4.5: The Nurse’s Role in the Promotion of Gut Health
By the end of this section, you should be able to:
- 4.4.1 Describe the homeostasis of gut health.
- 4.4.2 Describe the impact of diet on gut health.
- 4.4.3 Identify the impact of pharmacology on gut health.
- 4.4.4 Identify pathophysiology and gut health.
Gut Health Homeostasis
For healthy digestion and absorption of food and nutritional health, the gut must maintain a healthy homeostasis , the equilibrium maintained in the body through physiological processes and evidenced by intestinal microbiota growth. The nurse plays an important role in client education that leads to optimal nutritional health.
The composition of the gut’s microbial growth has a strong impact on the homeostasis of the gut . When the gut is not in a state of homeostasis, the imbalance that results can cause dysbiosis , a disruption to the gut microbiome. Many diseases have been linked to dysbiosis, including anxiety, depression, hypertension, cardiovascular disease, cancer, inflammatory bowel disease, obesity, and diabetes, so the impact of an unhealthy gut surpasses just gastrointestinal health (Afzaal et al., 2022).
To maintain proper gut homeostasis , bacterial growth requires energy in the form of adenosine triphosphate and oxygen (Lee et al., 2022). Dietary factors, antibiotics, and underlying host conditions can reduce oxygen consumption in the colonic epithelium, thereby reducing the available oxygen to the indigenous microbes.
The microbiota of the gut has many functions in the body. These bacterial residents help control nutrient metabolism, defend against pathogens , contribute to carbohydrate digestion, synthesize methane from hydrogen processed by bacterial metabolism, neutralize carcinogens , and synthesize biotin , folate , and vitamin K (Afzaal et al., 2022).
Diet and Gut Health
Diet directly impacts the health of microbial growth (Lee et al., 2022). There are numerous foods that have a positive impact on gut health . For example, fiber, water intake, fruits and vegetables, whole grains, and an overall well-balanced diet have shown positive effects. Yogurt is known to have a positive impact because it contains active bacterial cultures, which have been shown to be supportive of the gut microbiome environment (Stanford Medicine Children’s Health, 2023a). High-fat diets , in contrast, have been shown to have a negative impact. They tend to contribute to diabetes , obesity , metabolic syndrome , and colon cancer (Afzaal et al., 2022; Baton Rouge General, 2023). In addition, high sodium intake and alcohol use can contribute to dysbiosis and have been linked to hypertension .
A nurse can make a direct impact on client health by educating clients on why diet, gut microbiota, and homeostasis are important. The nurse should speak to the specific reasons why fiber, water, probiotics , and whole grains are valuable in a diet and why saturated fat s, alcohol , and sodium are areas of concern. The nurse should discuss with the client so they understand which types of food to increase or decrease. Table 4.3 lists common foods and their impact on the gastrointestinal system.
| Food | Impact on Gastrointestinal System |
|---|---|
| Fresh fruits and vegetables, lentils, kidney beans, split peas, chickpeas, quinoa, oats, popcorn, almonds, chia seeds, sweet potatoes, and dark chocolate | These foods are high in fiber and help maintain gut homeostasis , reduce cholesterol , promote healthy weight through satiety , reduce constipation by adding bulk to stool, and possibly reduce gastrointestinal cancer risks due to potential antioxidant properties (Gunnars, 2020; Moffitt Cancer Center, 2023). |
| Water | Increasing fiber requires increasing water in the diet to ensure there is sufficient water in the stool to account for the added bulk. |
| Saturated fats (found in fatty cuts of meat, pies, sausage, butter, lard, cheddar cheese, ice cream, sour cream, biscuits, cakes, pastries, milk chocolate confectionaries, palm oil, coconut oil, and coconut cream) | Saturated fats can lead to gut imbalance (Afzaal et al, 2022), raise bad cholesterol levels (NHS, 2023), and increase colon cancer risks (Baton Rouge General, 2023). |
| High-sodium foods (breads; pizza; sandwiches; deli meats and cured meats; cheese; soups; burritos and tacos; snacks like chips, popcorn, and pretzels; chicken; and omelets) | High amounts of sodium can upset the homeostasis of the gut by increasing dehydration . |
Pharmacology and Gut Health
Pharmacology and nutrition are interdependent because they rely on the same body processes. When the body lacks nutrients, the body’s ability to utilize certain drugs is impacted (Bhupathiraju & Hu, 2023). A lack in nutrients also impacts the body’s ability to properly dispose of toxins from drugs and digestion waste. The nutritional state of a client greatly impacts the ability of their medications to be properly metabolized, used, and excreted.
Probiotics, such as Lactobacillus , are supplements that are known to have positive impacts on the gut microbiome. These are useful in helping clients with dietary restrictions or who are taking certain medications that interfere with gut health. These supplements should be taken in combination with a healthy diet because they do not prevent gastrointestinal issues on their own but rather support healthy practices.
There are medications that can help and hurt the microbiota of the gut. Antibiotics are a well-known culprit for altering microbial growth . Long-term antibiotic use has been linked to Clostridium difficile ( C. diff ) infection and colon cancer. When a client needs antibiotics for longer than three days, it is common practice that they also be given a probiotic to counteract some side effects and the possibility of infection.
