Lesson 4: More Operations on Fractions

New Concept

This lesson introduces complex fractions—fractions containing other fractions, like $\frac{x + \frac{1}{2}}{x - 1}$. You'll learn powerful techniques, such as using the LCD and polynomial division, to simplify these expressions into a single, manageable fraction.

What’s next

You'll start by simplifying complex fractions using our interactive examples. Then, you will test your understanding with a series of practice cards.

Property

A fraction that contains one or more fractions in either its numerator or its denominator or both is called a complex fraction. Like simple fractions, complex fractions represent quotients. For example, $\frac{\frac{2}{3}}{\frac{5}{6}} = \frac{2}{3} \div \frac{5}{6}$.

To simplify a complex fraction:

1. Find the LCD of all the fractions contained in the complex fraction.
2. Multiply the numerator and the denominator of the complex fraction by the LCD.
3. Reduce the resulting simple fraction, if possible.

Examples

• To simplify $\frac{\frac{3}{4}}{\frac{7}{8}}$, we can treat it as a division: $\frac{3}{4} \div \frac{7}{8} = \frac{3}{4} \cdot \frac{8}{7} = \frac{24}{28} = \frac{6}{7}$.

Property

Algebraic fractions are sometimes written using negative exponents. To simplify, first convert the negative exponents into fractions. For example, $x^{-1} - y^{-1} = \frac{1}{x} - \frac{1}{y}$. Then combine the fractions using a common denominator.

Caution:

1. A difference of fractions is not the fraction of the difference: $\frac{1}{x} - \frac{1}{y} \neq \frac{1}{x - y}$.
2. The power of a sum is not the sum of the powers: $(a + b)^n \neq a^n + b^n$. Therefore, $(x^{-2} + y^{-2})^{-1} \neq x^2 + y^2$.

Examples

• To simplify $a^{-1} + 3b^{-1}$, first rewrite as fractions: $\frac{1}{a} + \frac{3}{b}$. The LCD is $ab$, so we get $\frac{b}{ab} + \frac{3a}{ab} = \frac{b + 3a}{ab}$.

Property

The average speed for a trip is the total distance divided by the total time. For a two-part trip with distances $d_1$ and $d_2$ traveled at rates $r_1$ and $r_2$, the times for each part are $t_1 = \frac{d_1}{r_1}$ and $t_2 = \frac{d_2}{r_2}$.

The average speed for the entire trip is given by the formula:

Examples

• A car travels 150 miles at 50 mph and returns the same distance at 30 mph. Total distance is 300 miles. Total time is $\frac{150}{50} + \frac{150}{30} = 3 + 5 = 8$ hours. Average speed is $\frac{300}{8} = 37.5$ mph.

Property

An algebraic fraction is 'improper' if the degree of the numerator is greater than or equal to the degree of the denominator. We can simplify it by division. The result is the sum of a polynomial and a simpler fraction.

If the divisor is a monomial, divide it into each term of the numerator:

If the divisor is a polynomial, use a method similar to long division. The final answer is expressed as: