Section 8.3 Operations on Algebraic Fractions
Subsection 8.3.1 Products of Fractions
To multiply two fractions together, we multiply their numerators together and then multiply their denominators together.
Product of Fractions.
If \(b \ne 0\) and \(d \ne 0\text{,}\) then
If a common factor occurs in a numerator and a denominator of either fraction, we can divide it out either before or after multiplying. For example,
or
It is usually easier to cancel any common factors before multiplying.
Example 8.3.1.
Multiply \(~\dfrac{2x4}{3x+6} \cdot \dfrac{6x+9}{x2}\)
First, we factor each numerator and denominator. Then we divide numerator and denominator by any common factors.
Checkpoint 8.3.2. Practice 1.
To multiply a fraction by a whole number, we write the whole number with 1 as denominator.
The same applies to the product of an algebraic fraction and any nonfractional expression. For example,
We summarize the procedure for multiplying algebraic fractions as follows.
To multiply algebraic fractions:.
Factor each numerator and denominator.
If any factor appears in both a numerator and a denominator, divide out that factor.
Multiply the remaining factors of the numerator and the remaining factors of the denominator.
Reduce the product if necessary.
Subsection 8.3.2 Quotients of Fractions
To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. For example,
We express this rule in symbols as follows.
Quotient of Fractions.
If \(b, c, d \ne 0\text{,}\) then
Thus, to divide two algebraic fractions, we take the reciprocal of the divisor and then follow the rules for multiplying algebraic fractions.
Example 8.3.3.
Divide: \(~\dfrac{a2}{6a^2} \div \dfrac{4a^2}{4a^22a}\)
First, we change the operation to multiplication by taking the reciprocal of the divisor.
Now we follow the rules for multiplication: First, we factor each numerator and denominator.
Note 8.3.4.
When working with algebraic fractions, we often leave the denominators in factored form. This makes it easier to add and subtract fractions, and to check whether they can be reducecd.
Checkpoint 8.3.5. QuickCheck 1.
Fill in the blanks.
The first step in multiplying fractions is to
add together the numerators
factor each numerator and denominator
find the reciprocal of each fraction
remove any exponents
It is usually easier to cancel common factors
after
before
To divide two fractions, we multiply the first fraction by the
GCF
LCD
opposite
reciprocal
We can write a whole number as a fraction with denominator .
We summarize the procedure for dividing algebraic fractions as follows.
To divide algebraic fractions:.
Take the reciprocal of the second fraction and change the operation to multiplication.
Follow the rules for multiplication of fractions.
Checkpoint 8.3.6. Practice 2.
Subsection 8.3.3 Adding and Subtracting Like Fractions
Fractions with the same denominator are called like fractions. For example,
are like fractions, while
are unlike fractions. We can add or subtract like fractions for the same reason that we can add like terms. Just as
we can think of the sum \(~\dfrac{3}{5}+\dfrac{4}{5}~\) as
or \(\dfrac{7}{5}\text{.}\) The denominators of the terms must be the same because they tell us what kind of quantity we are adding. We can add two quantities if they are the same kind.
When we add like fractions, we add their numerators and keep their denominators the same. For example,
The same holds true for all algebraic fractions.
Sum or Difference of Like Fractions.
If \(c \ne 0\text{,}\) then
Example 8.3.7.
Add.
\(\displaystyle \dfrac{2x}{9z^2}+\dfrac{5x}{9z^2}\)
\(\displaystyle \dfrac{2x5}{x+2}+\dfrac{x+4}{x+2}\)

Because these are like fractions, we add their numerators and keep the same denominator.
\begin{equation*} \dfrac{2x}{9z^2}+\dfrac{5x}{9z^2} = \dfrac{2x+5x}{9z^2} = \dfrac{7x}{9z^2} \end{equation*} 
We combine the numerators over a single denominator.
\begin{align*} \dfrac{2x5}{x+2}+\dfrac{x+4}{x+2} \amp = \dfrac{(2x5)+(x+4)}{x+2} \amp \amp \blert{\text{Add like terms in the numerator.}}\\ \amp = \dfrac{3x1}{x+2} \end{align*}
We summarize our procedure for adding or subtracting like fractions as follows.