There are other medications that are lesser known for being harmful. They include:
- Proton pump inhibitors to treat gastroesophageal reflux disease , which can cause enteric infections like C. diff (Weersma et al., 2020)
- Metformin , used to treat diabetes , which can increase Escherichia coli ( E. coli ) in the intestines and lower other bacteria, causing imbalances in the microbiome
- Statins to treat hyperlipidemia , which can cause constipation or diarrhea , upsetting the homeostasis in the gut
- Antidepressants , which can kill off beneficial gut bacteria and alter the normal microbiome of the gut
- Opioids to treat chronic pain, which can cause severe bouts of constipation, upsetting the microbiome and preventing the excretion of wastes from the body
- Laxatives , which can cause disruption of the microbiome through electrolyte disturbances , dehydration, and mineral deficiencies
Pathophysiology and Gut Health
Gut health and its impact on the body are often discussed in relation to bidirectional crosstalk at an axis point (the two-way communication that is focused at a specific intersection) between body systems that regulate health relative to one another (Tilg et al., 2022). The gut has many of these axes that connect it to the liver, brain, lung, nervous system, sensory system, etc. The following section will look at the gut and its connection with the liver, brain, and lung.
Gut-Liver, Gut-Brain, and Gut-Lung Axis
Gut-liver axis communication happens between nutrients, metabolites, microbial antigens, and bile acids within a microbial community (Ding et al., 2020; Tilg et al., 2022). A disturbance in this gut-liver network has been linked to an increase in hepatic inflammation and hepatic encephalopathy and decompensation of liver cirrhosis (Klein, 2022).
The gut-brain axis has been shown to have many devastatingly negative effects. The gut has been shown to produce neuroactive compounds that modulate metabolism and immunity (Benakis & Liesz, 2022), which are essential components to healing after a stroke . Because a stroke leads to intestinal dysmotility and permeability of the intestinal barrier, and thus decreased metabolism and depressed immunity, stroke victims often have impaired healing. Table 4.4 outlines some other diseases connected to the gut-brain connection.
| Disease | Impact |
|---|---|
| Parkinson’s disease | Overgrowth of intestinal bacteria results in an increase in opportunistic pathogens, causing the neurotransmitter L-dopa to be decreased (Wallen et al., 2022). Parkinson’s disease is directly impacted by the availability of L-dopa, and symptoms are worsened when the availability is decreased. |
| Multiple sclerosis | Overgrowth of certain gut bacteria causes an imbalance in homeostasis; exacerbations and advancement of the disease can be affected (eBioMedicine, 2022; French, 2022). |
| Alzheimer’s disease | Clients with Alzheimer’s disease are shown to demonstrate excessive production and aggregation of amyloid-beta peptides that lead to extracellular plaque formation (Sidharthan, 2022). Gut microbiota health impacts the disease because when it is unbalanced, it releases more of certain by-products that can alter inflammatory cytokine-signaling pathways, leading to amyloid-beta peptide accumulation. |
| Mental health | A link has been found between systemic inflammations and anxiety, depression, and other mental illnesses (GI Associate and Endoscopy Center, 2022a). The health of the gut has a large impact on the inflammatory state of the body. It also has a key function in the release of many neurotransmitters, one of which is serotonin, an important mood neurotransmitter. |
| Huntington’s disease | When the gut microbiota is unbalanced, there is an influence on mood, cognition, and sleep, which are already negatively impacted by this disease (Wasser et al., 2020). |
Due to the recent COVID-19 pandemic, the gut-lung axis has been more recently explored. Overall, the gut’s role in the health of the body’s overall immunity and the reduction of systemic inflammation has been applied to COVID-19 infection. The resulting acute respiratory distress syndrome that is a life-threatening effect of the virus is caused by an autoimmune-response-mediated inflammation in the lungs. Overall rates of mortality with COVID-19 are markedly higher for those admitted to the intensive care unit, and those who have gastrointestinal symptoms in conjunction with their respiratory symptoms are much more likely to need intensive care, making gastrointestinal issues a determinant of prognosis (Chen et al., 2020; Cleveland Clinic, 2022b; Ye et al., 2020; Ouali et al., 2021). Long-haul COVID-19 has also become more prevalent. Because clients who have gastrointestinal symptoms with COVID-19 infection have a longer detoxification time after infection, they are at a higher risk for developing long-haul COVID than those without gastrointestinal symptoms.


6.1: BEFORE YOU MOVE ON
Key Concepts
We have observed in this chapter that art is like a mirror reflecting, communicating, and interpreting self, individuals, and society. Throughout history from primitive to modern, humans have been able to express a variety of ideas and feelings and even to evoke responses from neighbors through artistic markings and with the creation of structures. Those artistic expressions have been a major source in understanding each other and the world we live in. It has communicated in many different ways and styles the practical and abstraction, the cultural and the aesthetics of a people. As we have previously noted, Immanuel Kant characterized beauty or aesthetics and the practicality of it as a systematic way in understanding the range of the arts. We have noted that art can be an instrumental discipline, a powerful social or political force by which society interprets, controls, modifies, or adapts to their environment or to their personal taste and/or beliefs. Examples include the political and social statements of Jaune Quick—To—See Smith’s “the Quincentenary Non-Celebration” or Jona Cerwinske’s graffiti and murals, or the romantic and sacred aesthetic styles of Albert Bierstadt and Maurice Denis; the genre representation of cultural identity in the Ashante Kente cloth and the hyperrealist works of sculptor Ron Mueck; and in earlier years, a holy site like the Islamic structure The Dome of the Rock that is identified and recognized as a holy place by several diverse religious groups: Muslims, Jews and Catholics, thus representing several diverse groups all of which communicate powerful artistic messages. Each and all bring people together with like beliefs or views through an artistic structure of communicating: creativity (a substance of inventive, original , imaginative ideas); disposition (the character, temperament, formal structure qualities, and sequence); and style (communicating and delivering specific resources and physical attributes that send off a reaction).
