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# Complex Fractions

## Solve fractions with fractional numerators and/or denominators.

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Complex Fractions

Gupta knows the area and width of a rectangle. He comes up with this equation for the length of the rectangle 2x212xx+1\begin{align*}\frac{\frac{2}{x^2-1}}{\frac{2x}{x+1}}\end{align*}. 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 9xx+23x24\begin{align*}\frac{\frac{9x}{x+2}}{\frac{3}{x^2-4}}\end{align*}.

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

9xx+23x24=9xx+2÷3x24
.

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 \begin{align*}\frac{\frac{1}{x} + \frac{1}{x+1}}{4- \frac{1}{x}}\end{align*}.

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 \begin{align*}x(x+1)\end{align*} and the denominator is just \begin{align*}x\end{align*}.

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

#### Example C

Simplify \begin{align*}\frac{\frac{5-x}{x^2+6x+8} + \frac{x}{x+4}}{\frac{6}{x+2} - \frac{2x+3}{x^2-3x-10}}\end{align*}.

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 \begin{align*}\frac {1}{x^2-x}\end{align*}.

### Guided Practice

Simplify the complex fractions.

1. \begin{align*}\frac{\frac{5x-20}{x^2}}{\frac{x-4}{x}}\end{align*}

2. \begin{align*}\frac{\frac{1-x}{x} - \frac{2}{x-1}}{1 + \frac{1}{x}}\end{align*}

3. \begin{align*}\frac{\frac{\text{-}3}{\text{-}4x^2-5x+6} + \frac{\text{-}4}{\text{-}4x+3}}{\text{-}\frac{1}{\text{-}4x^2-5x+6} + \frac{2}{\text{-}4x+3}} \end{align*}

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

\begin{align*}\frac{\frac{5x-20}{x^2}}{\frac{x-4}{x}} = \frac{5x-20}{x^2} \div \frac{x-4}{x} = \frac{5 \cancel{(x-4)}}{x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{x-4}} = \frac{5}{x}\end{align*}

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

\begin{align*}\frac{\frac{1-x}{x} - \frac{2}{x-1}}{1+\frac{1}{x}} = \frac{\frac{x-1}{x-1} \cdot \frac{1-x}{x} - \frac{2}{x-1} \cdot \frac{x}{x}}{\frac{x}{x} \cdot 1+ \frac{1}{x}} = \frac{\frac{(x-1)(1-x)-2x}{x(x-1)}}{\frac{x+1}{x}} = \frac{\frac{-x^2+1}{x(x-1)}}{\frac{x+1}{x}} \end{align*}

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

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

### Explore More

Simplify the complex fractions.

1. \begin{align*}\frac{\frac{2x}{5}}{\frac{8}{7}}\end{align*}
2. \begin{align*}\frac{\frac{4}{x^2-9}}{\frac{6x}{x+3}}\end{align*}
3. \begin{align*}\frac{\frac{7x^3}{x^2+5x+6}}{\frac{35x^2}{x+2}}\end{align*}
4. \begin{align*}\frac{\frac{24x+3}{3x+1}}{\frac{16x+2}{6x^2-13x-5}}\end{align*}
5. \begin{align*}\frac{\frac{4}{x-1} + \frac{1}{x}}{\frac{1}{x} -5}\end{align*}
6. \begin{align*}\frac{\frac{3x}{x+4} - \frac{1}{x}}{\frac{3x-4}{x^2+6x+8}}\end{align*}
7. \begin{align*}\frac{8- \frac{3x}{x+5}}{\frac{10}{x+5} + \frac{5}{x+1}}\end{align*}
8. \begin{align*}\frac{\frac{x}{x+3} - \frac{4}{2x+1}}{\frac{3}{2x+1} + \frac{6}{x^2-9}}\end{align*}
9. \begin{align*}\frac{\frac{x+3}{x} + \frac{2x}{5-x}}{\frac{3}{2x} - \frac{4x}{x-5}}\end{align*}
10. \begin{align*}\frac{\frac{2x}{5x^2-13x-6} + \frac{1}{x-3}}{\frac{4}{5x+2} - \frac{5x}{5x^2-3x-2}}\end{align*}
11. \begin{align*}\frac{\frac{3x}{x^2-4} + \frac{x+4}{x^2+3x+2}}{\frac{x+1}{x^2-x-2} - \frac{2x}{x^2+2x+1}}\end{align*}

Use the following pattern to answer the next four questions.

\begin{align*}2+\frac{1}{1+\frac{1}{2}}, \ 2+\frac{1}{1+\frac{1}{2+\frac{2}{3}}}, \ 2 + \frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{4}}}}\end{align*}

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.13.

### Vocabulary Language: English

Complex Fraction

Complex Fraction

A fraction with rational expression(s) in the numerator and denominator (a fraction composed of other fractions) is known as a complex fraction.