To add or subtract like fractions.
Add or subtract the numerators.
Keep the same denominator.
Reduce the sum or difference if necessary.
Checkpoint 8.3.8. Practice 3.
We must be careful when subtracting algebraic fractions: A subtraction sign in front of a fraction applies to the entire numerator.
Example 8.3.9.
Subtract: \(~\dfrac{x3}{x1}\dfrac{3x5}{x1}\)
We combine the numerators over a single denominator. We use parentheses around \(3x5\) to show that the subtraction applies to the entire numerator.
We should always check to see whether the fraction can be reduced. Factor the numerator to find
Caution 8.3.10.
In the Example above, the subtraction symbol in the numerator \((x3)(3x5)\) applies to both terms of \((3x5)\text{.}\) That is why
Checkpoint 8.3.11. QuickCheck 2.
True or False.
Like fractions have the same denominator.
True
False
To add like fractions, we add their numerators and add their denominators.
True
False
A negative sign in front of a fraction changes the sign of both numerator and denominator.
True
False
When we subtract fractions, the subtraction symbol applies only to the first term of the numerator.
True
False
Checkpoint 8.3.12. Practice 4.
Subsection 8.3.4 Unlike Fractions
Let's review the steps for adding or subtracting unlike fractions. For example, to compute the sum
we first find the lowest common denominator, or LCD, for the two fractions: The smallest number that is a multiple of both 3 and 4 is 12.
Next, we use the fundamental principle of fractions to build each fraction to an equivalent one with denominator 12:
There is noting mysterious about the building process; we are breaking up each fraction into smaller pieces of the same size so that we can add them together.
Finally, the new fractions are like fractions, and we can add them by combining their numerators.
We add or subtract algebraic fractions with the same three steps.
To add or subtract algebraic fraction.
Find the lowest common denominator (LCD) for the fractions.
Build each fraction to an equivalent one with the same denominator.
Add or subtract the resulting like fractions: Add or subtract their numerators, and keep the same denominator.
Reduce the sum or difference if necessary.
Example 8.3.13.
Subtract: \(~\dfrac{x+2}{6}\dfrac{x1}{15}\)
Step 1: We find the LCD: The smallest multiple of both 6 and 15 is 30.
Step 2: We build each fraction to an equivalent one with denominator 30. The building factor for the first fraction is \(\alert{5}\text{,}\) and for the second fraction the building factor is \(\alert{2}\text{.}\)
Step 3: We subtract the resulting like fractions to obtain
Step 4: Finally, we reduce the fraction.
Checkpoint 8.3.14. Practice 5.
Checkpoint 8.3.15. QuickCheck 3.
Fill in the blanks.
To add unlike fractions, we must first convert them into
decimal
factored
like
reduced
Building a fraction is an application of the
distributive law
fundamental principle of fractions
Pythagorean Theorem
zero factor principle
The LCD is the smallest
divisor
factor
multiple
The last step in adding fractions is to
check the solution in the equation
factor the numerator
reduce the fraction
Subsection 8.3.5 Finding the Lowest Common Denominator
The lowest common denominator for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator. If neither denominator can be factored, then their LCD is just the product of the two expressions.
Example 8.3.16.
Add: \(~\dfrac{6}{x} + \dfrac{x}{x2}\)
Step 1: The LCD for these fractions is just the product of their denominators, \(x(x2)\)
Step 2: We build each fraction to an equivalent one with denominator \(x(x2)\text{.}\) The building factor for \(\dfrac{6}{x}\) is \(\alert{(x2)}\) and the building factor for \(\dfrac{x}{x2}\) is \(\alert{x}\text{.}\) We multiply numerator and denominator of each fraction by its building factor.
Step 3: We combine the resulting like fractions.
Step 4: The numerator of this fraction cannot be factored, so the sum cannot be reduced.
