9.5: Division of Square Root Expressions
The Division Property of Square Roots
In our work with simplifying square root expressions, we noted that
\[\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}} \nonumber\]
Since this is an equation, we may write it as:
\[\dfrac{\sqrt{x}}{\sqrt{y}} = \sqrt{\dfrac{x}{y}} \nonumber\]
To divide two square root expressions, we use the division property of square roots:
\[\dfrac{\sqrt{x}}{\sqrt{y}} = \sqrt{\dfrac{x}{y}}\]
The quotient of the square root is the square root of the quotient.
Rationalizing the Denominator
As we can see by observing the right side of the equation governing the division of square roots, the process may produce a fraction in the radicand. This means, of course, that the square root expression is not in simplified form. It is sometimes more useful to rationalize the denominator of a square root expression before actually performing the division.
Sample Set A
Simplify the square root expressions.
\[\sqrt{\dfrac{3}{7}} \nonumber\]
This radical expression is not in simplified form since there is a fraction under the radical sign. We can eliminate this problem using the division property of square roots.
\[\sqrt{\dfrac{3}{7}} = \dfrac{\sqrt{3}}{\sqrt{7}} = \dfrac{\sqrt{3}}{\sqrt{7}} \cdot \dfrac{\sqrt{7}}{\sqrt{7}} = \dfrac{\sqrt{3} \sqrt{7}}{7} = \dfrac{\sqrt{21}}{7} \nonumber\]
\(\dfrac{\sqrt{5}}{\sqrt{3}}\)
A direct application of the rule produces \(\sqrt{\dfrac{5}{3}}\), which must be simplified. Let us rationalize the denominator before we perform the division.
\(\dfrac{\sqrt{5}}{\sqrt{3}}=\dfrac{\sqrt{5}}{\sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{\sqrt{5} \sqrt{3}}{3}=\dfrac{\sqrt{15}}{3}\)
\(\dfrac{\sqrt{21}}{\sqrt{7}} = \sqrt{\dfrac{21}{7}} = \sqrt{3}\)
The rule produces the quotient quickly. We could also rationalize the denominator first and produce the same result.
\(\dfrac{\sqrt{21}}{\sqrt{7}}=\dfrac{\sqrt{21}}{7} \cdot \dfrac{\sqrt{7}}{\sqrt{7}}=\dfrac{\sqrt{21 \cdot 7}}{7}=\dfrac{\sqrt{3 \cdot 7 \cdot 7}}{7}=\dfrac{\sqrt{3 \cdot 7^{2}}}{7}=\dfrac{7 \sqrt{3}}{7}=\sqrt{3}\)
\(\dfrac{\sqrt{80 x^{9}}}{\sqrt{5 x^{4}}}=\sqrt{\dfrac{80 x^{9}}{5 x^{4}}}=\sqrt{16 x^{5}}=\sqrt{16} \sqrt{x^{4} x}=4 x^{2} \sqrt{x}\)
\(\dfrac{\sqrt{50 a^{3} b^{7}}}{\sqrt{5 a b^{5}}}=\sqrt{\dfrac{50 a^{3} b^{7}}{5 a b^{5}}}=\sqrt{10 a^{2} b^{2}}=a b \sqrt{10}\)
\(\dfrac{\sqrt{5a}}{\sqrt{b}}\)
Some observation shows that a direct division of the radicands will produce a fraction. This suggests that we rationalize the denominator first.
\(\dfrac{\sqrt{5a}}{\sqrt{b}} = \dfrac{\sqrt{5a}}{\sqrt{b}} \cdot \dfrac{\sqrt{b}}{\sqrt{b}} = \dfrac{\sqrt{5a} \sqrt{b}}{b} = \dfrac{\sqrt{5ab}}{b}\)
\(\dfrac{\sqrt{m-6}}{\sqrt{m+2}} = \dfrac{\sqrt{m-6}}{\sqrt{m+2}} \cdot \dfrac{\sqrt{m+2}}{\sqrt{m+2}} = \dfrac{\sqrt{m^2 - 4m - 12}}{m + 2}\)
\(\dfrac{\sqrt{y^{2}-y-12}}{\sqrt{y+3}}=\sqrt{\dfrac{y^{2}-y-12}{y+3}}=\sqrt{\dfrac{(y+3)(y-4)}{(y+3)}}=\sqrt{\dfrac{\cancel{(y+3)}(y-4)}{\cancel{(y+3)}}}=\sqrt{y-4}\)
Practice Set A
Simplify the square root expressions.
\(\dfrac{\sqrt{26}}{\sqrt{13}}\)
- Answer
-
\(\sqrt{2}\)
\(\dfrac{\sqrt{7}}{\sqrt{3}}\)
- Answer
-
\(\dfrac{\sqrt{21}}{3}\)
\(\dfrac{\sqrt{80m^5n^8}}{\sqrt{5m^2n}}\)
- Answer
-
\(4mn^3 \sqrt{mn}\)
\(\dfrac{\sqrt{196(x+7)^8}}{\sqrt{2(x+7)^3}}\)
- Answer
-
\(7(x+7)^2 \sqrt{2(x+7)}\)
\(\dfrac{\sqrt{n+4}}{\sqrt{n-5}}\)
- Answer
-
\(\dfrac{\sqrt{n^2 - n - 20}}{n-5}\)
\(\dfrac{\sqrt{a^2 - 6a + 8}}{\sqrt{a-2}}\)
- Answer
-
\(\sqrt{a-4}\)
\(\dfrac{\sqrt{x^{2n}}}{\sqrt{x^n}}\)
- Answer
-
\(x^n\)
\(\dfrac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}\)
- Answer
-
\(a^{m-2}\)
Conjugates and Rationalizing the Denominator
To perform a division that contains a binomial in the denominator, such as \(\dfrac{3}{4 + \sqrt{6}}\), we multiply the numerator and denominator by a conjugate of the denominator.
A conjugate of the binomial \(a + b\) is \(a-b\). Similarly, a conjugate of \(a-b\) is \(a + b\).
Notice that when the conjugates \(a + b\) and \(a - b\) are multiplied together, they produce a difference of two squares.
\((a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2\)
This principle helps us eliminate square root radicals, as shown in these examples that illustrate the produce of conjugates.
\(\begin{array}{flushleft}
(5 + \sqrt{2})(5 - \sqrt{2}) &= 5^2 - (\sqrt{2})^2\\
&= 25 - 2\\
&= 23
\end{array}\)
\(\begin{array}{flushleft}
(\sqrt{6} - \sqrt{7})(\sqrt{6} + \sqrt{7}) &= (\sqrt{6})^2 - (\sqrt{7})^2\\
&= 6 - 7\\
&= -1
\end{array}\)
Sample Set B
Simplify the following expressions.
\(\dfrac{3}{4 + \sqrt{6}}\)
The conjugate of the denominator is \(4 - \sqrt{6}\). Multiply the fraction by \(1\) in the form of \(\dfrac{4 - \sqrt{6}}{4 - \sqrt{6}}\)
\(\begin{array}{flushleft}
\dfrac{3}{4 + \sqrt{6}} \cdot \dfrac{4 - \sqrt{6}}{4 - \sqrt{6}} &= \dfrac{3(4 - \sqrt{6})}{4^2 - (\sqrt{6})^2}\\
&= \dfrac{12 - 3\sqrt{6}}{16 - 6}\\
&= \dfrac{12 - 3\sqrt{6}}{10}
\end{array}\)
\(\dfrac{\sqrt{2x}}{\sqrt{3} - \sqrt{5x}}\)
The conjugate of the denominator is \(\sqrt{3} + \sqrt{5x}\). Multiply the fraction by \(1\) in the form of \(\dfrac{\sqrt{3} + \sqrt{5x}}{\sqrt{3} + \sqrt{5x}}\).
\(\begin{array}{flushleft}
\dfrac{\sqrt{2 x}}{\sqrt{3}-\sqrt{5 x}} \cdot \dfrac{\sqrt{3}+\sqrt{5 x}}{\sqrt{3}+\sqrt{5 x}} &=\dfrac{\sqrt{2 x}(\sqrt{3}+\sqrt{5 x})}{(\sqrt{3})^{2}-(\sqrt{5 x})^{2}} \\
&=\dfrac{\sqrt{2 x} \sqrt{3}+\sqrt{2 x} \sqrt{5 x}}{3-5 x} \\
&=\dfrac{\sqrt{6 x}+\sqrt{10 x^{2}}}{3-5 x} \\
&=\dfrac{\sqrt{6 x}+x \sqrt{10}}{3-5 x}
\end{array}\)
Practice Set B
Simplify the following expressions.
\(\dfrac{5}{9 + \sqrt{7}}\)
- Answer
-
\(\dfrac{45 - 5\sqrt{7}}{74}\)
\(\dfrac{-2}{1 - \sqrt{3x}}\)
- Answer
-
\(\dfrac{-2 - 2\sqrt{3x}}{1 - 3x}\)
\(\dfrac{\sqrt{8}}{\sqrt{3x} + \sqrt{2x}}\)
- Answer
-
\(\dfrac{2\sqrt{6x} - 4\sqrt{x}}{x}\)
\(\dfrac{\sqrt{2m}}{m - \sqrt{3m}}\)
- Answer
-
\(\dfrac{\sqrt{2m} + \sqrt{6}}{m - 3}\)
Exercises
For the following problems, simplify each expression.
\(\dfrac{\sqrt{28}}{\sqrt{2}}\)
- Answer
-
\(\sqrt{14}\)
\(\dfrac{\sqrt{200}}{\sqrt{10}}\)
\(\dfrac{\sqrt{28}}{\sqrt{7}}\)
- Answer
-
\(2\)
\(\dfrac{\sqrt{96}}{\sqrt{24}}\)
\(\dfrac{\sqrt{180}}{\sqrt{5}}\)
- Answer
-
\(6\)
\(\dfrac{\sqrt{336}}{\sqrt{21}}\)
\(\dfrac{\sqrt{162}}{\sqrt{18}}\)
- Answer
-
\(3\)
\(\sqrt{\dfrac{25}{9}}\)
\(\sqrt{\dfrac{36}{35}}\)
- Answer
-
\(\dfrac{6\sqrt{35}}{35}\)
\(\sqrt{\dfrac{225}{16}}\)
\(\sqrt{\dfrac{49}{225}}\)
- Answer
-
\(\dfrac{7}{15}\)
\(\sqrt{\dfrac{3}{5}}\)
\(\sqrt{\dfrac{3}{7}}\)
- Answer
-
\(\dfrac{\sqrt{21}}{7}\)
\(\sqrt{\dfrac{1}{2}}\)
\(\sqrt{\dfrac{5}{2}}\)
- Answer
-
\(\dfrac{\sqrt{10}}{2}\)
\(\sqrt{\dfrac{11}{25}}\)
\(\sqrt{\dfrac{15}{36}}\)
- Answer
-
\(\dfrac{\sqrt{15}}{6}\)
\(\sqrt{\dfrac{5}{16}}\)
\(\sqrt{\dfrac{7}{25}}\)
- Answer
-
\(\dfrac{\sqrt{7}}{5}\)
\(\sqrt{\dfrac{32}{49}}\)
\(\sqrt{\dfrac{50}{81}}\)
- Answer
-
\(\dfrac{5 \sqrt{2}}{9}\)
\(\dfrac{\sqrt{125x^5}}{\sqrt{5x^3}}\)
\(\dfrac{\sqrt{72m^7}}{\sqrt{2m^3}}\)
- Answer
-
\(6m^2\)
\(\dfrac{\sqrt{162a^{11}}}{\sqrt{2a^5}}\)
\(\dfrac{\sqrt{75y^{10}}}{\sqrt{2a^5}}\)
- Answer
-
\(5y^3\)
\(\dfrac{\sqrt{48x^9}}{\sqrt{3x^2}}\)
\(\dfrac{\sqrt{125a^{14}}}{\sqrt{5a^5}}\)
- Answer
-
\(5a^4 \sqrt{a}\)
\(\dfrac{\sqrt{27a^{10}}}{\sqrt{3a^5}}\)
\(\dfrac{\sqrt{108x^{21}}}{\sqrt{3x^4}}\)
- Answer
-
\(6x^8 \sqrt{x}\)
\(\dfrac{\sqrt{48x^6y^7}}{\sqrt{3xy}}\)
\(\dfrac{\sqrt{45a^3b^8c^2}}{\sqrt{5ab^2c}}\)
- Answer
-
\(3ab^3 \sqrt{c}\)
\(\dfrac{\sqrt{66m^{12}n^{15}}}{\sqrt{11mn^8}}\)
\(\dfrac{\sqrt{30p^5q^{14}}}{\sqrt{5q^7}}\)
- Answer
-
\(p^2q^3 \sqrt{6pq}\)
\(\dfrac{\sqrt{b}}{\sqrt{5}}\)
\(\dfrac{\sqrt{5x}}{\sqrt{2}}\)
- Answer
-
\(\dfrac{\sqrt{10x}}{2}\)
\(\dfrac{\sqrt{2a^3b}}{\sqrt{14a}}\)
\(\dfrac{\sqrt{3m^4n^3}}{\sqrt{6mn^5}}\)
- Answer
-
\(\dfrac{m \sqrt{2m}}{2n}\)
\(\dfrac{\sqrt{5(p-q)^6(r+s)^4}}{\sqrt{25(r+s)^3}}\)
\(\dfrac{\sqrt{m(m-6)-m^2 + 6m}}{\sqrt{3m - 7}}\)
- Answer
-
\(0\)
\(\dfrac{\sqrt{r+1}}{\sqrt{r-1}}\)
\(\dfrac{\sqrt{s+3}}{\sqrt{s-3}}\)
- Answer
-
\(\dfrac{\sqrt{s^2-9}}{s-3}\)
\(\dfrac{\sqrt{a^2 + 3a + 2}}{\sqrt{a + 1}}\)
\(\dfrac{\sqrt{x^2 - 10x + 24}}{\sqrt{x-4}}\)
- Answer
-
\(\sqrt{x-6}\)
\(\dfrac{\sqrt{x^2 - 2x - 8}}{\sqrt{x + 2}}\)
\(\dfrac{\sqrt{x^2 - 4x + 3}}{\sqrt{x-3}}\)
- Answer
-
\(\sqrt{x-1}\)
\(\dfrac{\sqrt{2x^2 - x - 1}}{\sqrt{x - 1}}\)
\(\dfrac{-5}{4 + \sqrt{5}}\)
- Answer
-
\(\dfrac{-20 + 5\sqrt{5}}{11}\)
\(\dfrac{1}{1 + \sqrt{x}}\)
\(\dfrac{2}{1 - \sqrt{a}}\)
- Answer
-
\(\dfrac{2(1 + \sqrt{a})}{1 - a}\)
\(\dfrac{-6}{\sqrt{5} - 1}\)
\(\dfrac{-6}{\sqrt{7} + 2}\)
- Answer
-
\(-2(\sqrt{7} - 2)\)
\(\dfrac{3}{\sqrt{3} - \sqrt{2}}\)
\(\dfrac{4}{\sqrt{6} + \sqrt{2}}\)
- Answer
-
\(\sqrt{6} - \sqrt{2}\)
\(\dfrac{\sqrt{5}}{\sqrt{8} - \sqrt{6}}\)
\(\dfrac{\sqrt{12}}{\sqrt{12} - \sqrt{8}}\)
- Answer
-
\(3 + \sqrt{6}\)
\(\dfrac{\sqrt{7x}}{2 - \sqrt{5x}}\)
\(\dfrac{\sqrt{6y}}{1 + \sqrt{3y}}\)
- Answer
-
\(\dfrac{\sqrt{6y} - 3y\sqrt{2}}{1 - 3y}\)
\(\dfrac{\sqrt{2}}{\sqrt{3} - \sqrt{2}}\)
\(\dfrac{\sqrt{a}}{\sqrt{a} + \sqrt{b}}\)
- Answer
-
\(\dfrac{a - \sqrt{ab}}{a - b}\)
\(\dfrac{\sqrt{8^3b^5}}{4 - \sqrt{2ab}}\)
\(\dfrac{\sqrt{7x}}{\sqrt{5x} + \sqrt{x}}\)
- Answer
-
\(\dfrac{\sqrt{35} - \sqrt{7}}{4}\)
\(\dfrac{\sqrt{3y}}{\sqrt{2y} - \sqrt{y}}\)
Exercises for Review
Simplify \(x^8y^7 \dfrac{x^4y^8}{x^3y^4}\)
- Answer
-
\(x^9y^{11}\)
Solve the compound inequality \(-8 \le 7 - 5x \le -23\)
Construct the graph of \(y = \dfrac{2}{3}x - 4\)
- Answer
-
The symbol \(\sqrt{x}\) represents which square root of the number \(x, x \ge 0\)?
Simplify \(\sqrt{a^2 + 8a + 16}\)
- Answer
-
\(a + 4\)


2: Market Supply
The learning objectives for this chapter are as follows:
- Describe the firm’s production decision, specifically the conditions under which the firm will enter the market and the conditions under which the firm will exit the market.
- Express the firm’s profit maximization problem in terms of revenue and cost.
- Describe and explain the logic of the firm’s profit maximization condition (MR = MC).
- Explain the idea of opportunity cost.
- Identify variables that shift the supply schedule. Be able to predict the direction of the shift.
- Distinguish between competing products in production and joint products in production. Provide examples of each.
- Explain the derivation of input demands from profit maximizing input choices, and describe variables that shift demand schedules for inputs.
- Explain producer surplus and compute producer surplus given a linear supply schedule.


5.1: Tables, Charts, and Infographics
Learning Outcome
- Discuss the appropriate use of common tables, charts, and infographics
In this age of information, there are huge amounts of data to process and an equally impressive number of graphs and charts you can use to tell the data’s story. Here are some of the more commonly used graphs and the kinds of stories they can help you tell.
Numerical Data Charts
Let’s talk about a few of the most common types of numerical charts:
- Bar graphs
- Line graphs
- Pie charts
- Stacked comparison charts
Bar Graphs
Bar graphs are used to compare categories. The x -axis (the horizontal line at the bottom of a graph) is usually used to show the categories: in this case, countries with universal healthcare expenses for 2015. The y -axis (the vertical line) isn’t marked here, but it’s clearly showing dollars spent in millions. The taller the bar, the more dollars were spent. This chart clearly shows that the United States spends more on health care per citizen than other countries. This is clearly identified on this chart with bars in contrasting colors. You can’t help but notice it—it breaks pattern and it’s a bold color that draws the eye.
Line Graphs
Like bar graphs, line graphs compare categories, but they’re most often used to show trends. With the x -axis frequently showing the passage of time, the graphed data points will show an upward or downward trend of the categories in question.
The line graph above illustrates social media traffic trends. Each social media organization is represented by a different colored line. The x -axis shows the passage of time, and the y -axis shows the percentage of media traffic each organization is capturing. The graph shows that Facebook traffic is trending up, while Pinterest has experienced some ups and downs. Third-place Twitter traffic is relatively flat.
Pie Charts
Pie charts show the composition of data, or the pieces of a whole. It can be as simple as “the team here is composed of 50 percent men and 50 percent women” or “Our sales are made up of 30 percent fiction books and 70 percent non-fiction.”
In Figure 3, the whole pie (the whole circle) represents the total products sold at a store; the pieces of that pie show you the percentage of sales each department made. A chart like this makes it very easy to see that the clothing and accessory departments make up the largest section of sales, and fragrances the smallest.
Some comparison charts aren’t shaped like a circle. Sometimes they’re donuts, and other times they’re shown in bars, as we’ll see next.
Stacked Comparison Charts
You can compare categories with a “pie chart” approach, incorporating the composition factor in a variety of ways. The categories in this chart are represented by bars, but the bars themselves are composition charts. Each bar is valued at 100%, and the colored blocks represent different levels of pet ownership within the population.
Similarly, the line graph below is “stacked” to show the level of sales based on product type. You’re seeing a trend as well as a portion of a whole – comparing and viewing composition.
Non-Numerical Data Charts
The charts above are very handy when your data is numeric. But there are charts for other types of data. Let’s talk about a few of the most common types of non-numerical charts.
- Venn diagrams
- Flow charts
- Gantt charts
- Organizational charts
- Pictographs
- Infographics
Venn Diagrams
A Venn diagram shows a comparison of two different categories and the items they have in common. The diagram in Figure 6 shows us the different types of image types that can be safely shared on the Internet.
Wherever the circles overlap represents a shared characteristic. For example, JPG and PNG files both have a large color palette, so they overlap in that area. “Web safe” is a characteristic shared by all three, so it’s in the middle where all the circles overlap. Areas of each circle that do not overlap represent characteristics unique to each file type—something they don’t share with any other file type represented.
Flow Charts
Flow charts show a process. Flow charts document a sequence of events from start to finish so that the process can be documented, followed, and managed.
The flow chart in Figure 7 shows the process of how theories are created, spread, and accepted.
You can see at the top there is an idea that leads to an experiment, and then the results determine which path is taken. If the experiment is successful or unsuccessful, different paths will be chosen. Along the path of the flow chart, all alternatives are presented and choices are made between them. Your eye follows the path from start to finish for every part of the scenario.
Gantt Chart
A Gantt chart is a timeline. Multiple projects can be added to the timeline with start and finish dates, and milestones and deadlines are also reflected. This chart is used to determine how long a project will take, the resources needed, and the order in which tasks need to be completed.
In Figure 8, you have two different teams running one project. The Market Team (red) completes the market research and defines specifications by the week of July 23rd (the date of the first milestone). Then, the Planning Team (blue) takes over with the overall architecture and project planning and is responsible for hitting that second milestone on August 6th. As you can see, the Market and Planning Teams have additional work to contribute even after their milestones are hit, and the project is not complete until the end of November.
Organizational Charts
Organizational charts (sometimes call hierarchy charts) show the people in an organization and their reporting relationships. Usually, the organizational chart will have a chairman or CEO at the top, followed by a team of presidents and vice presidents, and then their direct reports, and so on. An organizational chart is usually created and maintained by human resource professionals who wants a visual view of their organization’s structure and reporting relationships so they can make better decisions about leveraging the company’s talent.
The organizational chart in Figure 8 shows a chairman at the top of the hierarchy and a managing director, quality assurance leader and a secretary reporting directly to the chairman. Each of those direct reports has direct reports of his or her own, and so on.
Pictographs
Pictographs use symbols and images to convey data, information, or ideas. The pictographs in Figure 9 are actually part of a resume and signal what kind of information can be found next to each pictograph (phone number, address, Twitter handle, and email respectively).
This makes it easier for a recruiter to view and pick out the necessary contact information. Other pictograms can convey processes (like the instructions to build IKEA furniture) or information (like when the weather app on your phone tells you it’s going to rain today by posting a picture of a rain cloud).
Infographics
Similarly, infographics use pictures, but they also incorporate data and words, all to explain a single point. Infographics can be used to make a complex subject a little simpler to understand, particularly when there are more than two or three factors to consider. They can also be used to compare two or more categories and make your data more interesting and eye catching.
This infographic shows how chicken consumption relates to the Superbowl. You can see how consumption and the price per pound of chicken has increased over time. It also describes how chicken wings in particular have become more popular over time, all with a an eye-catching and entertaining quality that’s engaging and interesting to the audience.
This is not an all-inclusive list of the kinds of charts and graphs available. If you work in the science arena, you may find yourself using scatter charts or heat maps. Statisticians might be reaching for a trellis chart or even a function graph. Learn what graphs your industry relies on most and take the time to familiarize yourself with them so you can use these important visual communication tools to your advantage.
As we mentioned earlier in this module, all graphs and charts that you see on the internet are someone else’s property, and using them in your work is stealing. But there’s little reason to use someone else’s chart—you’re creating your own chart to communicate your own data! Here are a few tools that can help you create something great:
- MS Excel. You’ve probably stumbled upon the charts and graph tools in your Microsoft program long ago, and if it’s simple graphs you want, that’s an easy tool to use to create them. Google Sheets and Apple’s Pages offer similar options. See Module 4: Research for more on making charts and graphs in Excel.
- Lucidchart . If it’s flowcharts you’re looking for, Lucidchart is the right place to start. This free, easy-to-use resource allows you to create flowcharts, organizational charts and diagrams that look smart.
- OnlineChartTool . If you’re looking for something different in terms of bar charts and graphs, and you’re tired of the graphics Microsoft Excel provides, take a look at OnlineChartTool. It might have what you’re looking for.
- Beam . This tool allows you to make engaging graphs and charts, even on your mobile phone.
- BeFunky , Visme , and Canva. Don’t let infographics intimidate you! Infographics are easily created with the BeFunky, Visme and Canva tools. These tools have free versions and include hundreds of images and templates to help you make a stunning visual.
Tables
A communicator can also use tables to display data. Tables can be formatted for words or for numerical data. They can be used for comparing data, or when one item has several data points associated with it.
Table 1 is a simple table of three people who are running races. George, Alana, and Sebastián each have four data points associated with them. By using a simple table, you can put all of this data in front of your audience. They can compare best times and review rankings of each runner.
| Table 1. Running a Race | ||||
|---|---|---|---|---|
| Participants | Races Run | Best Time | Average Time | Ranking |
| George | 2 | 3:23 | 3:47 | 3 |
| Alana | 3 | 2:56 | 3:12 | 1 |
| Sebastián | 2 | 2:54 | 3:36 | 2 |
Table 2 is a bit more complex, and because of that, the data is not quite as accessible to the reader. But the data is complex as well, and if it’s going to be displayed for ease of review, this seems like a decent choice. Table 2 shows the results of a survey where participants were asked to rank eight speakers (a male and a female of different English Language dialects) according to their professionalism, intelligence, education, friendliness, and sociability. The response rates are shown for male and female speakers, and then an average is calculated for each dialect.
| Table 2. Average Perceptions of English Speakers* | |||||
|---|---|---|---|---|---|
| Standard American English | |||||
| Gender | Professional | Intelligent | Educated | Friendly | Extroverted |
| Female Speaker | 5.83 | 5.83 | 5.75 | 5.42 | 4.92 |
| Male Speaker | 6.92 | 6.67 | 6.75 | 6.42 | 6.33 |
| Southern American English | |||||
| Gender | Professional | Intelligent | Educated | Friendly | Extroverted |
| Female Speaker | 5.75 | 5.17 | 5.00 | 7.25 | 7.00 |
| Male Speaker | 4.33 | 4.17 | 3.75 | 5.92 | 6.42 |
| British English | |||||
| Gender | Professional | Intelligent | Educated | Friendly | Extroverted |
| Female Speaker | 7.50 | 7.33 | 7.33 | 5.50 | 5.25 |
| Male Speaker | 6.50 | 6.25 | 6.17 | 5.17 | 4.92 |
| Australian English | |||||
| Gender | Professional | Intelligent | Educated | Friendly | Extroverted |
| Female Speaker | 7.00 | 6.92 | 7.08 | 6.25 | 6.42 |
| Male Speaker | 6.92 | 6.92 | 6.75 | 6.17 | 6.00 |
| *Participants in this survey were asked to rate speakers on a scale of 1–10. |
Tables help you manage more complex sets of data. A table can be used if you’re looking to display individual values, if values are being compared, or if data is going to be shown and then summarized. They won’t convey your story to the reader as quickly as a graph might, but you will still be conveying a large amount of information in an easy-to-understand way.
Contributors and Attributions
- Tables, Charts, and Infographics: Overview. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Complex Table. Provided by : Lumen Learning. License : CC BY: Attribution
- Stacked Line Graph. Provided by : Lumen Learning. License : CC BY: Attribution
- Stacked Bar Chart. Provided by : Lumen Learning. License : CC BY: Attribution
- Pie chart. Provided by : Lumen Learning. License : CC BY: Attribution
- Pictographs. Provided by : Lumen Learning. License : CC BY: Attribution
- Organizational Hierarchy Chart. Provided by : Lumen Learning. License : CC BY: Attribution
- Line Graph. Provided by : Lumen Learning. License : CC BY: Attribution
- Bar chart comparing healthcare costs as percentage of GDP across OECD countries. Authored by : Farcaster. Provided by : Wikimedia. Located at : https://commons.wikimedia.org/wiki/File:Healthcare_costs_to_GDP_OECD_2015_v1.png . License : CC BY-SA: Attribution-ShareAlike
- Venn Diagram. Authored by : Dflamini. Provided by : Pixabay. Located at : pixabay.com/en/infographic-web-safe-venn-2722977/. License : CC0: No Rights Reserved
- Pert example gantt chart. Authored by : Dbsheajr. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/File:Pert_example_gantt_chart.gif . License : CC BY-SA: Attribution-ShareAlike
- Super Bowl infographic and chicken wing facts. Authored by : U.S. Department of Agriculture. Provided by : Flickr. Located at : https://www.flickr.com/photos/usdagov/24700496851 . License : CC BY: Attribution
- Flowchart. Authored by : Wellington Grey. Provided by : Wikimedia. Located at : https://commons.wikimedia.org/wiki/File:Scientific_Theory_Flowchart.png . License : Public Domain: No Known Copyright


9.13B: Viral Roles in Ecosystems
- Show the roles viruses play on ecosystems
Metagenomics is a relatively recent field of study that tries to understand the diversity—especially microbial—of the world around us. Through these studies it is now known that the number of viral particles and number of different viral species in almost every environment on Earth is immense. It is largely believed that viruses by species are the most numerous of any biological entity on earth. This is typified by the role of viruses in marine ecology. A teaspoon of seawater contains about one million viruses.
Viruses are essential to the regulation of saltwater and freshwater ecosystems. Most of these viruses are bacteriophages, which are harmless to plants and animals. They infect and destroy the bacteria in aquatic microbial communities, comprising the most important mechanism of recycling carbon in the marine environment. The organic molecules released from the bacterial cells by the viruses stimulate fresh bacterial and algal growth.
Microorganisms constitute more than 90% of the biomass in the sea. It is estimated that viruses kill approximately 20% of this biomass each day, and that there are 15 times as many viruses in the oceans as there are bacteria and archaea. Viruses are the main agents responsible for the rapid destruction of harmful algal blooms, which often kill other marine life. The number of viruses in the oceans decreases further offshore and deeper into the water, where there are fewer host organisms. The effects of marine viruses are far-reaching; by increasing the amount of photosynthesis in the oceans, viruses are indirectly responsible for reducing the amount of carbon dioxide in the atmosphere by approximately 3 gigatonnes of carbon per year. Like any organism, marine mammals are susceptible to viral infections. In 1988 and 2002, thousands of harbor seals were killed in Europe by the phocine distemper virus. Many other viruses, including caliciviruses, herpesviruses, adenoviruses, and parvoviruses, circulate in marine mammal populations.
As mentioned, marine viruses are mostly bacteriophages, or phages. Phages are obligate intracellular parasites, meaning that they are able to reproduce only while infecting bacteria. Phages therefore are found only within environments that contain bacteria. Most environments contain bacteria, including our own bodies (called normal flora). Often these bacteria are found in large numbers. As a consequence, phages are found almost everywhere. As a rule of thumb, many phage biologists expect that phage population densities will exceed bacterial densities by a ratio of 10:1 or more (VBR or virus-to-bacterium ratio).
Estimates of bacterial numbers on Earth reach approximately 10 30 ; consequently, there is an expectation that 10 31 or more individual virus (mostly phage) particles exist, making phages the most numerous category of “organisms” on our planet. Bacteria (along with archaea) appear to be highly diverse and there are possibly millions of species. Phage-ecological interactions therefore are quantitatively vast, with huge numbers of interactions. Phage-ecological interactions are also qualitatively diverse: there are huge numbers of environment types, bacterial-host types, and also individual phage types.
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Key Points
- The diversity and numbers of different viruses is incredible.
- In the world’s oceans, 90% of the biomass is microbial, with viruses turning over 20% of that daily. Without the turnover of biomass driven by viruses, many sources of food would not be present for other organisms.
- As bacteria are incredibly diverse, then the viruses that infect them, phages, are even more diverse. As bacteria inhabit almost every niche of the earth, so do viruses.
Key Terms
- ecological : Relating to ecology, the interrelationships of organisms and their environment.
- metagenomics : The study of genomes recovered from environmental samples; especially the differentiation of genomes from multiple organisms or individuals, either in a symbiotic relationship, or at a crime scene.
- intracellular : Inside or within a cell.


4.4: Alternating Groups
In this section, we describe a special class of permutation groups. To get started, let’s play with a few exercises.
Write down every permutation in \(S_3\) as a product of 2-cycles in the most efficient way you can find (i.e., use the fewest possible transpositions). Now, write every permutation in \(S_3\) as a product of adjacent 2-cycles, but don’t worry about whether your decompositions are efficient. Any observations about the number of transpositions you used in each case? Think about even versus odd.
If \(\alpha_1,\alpha_2,\ldots,\alpha_k\) is a collection of 2-cycles in \(S_n\) such that \(\alpha_1\alpha_2\cdots\alpha_k=(1)\) , then \(k\) must be even.
- Proof.
-
Suppose \(\alpha_1,\alpha_2,\ldots,\alpha_k\) is a collection of 2-cycles in \(S_n\) such that \(\alpha_1\alpha_2\cdots\alpha_k=(1)\) . We need to show that \(k\) is even. We proceed by strong induction. First, it is clear that the statement is not true when \(k=1\) , but is true when \(k=2\) .
Now, assume that \(k>2\) and if \(j\leq k-1\) and we have a product of \(j\) 2-cycles that equals the identity, then \(j\) is even. Consider \(\alpha_1\alpha_2\) . The only possibilities are:
- \(\alpha_1\alpha_2=(a,b)(a,b)\) ,
- \(\alpha_1\alpha_2=(a,b)(a,c)\) ,
- \(\alpha_1\alpha_2=(a,b)(c,d)\) ,
- \(\alpha_1\alpha_2=(a,b)(b,c)\) .
If case (i) happens, then \[(1)=\alpha_1\alpha_2\cdots\alpha_k=\alpha_3\alpha_4\cdots\alpha_k.\] Since the expression on the right consists of \(k-2\) factors, \(k-2\) must be even by induction, which implies that \(k\) is even. Now, suppose we are in one of cases (ii), (iii), or (iv). Observe that:
- \((a,b)(a,c)=(b,c)(a,b)\) ,
- \((a,b)(c,d)=(c,d)(a,b)\) ,
- \((a,b)(b,c)=(b,c)(a,c)\) .
In each case, we were able to move \(a\) from the original left 2-cycle to a new right 2-cycle. That is, we were able to rewrite \(\alpha_1\alpha_2\) so that \(a\) does not appear in the left 2-cycle. Systematically repeat this process for the pairs \(\alpha_2\alpha_3\) , \(\alpha_3\alpha_4\) ,…, \(\alpha_{k-1}\alpha_k\) . If we ever encounter case (i) along the way, then we are done by induction. Otherwise, we are able to rewrite \(\alpha_1\alpha_2\cdots\alpha_k\) so that \(a\) only appears in the rightmost 2-cycle. But this implies that \(\alpha_1\alpha_2\cdots\alpha_k\) does not fix \(a\) , which contradicts \(\alpha_1\alpha_2\cdots\alpha_k =(1)\) . This implies that at some point we must encounter case (i), and hence \(k\) is even by induction.
If \(\sigma\in S_n\) , then every transposition representation of \(\sigma\) has the same parity.
The previous theorem tells us that the following definition is well-defined.
A permutation is even (respectively, odd ) if one of its transposition representations consists of an even (respectively, odd) number of transpositions.
Classify all of the permutations in \(S_3\) as even or odd.
Classify all of the permutations in \(S_4\) as even or odd.
Determine whether \((1,4,2,3,5)\) is even or odd. How about \((1,4,2,3,5)(7,9)\) ?
Consider the arbitrary \(k\) -cycle \((a_1,a_2,\ldots, a_k)\) from \(S_n\) (with \(k\leq n\) ). When will this cycle be odd versus even? Briefly justify your answer.
Conjecture a statement about when a permutation will be even versus odd. Briefly justify your answer.
And finally, we are ready to introduce the alternating groups.
The set of all even permutations in \(S_n\) is denoted by \(A_n\) and is called the alternating group .
Since we referred to \(A_n\) as a group, it darn well better be a group!
The set \(A_n\) forms a group under composition of permutations and has order \(n!/2\) .*
- *
-
Hint: To show that \(A_n\) is a group, argue that \(A_n\) is a subgroup of \(S_n\) using the Two-Step Subgroup Test (see Theorem [thm:subgroup_criterion] ). To show that \(|A_n|=n!/2\) , prove that the number of even permutations in \(S_n\) is the same as the number of odd permutations in \(S_n\) . Here is one way to accomplish this. Define \(f:A_n\to S_n\setminus A_n\) via \(f(\sigma)=(1,2)\sigma\) . Note that \(S_n\setminus A_n\) is the set of odd permutations in \(S_n\) . Show that \(f\) is a bijection.
Find \(A_3\) . What group is \(A_3\) isomorphic to?
Find \(A_4\) and then draw its subgroup lattice. Is \(A_4\) abelian?
What is the order of \(A_5\) ? Is \(A_5\) abelian?
What orders of elements occur in \(S_6\) and \(A_6\) ? What about \(S_7\) and \(A_7\) ?
Does \(A_8\) contain an element of order 15? If so, find one. If not, explain why no such element exists.
Below are a few interesting facts about \(A_4\) and \(A_5\) , which we will state without proof.
- The group of rigid motion symmetries for a regular tetrahedron is isomorphic to \(A_4\) .
- You can arrange the Cayley diagram for \(A_4\) with generators \((1,2)(3,4)\) and \((2,3,4)\) on a truncated tetrahedron, which is depicted in Figure \(\PageIndex{1}\)a.
- You can arrange the Cayley diagram for \(A_5\) with generators \((1,2)(3,4)\) and \((1,2,3,4,5)\) on a truncated icosahedron, which is given in Figure \(\PageIndex{1}\)b. You can also arrange the Cayley diagram for \(A_5\) with generators \((1,2,3)\) and \((1,5)(2,4)\) on a truncated dodecahedron seen in Figure \(\PageIndex{1}\)c.


6.4: Mapping the Seafloor
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The previous sections included some information on the depths of the oceans in various places. So how are we able to map the ocean floor to ascertain these vast depths?
To map the ocean floor we need to know the depth at a number of places. The process of measuring the depths is known as bathymetry . These measurements were first made through soundings , where a weighted line (lead line) was let out by hand until it touched the bottom, and the depth could be recorded from the length of the line (Figure \(\PageIndex{1}\)). This technique led to the fathom as a unit of depth; as sailors hauled in the sounding line they would stretch it out to cover their arm span. The average arm span of a sailor was about six feet, so one fathom equals six feet, and the sailors could simply count the number of “arm spans” as they pulled in the line.
This technique had a number of drawbacks, and was usually limited to shallower water. It was very time consuming, and only gave depth data for a single point, so many individual soundings were needed to map an area. It could also be error-prone; in deep water it could be difficult to determine when the weight hit the bottom as the weight of the line itself could cause the line to keep sinking, and currents could deflect the line away from vertical, thus overestimating the depth. In later years, winches and heavy steel cables were used for deeper water, but this did not solve all of the problems inherent in the sounding method, and also added the constraint of excessive weight of the equipment.
In the 19th century, a number of modifications were made to this simple design. In 1802 the British clockmaker Edward Massey invented a mechanical device that was attached to the sounding line; as the device sank, a rotor turned a dial which locked in place when the line hit bottom (Figure \(\PageIndex{2}\)). The line could then be reeled in and the depth read from the dial. In 1853 American sailor John Mercer Brooke developed a cannonball weight attached to a twine reel. The cannonball was dropped over the side and allowed to free-fall to the bottom; by timing the fall rate (the rate at which the twine unspooled) and noting when the rate changed as the cannonball hit the bottom, the water depth could be calculated. When it hit bottom, the cannonball was released and the line could be hauled back in, bringing with it a sample of mud in the iron bar that held the cannonball, thus confirming that the bottom had been reached.
After the Titanic disaster in 1912, there was an effort to develop better methods of detecting icebergs from a ship. This led to the development of sonar (SOund Navigation And Ranging) technology, which was soon applied to mapping bathymetry. A sonar device called an echosounder sends out a pulse of sound, then listens for the returning echo. The timing of the returning echo is used to calculate depth. We know that the speed of sound in water is approximately 1500 m/s (see section 6.4). Since the returning echo traveled to the bottom and back, the water depth corresponds to half the time it takes for an echo to return, multiplied by the speed of sound in water (Figure \(\PageIndex{3}\)):
depth=1/2*(two-way travel time)*(speed of sound in water)
Echosounders allowed a fast, continuous record of bathymetry under a moving ship. However, they only give the depth directly under a ship’s path. Today, high resolution seafloor maps are made through multibeam or side scan sonar, either from a ship or from a towed transmitter (Figure \(\PageIndex{4}\)). Multibeam sonar produces a fan-shaped acoustic field allowing a much a wider area (>10 km wide) to be mapped simultaneously.
Large-scale mapping of the ocean floor is also carried out by satellites (originally SEASAT, then GEOSAT, now the Jason satellites) which use radio waves to measure the height of the sea surface (radar altimetry). The sea surface is not flat; gravity causes it to be slightly higher over elevated features on the ocean floor, and slightly lower over trenches and other depressions. Satellites send out radio waves, and similar to an echosounder, can use the returning waves to detect differences in sea surface height down to 3-6 cm (Figure \(\PageIndex{5}\)). These differences in sea surface heights allow us to determine the topography under the surface. Unlike the old lead line technology, where hundreds of soundings were necessary to map a small area, each taking an hour or more to complete, the current satellites can map over 90% of Earth’s ice-free sea surface every 10 days!
Additional links for more information:
- NOAA page on side scan sonar: http://www.nauticalcharts.noaa.gov/hsd/SSS.html
- NOAA page on exploring the ocean using satellite altimetry data: https://www.ngdc.noaa.gov/mgg/bathym...d/explore.HTML


8.1: Simple Interest and Discount
Learning Objectives
In this section, you will learn to:
- Use the simple interest formula to find an account balance, principal, rate, or time.
- Find discounts and proceeds.
Prerequisite Skills
Before you get started, take this prerequisite quiz.
1. Solve \(5=2(x+9)\).
- Click here to check your answer
-
\(x=-\dfrac{13}{2}\)
If you missed this problem, review Section 1.1 . (Note that this will open in a new window.)
2. Solve \(x=-3(7-10)\).
- Click here to check your answer
-
\(x=9\)
If you missed this problem, review Section 1.1 . (Note that this will open in a new window.)
3. Solve \(-12=x(3+8)\).
- Click here to check your answer
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\(-\dfrac{12}{11}\)
If you missed this problem, review Section 1.1 . (Note that this will open in a new window.)
4. Solve \(4=6(2-x)\).
- Click here to check your answer
-
\(\dfrac{4}{3}\)
If you missed this problem, review Section 1.1 . (Note that this will open in a new window.)
Simple Interest
It costs to borrow money. The rent one pays for the use of money is called the interest . The amount of money that is being borrowed or loaned is called the principal , also called the present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money. Although the interest rate is most commonly specified for a year, it may be specified for a week, a month, or a quarter, etc. Credit card companies often list their charges as monthly rates, sometimes it is as high as 1.5% a month.
Definition: Simple Interest
If an amount \(P\) is borrowed for a time \(t\) at an interest rate of \(r\) per time period, then the simple interest is given by
\[ I = P \cdot r \cdot t\]
Definition: Accumulated Value
The total amount \(A\), also called the accumulated value or the future value, is given by
\[ \begin{align*} A &= P + I \\[4pt] &= P + Prt \end{align*}\]
or
\[ A = P(1+rt) \label{simple3}\]
where interest rate \(r\) is expressed in decimals.
Example \(\PageIndex{1}\)
Ursula borrows $600 for 5 months at a simple interest rate of 15% per year. Find the interest, and the total amount she is obligated to pay?
Solution
The interest is computed by multiplying the principal with the interest rate and the time.
\[\begin{align*} \mathrm{I} &=\operatorname{Prt} \\[4pt] &=\$ 600(0.15) \frac{5}{12} \\[4pt] &=\$ 37.50 \end{align*} \]
The total amount is
\[\begin{align*} \mathrm{A} &=\mathrm{P}+\mathrm{I} \\[4pt] &=\$ 600+\$ 37.50 \\[4pt] &=\$ 637.50 \end{align*} \]
Incidentally, the total amount can be computed directly via Equation \ref{simple3} as
\[\begin{align*} A &=P(1+r t) \\[4pt] &=\$ 600[1+(0.15)(5 / 12)] \\[4pt] &=\$ 600(1+0.0625) \\[4pt] &=\$ 637.50 \end{align*} \]
Example \(\PageIndex{2}\)
Jose deposited $2500 in an account that pays 6% simple interest. How much money will he have at the end of 3 years?
Solution
The total amount or the future value is given by Equation \ref{simple3}.
\[\begin{align*} A &= P(1 + rt) \\[4pt] &=\$ 2500[1+(.06)(3)] \\[4pt] \mathrm{A} &=\$ 2950 \end{align*}\]
Example \(\PageIndex{3}\)
Darnel owes a total of $3060 which includes 12% simple interest for the three years he borrowed the money. How much did he originally borrow?