Test Yourself
- How did the development of photography impact social consciousness and awareness in the arts; cite examples. Discuss and show change and influence.
- Historically, markings have been a means of delivering a religious message to different cultures. Identify and discuss at least three different early written religious art forms used to communicate a message. Explain the message and how it is influenced by the artist style in written form or imagery.
- How have people used art to commemorate events in their lives throughout history? Show examples of images and elaborate on artist style and presentation of depicting the event.
KEY TERMS
aesthetics: the study of principles and appreciation of beauty.
Ara Pacis : an enclosed altar in Rome dedicated to Pax, the Roman goddess of Peace.
artifacts: a tool, weapon, or ornament created by humans that usually has historical significance.
avant-garde : works of art that are innovative, experimental, different from the norm or on the cutting edge.
avatar : physical form of the Hindu god Vishnu.
Bhudevi: a Hindu earth goddess and the divine wife of Varaha, an Avatar of Vishnu.
central-plan churches : are symbolic to reference the cross of Christ. Its round, cruciform, or polygonal design was popular in the West and East after the fourth century.
gui : a ritual scepter, held by a ruler during ceremonies as a symbol of rank and power.
hieroglyphs : abstracted pictures that represent words or sounds.
Kente cloth : woven silk and cotton wrap worn by Ashante kings during special ceremonies.
klismos chair: a curved-leg chair style popular in Ancient Greece.
les Nabis : a movement of Post –impressionist graphic and fine artists in France during the 1890s.
Neolithic period: known also as the Stone Age, is the last stage of prehistoric human cultural evolution. It is a period known for its polished stone tools, spread of architecture, megalithic architecture, and domestication of animals.
Palace of Knossos : the first Minoan monument located in Knossos. It was the residence of King Minos’s dynasty, where he ruled.
Shandong Longshan Culture: Central China’s Neolithic culture named after Longshan, Shandong Province. The culture is known for its production of black pottery.
Stele : grave marker.
Varaha: a Hindu god in the form of a boar during the Satya Yuga.


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blood urea nitrogen (BUN)
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a blood test that measures the amount of urea nitrogen in the blood; elevated levels can indicate decreased renal function
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Bowman’s capsule
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part of the nephron that contains the glomerulus
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cortical nephrons
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comprise a majority of the functional units of the kidney located in the outer renal cortex
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fenestrated capillaries
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small blood vessels in the glomerulus that filter waste
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glomerular filtration
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initial step in making urine involving the filtering of excess fluid and waste products from blood
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glomerular filtration rate (GFR)
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the rate at which the kidneys filter the blood, calculated using the results of blood tests that measure renal function
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glomerulus
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located in nephrons, a tuft of capillaries surrounded by the Bowman’s capsule that filters blood
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hilum
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indentation on the medial side of the kidney that allows blood vessels, lymphatic vessels, and nerves to enter and exit
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juxtamedullary nephrons
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nephrons with longer loops of Henle that extend into the medulla
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loop of Henle
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part of the nephron that contributes to absorption of sodium
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mesangial cells
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cells that increase the body surface area for glomerular filtration by limiting the size of molecules that can be filtered
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micturition
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the process through which urine is released from the bladder
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nephrons
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the functional units of the kidney that produce urine
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renal colic
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acute pain associated with kidney stones
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renal cortex
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outermost layer of the internal areas of the kidney containing nephrons
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renal medulla
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inner part of the kidney containing structures essential for regulating urine concentration
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renal pelvis
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funnel-shaped area of the kidney that collects urine and is connected to the ureter
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tubular necrosis
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damage to the tubule cells of the kidney typically resulting from reduced blood flow or chemical exposure
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vasa recta
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long, hairpin-shaped blood vessels that run parallel to the loops of Henle and facilitate water reabsorption
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vesicoureteral reflux
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abnormal backflow of urine from the bladder up through the ureters


7.1: Interview
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Closed-ended questions – Respondents' answers are limited to a fixed set of responses.
- Yes/no questions – The respondent answers with a "yes" or a "no".
- Multiple choice – The respondent has several options from which to choose.
- Scaled questions – Responses are graded on a continuum (e.g.: rate the appearance of the product on a scale from 1 to 10, with 10 being the most preferred appearance). Examples of types of scales include the Likert scale, semantic differential scale, and rank-order scale. (See scale for further information)
- Matrix questions – Identical response categories are assigned to multiple questions. The questions are placed one under the other, forming a matrix with response categories along the top and a list of questions down the side. This is an efficient use of page space and the respondents' time.
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Open-ended questions – No options or predefined categories are suggested. The respondent supplies their own answer without being constrained by a fixed set of possible responses. Examples include:
- Completely unstructured – For example, "What is your opinion on questionnaires?"
- Word association – Words are presented and the respondent mentions the first word that comes to mind.
- Sentence completion – Respondents complete an incomplete sentence. For example, "The most important consideration in my decision to buy a new house is..."
Prepare For the interview: It is extremely important to prepare for the interview because it is a very important meeting not just a casual conversation. It is a good practice for the interviewer to set an interview day and to sent an email to the interviewee or to set a reminder. The interview questions that was created should be sent the interviewee several days before the meeting. Keep in mind an interview is the interruption of someone else routine therefore an interview should be less than an hour.
