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Section 8.1 Algebraic Fractions

Subsection What is an Algebraic Fraction?

Variables may occur in the numerator or the denominator of a fraction, or both.

Example 8.1.

Choose variables and write a fraction for each situation.

  1. The five people in Tom's study group order Chinese food and agree to split the bill equally. How much will Tom's share be?
  2. Nurit has $800 to carpet a square bedroom in her house. What price per square foot can she afford?
  3. How long does it take an ocean liner to make a trans-Atlantic voyage?
Solution
  1. The amount of the bill is unknown. We let \(b\) represent the total bill. Then Tom's share is \(\dfrac{b}{5}\text{.}\)
  2. The dimensions of the room are unknown. We let \(L\) represent the length of bedroom. Then the price per square foot is \(\dfrac{800}{L^2}\text{.}\)
  3. The distance and the speed of the liner are unknown. We let \(D\) represent the distance traveled, and \(R\) the speed, or rate. Then the time for voyage is \(\dfrac{D}{R}\text{.}\)

An algebraic fraction (or rational expression, as they are sometimes called) is a fraction in which both numerator and denominator are polynomials.

Here are some examples:

\begin{equation*} \dfrac{3}{x},~~~~~~\dfrac{a^2+1}{a-2},~~~~~~\text{and}~~~~~~\dfrac{z-1}{2z+3} \end{equation*}
Look Closer.

We can evaluate algebraic fractions just as we do any other algebraic expression. But we cannot evaluate a fraction at any values of the variable that make the denominator equal to zero.

Example 8.2.
  1. Evaluate \(~~\dfrac{a^2+1}{a-2}~~\) for \(~a=4\text{.}\)
  2. For what value of \(a\) is the fraction undefined?
Solution
  1. Substitute \(\alert{4}\) for \(a\) in the fraction.

    \begin{align*} \dfrac{a^2+1}{a-2} \amp = \dfrac{\alert{4}^2+1}{\alert{4}-2} \amp \amp \blert{\text{Simplify numerator and denominator.}}\\ \amp = \dfrac{17}{2} \end{align*}
  2. Because we cannot divide by zero, a fraction is undefined if its denominator is zero. This fraction is undefined for \(a=\alert{2}\text{,}\) because \(~\dfrac{\alert{2}^2+1}{\alert{2}-2}=\dfrac{5}{0}\text{.}\)

Reading Questions Reading Questions

1.

What is an algebraic fraction? Give an example.

2.

What values of the variable must we exclude when working with algebraic fractions?

Subsection Applications of Algebraic Fractions

Envirogreen Technology, Inc. decides to produce a water filter for home use. They spend $5000 for startup costs, and each filter costs $50 to produce. To help her decide on a selling price for the filters, the marketing manager would like to know the average cost per filter if they produce \(x\) filters. She first computes the total cost of producing \(x\) filters.

\begin{align*} \text{Total Cost}~ \amp =~\text{Startup Cost}~+~\text{Cost of}~x~\text{Filters}\\ \amp = 5000 + 50x \end{align*}

Then, to find the average cost per filter, she divides the total cost by the number of filters produced.

\begin{equation*} \text{Average Cost}~=~\dfrac{\text{Total Cost}}{\text{Number of Filters}}~=~\dfrac{5000+50x}{x} \end{equation*}

This expression is an algebraic fraction.

Example 8.3.

Sketch a graph of the equation \(~~A=\dfrac{5000+50x}{x}\)

Solution

We evaluate the average cost for various production levels, \(x\text{,}\) and make a table of values. We plot the points and connect them with a smooth curve to obtain the graph shown below.

\(x\) \(A\)
\(50\) \(150\)
\(100\) \(100\)
\(200\) \(75\)
\(400\) \(62.50\)
\(500\) \(60\)
\(1000\) \(55\)
\(1250\) \(54\)
\(2000\) \(52.50\)
curve

Subsection Reducing Fractions

We can reduce a fraction if we can divide both numerator and denominator by a common factor. In algebra, it is helpful to think of factoring out the common factor first. For example,

\begin{equation*} \dfrac{27}{36} = \dfrac{\cancel{9} \cdot 3}{\cancel{9} \cdot 4} = \dfrac{3}{4} \end{equation*}

where we have divided both numerator and denominator by 9. The new fraction has the same value as the old one, namely 0.75, but it is simpler (the numbers are smaller.) Reducing is an application of the fundamental principle of fractions.

Fundamental Principle of Fractions.

We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

\begin{equation*} \blert{\dfrac{a \cdot c}{b \cdot c} = \dfrac{a}{b}~~~\text{if}~~~b,~c \not= 0} \end{equation*}
Example 8.4.

Reduce the fraction \(~~\dfrac{42ab^2}{35ab^3}\)

Solution

First we consider the numerical part of the fraction: we look for the largest common factor of 42 and 35. This factor is 7, so we write 42 and 35 in factored form:

\begin{equation*} \dfrac{42ab^2}{35ab^3} = \dfrac{7 \cdot 6~ab^2}{7 \cdot 5~ab^3} \end{equation*}

Next, we write the variable parts of the numerator and denominator in factored form. Remember that \(b^2\) means \(b \cdot b\) and \(b^3\) means \(b \cdot b \cdot b\) .

\begin{equation*} \dfrac{42ab^2}{35ab^3} =\dfrac{7 \cdot 6~a~b~b}{7 \cdot 5~a~b~b~b} \end{equation*}

Finally, we divide any common factors from the numerator and denominator.

\begin{equation*} \dfrac{42ab^2}{35ab^3} =\dfrac{\cancel{7} \cdot 6~\cancel{a}~\cancel{b}~\cancel{b}}{\cancel{7} \cdot 5~\cancel{a}~\cancel{b}~\cancel{b}~b} = \dfrac{6}{5b} \end{equation*}

The reduced fraction is \(\dfrac{6}{5b}\text{.}\)

Look Closer.

When we apply the fundamental principle, we often say that we are canceling common factors. In this context, "canceling" means dividing: Because division is the opposite operation for multiplication, we can cancel expressions that are multiplied together.

Example 8.5.

Reduce the fraction \(~~\dfrac{8x+4}{4}\)

Solution

We cannot cancel the 4's, because 4 is added to \(8x\text{,}\) not multiplied. Instead, we write the numerator as a product instead of a sum. To do that, we factor the numerator.

\begin{equation*} \dfrac{8x+4}{4} = \dfrac{4(2x+1)}{4} \end{equation*}

We see that 4 is a factor of the numerator, so we can divide top and bottom of the fraction by 4 to get

\begin{equation*} \dfrac{\cancel{4}(2x+1)}{\cancel{4}} = \dfrac{1(2x+1)}{1} = 2x+1 \end{equation*}

Reading Questions Reading Questions

3.

What does "canceling" mean when we reduce a fraction?

Caution 8.6.

We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted). Use your calculator to decide which calculation in each pair is correct. (In the second pair, choose a value for the variable and evaluate.)

    1. \(\dfrac{12}{8} = \dfrac{4 \cdot 3}{4 \cdot 2} \rightarrow \dfrac{3}{2}\)
    2. \(\dfrac{7}{6} = \dfrac{4 + 3}{4 + 2} \rightarrow \dfrac{3}{2}\)
    1. \(\dfrac{5x}{8x} \rightarrow \dfrac{5}{8}\)
    2. \(\dfrac{x+5}{x+8} \rightarrow \dfrac{5}{8}\)
Look Ahead.

Now we can apply our method to algebraic fractions. We must be very careful to cancel only common factors, never common terms. For this reason, we will think of reducing fractions in two steps: first factor the numerator and denominator completely, then divide by any common factors.

To Reduce an Algebraic Fraction.
  1. Factor numerator and denominator completely.
  2. Divide numerator and denominator by any common factors.
Example 8.7.
  1. Reduce \(~~\dfrac{3x^2+12x}{6x+24}\)
  2. Evaluate the fraction for \(x=4\text{.}\)
Solution
  1. We first factor the numerator and denominator completely.

    \begin{equation*} \dfrac{3x^2+12x}{6x+24} = \dfrac{3x(x+4)}{2 \cdot 3 (x+4)} \end{equation*}

    Then we divide numerator and denominator by any common factors.

    \begin{equation*} \dfrac{3x(x+4)}{2 \cdot 3 (x+4)} = \dfrac{\cancel{3}x\cancel{(x+4)}}{2 \cdot \cancel{3}\cancel{(x+4)}} = \dfrac{x}{2} \end{equation*}
  2. If we substitute \(x=\alert{4}\) into the original fraction and simplify, we find

    \begin{equation*} \dfrac{3(\alert{4})^2+12(\alert{4})}{6(\alert{4})+24} = \dfrac{48+48}{24+24} = \dfrac{96}{48} = 2 \end{equation*}

    However, the reduced fraction, \(\dfrac{x}{2}\text{,}\) is equivalent to the original one, so we should get the same answer if we substitute \(x=\alert{4}\) into the reduced fraction:

    \begin{equation*} \dfrac{x}{2} = \dfrac{\alert{4}}{2} = 2 \end{equation*}
Caution 8.8.