Solution
This time we are asked to compute the principal \(P\) via Equation \ref{simple3}.
\[\begin{align*} \$ 3060 &=\mathrm{P}[1+(0.12)(3)] \\[4pt] \$ 3060 &=\mathrm{P}(1.36) \\[4pt] \dfrac{\$ 3060}{1.36}&=\mathrm{P} \\[4pt] \$ 2250 &=\mathrm{P} \quad \text { Darnel originally borrowed \$2250. } \end{align*}\]
Example \(\PageIndex{4}\)
A Visa credit card company charges a 1.5% simple interest finance charge each month on the unpaid balance. If Martha owed $2350 and has not paid her bill for three months, how much does she owe now?
Solution
Before we attempt the problem, the reader should note that in this problem the rate of finance charge is given per month and not per year.
The total amount Martha owes is the previous unpaid balance plus the finance charge.
\[ A=\$ 2350+\$ 2350(.015)(3)=\$ 2350+\$ 105.75=\$ 2455.75 \nonumber\]
Alternatively, again, we can compute the amount directly by using formula \(A = P(1 + rt)\)
\[A=\$ 2350[1+(.015)(3)]=\$ 2350(1.045)=\$ 2455.75 \nonumber\]
Discount and Proceeds
Banks often deduct the simple interest from the loan amount at the time that the loan is made. When this happens, we say the loan has been discounted. The interest that is deducted is called the discount , and the actual amount that is given to the borrower is called the proceeds . The amount the borrower is obligated to repay is called the maturity value.
Discount and Proceeds
If an amount \(M\) is borrowed for a time \(t\) at a discount rate of \(r\) per year, then the discount \(D\) is
\[ D = M \cdot r \cdot t \]
The proceeds \(P\), the actual amount the borrower gets, is given by
\[\begin{align*} P &=M-D \\[4pt] &=M-M r t \end{align*}\]
or
\[P = M(1-rt)\]
where interest rate \(r\) is expressed in decimals.
Example \(\PageIndex{5}\)
Francisco borrows $1200 for 10 months at a simple interest rate of 15% per year. Determine the discount and the proceeds.
Solution
The discount \(D\) is the interest on the loan that the bank deducts from the loan amount.
\[\begin{array}{l}
\mathrm{D}=\mathrm{Mrt} \\
\mathrm{D}=\$ 1200(0.15)\left(\frac{10}{12}\right)=\$ 150
\end{array} \nonumber\]
Therefore, the bank deducts $150 from the maturity value of $1200, and gives Francisco $1050. Francisco is obligated to repay the bank $1200.
In this case, the discount \(D\) = $150, and the proceeds
\[P = \$1200 - \$150 = \$1050. \nonumber\]
Example \(\PageIndex{6}\)
If Francisco wants to receive $1200 for 10 months at a simple interest rate of 15% per year, what amount of loan should he apply for?
Solution
In this problem, we are given the proceeds \(P\) and are being asked to find the maturity value \(M\).
We have \(P\) = $1200, \(r = 0.15\), \(t\) = 10/12 . We need to find \(M\).
We know \(P = M - D\)
but also \(D = Mrt\)
therefore
\[\begin{align*} \mathrm{P} &=\mathrm{M}-\mathrm{Mrt} \\[4pt] &=\mathrm{M}(1-\mathrm{rt}) \\[4pt] \$ 1200 &=\mathrm{M}\left[1-(0.15)\left(\dfrac{10}{12}\right)\right] \end{align*}\]
We need to solve for \(M\).
\[\begin{align*}
\$ 1200 &=\mathrm{M}(1-0.125) \\[4pt]
\$ 1200 &=\mathrm{M}(0.875) \\[4pt]
\dfrac{\$ 1200}{0.875} &=\mathrm{M} \\[4pt]
\$ 1371.43 &=\mathrm{M}
\end{align*}\]
Therefore, Francisco should ask for a loan for $1371.43.
The bank will discount $171.43 and Francisco will receive $1200.
Summary
Below is a summary of the formulas we developed for calculations involving simple interest:
Simple interest
If an amount \(P\) is borrowed for a time \(t\) at an interest rate of \(r\) per time period, then the simple interest is given by
\[ I = P \cdot r \cdot t \nonumber\]
The total amount \(A\), also called the accumulated value or the future value, is given by
\[ A = P + I = P + Prt \nonumber\]
or
\[ A = P(1+rt) \nonumber\]
where interest rate \(r\) is expressed in decimals.
Discount and Proceeds
If an amount \(M\) is borrowed for a time \(t\) at a discount rate of \(r\) per year, then the discount \(D\) is
\[ D = M \cdot r \cdot t \nonumber\]
The proceeds \(P\), the actual amount the borrower gets, is given by
\[\begin{array}{l}
P=M-D \\
P=M-M r t
\end{array} \nonumber\]
or \[P = M(1-rt) \nonumber\]
where interest rate \(r\) is expressed in decimals.
At the end of the loan's term, the borrower repays the entire maturity amount \(M\).


Friedrich Nietzsche
1. Introduction
Each of us has, at one time or another, talked about our conscience. We might have been “pricked by our conscience” or our conscience might have “butted in” when it was not wanted. We might be going on quite happily telling a lie to a friend, or might have accidentally walked out of a shop without paying for something and it is our conscience that makes us confess or stops us in our tracks spins us on our heels and takes us back into the shop.
People from different walks of life talk of the “conscience”, from the
religious believer, the politician, the celebrity, to every day folk; we might hear someone berate their conscience for nagging them to do something they do not want to. People might be labelled “conscientious objectors” because they feel their conscience is telling them to object to certain political actions, e.g. war. A protester might lament the erosion of their “freedom of conscience”. And we can find concepts very similar to “conscience” in many non-Christian religious traditions both Eastern and Western throughout history and from around the globe.
1
However, the nature of conscience is obscure and consequently the philosophical discussion of conscience is complex and has a long history. It draws on issues in philosophical psychology, philosophy of religion, epistemology, philosophy of mind, applied ethics, normative ethics and Metaethics.
In this chapter we’ll give a general overview of two theories of conscience. One draws on Aquinas’s account; the other
Sigmund Freud
’s (1856–1939). Although Freud is not typically seen as a philosopher (he’s a psychologist) his account will provide us with some insights which allows us to think philosophically about this thing we call “the conscience”.
2. The History of Conscience
In the twenty-first century conscience is not thought of as solely a religious idea. However talk of “conscience” was popularised, at least in “the West”, due to its adoption by both Protestant and Catholic traditions. In this section we’ll look how “conscience” is, and has been, used in order to draw out some general features.
“Conscience” played a role in one of the most famous speeches in the protestant reformation. Martin Luther (1483–1546), being charged with heresy and being forced to recant by Charles V, stands his grounds and says “Here I stand, I can do no other” and “I cannot nor will I retract anything, since it is never safe nor virtuous to go against conscience”. Luther believed that his God-given conscience was not allowing him to recant, not even under the considerable pressure by the powerful people before him. Or consider a more recent example.
In the midst of political turmoil of the civil rights movement Martin Luther King Jr., who was under threat and constant pressure to change his views said:
But, conscience asks the question, is it right? And there comes a time when we must take a position that is neither safe, nor politic, nor popular, but one must take it because it is right . 2
Conscience is, then, powerful. It seems that it can move a person to put themselves in mortal danger, to “stand up and be counted”, to act contrary to self-interest.
But it is not just saints and heroes that talk of conscience, conscience has been cited by the most repugnant and morally abhorrent people who have ever lived, racists, murderers, tyrants, dictators. For example as Bettina Stangneth’s states in
Eichmann Before Jerusalem: The Unexamined Life of a Mass Murderer
,
3
a discussion of the inner life of one of the Nazi’s most notorious officers: “Conscience was simply the ‘morality of the Fatherland that dwells within’ a person, which Eichmann also termed ‘the voice of the blood’”.
Conscience can be male or female or both or neither, it can be one voice or many, it can echo religious ideas, social ideas, racist ideas, lofty ideas or ideas found in the filth of human corruption. Conscience can develop at any particular age and dissipates at any age. It does not “speak”, and it does “speak”, and does not have a language of choice. All of these observations then leave a number of observations and questions.
There seem to be (at least) three related functions that we think the conscience plays. First,
it tells us what we ought to do
as a guide for our lives. Second, it is a source of moral
knowledge
. That is, we might say “I
know
that stealing a pen is wrong because my conscience told me”. Third it might be thought of as a
motivation
. That is, it might be the thing that actually gets us up out of our seat to act in certain ways, even when things are difficult or even life threatening.
Just to clarify, we can see the difference in the first two of these functions if we think of a tyrant who says for example: “my conscience tells me I have to kill all mentally ill people to help the country”. Clearly this is a case where her conscience is telling her how she
ought
to behave. But, given that we think that killing the mentally ill is morally wrong, we do not want to say that in this case her conscience gives her
knowledge
of what is right and wrong. So it might be true that the conscience gives us guidance but not knowledge.
Equally the opposite seems true, that we might know what is right and wrong yet fail to be guided to do it. This predicament is what Shakespeare captured in this famous quotation: “conscience doth make cowards of us all” (Hamlet 3.1.78–82).
Consider another point. Conscience is
subjective
in that it is about one reflecting inwards on oneself, on how one might “feel” about certain things. It is not about looking
out
into the world, at a set of rules or laws. We experience the conscience differently than we would if a friend, priest, politician or Imam was telling us what to do. Of course, although conscience is “inward looking”, that is not the same as saying that we just make up what the conscience allegedly tells us. For instance, we might think that what is right and wrong is dependent on God but also think that we come to know what is right and wrong through our conscience.
Finally it is worth noting that the term “conscience” was only formalized in its modern moral meaning in the mid eighteenth century (e.g. neither Plato nor Aristotle talk of conscience). However, note that just because a term is modern, or just because there is disagreement with how a term is used, that does not mean that the ideas themselves are new.
Consider the point that the terms “molecules” and “atoms” were recent inventions, and that in their development they might be used to talk about different things, and they engendered disagreement within the scientific community. This in itself does not lead us to the conclusion that there are no molecules and no atoms. So the lack of term “conscience”, and disagreement about what “conscience” means does not mean that conscience is
merely
an “invention”. With all these points in mind let’s consider one of the key thinkers in relation to conscience, Aquinas.
3. Aquinas on Conscience
If you recall from Chapter 4, Aquinas developed a Natural Law theology. The basic idea is that through reason (what he calls ratio ) we can come to recognize certain precepts that we ought to live by. Aquinas thinks that this reliance on thinking and reflection is revealed in the Bible:
They show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts sometimes accusing them and at other times even defending them . 4
Notice then that for Paul — and Aquinas — the “conscience”
bears witness
sometimes accusing the person, sometimes defending them. For Aquinas
conscience is morally neutral
, it simply “bears witness”, it is a “sign-post” and after all signposts do not opinions on things (see Aquinas,
Summa
, Part 1, Question 79, Article 13).
To be clear then Aquinas
did not take conscience to be a source of moral knowledge
but as a guide. This means that Aquinas, unlike Luther and post-reformation thinkers, took conscience to be
fallible
. For Aquinas we may be wrong in following our conscience as it can move us in the wrong direction/mislead us
For Aquinas the conscience is the act of applying the universal principles (the Eternal/Divine law) to actual real life situations.
Aquinas explicitly defines “conscience” as the “application of knowledge to activity” (
Summa Theologica
, I–II, I). So, if conscience for Aquinas is about the application of
knowledge
to activity, this raises the question how we get this knowledge? This is where another key technical term is introduced. The
synderesis
. Synderesis is not the same as conscience but is the innate ability of the mind — what he calls a
habit of the mind
— to apprehend the eternal/divine laws. The role of conscience is to apply the primary precepts discovered as the content of synderesis.