Conduct the Interview: conducting the interview is the main step of the planning. When conducting the interview, make sure you introduce yourself, describe the project, and explain the objectives of the interview and why it is important. Make sure you can see the interviewee either on camera or face to face so you can observe his or her body language. make the person been interview feel comfortable and at ease. Give him or her enough time to respond and listen carefully. After each response mark a very short break of three to five second then move on to the next question. Before finishing the interview summarize the session and seek the confirmation of the interviewee, by doing so that will give him or her the opportunity to correct you if there is any mistake.
Document the Interview: It is a good practice to take some note while conducting the interview so you can jog your memory after the interview. Keep the note taking to a minimum to avoid distracting the interviewee. Record the information gather immediately after the you finished the interview to remember all the information provided. Therefore, it is not a good idea to conduct back to back interviews. They use of devices as a recorder during the interview should be discuss and approve by the interviewee and make sure the content is deleted right after you finish to record the interview. Whether the interview is recorder or not, you should listen carefully to the interviewee in order to ask good follow up question. After the conversation send a memo to the interviewee express your appreciation, in the memo you should include the time, date, location, reason for the interview, and all the main points discuss during the meeting so that the interviewee can make correction or give more details if necessary.
Evaluate the Responses: while recording the interview it is very important to evaluate the interviewee responses to identify any biases. For instance, an interviewee who has a strong opinion about the current or future system might give incomplete answers or refrain the information. Also, an interviewee might provide answers to be helpful even though they do not have enough experience to give accurate information.


22: Stars from Adolescence to Old Age
The Sun and other stars cannot last forever. Eventually they will exhaust their nuclear fuel and cease to shine. But how do they change during their long lifetimes? And what do these changes mean for the future of Earth?
We now turn from the birth of stars to the rest of their life stories. This is not an easy task since stars live much longer than astronomers. Thus, we cannot hope to see the life story of any single star unfold before our eyes or telescopes. To learn about their lives, we must survey as many of the stellar inhabitants of the Galaxy as possible. With thoroughness and a little luck, we can catch at least a few of them in each stage of their lives. As you’ve learned, stars have many different characteristics, with the differences sometimes resulting from their different masses, temperatures, and luminosities, and at other times derived from changes that occur as they age. Through a combination of observation and theory, we can use these differences to piece together the life story of a star.
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- 22.2: Evolution from the Main Sequence to Red Giants
- When stars first begin to fuse hydrogen to helium, they lie on the zero-age main sequence. The amount of time a star spends in the main-sequence stage depends on its mass. More massive stars complete each stage of evolution more quickly than lower-mass stars. The fusion of hydrogen to form helium changes the interior composition of a star, which in turn results in changes in its temperature, luminosity, and radius.
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- 22.3: Star Clusters
- Star clusters provide one of the best tests of our calculations of what happens as stars age. The stars in a given cluster were formed at about the same time and have the same composition, so they differ mainly in mass, and thus, in their life stage. There are three types of star clusters: globular, open, and associations. Globular clusters have diameters of 50–450 light-years, contain hundreds of thousands of stars, and are distributed in a halo around the Galaxy.
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- 22.4: Checking Out the Theory
- The H–R diagram of stars in a cluster changes systematically as the cluster grows older. The most massive stars evolve most rapidly. In the youngest clusters and associations, highly luminous blue stars are on the main sequence; the stars with the lowest masses lie to the right of the main sequence and are still contracting toward it. With passing time, stars of progressively lower masses evolve away from (or turn off) the main sequence.
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- 22.5: Further Evolution of Stars
- After stars become red giants, their cores eventually become hot enough to produce energy by fusing helium to form carbon (and sometimes a bit of oxygen.) The fusion of three helium nuclei produces carbon through the triple-alpha process. The rapid onset of helium fusion in the core of a low-mass star is called the helium flash. After this, the star becomes stable and reduces its luminosity and size briefly.
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- 22.6: The Evolution of More Massive Stars
- In stars with masses that are >8 solar masses, nuclear reactions involving carbon, oxygen, and still heavier elements can build up nuclei as heavy as iron. The creation of new chemical elements is called nucleosynthesis. The late stages of evolution occur very quickly. Ultimately, all stars must use up all of their available energy supplies. In the process of dying, most stars eject some matter, enriched in heavy elements, into interstellar space where it can be used to form new stars.
Thumbnail: During the later phases of stellar evolution, stars expel some of their mass, which returns to the interstellar medium to form new stars. This Hubble Space Telescope image shows a star losing mass. Known as Menzel 3, or the Ant Nebula, this beautiful region of expelled gas is about 3000 light-years away from the Sun. We see a central star that has ejected mass preferentially in two opposite directions. The object is about 1.6 light-years long. The image is color coded—red corresponds to an emission line of sulfur, green to nitrogen, blue to hydrogen, and blue/violet to oxygen. (credit: modification of work by NASA, ESA and The Hubble Heritage Team (STScI/AURA))


4.7: Terry Stops
Terry v. Ohio is the Supreme Court Case that provides for police officers to use their training, experience, knowledge, skill, and observation to intercede on behalf of the public into criminal conduct. None the less it is a court-sanctioned form of profiling, specifically criminal profiling that as contended early on in this discussion is the basis of other forms of profiling, both positive and negative. In brief, an experienced Cleveland police officer (detective) observed the conduct of three men that rose to the level of casing a store in preparation to commit a robbery. The police officer acting on reasonable suspicion removed the trio from an automobile and patted them down. The pat-down revealed what appeared to be weapons on two of the culprits and resulted in a search that confirmed the officer’s suspicions. The court held that in these circumstances an over-the-clothes pat down to provide safety for the officer is sufficient and does not rise to the level of a seizure but rather a stop and frisk is not a search (White, 2007).