In Example 8.7, it would be incorrect to "cancel" terms of the numerator and denominator separately, as shown below:

\begin{equation*} \dfrac{\cancel{3x^2}+\bcancel{12x}}{\cancel{6x}+\bcancel{24}} \rightarrow\dfrac{x+x}{2+2}~~~~~~~\alert{\text{Incorrect!}} \end{equation*}

We must always factor the numerator and denominator before attempting to cancel.

Look Closer.

Suppose we'd like to evaluate the fraction in Example 8.7 for \(x=478\text{.}\) Instead of substituting 478 into the original fraction, it is much simpler to use the reduced fraction, and get

\begin{equation*} \dfrac{x}{2}=\dfrac{\alert{478}}{2} = 239 \end{equation*}

This the whole point of reducing a fraction: to get a simpler but equivalent expression.

Reading Questions Reading Questions

4.

Why is it a good idea to reduce an algebraic fraction when possible?

Subsection Negative of a Binomial

How can we deal with negative factors in a fraction? First, we consider some facts about the negative, or opposite, of an expression.

A fraction is a negative number if either its numerator or its denominator is negative, but not both. For example,

\begin{equation*} -\dfrac{2}{3}=\dfrac{-2}{3}=\dfrac{2}{-3} \end{equation*}

However, the fraction \(\dfrac{-2}{-3}\) is a positive number, because the quotient of two negative numbers is positive.

Look Closer.

In algebra, we prefer to write such a fraction with the negative sign in the numerator, so that the standard form for the opposite of \(~\dfrac{2}{3}~\) is \(~\dfrac{-2}{3}\text{.}\)

Reading Questions Reading Questions

5.

Does a negative sign in front of a fraction apply to the numerator, the denominator, or both?

To find the opposite, or negative, of a binomial (or any other algebraic expression) we multiply the expression by \(-1\text{.}\)

Negative of a Binomial.

The opposite of \(a-b\) is

\begin{equation*} \blert{-(a-b) = -a+b = b-a} \end{equation*}
Example 8.9.

Find the opposite of each binomial.

  1. \(2a-3b\)
  2. \(-x-1\)
Solution
  1. The opposite of \(~2a-3b~\) is

    \begin{equation*} -(2a-3b) = -2a+3b~~~~ \text{or}~~~~3b-2a \end{equation*}
  2. The opposite of \(~-x-1~\) is

    \begin{equation*} -(-x-1) = x+1 \end{equation*}
Look Closer.

Recall that any number (except zero) divided by itself is 1, and any number divided by its opposite is \(-1\text{.}\) For example,

\begin{equation*} \dfrac{8}{-8} = -1~~~~~~\text{and}~~~~~~\dfrac{-5}{5} = -1 \end{equation*}

The same is true for binomials and other algebraic expressions, so that

\begin{equation*} \dfrac{b-a}{a-b} = \dfrac{-(a-b)}{a-b} = -1 \end{equation*}

Here is an example of how opposites can arise in algebraic fractions.

Example 8.10.

Reduce \(~~\dfrac{2x-4y}{6y-3x}\)

Solution

First, we factor the numerator and denominator.

\begin{equation*} \dfrac{2x-4y}{6y-3x} = \dfrac{2(x-2y)}{3(2y-x)} \end{equation*}

We see that \(x-2y\) is the opposite of \(2y-x\text{,}\) that is, \(x-2y = -(2y-x)\text{.}\) Thus,

\begin{equation*} \dfrac{2(x-2y)}{3(2y-x)} = \dfrac{-2\cancel{(2y-x)}}{3\cancel{(2y-x)}} = \dfrac{-2}{3} \end{equation*}

Reading Questions Reading Questions

6.

What do we get when we divide an expression by its opposite?

Subsection Skills Warm-Up

Exercises Exercises

Factor completely.

