Topic: Rational Expressions
Subtopic: Compound Fractions
Overview
A compound fraction is a fraction within a fraction (yikes!). There are two types. Type 1 has a single fraction in the numerator and a single fraction in the denominator. This type is best simplified using the method of flip 'n multiply where you flip the denominator and multiply it by the numerator. Type 2 has more than one fraction being added in the numerator or in the denominator or both, basically little fractions all over the place. The fastest way to eliminate all the little fractions is to multiply through by 1 in the form of the LCD/LCD. This method clears the little fractions in one fell swoop. Some people prefer though to convert type 2's into type 1's and then use flip 'n multiply. Let's compare these two approaches on the DB.
This section is quite involved and can be a bit overwhelming. Try to treat it as an introduction to compound fractions and don’t get too bogged down in the details. You will see them again in a 100-level math course where the processes will become more fluid and you will be motivated by real-world applications. For now, just try to get the main ideas and processes down.
Objectives
By the end of this topic you should know and be prepared to be tested on:
- 8.4.1 Simplify "type 1" compound fractions by the flip 'n multiply method
- 8.4.2 Simplify "type 2" compound fractions by the LCD method
- 8.4.3 Evaluate expressions/functions containing compound fractions
Terminology
Define: compound fraction (a.k.a. complex fraction), the difference between "type 1" and "type 2" compound fractions
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 7.5
- Examples 1-3 treat the numerator and denominator separately at first. For each they add/subtract the terms to combine into a single fraction. Then put these two fractions back into the compound fraction so that it is a (type 1) single fraction over single fraction. This can then be treated as a division problem or better yet the top fraction multiplied by the flip of the bottom fraction.
- Examples 4-6 eliminate the denominators of the little fractions all at once by multiplying the numerator and denominator of the big compound fraction by the LCD of all the little fractions. This is the preferred way to work all type 2 compound fractions because there are fewer steps overall. But if this way doesn't make sense to you, feel free to do them all by the example 1-3 way.
Supplementary Resources
The compound fractions we work with at this level are "terminate", but there are ones that don't. They are called continued fractions and they are fractions inside fractions inside fractions going on forever. Remarkably these continued fractions can have "nice" forms. For instance these three representations of continued fractions each of whom exactly equal the irrational number pi:
One was found hundreds of years ago by Leonhard Euler (pronounced "oiler"), one of the great mathematicians, and one was a fairly recent find. Mathematics is still growing!