Checkpoint 8.3.17. Practice 6.
If the denominators contain any common factors, the LCD is not simply their product. For example, the LCD for
is not \(12(18) = 216\text{.}\) It is true that 216 is a multiple of both 12 and 18, but it is not the smallest one! We can find a smaller common denominator by factoring each denominator.
To find a number that both 12 and 18 divide into evenly, we need only enough factors to cover each of them. In this case two 2'sand two 3'sare sufficient, so the LCD is
You can check that both 12 and 18 divide evenly into 36.
In general, we can find the LCD in the following way.
To Find the LCD.
Factor each denominator completely.
Include each different factor in the LCD as many times as it occurs in any one of the given denominators.
Example 8.3.18.
Find the LCD for the fractions \(~\dfrac{2x}{x^21}~\) and \(~\dfrac{x+3}{x^2+x}~\)
We factor the denominators of each of the given fractions.
The factor \((x1)\) occurs once in the first denominator, the factor \(x\) occurs once in the second denominator, and the factor \((x+1)\) occurs once in each denominator. Therefore we include in our LCD one copy of each of these factors. The LCD is \(x(x1)(x+1)\text{.}\)
Caution 8.3.19.
In the previous Example, we do not include two factors of \((x+1)\) in the LCD. We need only one factor of \((x+1)\text{,}\) because \((x+1)\) occurs only once in each denominator. You can check that each original denominator divides evenly into our LCD, \(x(x1)(x+1)\text{.}\)
Checkpoint 8.3.20. Practice 7.
Subsection 8.3.6 Adding and Subtracting Unlike Fractions
After finding the LCD, we build each fraction to an equivalent one with the LCD as its denominator. The new fractions will be like fractions, so we can combine their numerators.
Building a fraction is an application of the fundamental principle of fractions,
It is the opposite of reducing a fraction, because we multiply, rather than divide, the numerator and denominator by an appropriate factor. To find the building factor, we compare the factors of the original denominator with those of the desired common denominator.
Example 8.3.21.
Add: \(~\dfrac{x4}{x^22x} + \dfrac{4}{x^24}\)
Step 1: Find the LCD. We factor each denominator completely.
The LCD is \(~x(x2)(x+2)\text{.}\)
Step 2: We build each fraction to an equivalent one with the LCD as its denominator. The building factor for \(\dfrac{x4}{x(x2)}\) is \(\alert{(x+2)}\text{.}\) We multiply the numerator and denominator of the first fraction by \(\alert{(x+2)}\text{:}\)
The building factor for \(\dfrac{4}{(x2)(x+2)}\) is \(\alert{x}\text{.}\)
Step 3: The fractions are now like fractions, so we add them by combining their numerators.
Step 4: Finally, we reduce the fraction.
Caution 8.3.22.
Do not reduce the builtup fractions in Step 3  you will just get back to the original problem. When adding fractions, we have to make the fractions "harder" before we can combine them. Don't reduce until the last step of the problem.
Checkpoint 8.3.23. QuickCheck 4.
True or False.
We can always find the LCD by multiplying together the denominators of the fractions.
True
False
If each denominator includes one factor of \(x3\text{,}\) we include two factors of \(x3\) in the LCD.
True
False
We find the building factor for each fraction by comparing its denominator to the LCD.
True
False
When adding fractions, it is best to reduce the fractions at each step.
True
False
Checkpoint 8.3.24. Practice 8.
Subsection 8.3.7 Applications
Recall that the formulas for rational functions are algebraic fractions. It is often useful to simplify the formula for a function before using it.
Example 8.3.25.
When estimating their travel time, pilots must take into account the prevailing winds. A tail wind adds to the plane's ground speed, while a head wind decreases the ground speed. Skyhigh Airlines is setting up a shuttle service from Dallas to Phoenix, a distance of 800 miles.
Express the time needed for a oneway trip, without wind, as a function of the speed of the plane.
Suppose there is a prevailing wind of 30 miles per hour blowing from the west. Write expressions for the flying time from Dallas to Phoenix and from Phoenix to Dallas.
