Unit 4: Solving Rational Equations and Their Applications This unit will continue with an application of the quadratic equation called frame problems. You will also be introduced to techniques for solving rational equations and some relevant applications. For example, you will study how rational equations can solve work related problems. As an example, Mary can clean her house in 3 hours. Her teenage daughter can clean the house in 5 hours. How long will it take to clean the house if Mary and her daughter work together? You will also learn how rational equations can be used to solve revenue questions like the following: you’re planning a community party with your neighbors and the cost will be $1000. Two of your neighbors back out because of emergencies. The remaining members had to pay an additional $8. How many neighbors are involved with the party? Lastly, you will practice using rational equations to solve distance problems using the formula d = rt, where d is distance, r is the rate of travel, and t is the time traveled.
Unit 4 Time Advisory
Completing this unit should take approximately 21 hours.
☐ Subunit 4.1: 4 hours
☐ Subunit 4.2: 4.5 hours
☐ Subunit 4.3: 4.5 hours
☐ Subunit 4.4: 2 hours
☐ Subunit 4.5: 6 hours
Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- solve another application of the quadratic equation: frame problems;
- solve rational equations and simultaneous product equations; and
- solve applications of rational equations including revenue,
distance, and work problems.
4.1 Frames Note: This reading was assigned in Subunit 3.6, but we will extend the application to frames.
Reading: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 9, Section 9.7: Application: Rectangles” Link: Washington State Board for Community and Technical Colleges: Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd Edition: “Chapter 9, Section 9.7: Application: Rectangles” (PDF)
Instructions: Please review Section 9.7 in Chapter 9 of your textbook, pages 357–363, to review applications of the quadratic equation. Pay close attention to the application that is described about a picture frame. Note that this reading covers the topics in Subunits 4.1.1–4.1.4.
Reading this section and taking notes should take approximately 2 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Frames” Link: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Frames” (PDF)
Instructions: Complete pages 93–94 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 4.1.1–4.1.4, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
4.1.1 Picture Frames (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Picture Frames (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Picture Frames (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses an interesting application of quadratics, a picture frame. Basically, if you know the area of the frame and the dimensions of the picture and the picture is positioned uniformly (same distance from the height and width of the frame), then you can determine the dimensions of the frame.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.1.2 Picture Frames (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Picture Frames (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Picture Frames (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of the picture in a frame problem.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.1.3 Percent Frames (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Percent Frames (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Percent Frames (Part
1)” (YouTube)
Instructions: Watch the video linked above, which continues the discussion of the frame application. While not a picture frame, you can consider mowing your yard in a spiral fashion as a similar concept. This application is slightly different because it uses a percentage of completion to determine the picture size.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.1.4 Percent Frames (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.1.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Percent Frames (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Frames –
Percent Frames (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example similar to the previous video in which you use a percentage to solve the equation.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.2 Rational Equations
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 7, Section 7.7: Solving Rational
Equations”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 7, Section 7.7: Solving Rational
Equations” (PDF)
Instructions: Read Section 7.7 in Chapter 7 of your textbook, pages
274–278, to learn techniques to solve rational equations. Rational
equations are equations that contain rational expressions
(fractions) with algebraic expressions in the denominator. Note that
this reading covers the topics in Subunits 4.2.1–4.2.3.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Rational Equations”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab Notebook: “Rational
Equations”
(PDF)
Instructions: Complete pages 81–83 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 4.2.1–4.2.3, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
4.2.1 Rational Equations with No Factoring of the Denominators Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational
Equations – Clear Denominator”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational
Equations – Clear
Denominator”
(YouTube)
Instructions: Watch the video linked above, which discusses solving rational equations. The method is identical to solving equations with fractions (multiply both sides of the equation by the lowest common denominator – LCD) except the denominator has an algebraic expression (consists of an unknown).