The preceding is an example of descriptive common sense police work that orchestrated profiling at differing levels. In recent years Racial Profiling is on highways and streets solely based on race which is a form of police misconduct. However, an officer acting on observed conduct that raises the suspicions of the police that criminal activity may be afoot is criminal profiling and occurs hundreds if not thousands a time daily across this country. If not for profiling, what would constitute a reason for a stop? How much police work would be done? It is in this manner that criminal thinking is apparent in behaviorism or mannerisms that create suspicions. Further criminal thinking is important in all fashions of profiling however the type and manner of Terry Stops are critical to day-to-day operations.
Since Terry, the court rendered rulings regarding stops based on race, ethnic origin, or minority status. In 1975, U.S. v. Brignoni-Ponce was decided. The facts of the case relate that Felix Humberto Brignoni-Ponce was traveling in his vehicle and was stopped by border patrol agents because he appeared to be Mexican (Oyez.org, 1974). The questioning by agents and other passengers revealed the illegal status of the occupants. The court ruled that the information gleaned from the interviews was inadmissible due to lack of probable cause by agents in the initial stop. The actions of the agents were a violation of the Fourth Amendment.
However, in subsequent matters, the court ruled to the contrary. The U.S. Supreme Court determined in 1993 that disparity in conviction rates is not necessarily unconstitutional unless data demonstrates that defendants of another race similarly situated are disparately impacted (U.S. v. Armstrong, 1993). In another case, Whren v United States (1996), police stopped a truck for failing to use a turn signal, and upon approaching the vehicle police observed Whren in possession of crack cocaine. The court sustained the prosecution finding that police did not violate the Fourth Amendment, regardless of the pretext of the officers. An observable traffic infraction provided license and privilege for police to act.
Having stated the purpose of this article it is essential to provide a description of criminal profiling as detailed in current research and then the brief discussion of Terry Stops criminal profiling. According to Hatch-Maillette, Scalora, Huss & Baumgartner (2001):
Individuals who demonstrate heavy involvement with the legal system are an undeniably important population to consider in terms of economic, social, and ethical issues. To the extent that knowledge of these people can be amassed, researchers and clinicians can begin to address the financial and societal burdens experienced by those who are indirectly or directly affected by our criminal justice system. One way in which our understanding can increase is by systematically examining the thought content of criminal offenders in hopes of detecting patterns in specific categories of cognitions that are indigenous to the type of crime committed (p.115).
Learn more about the court case that created the precedent behind stop-and-frisk in this video: stop and frisk .
*If you are accessing a print version of this book, type the following short url into your browser to visit this source: bit.ly/3zBeQWg


12.7: Multicultural Teams
- What are some challenges and best practices for managing and working with multicultural teams?
As globalization has increased over the last decades, workplaces have felt the impact of working within multicultural teams. The earlier section on team diversity outlined some of the highlights and benefits of working on diverse teams, and a multicultural group certainly qualifies as diverse. However, some key practices are recommended to those leading multicultural teams so that they can parlay the diversity into an advantage and not be derailed by it.
People may assume that communication is the key factor that can derail multicultural teams, as participants may have different languages and communication styles. In the Harvard Business Review article “Managing Multicultural Teams,” the authors point out four key cultural differences that can cause destructive conflicts in a team (Brett et al., 2020). The first difference is direct versus indirect communication . Some cultures are very direct and explicit in their communication, while others are more indirect and ask questions rather than pointing out problems. This difference can cause conflict because, at the extreme, the direct style may be considered offensive by some, whereas the indirect style may be perceived as unproductive and passive-aggressive in team interactions.
The second difference that multicultural teams may face is trouble with accents and fluency . When team members don’t speak the same language, there may be one language that dominates the group interaction—and those who don’t speak it may feel left out. The speakers of the primary language may feel that those members don’t contribute as much or are less competent. The next challenge is when there are differing attitudes toward hierarchy . Some cultures are very respectful of the hierarchy and will treat team members based on that hierarchy. Other cultures are more egalitarian and don’t observe hierarchical differences to the same degree. This may lead to clashes if some people feel they are being disrespected and not treated according to their status. The final difference that may challenge multicultural teams is conflicting decision-making norms . Different cultures make decisions differently, and some will apply significant analysis and preparation beforehand. Those cultures that make decisions more quickly (and need just enough information to make a decision) may be frustrated with the slow response and relatively longer thought process.
These cultural differences are good examples of how everyday team activities (decision-making, communication, interaction among team members) may become points of contention for a multicultural team if there isn’t adequate understanding of everyone’s culture. The authors propose several potential interventions to try if these conflicts arise. One simple intervention is adaptation, which is working with or around differences. This is best used when team members are willing to acknowledge the cultural differences and learn how to work with them. The next intervention technique is structural intervention or reorganizing to reduce friction on the team. This technique is best used if unproductive subgroups or cliques within the team need to be moved around. Managerial intervention is the technique of making decisions by management and without team involvement. This technique should be used sparingly, as it essentially shows that the team needs guidance and can’t move forward without management getting involved. Finally, exit is an intervention of last resort and is the voluntary or involuntary removal of a team member. If the differences and challenges have proven so great that an individual on the team can no longer work with the team productively, then removing the team member in question may be necessary.