Write an expression for the round trip flying time with a 30mileperhour wind from the west, as a function of the plane's speed. Simplify your expression.

Recall that \(\text{time} = \dfrac{\text{distance}}{\text{rate}}\text{.}\) If we let \(r\) represent the speed of the plane in still air, then the time required for a oneway trip is
\begin{equation*} f(r) = \dfrac{800}{r} \end{equation*} 
On the trip from Dallas to Phoenix the plane encounters a head wind of 30 miles per hour, so its actual ground speed is \(r30\text{.}\) On the return trip the plan enjoys a tail wind of 30 miles per hour, so its actual ground speed is \(r+30\text{.}\) Therefore, the flying times are
\begin{gather*} \text{Dallas to Phoenix:}~~~\dfrac{800}{r30}\\ \text{Phoenix to Dallas:}~~~\dfrac{800}{r+30} \end{gather*} 
The roundtrip flying time from Dallas to Phoenix and back is
\begin{equation*} F(r) = \dfrac{800}{r30} +\dfrac{800}{r+30} \end{equation*}The LCD for these fractions is \((r30)(r+30)\text{.}\) Thus,
\begin{align*} \dfrac{800}{r30} +\dfrac{800}{r+30} \amp = \dfrac{800\alert{(r+30)}}{(r30)\alert{(r+30)}} +\dfrac{800\alert{(r30)}}{(r+30)\alert{(r30)}}\\ \amp = \dfrac{(800r+24,000)+(800r24,000)}{(r+30)(r30)}\\ \amp = \dfrac{1600r}{r^2900} \end{align*}
Checkpoint 8.3.26. Practice 9.
A rowing team can maintain a speed of 15 miles per hour in still water. The team’s daily training session includes a 5mile run up the Red Cedar River and the return downstream.
Express the team’s time on the upstream leg as a function of the speed of the current.
Write a function for the team’s time on the downstream leg.
Write and simplify an expression for the total time for the training run as a function of the current’s speed.
Exercises 8.3.8 Problem Set 8.3
Warm Up
Exercise Group.
In Problems 1–6, we review the operations on arithmetic fractions.
1.
\(\dfrac{5}{8} \cdot \dfrac{3}{4} = \dfrac{15}{32}\)
\(\blert{\text{Multiply the numerators together; multiply the denominators together.}}\)
\(\displaystyle \dfrac{2}{5} \cdot \dfrac{2}{3}\)
\(\displaystyle \dfrac{2}{w} \cdot \dfrac{z}{3}\)
\(\displaystyle \dfrac{4}{15} \)
\(\displaystyle \dfrac{2z}{3w}\)
2.
\(\dfrac{3}{4} \cdot \dfrac{5}{6} = \dfrac{\cancel{3}}{4} \cdot \dfrac{5}{2 \cdot \cancel{3}} = \dfrac{5}{8}\)
\(\blert{\text{Divide out common factors first, then multiply.}}\)
\(\displaystyle \dfrac{5}{4} \cdot \dfrac{8}{9}\)
\(\displaystyle \dfrac{5}{2a} \cdot \dfrac{4a}{9}\)
3.
\(\dfrac{3}{4} \cdot 6 = \dfrac{3}{4} \cdot \dfrac{6}{1} = \dfrac{3}{2 \cdot \cancel{2}} \cdot \dfrac{\cancel{2} \cdot 3}{1} = \dfrac{9}{2}\)
\(\blert{\text{Write 6 as}~\dfrac{6}{1}}\)
\(\displaystyle \dfrac{7}{3} \cdot 12\)
\(\displaystyle \dfrac{7}{x} \cdot 4x\)
\(\displaystyle 28\)
\(\displaystyle 28\)
4.
\(\dfrac{12}{5} \div \dfrac{8}{5} = \dfrac{12}{5} \cdot \dfrac{5}{8} = \dfrac{\cancel{4} \cdot 3}{\cancel{5}} \cdot \dfrac{5\cancel{5}}{\cancel{4} \cdot 2} = \dfrac{3}{2}\)
\(\blert{\text{Take the reciprocal of the second fraction; then change to multiplication.}}\)
\(\displaystyle \dfrac{8}{3} \div \dfrac{2}{9}\)
\(\displaystyle \dfrac{4a}{3b} \div \dfrac{2a}{3}\)
5.
\(\dfrac{3}{5} \div 6 = \dfrac{3}{5} \cdot \dfrac{1}{6} = \dfrac{\cancel{3}}{5} \cdot \dfrac{1}{\cancel{3} \cdot 2} = \dfrac{1}{15}\)
\(\blert{\text{Take the reciprocal of the second fraction; then change to multiplication.}}\)
\(\displaystyle \dfrac{2}{3} \div 4\)
\(\displaystyle \dfrac{2}{3y} \div (4y)\)
\(\displaystyle \dfrac{1}{6}\)
\(\displaystyle \dfrac{1}{6y^2}\)
6.
\(\dfrac{5}{4} + \dfrac{3}{4} = \dfrac{5+3}{4} = \dfrac{8}{4} = 2\)
\(\blert{\text{Combine the numerators; keep the same denominator.}}\)
\(\displaystyle \dfrac{13}{6}  \dfrac{5}{6}\)
\(\displaystyle \dfrac{13k}{n}  \dfrac{5k}{n}\)
Skills Practice
7.
Multiply.
\(\displaystyle \dfrac{24ab}{5b} \cdot \dfrac{15ab}{14}\)
\(\displaystyle \dfrac{2b}{3} \cdot \dfrac{4}{b+1}\)
\(\displaystyle \dfrac{v}{v+1} \cdot \dfrac{v}{v1}\)
\(\displaystyle \dfrac{36a^2}{7} \)
\(\displaystyle \dfrac{8b}{3b+3}\)
\(\displaystyle \dfrac{v^2}{1v^2}\)
8.
Write each product as a fraction.
\(\displaystyle \dfrac{3}{4}(ab)\)
\(\displaystyle 4a^2b\dfrac{2a+b}{6ab}\)
\(\displaystyle 15x^2y\dfrac{3}{35xy^2}\)
Exercise Group.
For Problems 9–14, multiply.
9.
\(\dfrac{3y}{4xy6y^2} \cdot \dfrac{2x3y}{12x}\)
\(\dfrac{1}{8x}\)
10.
\(\dfrac{3x9}{5x15} \cdot \dfrac{10x5}{8x4}\)
11.
\(\dfrac{4a^21}{a^216} \cdot \dfrac{a^24a}{2a+1}\)
\(\dfrac{a(2a1)}{a+4}\)
12.
\(\dfrac{2x^2x6}{3x^2+4x+1}\cdot \dfrac{3x^2+7x+2}{2x^2+7x+6}\)
13.
\(\dfrac{3x^448}{x^44x^232} \cdot \dfrac{4x^48x^3+4x^2}{2x^4+16x}\)
\(\dfrac{6x(x2)(x1)^2}{(x^28)(x^22x+4)}\)
14.
\(\dfrac{x^43x^3}{x^4+6x^227} \cdot \dfrac{x^481}{3x^481x}\)
Exercise Group.
For Problems 15–24, divide.
15.
\(\dfrac{12a^4}{21b^4} \div \dfrac{24ab^2}{27b^3}\)
\(\dfrac{9a^3}{14b^3}\)
16.
\(6x^2y \div \dfrac{3x}{2y^3}\)
17.
\(\dfrac{a^2ab}{ab} \div \dfrac{2a2b}{3ab}\)
\(\dfrac{3a}{2}\)
18.
\(\dfrac{x^2+3x}{2y} \div (3x)\)
19.
\(\dfrac{3xy+x}{y^2y} \div \dfrac{3y+1}{xy}\)
\(\dfrac{x^2}{y1}\)
20.
\(\dfrac{6a^212a}{3a+9} \div \dfrac{8a^24a^3}{15+5a}\)
21.