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.2.2 Rational Equations with Factoring the Denominators First Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational
Equations – Factoring Denominator”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational
Equations – Factoring
Denominator”
(YouTube)
Instructions: Watch the video linked above, which discusses solving rational equations. Again, this is very similar to solving equations with fractions. If 4 and 6 were in the denominator, then the LCD is 12 because 4 = 2^{2} and 6 = 2∙3. So you need 2^{2}∙3. If x^{2} and x^{2} - x were your denominators, then the LCD is x^{2}(x - 1) because x^{2} = x∙x and x^{2} - x = x(x - 1).
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.2.3 Rational Equations That Have Extraneous Solutions Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.2.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Rational
Equations – Extraneous Solutions”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Rational
Equations – Extraneous
Solutions” (YouTube)
Instructions: Watch the video linked above, which discusses solutions to rational equations. When an equation is multiplied by a non-zero number, then the solution set is the same. With rational equations, since you are multiplying by an algebraic expression, you do not know if you are multiplying by 0 until the unknown variable is known. If the expression being multiplied is 0, then you will pick up what is known as extraneous solutions. You must always check your answer.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.3 Work Problems
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.8: Application:
Teamwork”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.8: Application
Teamwork”
(PDF)
Instructions: Read Section 9.8 in Chapter 9 of your textbook, pages
364–369, to learn an application of rational equations and
quadratics called teamwork. As an example, Mary can clean her house
in 3 hours. Her teenage daughter can clean the house in 5 hours. How
long will it take to clean the house if Mary and her daughter work
together? The solution to this problem is to set up the proper
rational equation. Suppose you know that when working together it
takes your two children 4 hours to clean the house and one child
claims it takes the other child 2 hours longer than the first child
to clean the house individually. How long will it take each child to
clean the house? The solution to this problem is to set up a
rational equation. Multiplying by the lowest common denominator
(LCD) will result in a quadratic equation to be solved. Note that
this reading also covers the topics in Subunits 4.3.1–4.3.3.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Work Problems”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab Notebook: “Work
Problems”
(PDF)
Instructions: Complete pages 84–85 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 4.3.1–4.3.3, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
4.3.1 Work Problems with One Unknown Time (Linear) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Work
Problems – One Unknown Time”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Work Problems
– One Unknown Time”
(YouTube)
Instructions: Watch the video linked above, which discusses how to set up the rational equation for a work problem. Since there is one unknown, solving this problem will result in solving a linear equation.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.3.2 Work Problems with Two Unknown Times (Quadratic) (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Work
Problems – Two Unknown Times (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Work Problems
– Two Unknown Times (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses how to set up the rational equation for a work problem. This example will result in a quadratic equation when solving since the times are related by a constant. The time each person will take to complete the task will be the reciprocal of x and x + or - a constant. Multiplying by the lowest common denominator (LCD) will result in a quadratic equation.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.3.3 Work Problems with Two Unknown Times (Quadratic) (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.3.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Work
Problems – Two Unknown Times (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series: “Work Problems
– Two Unknown Times (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of how to set up the rational equation for a work problem that will result in a quadratic equation. This video gives additional examples similar to Subunit 4.3.2.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.4 Simultaneous Products
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.9: Simultaneous
Products”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.9: Simultaneous
Products”
(PDF)
Instructions: Read Section 9.9 in Chapter 9 of your textbook, pages
370–372, to learn how to solve for a variable if given a product.
This is another application of the quadratic equation. As an
example, suppose Julie has a daughter and a son. She tells you that
the product of their ages is 72. If you subtract 2 from the
daughter’s age and add 3 to the son’s age and then multiply these
numbers, the result is 45. What are their ages?
Reading this section and taking notes should take approximately
1.25 hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
Assessment: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Simultaneous Products” Link: Big Bend Community College: Tyler Wallace’s Intermediate Algebra Lab Notebook: “Simultaneous Products” (PDF)
Instructions: Complete page 86 of Wallace’s workbook. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.
Completing this assessment should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Simultaneous Products” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Simultaneous Products” (YouTube)
Instructions: Watch the video linked above, which gives an example of setting up and solving an equation involving a simultaneous product. For instance, suppose the product of two integers = 12 (xy = 12); and if you add 3 to the first integer, then the product is 24 ((x + 3)y = 24). What are the two integers?