To get a better understanding of synderesis consider someone trying to work out the quickest way to get between two points. Through rational reflection they will see that it is the straight line. This “coming to recognize through reflection” is what Aquinas has in mind when he talks about synderesis. For Aquinas, unlike conscience,
synderesis is never mistaken
. Humans do wrong, thinks Aquinas, when
conscience
(and not synderesis) makes a mistake. This means that a
failure of conscience
needs to be clearly thought through on Aquinas’s account.
For Aquinas, conscience errs because of
ignorance
about how to apply the eternal/divine laws, of which there are
two types
. Ignorance that
can be
overcome by using one’s reason (
vincible ignorance
), and ignorance that
cannot be
overcome by using one’s reason (
invincible ignorance
). Invincible ignorance is doing something wrong when one could not have known better; vincible ignorance is doing wrong when one
ought
to have known better. But how might this relate to conscience?
Imagine two people going into a gun shop. The first person has no criminal record, has never been in trouble with the police nor at school and they have no record of mental illness. He is, for all intents and purposes, a model citizen.  The owner of the shop, by following her conscience, has not done something morally wrong because her
ignorance is invincible
; there was no indication that this would have been a likely outcome.
This contrasts to the person who is sold a gun even though he has a violent criminal record which would have shown up on a basic background check. In this case, the owner of the gun shop following her conscience has done something morally wrong because in this case her ignorance is vincible.
To conclude, Aquinas thinks all of us can know infallibly what is right and wrong through synderesis. However, even though we are infallible about this, we can, and do, make mistakes in applying this knowledge. It is our conscience (
conscientia
) which tells us how to apply this knowledge and moves us to act. It can go wrong through ignorance. Ignorance which could have been avoided (vincible) means our action is morally wrong. Mistakes deriving from ignorance we could not have avoided (invincible) means our action is not morally wrong.
In the next section, we will consider what Freud has to say about conscience, and explain how he reconceptualises it as a psychological and not theological concept, and in doing so argues we should not accept it as an inherently good notion.
4. Freud and the Conscience
Freud is best known as a psychologist and the architect of psychoanalysis. He is controversial and most philosophers and psychologists reject the ideas he presents. However, his ideas have been incredibly influential, and indeed his name has entered our everyday talk in the form of a “Freudian slip”. Among Freud’s many ideas his conceptualization of the structure of the mind is key to his views on conscience. He thinks the mind can be thought of as containing three parts: the
id
, the ego and the super-ego.
Freud’s account of conscience is understood as the relation between these
.
For Freud the
id
is the collection of our primal drives, e.g. the basic desires for food, sex, drink and is the oldest part of the mind. The
id
cannot be properly formalized or understood and Freud likens it to chaos. It is instinctive, emotional and illogical. We cannot list all the drives that make up the
id
as they are inaccessible to us. Freud has a nice way of describing the
id
; he calls it: “…a cauldron full of seething excitations…”. (SE, XXII.73). Although we can say very little about the content of the
id
, Freud did think that there was a general principle to help us understand the drives in the
id
, what he calls the “pleasure principle”. This is the claim that what identifies and unifies the drives of the
id
is the avoidance of pain and pursuit of pleasure.
Now, as a very young child it may be OK to be driven by the pleasure principle; they crawl single-mindedly after the chocolate buttons to put in their mouth, they crave their mother’s milk irrespective of anything else. However, as we develop we soon realize that we cannot simply act on the primal instincts of the
id
as we have to navigate ourselves in the social spaces we inhabit! We have to understand boundaries, sanctions and consequences. To successfully operate in the world, we need to consciously reflect and reason and ultimately, we have to delay instinctive behaviour and “weigh-up” the situation. Put bluntly someone whose
id
is unchecked would cease to be acceptable in society and find themselves physically, socially and emotionally isolated. It is what Freud calls the “ego” which plays this policing role.
But if we only have the
id
and the
ego
then it is unclear why we would
not simply follow the pleasure principle. That is, although the ego rationally
reflects, it needs something to weigh-up against the
id
. We need some
authority
that monitors
what the ego is doing. This authority is what Freud calls the
super-ego
.
Early in our life our parents (as well as society, religious leaders etc.) tell us what we can and cannot do and chastise us for breaking rules, and as we grow older we internalize these things and “hear them” as a voice of authority. Imagine that your mum has always told you not to sit with your elbows on the table then you internalize this rule. So when you are much older and not living with your mum any longer the voice of your “super-ego” speaks with authority — “take your elbows off the table!” These are the very basics of Freud structure of the mind. Our ego balances the primal drives of the id with the voice of authority from the super-ego.
Where does the conscience come in? For Freud the
conscience is the form that the super-ego takes in addressing the ego
. When the internalized authority derived from parental (social/religious) rules and regulations controls the ego is it is understood as “the conscience”. In our last example it is our “conscience” that tells us to remove our elbows from the table.
Notice then that our conscience often requires certain things from us which we fail to achieve and this gives rise to
guilt
. For Freud, the conscience can be thought of as synonymous with the “guilty conscience”. Our ego is punished through guilt by the form of the super-ego we call conscience. Furthermore, Freud says that when the super-ego fails to deal properly with the
id
— when the pleasure principle is repressed — this forms what he calls
neurosis
.
You can also hopefully see the differences between Aquinas and Freud. First, the obvious point is that for Freud the conscience is
not
the voice of God. Second, unlike Aquinas, Freud thinks that the conscience could be bad, destructive and unhelpful. The conscience is the way the ego experiences the authority of the super-ego. But the super-ego is arrived at through the experiences we have. And, of course, we might have had really
bad
experiences growing up where parents are stifling, overly authoritarian, distant, cold, hard, violent, abusive etc. In these sorts of cases the conscience would be stifling, overly authoritarian, distant, etc. This means that although Freud does not think we can, or should, get rid of the conscience he does think we should treat it with a healthy dose of scepticism and hence not be kowtowed by the “guilt” that is our ego’s punishment for falling short of the super-ego; conscience is the product of our often non-ideal upbringing rather than a divinely-inspired force for good.
5. Freud’s Psychosexual Development Theory
Psychosexual Development Theory is a theory of sexual development from birth to death. Freud was the first thinker to look at the entire lifespan in terms of development. Freud thought that as we develop we move through different stages. At each stage our libido (sex drive) is focused towards different things. If we fail to move through a stage completely, or return to a stage, then problems arise and we might become fixated with the area associated with that stage. This can be a serious problem for our relationships and could be an underlying cause of mental illness.
The first stage is the oral stage from birth to about one and a half. This stage is where babies get pleasure through putting things in their mouth, pleasure in biting, chewing and sucking. For example, babies soon after they are born are breastfeeding and as the baby develops they navigate and explore the world through putting things in their mouth. Notice that during this phase babies are very dependent on others. According to Freud at this stage not only do we get information about the world, but we also fulfil the
id
. Babies who can bite, chew and such as much as they want are being guided by the
id
. Freud explains behaviours like smoking, chewing gum, overeating, with failure to move properly through this stage which prevented the successful development of the
id
.
The next stage, from about one and a half to three years, is the
anal stage
. Here pleasure is gained through
controlling going to the toilet
. This stage is about gaining control of one’s body, and it starts with controlling the bladder and bowels (being potty trained). It is around this time that the ego develops. This control of their bodies is a source of pride and pleasure for children. Agents who fail to properly move through this stage are what are sometimes called “anally retentive”. That is, someone who is overly controlling or out of control and messy, because — according to Freud — they do not want to let go of their waste, or do not care where or when they let go of their waste.
The next phase of development, from about three to six years, is
the phallic stage
in which a child discovers one’s genitals, and importantly that they are different in men and women. This stage is where Freud thinks we develop the Oedipus and the Electra complex. A problematic phallic stage will cause problems with intimacy in later life.
The next stage,
the latency stage
, is from six years to the onset of puberty. This stage is not about pleasure in the body as the libido is “latent” or hidden — this is the stage where sexual desire is repressed and no new sexual desires emerge. At this stage girls play with girls in order to learn the role of a girl and boys play with boys in order to learn about the role of boys.. The child learns how to navigate the social world. A difficult latency stage leads to relational problems and understanding one’s gender.
The libido then reappears in the final stage which lasts to our death and which Freud calls the
mature genital stage
. This is where the individual not only recognises the difference between men and women but also shows a desire to engage in a sexual relationship and, more generally, a pursuit of pleasure and happiness. People become sexually active, fall in love and get married. This is the stage where we acquire a fully developed conscience.
SUMMARY
|
The notion of “conscience” has appeared for thousands of years in different cultures, even though it has not always been called “conscience”. Modern Christian orthodoxy popularised it and characterised it in relation to God’s voice, and guidance. Aquinas thought that conscience is the way we understand how to apply what we know. In Aquinas’s view, our conscience is fallible and might guide us wrongly. When our conscience “gets it wrong” we can be either culpable — through vincible ignorance — or not culpable — through invincible ignorance.
Freud is less convinced that conscience is a force for good, and he is certain that it has not got anything to do with God. For Freud conscience can be either a good or bad. We can think of our mind as having three parts, the id, ego and super-ego. The conscience for Freud is the form the super-ego takes when it is trying to keep the ego in line. It is internalized as the voice of authority. The superego is about following rules but those rules do not come from “on high”, they derive from the upbringing we have had. So if we have had a repressive upbringing then the super-ego — the voice of conscience — will be repressive. How we develop these three features of the mind is through what Freud calls Psychosexual Development; if we do not develop correctly then we become fixated and repressive, form a neurosis and ultimately become mentally ill. Freud thought that this could be avoided by working through the Psychosexual Stages in the normal way, and can be treated through psychosexual counselling. |
COMMON STUDENT MISTAKES
- Writing conscious when meaning conscience.
- Confuse synderesis and conscience.
- Confusing conscience as guidance with conscience as the source of
- knowledge.
- Believe Freud thinks the conscience is always bad.
- Thinking that for Aquinas conscience is way of knowing what is right
- and wrong.
- Thinking that because the term conscience is new, conscience itself is a
- modern invention.
ISSUES TO CONSIDER
- Do you think you have a conscience? What does it tell you?
- What is the difference between synderesis and conscience?
- Do you think that everyone ultimately knows — if they reason correctly — what is right and wrong?
- What is the difference between vincible and invincible? Is not most of the supposedly invincible knowledge, really vincible? We just need to try harder?
- What are the possible different roles for the conscience?
- Could the conscience be a morally bad thing?
- Why does Freud think we need to be cautious about listening to our conscience?
- How does Freud’s account of conscience relate to his Psychosexual Development Theory?
- What do you think about Freud’s Psychosexual Development Theory?
- Draw up a table of the key stages and accompanying characteristics of Freud’s Psychosexual Development Theory.
- Could it ever make sense to talk about animals/robots having a conscience? If not, why not?
- Do you think conscience will still shape our lives in one thousand years?
KEY TERMINOLOGY
Pleasure principle
Id
Ego
Super-ego
Synderesis
Vincible ignorance
Invincible ignorance
Psychosexual Development Theory (oral, anal, phallic, latency and mature genital phases)
References
Aquinas, Thomas, Summa Theologica, freely available at http://www.newadvent.org/summa/
―, Romans (Commentary on the Letter of Saint Paul to the Romans).