\(\dfrac{2z^2+3z2}{2z^23z2} \div \dfrac{2z^3z^2}{z^24}\)
\(\dfrac{(z+2)^2}{z^2(2z1)}\)
22.
\(\dfrac{a^2a6}{a^2+2a15} \div \dfrac{a^24}{a^2+6a+5}\)
23.
\(\dfrac{8x^3y^3}{x+y} \div \dfrac{2xy}{x^2y^2}\)
\((xy)(4x^2+2xy+y^2)\)
24.
\((x^2+5x4) \div \dfrac{x^21}{x^2}\)
Exercise Group.
For Problems 25–28, add or subtract.
Exercise Group.
For Problems 29–36, find the lowest common denominator for the fractions, then add or subtract.
29.
\(\dfrac{5}{2x} + \dfrac{3}{4x^2}\)
\(\dfrac{10x+3}{4x^2}\)
30.
\(\dfrac{2z3}{8z} + \dfrac{z2}{6z}\)
31.
\(\dfrac{3}{2ab} + \dfrac{1}{8a4b}\)
\(\dfrac{13}{8a4b}\)
32.
\(\dfrac{3}{x}  \dfrac{2}{x+1}\)
33.
\(h  \dfrac{3}{h+2}\)
\(\dfrac{h^2+2h3}{h+2}\)
34.
\(\dfrac{3}{n+3}  \dfrac{4n}{n3}\)
35.
\(\dfrac{2}{x} + \dfrac{x}{x2}  2\)
\(\dfrac{6xx^24}{x(x2)}\)
36.
\(\dfrac{1}{x} + \dfrac{x}{y} + \dfrac{x}{z}\)
37.
\(\dfrac{5}{2p4}  \dfrac{2}{63p}\)
\(\dfrac{19}{6(p2)}\)
38.
\(\dfrac{2}{m^23m} + \dfrac{1}{m^29}\)
39.
\(\dfrac{4}{k^23k} + \dfrac{1}{k^2+k}\)
\(\dfrac{5k+1}{k(k3)(k+1)}\)
40.
\(\dfrac{x1}{x^2+3x} + \dfrac{x}{x^2+6x+9}\)
41.
\(\dfrac{y}{2y1}  \dfrac{2y}{y+1}\)
\(\dfrac{3y3y^2}{(y+1)(2y1)}\)
42.
\(\dfrac{y1}{y+1}  \dfrac{y2}{2y3}\)
Exercise Group.
For Problems 43–46, find the lowest common denominator.
Applications
Exercise Group.
For Problems 47–54, multiply.
47.
\(\dfrac{4V}{D} \cdot \dfrac{LR}{DV}\)
\(\dfrac{4LR}{D^2}\)
48.
\(\dfrac{1}{2}MR^2 \cdot \dfrac{a}{R}\)
49.
\(\dfrac{2L}{c} \left(1+\dfrac{V^2}{c^2}\right)\)
\(\dfrac{2L}{c}+\dfrac{2LV^2}{c^3}\)
50.
\(\dfrac{4\pi}{c^2}\left(\dfrac{c}{4\pi}H+cM\right)\)
51.
\(\dfrac{q}{8 \pi} \left(\dfrac{3}{R}\dfrac{a^2}{R^3}\right)\)
\(\dfrac{3q}{8\pi R}\dfrac{a^2q}{8 \pi R^3}\)
52.
\(\dfrac{a^2}{d} \left(1\dfrac{at}{2d}\right)\)
53.
\(\dfrac{2}{t^2}\left(4t^3\dfrac{t^2}{8}+\dfrac{3t}{2}\right)\)
\(8t +\dfrac{1}{4}  \dfrac{3}{t}\)
54.
\(\dfrac{4}{3}v\left(\dfrac{2}{3}v\dfrac{6}{v^2}\dfrac{3}{v}\right)\)
Exercise Group.
Write an algebraic expression for each phrase Problems 55–57, and simplify.
55.
