Simplifying Complex Fractions

When a “normal” fraction contains fractions in either the numerator or denominator or both, then we consider it to be a complex fraction. This type of fraction is also known as a compound fraction.

There are two methods used to simplify such kind of fraction.

Method 1

Key steps:

  • Create a single fraction in the numerator and denominator.
  • Apply the division rule of fractions by multiplying the numerator by the reciprocal or inverse of the denominator.
  • Simplify, if necessary.

Method 2

Key steps:

  • Find the Least Common Denominator (LCD) of all the denominators in the complex fractions.
  • Multiply this LCD to the numerator and denominator of the complex fraction.
  • Simplify, if necessary.

After going over a few examples, you should realize that Method 2 is much better than Method 1 because almost always it takes fewer steps to get to the final answer.


Examples of How to Simplify Complex Fractions

Example 1: Simplify the complex fraction below.

[(2x^3)/(3y)]/[(x^2)/(6y)]
  • Using Method 1

Both the numerator and denominator of the complex fraction are already expressed as single fractions. This is great!

[(2x^3)/(3y)]/[(x^2)/(6y)]

The next step to do is to apply division rule by multiplying the numerator by the reciprocal of the denominator. Finish off by canceling out common factors to get the final answer.

The complex fraction [(2x^3)/(3y)]/[(x^2)/(6y)] can be simplified as follows: [(2x^3)/(3y)]/[(x^2)/(6y)] = [(2x^3)/(3y)] * [(6y)/(x^2)]=[(2x)/(1)] * (1/2) = 4x. Therefore, the final simplified answer is 4x. Make sure to check your work.
  • Using Method 2

Find the LCD of the entire problem, that is, the LCD of the top and bottom denominators.

The second method requires to find the least common denominator (LCD) of the denominators of top fraction and bottom fraction. The top fraction has a denominator of 3y while the bottom fraction has a denominator of 6y. The least common denominator (LCD) of 3y and 6y is 6y.

Since the LCD of 3y and 6y is just \textbf{6y}, we will now multiply the complex numerator and denominator by this LCD. After doing so, we can expect the problem to be reduced to a single fraction which can be simplified as usual.

After figuring out that the least common denominator is 6y, we will use this to multiply both the top and bottom fractions of the given complex fraction. Thus, [(2x^3/3y)*(6y)]/[(x^2/6y)*(6y)]=(4x^3)/(x^2)=4x.

Example 2: Simplify the complex fraction below.

The complex fraction to simplify is [1+(1/x)]/[1-(1/x^2)].
  • Using Method 1

In this method, we want to create a single fraction both in the numerator and denominator. Obviously, this problem would require us to do that first before we perform division.

Add the fractions in the numerator, and subtract the ones in the denominator.

Now we simplify the complex fraction [1+(1/x)]/[1-(1/x^2)]. Here are the steps: [1+(1/x)]/[1-(1/x^2)] = [(x/x)+(1/x)]/[(x^2/x^2)-(1/x^2)]=[(x+1)/x]/[(x^2-1)/x^2]=[(x+1)/x]*[(x^2)/(x^2-1)]=x/(x-1). That makes our final answer to be x/(x-1). This can be described as the quotient of x, and the difference of x and 1.
  • Using Method 2

Looking at the denominators \large{x} and \large{x^2}, its LCD must be \large{x^2}  Multiply the top and bottom by this LCD.

The second method to simply the complex fraction is to multiply the upper and lower fractions by the least common denominator of their denominators. In this example, it is x^2. So we have [(1+1/x)]/[1-(1/x^2)] * (x^2/x^2) = (x^2+x)/(x^2-1)=x/(x-1). The final answer is x/(x-1).

Example 3: Simplify the complex fraction below.

Reduce this complex fraction into its lowest form: [(2/x)-2/(3x)]/[(1/x)-5/(6x)]
  • Using Method 1

Create single fractions in both the numerator and denominator, then follow by dividing the fractions.

Using the first method of trial and error by finding the multiplier to make the denominators the same, we have [(2/x)-2/(3x)]/[(1/x)-5/(6x)]=[(2/x)*(3/3)-2/(3x)]/[(1/x)*(6/6)-5/(6x)]=[6/(3x)-2/(3x)]/[6/(6x)-5/(6x)]=[4/(3x)]/[1/(6x)=[4/(3x)]*[(6x)/1]=8. The final answer is just 8.
  • Using Method 2

The overall LCD of the denominators is \color{red}6x. Use this to multiply through the top and bottom expressions.

Since the least common denominator (LCD) of the denominators of the upper and lower fractions is 6x, we multiply the top and bottom fractions by 6x to simplify the complex fraction. Therefore, [(2/x)-2/(3x)]/[(1/x)-(5/)6x)] * (6x/6x) = (12-4)/(6-5)=8/1=8. The final answer is also 8 which matches the first method.

Example 4: Simplify the complex fraction below.

[(x+1)/(x-1)]-[(x-1)/(x+2)] / [(15x+3)/(x^2-1)]

For this problem, we are going to use Method 1 only.

The problem requires you to apply the FOIL method (multiplication of two binomials) and a simple factorization of trinomial. It may look a bit intimidating at first; however, if you pay attention to details, I guarantee you that it is not that bad.

If you observe, the complex denominator is already in the form that we want – having one fractional symbol. This means we have to work a bit on the complex numerator. Our next step would be to transform the complex numerator into a “simple” or single fraction.

[(x+1)/(x-1)] * [(x+2)/(x+2)]-[(x-1)/(x+2)] * [(x-1)/(x-1)] / [(15x+3)/(x^2-1)] = {(x^2+3x+2)/[(x-1)(x+2)]- (x^2-2x+1)/[(x-1)(x+2)]}/[(15x+3)/(x^2-1)]= (5x+1)/[(x-1)(x+2)] * [(x-1)(x+1)]/[3(5x+1)] = (x+1)/[3(x+2)]

Example 5: Simplify the complex fraction below.

We will simplify this complex fraction using the LCD method: [1/(4x) + 1/(3x)] / [ 1/(4x) - 1/(3x) ]

For this problem, we are going to use Method 2 only.

Observe that the LCD of all the denominators is just \color{red}12x. Use this as the common multiplier for both top and bottom expressions.

Multiplying the top and bottom fractions by the least common denominator of 12, we get the following solution: [1/(4x) + 1/(3x)] / [ 1/(4x) - 1/(3x) ] = [1/(4x) + 1/(3x)] / [ 1/(4x) - 1/(3x) ] * (12x)/(12x) = (3+4)/(3-4) = 7/(-1) = -7. The final answer is -7.

Practice with Worksheets

You might also be interested in:

Multiplying Complex Fractions
Dividing Complex Numbers