Some people seem to be innately aware of and able to work with cultural differences on teams and in their organizations. These individuals might be said to have cultural intelligence. Cultural intelligence is a competency and a skill that enables individuals to function effectively in cross-cultural environments. It develops as people become more aware of the influence of culture and more capable of adapting their behavior to the norms of other cultures. In the IESE Insight article entitled “Cultural Competence: Why It Matters and How You Can Acquire It” the authors Lee and Liao (2015) assert that “multicultural leaders may relate better to team members from different cultures and resolve conflicts more easily. Their multiple talents can also be put to good use in international negotiations.” Multicultural leaders don’t have a lot of “baggage” from any one culture and so are sometimes perceived as being culturally neutral. They are very good at handling diversity, which gives them a great advantage in their relationships with teammates.
To help employees become better team members in an increasingly multicultural world, the authors recommend a few best practices for honing cross-cultural skills. The first is to “broaden your mind”—expand your own cultural channels (travel, movies, books) and surround yourself with people from other cultures. This helps raise your awareness of the cultural differences and norms you may encounter. Another best practice is to “develop your cross-cultural skills through practice” and experiential learning. You may have the opportunity to work or travel abroad—but if you don’t, getting to know some of your company’s cross-cultural colleagues or foreign visitors will help you practice your skills. Serving on a cross-cultural project team and getting to know and bond with your global colleagues is an excellent way to develop skills. In my own “past life,” I led a global human resources organization, and my team included employees from China, India, Brazil, Hungary, the Netherlands, and the United States. We would have annual meetings as a global HR team, and it was so rewarding to share and learn about each other’s cultures. We would initiate the week with a gift exchange in a “show and tell” format from our various countries, so that everyone would learn a little bit more about the cultures in which our colleagues were working. This type of interaction within a global team is a great way to facilitate cross-cultural understanding and communication and to sharpen everyone’s cultural intelligence.
 You first may have to juggle time zone differences to find an adequate meeting time that suits all team members. (I used to have a team call with my Chinese colleagues at 8 p.m. my time so that I could catch them at 8 a.m. in China the next day) Language challenges can also pose a problem. In many countries, people are beginning to learn English as one of the main business languages. However, as I have experienced, people don’t always speak their language like you might learn it in a book. There are colloquialisms, terms, and abbreviations of words you can’t learn in a classroom—you need to experience how people speak in their native countries.
You also need to be open-minded and look at situations from the perspective of your colleagues’ cultures, just as you hope they will be open-minded about yours. This is referred to as cultural intelligence. Whenever I would travel globally to visit my colleagues in other countries, I would see foods, traditions, situations, and behaviors that were very “foreign” to me. Although my first response to experiencing these might be to think, “Wow, that’s strange,” I would try to think about what some of my global colleagues find “foreign” when they come to visit me in the United States. For example, my travel to China would put me in contact with chicken feet, a very popular food in China and one that I dislike immensely. Whenever I was offered chicken feet, I would turn them down in the most polite way possible and take another food offered instead. I started to wonder about what my Chinese colleagues thought about the food when they’d come to visit me in the United States. Every year, I would host a global HR meeting in the United States, and a bit part of that meeting was the camaraderie and the sharing of various meals together. When I asked my Chinese colleagues what foods they thought were unpleasant, they mentioned cheese and meat. I was surprised about the meat, and when I asked, they said that it wasn’t the meat itself necessarily, but it was the giant portions of meat that Americans will eat that, to them, is pretty unappetizing. Again, it is so important to check yourself and your own culture every so often and to think about those elements that we take for granted (e.g., gigantic meat portions) and try to look at them from the eyes of another culture. It really makes us smarter and better partners to our global colleagues around the world.
In the HBR article “Getting Cross-Cultural Teamwork Right,” the author states that three key factors—mutual learning, mutual understanding, and mutual teaching—build trust with cross-cultural colleagues as you try to bridge cultural gaps. With mutual learning, global colleagues learn from each other and absorb the new culture and behaviors through listening and observation. In mutual understanding, you try to understand the logic and cultural behaviors of the new culture to understand why people are doing what they do. This, of course, requires suspending judgment and trying to understand and embrace the differences. Finally, mutual teaching involves instructing and facilitating. This means trying to bridge the gap between the two cultures and helping yourself and others see where different cultures are coming from to resolve misunderstandings.
Understanding and finding common ground with your global colleagues isn’t easy; it takes patience and continuous improvement. In the end, however, I think that you will find it one of the most rewarding and enlightening things you can do. The more we work to close the multicultural “gap” and make it a multicultural advantage, the better off we will be as professionals and people.
- What are some multicultural experiences you’ve had in which you feel there was a very wide gap between you and an individual from another culture? How did you handle it?
- Has economic globalization helped people to bridge these cultural gaps? Why or why not?
Once you have a sense of the different cultures and have started to work on developing your cross-cultural skills, another good practice is to “boost your cultural metacognition” and monitor your own behavior in multicultural situations. When you are in a situation in which you are interacting with multicultural individuals, you should test yourself and be aware of how you act and feel. Observe your positive and negative interactions with people and learn from them. Developing “cognitive complexity” is the final best practice for boosting multicultural skills. This is the most advanced, and it requires viewing situations from more than one cultural framework. To see things from another perspective, you need to have a strong sense of emotional intelligence, empathy, and sympathy and be willing to engage in honest communications.