Benhabib, Seyla, ‘Who’s on Trial, Eichmann or Arendt?’, The New York Times (21 September 2014), freely available at http://opinionator.blogs.nytimes.com...ann-or-anrendt
Freud S. and Freud A., Complete Psychological Works of Sigmund Freud (New York: Random House, 2001).
Giubilini, Alberto, ‘Conscience’, The Stanford Encyclopedia of Philosophy, Winter 2016 ed., edited by Edward N. Zalta, freely available at https://plato.stanford.edu/archives/...es/conscience/
King, Martin Luther, ‘A Proper Sense of Priorities’, 6 February 1968, Washington, D.C., freely available at http://www.aavw.org/special_features...ch_king04.html
Strohm, Paul, Conscience: A Very Short Introduction (Oxford: Oxford University Press, 2011), vol. 273, https://doi.org/10.1093/actrade/9780...69694.001.0001
|
1 See P. Strohm, Conscience , p. 18, for a good overview of this. 2 M. L. King Jr., ‘A Proper Sense of Priorities’, http://www.aavw.org/special_features/speeches_speech_ king04.html 3 S. Benhabib, ‘Who’s on Trial, Eichmann or Arendt?’, http://opinionator.blogs.nytimes.com...ann-or-anrendt 4 T. Aquinas, Romans , 2:15. |


6.3: Immunosuppressants, Biologics, Monoclonal Antibodies, and Biosimilar Drugs
By the end of this section, you should be able to:
- 6.3.1 Identify the characteristics of immunosuppressants, biologics, monoclonal antibodies, and biosimilar drugs.
- 6.3.2 Explain the indications, actions, adverse reactions, and interactions for immunosuppressant, biologics, monoclonal antibodies, and biosimilar drugs.
- 6.3.3 Describe the nursing implications related to immunosuppressants, biologics, monoclonal antibodies, and biosimilar drugs.
- 6.3.4 Explain the client education related to immunosuppressants, biologics, monoclonal antibodies, and biosimilar drugs.
This section of the chapter will delve into the realm of therapeutic agents used to manage various autoimmune diseases and conditions. Immunosuppressants are drugs that suppress the immune system, helping to prevent organ rejection in transplant recipients and manage autoimmune disorders. Biologic drugs , including monoclonal antibodies , represent a class of medications derived from living organisms that target specific molecules or cells involved in autoimmune processes, offering highly targeted and personalized treatment options. Biosimilar drugs , which are biologic medications designed to be highly similar to already approved biologics, provide cost-effective alternatives with comparable efficacy and safety profiles.
Immunosuppressant Drugs
Immunosuppressant is a class of medicines that inhibit or decrease the intensity of the immune response in the body. They are commonly prescribed to clients who have received an organ transplant and those with autoimmune disorders such as rheumatoid arthritis and systemic lupus erythematosus. While these medications can effectively prevent rejection of transplanted organs and manage symptoms of autoimmune diseases, they also come with potential adverse effects. Clients taking immunosuppressants may be at higher risk for infections, including opportunistic infections that can be life-threatening.
It is important for clients who are taking immunosuppressant drugs to work closely with their health care provider to monitor any potential side effects and receive regular follow-ups. Additionally, steps should be taken to minimize the risk of infection, such as practicing good hygiene and avoiding close contact with sick clients. Overall, while immunosuppressant drugs can be beneficial in certain medical conditions, it is important for clients to weigh the potential benefits against the risks and work closely with their health care provider to ensure safe and effective use.
Glucocorticoids
Glucocorticoids are a steroid hormone produced by the kidneys’ adrenal cortexes that regulate metabolism and the immune system. They have anti-inflammatory and immunosuppressive effects, making them essential in treating inflammatory and autoimmune disorders. Commonly used synthetic glucocorticoids include prednisone and dexamethasone. Glucocorticoids are typically reserved for short-term or last-resort use due to potential toxic effects. Careful tapering is necessary to address potential rebound symptoms resulting from prolonged use (see Table 6.2 for dosing information and Hypothalamus, Pituitary, and Adrenal Disorder Drugs for additional information on glucocorticoids).
Azathioprine
Azathioprine is an immunosuppressant used to treat autoimmune diseases such as rheumatoid arthritis. Azathioprine works by suppressing the immune system, reducing inflammation, and preventing tissue damage. Long-term use has been associated with an increased risk of certain types of cancer. Regular blood tests, such as a complete blood cell count to detect bone marrow suppression or infection, liver function tests to ensure no abnormal liver function is occurring, and serum creatinine to assess proper kidney function, may also be necessary to ensure that the medication is not causing any adverse or harmful effects on the body. See Table 6.3 for additional information on azathioprine.
Mycophenolate
Mycophenolate (also referred to as mycophenolic acid) is an immunosuppressive drug used to prevent organ rejection in clients who have undergone organ transplantation, such as kidney, heart, or liver transplants. It works by inhibiting the proliferation of T and B immune cells, thus suppressing the immune system’s response against the transplanted organs. Mycophenolic acid is commonly used in combination with other immunosuppressants to maintain the function of the transplanted organ and prevent rejection episodes.
Table 6.2 lists common immunosuppressants and typical routes and dosing for adult and pediatric clients.
| Drug | Routes and Dosage Ranges |
|---|---|
| Prednisone |
Dosage requirements are variable and individualized based on disease and response of the client; 5–60 mg orally daily.
In situations of less severity, lower doses will generally suffice, while in select clients, higher initial doses may be required. |
|
Dexamethasone
( Decadron ) |
Dosage requirements are variable and individualized based on disease and response of the client; 0.75–9 mg orally daily. |
|
Azathioprine
( Imuran ) |
For rheumatoid arthritis:
1 mg/kg (50–100 mg) orally given as a single dose or twice daily. The dose may be increased, beginning at 6–8 weeks and thereafter in 4-week intervals. Increments should be 0.5 mg/kg orally daily, up to a maximum dose of 2.5 mg/kg orally per day if there are no serious toxicities and if initial response is unsatisfactory.
For renal homotransplantation: Dosing is individualized and based on requirements to prevent rejection and minimize toxicity. Initial dose is 3–5 mg/kg orally daily, beginning at the time of transplant, with dose reduction to maintenance levels of 1–3 mg/kg orally daily. |
|
Mycophenolate
( Cellcept ) |
Children with a BSA 1.25m
2
to <1.5 m
2
: 750 mg orally twice daily (1.5 g total daily dose).
Children with a BSA ≥1.5 m 2 : 1 g orally twice daily (2 g total daily dose). Adults: 1.5 g orally twice daily (total daily dose of 3 g). |
Adverse Effects and Contraindications
Common adverse effects for glucocorticoids include weight gain, increased appetite, mood swings, insomnia, fluid retention, and elevated blood sugar levels. Long-term use can lead to more severe adverse effects, such as osteoporosis, high blood pressure, cataracts, and adrenal insufficiency. Contraindications include systemic fungal infections and live virus vaccines.
Azathioprine adverse effects include nausea, vomiting, diarrhea, and a higher risk of infections due to immune suppression. More severe adverse effects include bone marrow suppression, pancreatitis, and liver toxicity. Contraindications include hypersensitivity to the drug or any of its components and in clients with a genetic deficiency of enzyme TPMT (thiopurine methyltransferase).
Mycophenolate adverse effects include gastrointestinal disturbances (nausea, diarrhea), headaches, and an increased susceptibility to infections. It may also be associated with bone marrow suppression and increased risk of birth defects, making it contraindicated during pregnancy.
Table 6.3 is a drug prototype table for immunosuppressants and antimetabolites featuring azathioprine. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects and contraindications.
|
Drug Class
Immunosuppressant, antimetabolite Mechanism of Action Inhibits DNA synthesis by disrupting the replication and proliferation of T and B immune cells |
Drug Dosage
For rheumatoid arthritis: 1 mg/kg (50–100 mg) orally given as a single dose or twice daily. The dose may be increased, beginning at 6–8 weeks and thereafter in 4-week intervals. Increments should be 0.5 mg/kg orally daily, up to a maximum dose of 2.5 mg/kg orally per day if there are no serious toxicities and if initial response is unsatisfactory. For renal homotransplantation: Dosing is individualized and based on requirements to prevent rejection and minimize toxicity. Initial dose is 3–5 mg/kg orally daily, beginning at the time of transplant, with dose reduction to maintenance levels of 1–3 mg/kg orally daily. |
|
Indications
Management of active rheumatoid arthritis Prevention of rejection in renal homotransplantation Therapeutic Effects Decreases inflammation Suppresses the immune response |
Drug Interactions
Alkylating agents Xanthine oxidase inhibitors Febuxostat Aminosalicylates Angiotensin-converting enzyme (ACE) inhibitors Warfarin Ribavirin Food Interactions No significant interactions |
|
Adverse Effects
Leukopenia/thrombocytopenia/anemia Nausea/vomiting Fever Arthralgias/myalgias Reduction in sperm Hepatotoxicity Increased risk of infection Skin rash Alopecia |
Contraindications
Hypersensitivity Pregnancy and/or breastfeeding Caution: May increase risk of malignancy in clients previously treated with alkylating agents such as cyclophosphamide |
FDA Black Box Warning
Azathioprine
The use of azathioprine increases the risk of malignancy in humans, including post-transplant lymphoma and hepatosplenic T-cell lymphoma.
Mycophenolic Acid
The use of mycophenolic acid increases the risk of fetal toxicity (resulting in pregnancy loss or congenital malformations), malignancies (lymphomas and skin), and susceptibility to bacterial, viral, fungal, protozoal, and opportunistic infections.
Biologic Drugs and Monoclonal Antibodies
Biologic drugs , also known as biologics , are a class of medications derived from living sources such as cells, proteins, or tissues. They are used to treat various medical conditions, particularly autoimmune rheumatic diseases, and certain types of cancer (American Cancer Society, 2018; Drosos et al., 2021). Biologics are different from traditional chemical-based drugs because of their complex structure and the process used to manufacture them, which involves biotechnology and genetic engineering techniques.
Monoclonal antibodies are a type of biologic drug. They are engineered in the laboratory to target and bind to specific molecules or cells in the body, thus exerting precise and targeted effects (American Cancer Society, 2018). Monoclonal antibodies are designed to mimic the immune system’s natural ability to recognize and attack foreign invaders, such as bacteria, viruses, and cancer cells. These antibodies can be used to block certain pathways involved in disease processes, modulate immune responses, or deliver drugs directly to specific cells or tissues.
Both biologic drugs and monoclonal antibodies have revolutionized the treatment of various medical conditions, providing more personalized and effective therapeutic options for clients with complex and difficult-to-treat diseases. However, due to their complex manufacturing process and targeted actions, these medications can be relatively expensive compared to traditional drugs. Additionally, some clients may experience immune-related side effects, and close monitoring is often required during treatment.
Adalimumab
Adalimumab is a monoclonal antibody that targets and inhibits the action of tumor necrosis factor-alpha (TNF-alpha), a cytokine involved in inflammation. By blocking TNF-alpha, adalimumab reduces inflammation and alleviates symptoms associated with various autoimmune diseases. See Table 6.4 for dosing information.
Etanercept
Etanercept is another TNF inhibitor that acts as a soluble TNF receptor, binding to and neutralizing TNF-alpha. Like adalimumab, etanercept dampens the inflammatory response and provides relief from autoimmune-related inflammation. See Table 6.5 for additional information.
Infliximab
Infliximab is a chimeric monoclonal antibody that targets TNF-alpha, effectively neutralizing its activity. Infliximab is often used in conditions such as rheumatoid arthritis, psoriasis, psoriatic arthritis, ankylosing spondylitis, and inflammatory bowel diseases to manage inflammation and improve quality of life. See Table 6.4 for dosing information.
