Lesson 76: Complex Fractions

New Concept

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

What’s next

Next, we'll tackle worked examples that show how to simplify these fractions and convert tricky percentages into simple fractions.

Property

A complex fraction is a fraction that contains one or more fractions in the numerator or denominator, like this general form:

Examples

• Here is a fraction over a fraction: $\frac{\frac{1}{2}}{\frac{3}{4}}$
• Here is a mixed number over a whole number: $\frac{12\frac{1}{2}}{50}$
• Here is a whole number over a mixed number: $\frac{20}{5\frac{1}{4}}$

Explanation

Think of a fraction sandwich! Instead of just numbers, you have other fractions piled up in the numerator or denominator. It looks messy, but our job is to clean it up and find the simple fraction it represents. It is like a math puzzle waiting to be solved and simplified into a single, neat number.

Property

To simplify a complex fraction, multiply it by a clever form of 1. This special form of 1 is built by using the reciprocal of the denominator over itself, which makes the new denominator equal to 1.

Examples

• To solve, multiply by the reciprocal of the denominator: $\frac{\frac{3}{5}}{\frac{2}{3}} \cdot \frac{\frac{3}{2}}{\frac{3}{2}} = \frac{\frac{9}{10}}{1} = \frac{9}{10}$
• First, convert to improper fractions, then multiply: $\frac{15}{7\frac{1}{3}} = \frac{\frac{15}{1}}{\frac{22}{3}} \cdot \frac{\frac{3}{22}}{\frac{3}{22}} = \frac{\frac{45}{22}}{1} = 2\frac{1}{22}$
• Flip the bottom fraction and multiply: $\frac{\frac{1}{4}}{\frac{5}{8}} \cdot \frac{\frac{8}{5}}{\frac{8}{5}} = \frac{\frac{8}{20}}{1} = \frac{2}{5}$

Explanation

The ultimate trick is to make the denominator vanish by turning it into 1! Find the reciprocal of the bottom fraction (just flip it upside down) and multiply both the top and bottom of the big fraction by this flipped version. The bottom becomes 1, and the top becomes your new, simplified answer!

Property

A percent is just a fraction with a denominator of 100. To convert a percent that contains a fraction, simply write it over 100, like this:

Examples

• Convert the percent to a complex fraction, then simplify: $16\frac{2}{3}\% = \frac{16\frac{2}{3}}{100} = \frac{\frac{50}{3}}{\frac{100}{1}} = \frac{50}{3} \cdot \frac{1}{100} = \frac{50}{300} = \frac{1}{6}$
• A percent is a fraction over 100: $66\frac{2}{3}\% = \frac{\frac{200}{3}}{100} = \frac{200}{3} \cdot \frac{1}{100} = \frac{200}{300} = \frac{2}{3}$
• Write as a fraction, then multiply by the reciprocal of the denominator: $8\frac{1}{3}\% = \frac{\frac{25}{3}}{100} = \frac{25}{3} \cdot \frac{1}{100} = \frac{25}{300} = \frac{1}{12}$

Explanation

Don't let fractional percents scare you! Just remember that 'percent' means 'out of 100.' Put that funky fraction over 100 to create a complex fraction. Then, use your simplification superpowers to turn both the top and bottom into simple fractions and solve. It’s a two-step takedown: first make the complex fraction, then simplify it!