In the Harvard Business Review article “Cultural Intelligence,” the authors describe three sources of cultural intelligence that teams should consider if they are serious about becoming more adept in their cross-cultural skills and understanding. These sources, very simply, are head, body, and heart. One first learns about foreign cultures' beliefs, customs, and taboos via the head. Training programs are based on providing this type of overview information—which is helpful but obviously isn’t experiential. This is the cognitive component of cultural intelligence. The second source, the body, involves more commitment and experimentation with the new culture. This physical component (demeanor, eye contact, posture, accent) shows a deeper understanding of the new culture and its physical manifestations. The final source, the heart, deals with a person’s own confidence in their ability to adapt to and deal well with cultures outside of their own. Heart really speaks to one’s own level of emotional commitment and motivation to understand the new culture.
The authors have created a quick assessment to diagnose cultural intelligence based on these cognitive, physical, and emotional/motivational measures (i.e., head, body, heart).
Please refer to Table 12.1 for a short diagnostic that allows you to assess your cultural intelligence.
| Generally, scoring below 3 in any one of the three measures signals an area requiring improvement. Averaging over 4 displays strength in cultural intelligence. | |
| Adapted from “Cultural Intelligence,” Earley and Mosakowski, Harvard Business Review , October 2004 | |
| Give your responses using a 1 to 5 scale where 1 means strongly disagree and 5 means strongly agree with the statement. | |
| Before I interact with people from a new culture, I wonder to myself what I hope to achieve. | |
| If I encounter something unexpected while working in a new culture, I use that experience to build new ways to approach other cultures in the future. | |
| I plan on how I will relate to people from a different culture before meeting with them. | |
| When I come into a new cultural situation, I can immediately sense whether things are going well or going wrong. | |
| Add your total from the four questions above. | |
| Divide the total by 4. This is your Cognitive Cultural Quotient . | |
| It is easy for me to change my body language (posture or facial expression) to suit people from a different culture. | |
| I can alter my expressions when a cultural encounter requires it. | |
| I can modify my speech style by changing my accent or pitch of voice to suit people from different cultures. | |
| I can easily change the way I act when a cross-cultural encounter seems to require it. | |
| Add your total from the four questions above. | |
| Divide the total by 4. This is your Cognitive Physical Quotient . | |
| I have confidence in my ability to deal well with people from cultures different from mine. | |
| I am certain that I can befriend people of different cultural backgrounds than mine. | |
| I can adapt to the lifestyle of a different culture with relative ease. | |
| I am confident in my ability to deal with an unfamiliar cultural situation or encounter. | |
| Add your total from the four questions above. | |
| Divide the total by 4. This is your Emotional/Motivational Cognitive Quotient . |
Table 12.1
Cultural intelligence is an extension of emotional intelligence. An individual must have a level of awareness and understanding of the new culture to adapt to the style, pace, language, nonverbal communication, etc., and work together successfully with the new culture. A multicultural team can only find success if its members take the time to understand each other and ensure that everyone feels included. Multiculturalism and cultural intelligence are traits that are taking on increasing importance in the business world today (Earley & Mosakowski, 2004). By following best practices and avoiding the challenges and pitfalls that can derail a multicultural team, a team can find great success and personal fulfillment well beyond the boundaries of the project or work engagement.
- What are some of the challenges of a multicultural team?
- Explain the cultural intelligence techniques of head, body, and heart.


9.10: National Representation in Style and Instrumentation
In the previous section, we looked at four individual works that can each be understood to express something about national identity. All four of these works, however, belong generally to the tradition of Western art music. When Liszt and Bartók wrote for solo piano, they contributed to the repertoire for pianists trained in the classical style. When Dvořák and Beach composed symphonies, they adhered to norms developed by two centuries of European composers. In other cases, however, entirely new musical traditions come to represent a nation’s identity. Such traditions have unique characteristics and practices, and sometimes unique instruments as well. They become inextricably linked with national identity on an international level and can serve to further the political interests of a nation. Such is the case with the tradition of steelband music, which developed in and came to represent the nation of Trinidad and Tobago.


2.5E: Tangent Planes, Linear Approximations, and the Total Differential (Exercises)
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In exercises 1 - 2, find a unit normal vector to the surface at the indicated point.
1) \( f(x,y)=x^3,\quad (2,−1,8)\)
- Answer
- \( (\frac{\sqrt{145}}{145})(12\hat{\mathbf i}−\hat{\mathbf k})\)
2) \( \ln\left(\dfrac{x}{y−z}\right)=0\) when \( x=y=1\)
In exercises 3 - 7, find a normal vector and a tangent vector at point \( P\) .