1.
\(x^2-4\)
2.
\(x^2-4x\)
3.
\(x^2-4x+4\)
4.
\(4x^2-1\)
5.
\(4x^2-4x\)
6.
\(4x^2+4x+1\)

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 8.1

For Problems 1–2,

  1. Evaluate the fraction for the given values of the variable.
  2. For what values of the variable is the fraction undefined?
1.

\(\dfrac{x+1}{x-3},~~~~~x=\dfrac{1}{2},~{-4}\)

2.

\(\dfrac{2a-a^2}{a^2+1},~~~~~a=3,~{-1}\)

3.
  1. For what values of the variable is the fraction \(\dfrac{2x+1}{x^2-1}\) undefined?
  2. Find a value of \(x\) for which the fraction is equal to zero.

For Problems 4–11, reduce the fraction.

4.

\(\dfrac{15}{3x}\)

5.

\(\dfrac{24b}{14}\)

6.

\(\dfrac{-5z}{6z}\)

7.

\(\dfrac{5u}{120uv}\)

8.

\(\dfrac{16ab}{-10ab}\)

9.

\(\dfrac{3a^2}{27a}\)

10.

\(\dfrac{-9y^3z}{42yz}\)

11.

\(\dfrac{8u^3v^2}{12v^2w}\)

For Problems 12–17, reduce the fraction if possible. If the fraction cannot be reduced, state the reason.

12.
  1. \(\dfrac{a+4}{a+5}\)
  2. \(\dfrac{a \cdot 4}{a \cdot 5}\)
13.
  1. \(\dfrac{2 \cdot m}{4 \cdot n}\)
  2. \(\dfrac{2+m}{4+n}\)
14.
  1. \(\dfrac{z-3}{z+9}\)
  2. \(\dfrac{z(-3)}{z(+9)}\)
15.
  1. \(\dfrac{u(-v)}{u(v)}\)
  2. \(\dfrac{u-v}{u+v}\)
16.
  1. \(\dfrac{3+x+y}{2+x+y}\)
  2. \(\dfrac{3xy}{2xy}\)
17.
  1. \(\dfrac{x+4}{x+2}\)
  2. \(\dfrac{4x}{2x}\)

For Problems 18–22, reduce each fraction if possible, and select the correct response, (a) or (b).

18.

\(\dfrac{x+2}{y+2}\)

  1. \(\dfrac{x}{y}\)
  2. Cannot be reduced
19.

\(\dfrac{2x+4}{4}\)

  1. \(\dfrac{x+2}{2}\)
  2. \(2x\)
20.

\(\dfrac{y^2-1}{y-1}\)

  1. \(y+1\)
  2. \(y\)
21.

\(\dfrac{x^2+z^2}{x+z}\)

  1. \(x+z\)
  2. Cannot be reduced
22.

\(\dfrac{n^2+n}{n}\)

  1. \(n^2\)
  2. \(n+1\)
23.

\(\dfrac{z^2+2z+1}{z+1}\)

  1. \(z+1\)
  2. \(z^2+2\)
24.

\(\dfrac{m^2+4}{m+2}\)

  1. \(m+2\)
  2. Cannot be reduced

For Problems 25–30, reduce the fraction to lowest terms.

25.

\(\dfrac{3x+12}{6}\)

26.

\(\dfrac{b}{b^2-b}\)

27.

\(\dfrac{n^2+4n+4}{n^2-4}\)

28.

\(\dfrac{2a^2}{2a^2-6a}\)

29.

\(\dfrac{3(a+b)}{4(a+b)}\)

30.

\(\dfrac{x-4}{x^2-3x-4}\)

For Problems 31–34, decide whether the fraction is equivalent to 1, to \(-1\text{,}\) or cannot be reduced.

31.

\(\dfrac{x+4}{x-4}\)

32.

\(\dfrac{x+3z}{z+3x}\)

33.

\(\dfrac{x^2-4}{4-x^2}\)

34.

\(\dfrac{-(2-t)}{t-2}\)

For Problems 35–38, reduce the fraction if possible.

35.

\(\dfrac{b-2}{2-4b}\)

36.

\(\dfrac{a-b}{a^2-b^2}\)

37.

\(\dfrac{(3x+2y)^2}{4y^2-9x^2}\)

38.