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.5 Distance/Revenue
- Reading: Washington State Board for Community and Technical
Colleges: Tyler Wallace’s Beginning Algebra and Intermediate
Algebra, 2nd Edition: “Chapter 9, Section 9.10: Application:
Revenue and Distance”
Link: Washington State Board for Community and Technical Colleges:
Tyler Wallace’s Beginning Algebra and Intermediate Algebra, 2nd
Edition: “Chapter 9, Section 9.10: Application: Revenue and
Distance”
(PDF)
Instructions: Read Section 9.10 in Chapter 9 of your textbook,
pages 373–379, to learn the distance and revenue applications of
rational and quadratic equations. The distance applications use the
formula d = rt, where d is distance, r is the rate of travel, and t
is the time traveled. This formula leads to some interesting
applications that result in quadratic equations. As an example, if
you are planning a trip and know what your average speed will be,
what happens with the time if you can increase the speed slightly?
The revenue applications answer questions like the following:
you’re planning a community party with your neighbors and the cost
will be $1000. Two of your neighbors back out because of
emergencies. The remaining members had to pay an additional $8. How
many neighbors are involved with the party? Note that this reading
also covers the topics in Subunits 4.5.1–4.5.6.
Reading this section and taking notes should take approximately 2
hours.
Terms of Use: This resource is licensed under a Creative Commons
Attribution 3.0 Unported
License. It is
attributed to Tyler Wallace and the original version can be found
here.
- Assessment: Big Bend Community College: Tyler Wallace’s
Intermediate Algebra Lab Notebook: “Distance/Revenue”
Link: Big Bend Community College: Tyler Wallace’s Intermediate
Algebra Lab Notebook: “Distance
Revenue”
(PDF)
Instructions: Complete pages 87–92 of Wallace’s workbook. Try to complete this exercise before watching the videos in Subunits 4.5.1–4.5.6, and then review the worksheet as you follow along with the videos for solutions.
Completing this assessment should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Tyler Wallace and the original version can be found here.
4.5.1 Revenue Problems (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Revenue Problems (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Revenue Problems (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses the revenue application of the rational equation. Revenue means revenue from a joint venture. The video shows how to set up a problem using the formula R = n∙p, where R is revenue, n is the number of participants, and p is the price per participant.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.5.2 Revenue Problems (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Revenue Problems (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Revenue Problems (Part
2)”
(YouTube)
Instructions: Watch the video linked above, which presents a slightly different example of the revenue application of rational and quadratic equations.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.5.3 Distance Problems (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Distance Problems (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Distance Problems (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses the distance application of the rational equation. The distance application uses a fixed distance being traveled and modifies the parameters to determine the effects on the other variables. The video shows how to set up a problem using the formula d = rt, where d is distance, r is the rate, and t is the time.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.5.4 Distance Problems (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.5.
Lecture: YouTube: Tyler Wallace’s Math Lecture Series: “Distance/Revenue – Distance Problems (Part 2)” Link: YouTube: Tyler Wallace’s Math Lecture Series: “Distance/Revenue – Distance Problems (Part 2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of a distance problem.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.5.5 Up/Down Wind/Stream Problems (Part 1) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Streams and Wind (Part 1)”
Link: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Streams and Wind (Part
1)” (YouTube)
Instructions: Watch the video linked above, which discusses wind and stream problems. This is a variation of the distance, rate, time application where one is traveling downstream and then upstream with the water force providing additional assistance downstream and causing a drag going upstream.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
4.5.6 Up/Down Wind/Stream Problems (Part 2) Note: This subunit is also covered by the reading and assessment assigned in Subunit 4.5.
- Lecture: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Streams and Wind (Part 2)”
Link: YouTube: Tyler Wallace’s Math Lecture Series:
“Distance/Revenue – Streams and Wind (Part
2)” (YouTube)
Instructions: Watch the video linked above, which presents another example of traveling up- and downstream.
You may watch the video as often as you please. You may refer to the video when doing the assessment if necessary.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.