Dividing Fractions

The objective of this lesson is that you learn how to divide fractions correctly.

Numerator and Denominator

The number or algebraic expression that appears on the top line of a fraction is called the numerator of the fraction.

The number of algebraic expression that appears on the bottom line of a fraction is called the denominator of the fraction.

Dividing Fractions

Expressed in symbols, the rule for dividing one fraction by another is as follows:

The rule for dividing one fraction by another is usually called inverting and multiplying. To divide one fraction by another, you “flip” (or “invert”) the fraction on the bottom and then multiply.

Example

Work out each of the following divisions of fractions.

Solution

(a)

(b) 

When you get to the multiplication step of these calculations, you have to remember to multiply entire quantities by entire quantities (which will usually involve FOILing). For example, after the fraction has been inverted, the multiplication involves multiplying (x + 2) by (3·x +1). Multiplying these two quantities together will require you to FOIL the two brackets.

(c) 

Again, note the necessity for FOILing once the inversion has been carried out and the multiplication is begun.

Dividing Numbers by Fractions

It is sometimes confusing to work out the result when a number is divided by a fraction. The key thing to bear in mind here is that a number is the same as a fraction – you just use a denominator of “1.”

Example

Simplify: 

Solution

You can covert this to a fraction division problem by writing the fraction  instead of the “number” x. 

This is certainly simpler than it was, but is not the simplest format that is possible. Each of the terms in the numerator has a common factor of x, which can be factored out and canceled with the x in the numerator (provided x " 0). 

provided x ≠ 0.

Dividing Fractions by Numbers

You can also deal with potentially confusing problems in which fractions are divided by numbers by converting the numbers to fractions. This will convert the confusing problem into a fraction division problem.

Example

Simplify:

Solution

Again, the key is to write the “number” as the fraction  instead of just the “number” x + 9. This converts the problem into a fraction division problem that you can solve by “inverting and multiplying.”