fraction, which is 
. The division problem is now 
, which is solved just as described above.
is not in simplest form-- it can be reduced further to 
.
Another example: Divide 
.
Here we have both a whole number and a mixed number, which must both be rewritten as their equivalent improper fractions. The whole number 7 is equivalent to 
, and the mixed number 
can be rewritten as 
. Now we have a division problem consisting of two fractions, 
, which is solved as outlined above.
is an improper fraction, which can be rewritten as the mixed number 
. The fractional portion 
is in simplest form, and does not need further reduction.
End of Fractions Tutorial
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