Complex Fractions

Reference > Mathematics > Algebra > Algebraic Fractions

Sometimes you'll run across more complicated rational expressions to simplify - fractions within fractions, or even fractions within fractions within fractions. Expressions like:

a + 1

a² - a

a² - 1

These kinds of expressions can look scary, but they really aren't all that difficult to manage, as long as you remember that a fraction bar is really just a division symbol written a different way. Let's take a look at a simpler example, and then we'll come back to the one above.

Example #1

Solution #1

What does that fraction really mean? It means

divided by

, right? Because the fraction bar is division. So let's write it horizontally instead of vertically:

Now we can use our rules for dividing fractions. We'll turn it into a multiplication, and then simplify:

That wasn't so bad, right? So let's go back to our previous example.

Example #2

a + 1

a² - a

a² - 1

Solution #2

Rewrite this as division problem:

(a + 1)

(a² - a)

(a² - 1)

Now we factor and reduce, where possible:

(a + 1)

a(a - 1)

(a - 1)(a + 1)

(a + 1)

Now we rewrite as a multiplication problem:

(a + 1)

a(a + 1)

= 1

Isn't that funny - that ugly complicated expression is just equal to one!

Example #3

1 +

Solution #3

Oh boy! This one has multiple terms within the numerator and denominator, so we're going to have to be careful when we rewrite it; remember that the fraction bar is a grouping symbol, and once we exchange it for a division symbol (which is not a grouping symbol) we're going to have to insert parentheses to make sure we don't change the meaning of the expression:

(1 +

) ÷ (

)

Now we need to get common denominators for the first group, and also the second group:

(

) ÷ (

b²

a²

)

(a + b)

(b² - a²)

(a + b)

(b - a)(b + a)

(a + b)

(b - a)(b + a)

ab(a + b)

b((b - a)(b + a)

(b - a)

This kind of problem can be lengthy to work out, but as long as you remember that a fraction bar means division, and you don't forget parentheses to keep groupings together, you should be all set!