Onehalf of \(x\)
\(x\) divided by onehalf
Onehalf divided by \(x\)
\(\displaystyle \dfrac{x}{2}\)
\(\displaystyle 2x\)
\(\displaystyle \dfrac{1}{2x}\)
56.
Twothirds of \(y\)
\(y\) divided by twothirds
Twothirds divided by \(y\)
57.
The reciprocal of \(a+b\)
Threefourths of the reciprocal of \(a+b\)
The reciprocal of \(a+b\) divided by threefourths
\(\displaystyle \dfrac{1}{a+b}\)
\(\displaystyle \dfrac{3}{4(a+b)}\)
\(\displaystyle \dfrac{4}{3(a+b)}\)
58.
Simplify.
\(\displaystyle \left(\dfrac{1}{c} \cdot \dfrac{1}{5}\right) \div \dfrac{2}{5}\)
\(\displaystyle \dfrac{1}{c} \cdot \left(\dfrac{1}{5} \div \dfrac{2}{5}\right)\)
\(\displaystyle \dfrac{1}{c} \div \left(\dfrac{1}{5} \div \dfrac{2}{5}\right)\)
\(\displaystyle \left(\dfrac{1}{c} \div \dfrac{1}{5}\right) \div \dfrac{2}{5}\)
Exercise Group.
For Problems 59–64, add or subtract.
59.
\(\dfrac{H}{RT} + \dfrac{S}{R}\)
\(\dfrac{H+ST}{RT}\)
60.
\(\dfrac{q}{4 \pi r} + \dfrac{qa}{2 \pi r^2}\)
61.
\(\dfrac{1}{LC}  \left(\dfrac{R}{2L}\right)^2\)
\(\dfrac{4LR^2C}{4L^2C}\)
62.
\(\dfrac{L}{cV} + \dfrac{L}{c+v}\)
63.
\(\dfrac{2r^2}{a^2} + \dfrac{2r}{a} + 1\)
\(\dfrac{2r^2_2ra+a^2}{a^2}\)
64.
\(\dfrac{q}{ra}  \dfrac{2q}{r} + \dfrac{q}{r+a}\)
Exercise Group.
For Problems 65–68, write algebraic fractions in simplest.
65.
The dimensions of a rectangular rug are \(\dfrac{12}{x}\) feet and \(\dfrac{12}{x2}\) feet.
Write and sinplify an expression for the area of the rug.
Write and sinplify an expression for the perimeter of the rug.
\(\dfrac{144}{x^22x}\) sq ft
\(\displaystyle \dfrac{48x48}{x^22x} ft\)
66.
Colonial Airline has a commuter flight between Richmond and Washington, a distance of 100 miles. The plane flies at \(x\) miles per hour in still air. Today there is a steady wind from the north at 10 miles per hour.
How long will the flight from Richmond to Washington take?
How long will the flight from Washington to Richmond take?
How long will a round trip take?
Evaluate your fractions in parts (a)(c) for \(x=150\)
67.
Two pilots for the Flying Express parcel service receive packages simultaneously. Orville leaves Boston for Chicago at the same time Wilbur leaves Chicago for Boston. Each pilot selects an air speed of 400 miles per hour for the 900mile trip. The prevailing winds blow from east to west.
Express Orville's flying time as a function of the wind speed.
Write a function for Wilbur's flying time.
Who reaches his destination first? By how much time (in terms of wind speed)?
\(\dfrac{900}{400+w}\) hr
\(\dfrac{900}{400w}\) hr
Orville, by \(\dfrac{1800w}{160,000w^2}\) hr
68.
Francine's cocker spaniel eats a large bag of dog food in \(d\) days, and Delbert's sheep dog takes 5 fewer days to eat the same size bag.
What fraction of a bag of dog food does Francine's cocker spaniel eat in one day?
What fraction of a bag of dog food does Delbert's sheep dog eat in one day?
If Delbert and Francine get married, what fraction of a bag of dog food will their dogs eat in one day?
If \(d=25\text{,}\) how soon will Delbert and Francine have to buy more dog food?