Lesson 92: Simplifying Complex Fractions

New Concept

A complex fraction is a fraction that contains one or more fractions in the numerator or the denominator.

What’s next

Next, you'll learn powerful methods to simplify these expressions and apply them to solve real-world rate problems.

Property

A complex fraction is a fraction that contains one or more fractions in the numerator or the denominator. It can be written as division: $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d}$.

Explanation

Think of a complex fraction as a division problem in a clever disguise! To solve it, just rewrite the big fraction bar as a division symbol. Then, you can use the classic “keep, change, flip” trick by multiplying by the reciprocal of the denominator. It is that simple!

Examples

$\frac{\frac{a}{x}}{\frac{b}{a+x}} = \frac{a}{x} \div \frac{b}{a+x} = \frac{a}{x} \cdot \frac{a+x}{b} = \frac{a(a+x)}{xb}$
$\frac{\frac{3}{4}}{\frac{5}{7}} = \frac{3}{4} \div \frac{5}{7} = \frac{3}{4} \cdot \frac{7}{5} = \frac{21}{20}$

Property

To simplify complex fractions with polynomials, first factor out any Greatest Common Factors. Then rewrite as division, multiply by the reciprocal, and divide out common factors before the final multiplication step.

Explanation

Don’t rush to multiply! Be a math detective and factor everything you can first. This awesome trick reveals matching pieces on the top and bottom that you can cancel out, making the problem way simpler to solve before you even multiply.

Examples

$\frac{\frac{3x}{6x+12}}{\frac{9}{x+2}} = \frac{3x}{6(x+2)} \cdot \frac{x+2}{9} = \frac{x}{18}$
$\frac{\frac{y-5}{y^2-25}}{\frac{10}{y+5}} = \frac{y-5}{(y-5)(y+5)} \cdot \frac{y+5}{10} = \frac{1}{10}$

Factoring can reveal hidden shortcuts; let's see how it simplifies this complex fraction. This first key idea of factoring to simplify is a foundational skill.

Example Problem
Simplify the expression $\frac{\frac{5x}{10x+30}}{\frac{15}{x+3}}$.

Step-by-Step

1. First, rewrite the complex fraction as a division problem to make it easier to work with.

2. Now, factor out the Greatest Common Factor (GCF) from the denominator of the first fraction and rewrite the division as multiplication by the reciprocal.

3. Identify and divide out common factors in the numerators and denominators. Here, $(x+3)$ cancels out, and $5$ and $15$ have a common factor of $5$.

4. Multiply the remaining factors to get the simplified result.

Property

If the numerator or denominator has addition or subtraction, combine them into one fraction first. Then, simplify the resulting complex fraction using division.

Explanation

Is there addition or subtraction inside your big fraction? Pause! Clean up that part first by making it a single, tidy fraction. Once each part is a single fraction, you can go back to the familiar “flip and multiply” method to solve.

Examples

$\frac{\frac{1}{x}}{1-\frac{1}{x}} = \frac{\frac{1}{x}}{\frac{x-1}{x}} = \frac{1}{x} \cdot \frac{x}{x-1} = \frac{1}{x-1}$
$\frac{3+\frac{1}{a}}{\frac{2}{a}} = \frac{\frac{3a+1}{a}}{\frac{2}{a}} = \frac{3a+1}{a} \cdot \frac{a}{2} = \frac{3a+1}{2}$

What do you do when the denominator itself needs simplifying first? Let's tackle it. The second key idea is simplifying parts of the complex fraction first.

Example Problem
Simplify the expression $\frac{\frac{3}{a}}{1-\frac{3}{a}}$.

Step-by-Step

1. The denominator contains a subtraction. To simplify it, we first need to combine $1$ and $\frac{3}{a}$ into a single fraction. The least common denominator is $a$.

2. Perform the subtraction in the denominator.

3. Now that we have a single fraction over another, we can rewrite the expression as a division problem.

4. To divide, we multiply by the reciprocal of the second fraction.

5. Divide out the common factor $a$ from the numerator and denominator.

6. The final simplified expression is what remains.