# 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.

**Using Method 1**

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

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.

**Using Method 2**

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

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.

**Example 2:** Simplify the complex fraction below.

**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.

**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.

**Example 3:** Simplify the complex fraction below.

**Using Method 1**

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

**Using Method 2**

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

**Example 4:** Simplify the complex fraction below.

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.

**Example 5:** Simplify the complex fraction below.

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.

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