\(\dfrac{3a-a^2}{a^2-2a-3}\)

For Problems 39–42,

  1. Choose a variable to represent the unknown quantity,
  2. then write an algebraic fraction.
39.

There are 84 first-graders at Bonnair Elementary School. What fraction of the first-graders have been immunized against German measles?

40.

One hundred thirty-eight of the employees at Digitronics enrolled in a bonus incentive plan. What fraction of the employees enrolled?

41.

Gladys filled her car with gas three times while driving to Yellowstone National Park, a distance of 480 miles. What was her car's fuel efficiency during the trip?

42.

Write a fraction that gives the length of a rectangle in terms of its area and its width.

For Problems 43–45, write algebraic fractions.

43.

Sharelle's car still had 4 gallons in the gas tank when she filled up for $18.

  1. If the gas tank holds \(x\) gallons, what was the price per gallon of the gasoline?
  2. Evaluate your fraction for \(x=14\text{.}\) What does your answer mean in this context?
44.

Morgan drove across country in \(h\) hours, but he estimates that he spent 10 hours stopped for rest and meals.

  1. If he drove a total of 2800 miles, what was his average speed on the road?
  2. Evaluate your fraction for\(h=50\text{.}\) What does your answer mean in this context?
45.

The volume of a test tube is given by its height times the area of its cross-section. A test tube that holds 200 cubic centimeters is \(2x-1\) centimeters long.

  1. What is the area of its cross-section?
  2. Evaluate your fraction for \(x=13\text{.}\) What does your answer mean in the context of the problem?
46.

Ernestine wants to make a trip of 12 miles on her bicycle. If her trip takes a total of \(t\) hours, her average speed will be \(v\) miles per hour, where \(v\) is given by \(v = \dfrac{12}{t}\text{.}\)

  1. Complete the table and graph the equation on the grid below.
  2. What will be Ernestine's average speed if the trip takes her 2 hours? Locate the corresponding point on your graph.
  3. How long will it take Ernestine to finish the trip if she maintains an average speed of 18 miles per hour? Locate the corresponding point on the graph.
\(t\) \(v\)
\(0.75\) \(\hphantom{0000}\)
\(1\) \(\hphantom{0000}\)
\(1.2\) \(\hphantom{0000}\)
\(1.8\) \(\hphantom{0000}\)
\(2\) \(\hphantom{0000}\)
\(2.5\) \(\hphantom{0000}\)
\(3\) \(\hphantom{0000}\)
\(4\) \(\hphantom{0000}\)
grid
47.

The crew team can row at a steady pace of 10 miles per hour in still water. Every afternoon, their training includes a five-mile row upstream on the river. If the current in the river on a given day is \(v\) miles per hour, then the time required for this exercise, in minutes, is given by

\begin{equation*} t=\dfrac{300}{10-v} \end{equation*}

Use the graph of this equation shown in the figure to answer questions (a) and (b).

curve
  1. How long does the exercise take if there is no current in the river?
  2. How long will it take if the current is 4 miles per hour?
  3. Find an exact answer for part (b) by using the equation.
  4. If the exercise took 2 hours, what was the current in the river?
  5. As the speed of the current increases, what happens to the time needed for the exercise?

Mental Exercise: For Problems 48–50, decide which expressions are equivalent to the given fraction. Do not use pencil, paper, or calculator!

48.

\(\dfrac{2}{3}\)

  1. \(\dfrac{2+4}{3+4}\)
  2. \(\dfrac{2 \cdot 4}{3 \cdot 4}\)
  3. \(\dfrac{22}{33}\)
  4. \(\dfrac{2 \div 4}{3 \div 4}\)
49.

\(\dfrac{3}{5}\)

  1. \(\dfrac{3x}{5x}\)
  2. \(\dfrac{3-x}{5-x}\)
  3. \(\dfrac{30+x}{x+50}\)
  4. \(\dfrac{30x}{50x}\)
50.

\(\dfrac{2w-6}{w}\)

  1. \(-4\)
  2. \(\dfrac{6-2w}{-w}\)
  3. \(2-\dfrac{6}{w}\)
  4. \(-\dfrac{2w+6}{w}\)

For Problems 51–52, explain the error in each calculation.

51.

\(\dfrac{3x+4}{3}~ \rightarrow~x+4\)

52.

\(\dfrac{2(5z-4)}{5z}~ \rightarrow~\dfrac{2(-4)}{1} = -8\)