Rituximab
Rituximab is a monoclonal antibody that targets a specific protein known as CD20 found on the surface of B cells. It is primarily used to treat certain types of non-Hodgkin’s lymphoma and chronic lymphocytic leukemia. Rituximab works by binding to CD20-positive B cells, leading to the destruction and reduction in cancerous cell growth. It is also used to treat certain autoimmune diseases like rheumatoid arthritis and systemic lupus erythematosus. See Table 6.4 for dosing information.
Bevacizumab
Bevacizumab (Avastin) is a monoclonal antibody that targets vascular endothelial growth factor (VEGF). It is used to treat various types of cancer, including colorectal cancer, lung cancer, kidney cancer, and glioblastoma multiforme (a type of brain tumor). By inhibiting VEGF, bevacizumab disrupts the formation of new blood vessels that tumors need to grow and spread, ultimately slowing down tumor growth and improving outcomes.
Table 6.4 lists common biologic drugs and monoclonal antibodies and typical routes and dosing for adult clients.
| Drug | Routes and Dosage Ranges |
|---|---|
|
Adalimumab
( Humira ) |
RA, AS, PsA 10–<15 kg:
10 mg subcutaneously every other week.
RA, AS, PsA 15–<30 kg: 20 mg subcutaneously every other week. RA, AS, PsA ≥30 kg: 40 mg subcutaneously every other week. |
|
Etanercept
( Enbrel ) |
Adult RA, AS, PsA:
50 mg subcutaneously weekly.
Adult PsO: Starting dose 50 mg subcutaneously twice weekly for 3 months. Maintenance dose: 50 mg subcutaneously once weekly. |
| Infliximab ( Remicade ) | 5 mg/kg given as an intravenous induction regimen at 0, 2, and 6 weeks followed by a maintenance regimen of 5 mg/kg every 8 weeks thereafter. |
|
Rituximab
( Rituxan ) |
First infusion:
50 mg/hour intravenously. In the absence of infusion toxicity, increase infusion rate by 50 mg/hour increments every 30 minutes, to a maximum of 400 mg/hour.
Subsequent i nfusion: Initiate infusion at a rate of 100 mg/hr. In the absence of infusion toxicity, increase rate by 100 mg/hour increments at 30-minute intervals, to a maximum of 400 mg/hour. |
|
Bevacizumab
( Avastin ) |
Dosing is individualized and depends on the type of cancer being treated and the combination of other chemotherapy drugs being used.
5–15 mg/kg intravenously every 2–3 weeks. |
Adverse Effects and Contraindications
Common adverse effects of biologic drugs and monoclonal antibodies include increased risk of infections (bacterial, viral, fungal, and opportunistic infections), injection site reactions (erythema, itching, pain, and swelling), diarrhea, rash, pruritis, anemia, leukopenia, thrombocytopenia, congestive heart failure, lymphomas and certain types of skin cancer (melanoma and non-melanoma), convulsions, headache, interstitial lung disease, cutaneous vasculitis, Stevens–Johnson syndrome, and toxic epidermal necrolysis. Contraindications include hypersensitivity to the drug or any of its components, active infection, sepsis, pregnancy, and breastfeeding.
Table 6.5 is a drug prototype table for biologic, TNF inhibitors featuring etanercept. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Biologic, TNF inhibitor Mechanism of Action Blocks the binding of TNF-α and TNF-β (lymphotoxin-α [LT-α]) to cell surface TNF receptors, resulting in the inactivation of TNF’s biological effects |
Drug Dosage
Adult RA, AS, PsA: 50 mg subcutaneously weekly. Adult PsO: Starting dose 50 mg subcutaneously twice weekly for 3 months. Maintenance dose: 50 mg subcutaneously once weekly. |
|
Indications
Rheumatoid arthritis Polyarticular juvenile idiopathic arthritis Psoriatic arthritis Ankylosing spondylitis Plaque psoriasis Therapeutic Effects Decreases inflammation Suppresses the immune response |
Drug Interactions
Live vaccines Anakinra Cyclophosphamide Sulfasalazine Food Interactions No significant interactions |
|
Adverse Effects
Infections, including viral, bacterial, and fungal infection Injection site reactions (erythema, itching, pain, swelling) Diarrhea Rash Pruritis Hematologic (anemia, leukopenia, thrombocytopenia) Congestive heart failure Hepatotoxicity Lymphomas, melanoma, skin cancers, and Merkel cell carcinoma Convulsions Paresthesia Headache Interstitial lung disease Cutaneous vasculitis Stevens–Johnson syndrome Toxic epidermal necrolysis |
Contraindications
Hypersensitivity Active infection or sepsis Pregnancy and/or breastfeeding Caution: May increase risk of malignancy and can increase the risk of reactivating latent tuberculosis |
FDA Black Box Warning
Biologics and Monoclonal Antibodies
Biologic drugs and monoclonal antibodies increase the risk for infections including active tuberculosis and invasive fungal infections as well as bacterial, viral, and other opportunistic infections. They also can increase the risk for certain cancers including lymphomas, melanoma, non-melanoma skin cancer, and Merkel cell carcinoma.
Biosimilar Drugs
Biosimilar drugs are a type of biologic medication that is highly similar to an already approved reference biologic, also known as the original biologic or reference product. Biosimilars are designed to have similar efficacy, safety, and quality as the reference product (American Cancer Society, 2018; U.S. Food and Drug Administration, 2023). They are developed to be equivalent in terms of structure, biological activity, and clinical effects.
Unlike generic versions of traditional chemical-based drugs, which are identical to their reference products, biosimilars are not exact copies due to the complexity of biologics and the variability in the manufacturing process. Biosimilars undergo rigorous testing and comparison to the reference product to ensure they are highly similar and have no clinically meaningful differences in terms of safety and efficacy.
Biosimilars offer an opportunity to increase access to effective biologic treatments at potentially lower costs compared to the reference products. They play a crucial role in promoting competition and driving down prices, making biologic therapies more affordable for clients and health care systems.
Regulatory authorities, such as the U.S. Food and Drug Administration (FDA), have established robust guidelines for the approval of biosimilars, ensuring they meet stringent standards for quality, safety, and efficacy (American Cancer Society, 2018). This regulatory framework ensures that clients can have confidence in the safety and effectiveness of biosimilar medications.
Pegfilgrastim
Pegfilgrastim is a medication used to reduce the risk of infection in clients undergoing certain cancer treatments, such as chemotherapy, that may cause a decrease in the body's white blood cell count. It is a long-acting form of filgrastim, a synthetic version of a natural protein called granulocyte-colony stimulating factor (G-CSF).
Pegfilgrastim stimulates the bone marrow to increase white blood cell production, particularly neutrophils, which are essential for fighting infections. By increasing the white blood cell count, pegfilgrastim helps to reduce the likelihood of severe infections during cancer treatment. See Table 6.7 for additional information.
Table 6.6 lists common biosimilar drugs along with the typical routes and dosing for adult clients.
| Drug | Routes and Dosage Ranges |
|---|---|
|
Bortezomib
( Velcade ) |
Dosage requirements are variable and individualized based on disease and response of the client.
1.3 mg/m 2 intravenously at a concentration of 1 mg/mL in 3- to 5-second bolus. |
|
Pegfilgrastim
( Neulasta ) |
For clients receiving myelosuppression chemotherapy:
6 mg subcutaneously once per chemotherapy cycle.
For clients with hematopoietic subsyndrome of acute radiation syndrome: 2 doses 6 mg each subcutaneously 1 week apart. |
Adverse Effects and Contraindications
Adverse effects of bortezomib include GI effects (nausea, vomiting, diarrhea, constipation, fatigue, weakness, peripheral neuropathy, thrombocytopenia, anemia, neutropenia, fever, chills, and skin reaction). Contraindications include hypersensitivity to the drug or any of its components, severe liver impairment, and pregnancy and/or breastfeeding.
Adverse effects of pegfilgrastim also include bone pain, pain in the arms and legs, headache, fatigue and weakness, nausea, and injection site reactions (pain, erythema, and swelling). Contraindications include hypersensitivity to the drug or any of its components, pregnancy, and breastfeeding.
Table 6.7 is a drug prototype table for biosimilar drugs featuring pegfilgrastim. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Biosimilar, granulocyte-colony stimulating factors (G-CSFs) Mechanism of Action Binds to the precursor receptor cells on the surface of bone marrow cells, thereby signaling the bone marrow to increase the production and release of neutrophils into the bloodstream |
Drug Dosage
For clients receiving myelosuppression chemotherapy: 6 mg subcutaneously once per chemotherapy cycle. For clients with hematopoietic subsyndrome of acute radiation syndrome: 2 doses 6 mg each subcutaneously 1 week apart. |
|
Indications
Clients with cancer receiving myelosuppression chemotherapy Clients with hematopoietic subsyndrome of acute radiation syndrome Therapeutic Effects Increases production of white blood cells, particularly neutrophils |
Drug Interactions
No significant interactions Food Interactions No significant interactions |
|
Adverse Effects
Splenic rupture Bone pain Pain in extremities Headache Fatigue Nausea Injection site reactions (pain, erythema, and swelling) |
Contraindications
Hypersensitivity Pregnancy and/or breastfeeding |
Nursing Implications
The nurse should do the following for clients who are taking immunosuppressants, biologics, monoclonal antibodies, and biosimilar drugs:
- Review the client’s medical history to assess allergies and contraindications or potential drug interactions with current medications.
- Monitor complete blood counts for leukopenia, thrombocytopenia, and anemia; liver function tests for hepatotoxicity; renal functions tests for nephrotoxicity; and lipids for elevated lipid levels.
- Evaluate the client’s tuberculosis (TB) status prior to administering biologic drugs, as they can increase the risk of reactivating latent tuberculosis.
- Educate the client about the drug’s purpose, potential side effects, and benefits to help the client make an informed decision about the treatment plan.
- Ensure proper administration technique including the appropriate route, dosage, and injection site. Maintain proper aseptic technique throughout the procedure to prevent infections and ensure client safety.
- After administration, closely monitor the client for any immediate adverse reactions, such as allergic responses or infusion-related reactions. Vital signs, signs of infection, and changes in the client's condition should be regularly assessed, and any concerns should be promptly reported to the health care provider.
- Provide client teaching regarding the drug and when to call the health care provider. See below for client teaching guidelines.
Client Teaching Guidelines
The client taking an immunosuppressant, biologic drug, monoclonal antibody, or biosimilar drug should:
- Choose an injection site as recommended by the health care provider, avoiding areas that are bony, bruised, sore, red, scarred, or hard.
- Cleanse the injection area with an alcohol swab/pad and let dry for 30 seconds prior to administering the drug.
- Dispose of needles in an FDA-approved sharps disposal container after use.
- Keep a journal of symptoms and note improved or worsening symptoms.
- Practice good hygiene and avoid contact with sick individuals to decrease the risk of infection.
- Report symptoms of fever, chills, sore throat, and skin rash to the health care provider because these may indicate an adverse reaction to the drug.
- Attend all appointments, undergo recommended blood tests, and report any changes or concerns to the health care provider.
The client taking an immunosuppressant, biologic drug, monoclonal antibody, or biosimilar drug should not:
- Dispose of needles or sharps container in the household trash.
- Reuse needles.
- Stop taking the drug unless directed by the health care provider.
- Get a live vaccine such as measles, mumps, and rubella (MMR) or oral polio vaccine (OPV), as this increases the risk of developing the disease from the immunization.