Gupta knows the area and width of a rectangle. He comes up with this equation for the length of the rectangle . What is the length of the rectangle in simplified form?

### Guidance

A **complex fraction** is a fraction that has fractions in the numerator and/or denominator. To simplify a complex fraction, you will need to combine all that you have learned in the previous five concepts.

#### Example A

Simplify .

**Solution:** This complex fraction is a fraction divided by another fraction. Rewrite the complex fraction as a division problem.

.

Now, this is just like a problem from the *Dividing Rational Expressions* concept. Flip the second fraction, change the problem to multiplication and simplify.

#### Example B

Simplify .

**Solution:** To simplify this complex fraction, we first need to add the fractions in the numerator and subtract the two in the denominator. The LCD of the numerator is and the denominator is just .

This fraction is now just like Example A. Divide and simplify if possible.

#### Example C

Simplify .

**Solution:** First, add the fractions in the numerator and subtract the ones in the denominator.

Now, rewrite as a division problem, flip, multiply, and simplify.

**Intro Problem Revisit** This complex fraction is a fraction divided by another fraction. Rewrite the complex fraction as a division problem.

.

Flip the second fraction, change the problem to multiplication and simplify.

Therefore, the length of the rectangle in simplified form is .

### Guided Practice

Simplify the complex fractions.

1.

2.

3.

#### Answers

1. Rewrite the fraction as a division problem and simplify.

2. Add the fractions in the numerator and denominator together.

Now, rewrite the fraction as a division problem and simplify.

3. Add the numerator and subtract the denominator of this complex fraction.

Now, flip and multiply.

### Vocabulary

- Complex Fraction
- A fraction with rational expression(s) in the numerator and denominator.

### Practice

Simplify the complex fractions.

Use the following pattern to answer the next four questions.

- Find the next two terms in the pattern.
- Using your graphing calculator, simplify each term in the pattern to a decimal.
- Make a conjecture about this pattern and the number the terms appear to be approaching.
- Find the sixth term in the pattern. Does it support your conjecture?