3) \( x^2+xy+y^2=3,\quad P(−1,−1)\)
- Answer
- Normal vector: \( \hat{\mathbf i}+\hat{\mathbf j}\), tangent vector: \( \hat{\mathbf i}−\hat{\mathbf j}\)
4) \( (x^2+y^2)^2=9(x^2−y^2),\quad P(\sqrt{2},1)\)
5) \( xy^2−2x^2+y+5x=6,\quad P(4,2)\)
- Answer
- Normal vector: \( 7\hat{\mathbf i}−17\hat{\mathbf j}\), tangent vector: \( 17\hat{\mathbf i}+7\hat{\mathbf j}\)
6) \( 2x^3−x^2y^2=3x−y−7,\quad P(1,−2)\)
7) \( ze^{x^2−y^2}−3=0, \quad P(2,2,3)\)
- Answer
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Normal vector: \( -12\hat{\mathbf i}+12\hat{\mathbf j}-\hat{\mathbf k}\) or \( 12\hat{\mathbf i}-12\hat{\mathbf j}+\hat{\mathbf k}\),
Tangent vector: \( 0\hat{\mathbf i}+1\hat{\mathbf j}+12\hat{\mathbf k}\) or \( 1\hat{\mathbf i}+0\hat{\mathbf j}-12\hat{\mathbf k}\)
In exercises 8 - 19, find the equation for the tangent plane to the surface at the indicated point. (Hint: If the given function is not already solved for \(z\) , start by solving it for \( z\) in terms of \( x\) and \( y\) .)
8) \( −8x−3y−7z=−19,\quad P(1,−1,2)\)
9) \( z=−9x^2−3y^2,\quad P(2,1,−39)\)
- Answer
- \( −36x−6y−z=−39\)
10) \( x^2+10xyz+y^2+8z^2=0,\quad P(−1,−1,−1)\)
11) \( z=\ln(10x^2+2y^2+1),\quad P(0,0,0)\)
- Answer
- \( z=0\)
12) \( z=e^{7x^2+4y^2}, \quad P(0,0,1)\)
13) \( xy+yz+zx=11,\quad P(1,2,3)\)
- Answer
- \( 5x+4y+3z−22=0\)
14) \( x^2+4y^2=z^2,\quad P(3,2,5)\)
15) \( x^3+y^3=3xyz,\quad P(1,2,\frac{3}{2})\)
- Answer
- \( 4x−5y+4z=0\)
16) \( z=axy,\quad P(1,\frac{1}{a},1)\)
17) \( z=\sin x+\sin y+\sin(x+y),\quad P(0,0,0)\)
- Answer
- \( 2x+2y−z=0\)
18) \( h(x,y)=\ln\sqrt{x^2+y^2},\quad P(3,4)\)
19) \( z=x^2−2xy+y^2,\quad P(1,2,1)\)
- Answer
- \( −2(x−1)+2(y−2)−(z−1)=0\)
In exercises 20 - 25, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \( P_0(x_0,y_0,z_0)\) , and a vector \( \vecs v=⟨a,b,c⟩\) that is parallel to the line. Then the equations of the line are: \(\quad x=x_0+at,\quad y=y_0+bt, \quad z=z_0+ct.)\)
20) \( −3x+9y+4z=−4,\quad P(1,−1,2)\)
21) \( z=5x^2−2y^2,\quad P(2,1,18)\)
- Answer
- \( x=20t+2,y=−4t+1,z=−t+18\)
22) \( x^2−8xyz+y^2+6z^2=0,\quad P(1,1,1)\)
23) \( z=\ln(3x^2+7y^2+1),\quad P(0,0,0)\)
- Answer
- \( x=0,y=0,z=t\)
24) \( z=e^{4x^2+6y^2},\quad P(0,0,1)\)
25) \( z=x^2−2xy+y^2\) at point \( P(1,2,1)\)
- Answer
- \( x−1=2t;y−2=−2t;z−1=t\)
In exercises 26 - 28, use the figure shown here.
26) The length of line segment \( AC\) is equal to what mathematical expression?
27) The length of line segment \( BC\) is equal to what mathematical expression?
- Answer
- The differential of the function \( z(x,y)=dz=f_xdx+f_ydy\)
28) Using the figure, explain what the length of line segment \( AB\) represents.
29) Show that \( f(x,y)=e^{xy}x\) is differentiable at point \( (1,0).\)
- Answer
- Using the definition of differentiability, we have \( e^{xy}x≈x+y\).
30) Show that \( f(x,y)=x^2+3y\) is differentiable at every point. In other words, show that \( Δz=f(x+Δx,y+Δy)−f(x,y)=f_xΔx+f_yΔy+ε_1Δx+ε_2Δy\), where both \( ε_1\) and \( ε_2\) approach zero as \( (Δx,Δy)\) approaches \( (0,0).\)
- Answer
- \( Δz=2xΔx+3Δy+(Δx)^2.(Δx)^2→0\) for small \( Δx\) and \( z\) satisfies the definition of differentiability.
31) Find the total differential of each function:
- \( z=x^3 + y^3 - 5\)
- \( z=e^{xy}\)
- \( z=y\cos x+\sin y\)
- \(P = t^2 + 3t + tu^3\)
- \( w=e^y\cos(x)+z^2\)
- Answers:
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- \( dz = 3x^2\,dx +3y^2\,dy \)
- \( dz = ye^{xy}\,dx +xe^{xy}\,dy \)
- \( dz = -y\sin x\,dx +(\cos x + \cos y)\,dy \)
- \( dP = (2t + 3 + u^3)\, dt + 3t u^2 \,du \)
- \( dw = -e^y\sin(x)\,dx +e^y\cos(x)\,dy +2z\,dz\)
32) a. Find the total differential \(dz\) of the function \( z=\dfrac{xy}{y+x}\) and then
b. State its value where \( x\) changes from \( 10\) to \( 10.5\) and \( y\) changes from \( 15\) to \( 13\).
- Answer
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a. \( dz = \dfrac{y^2}{(x+y)^2}\, dx + \dfrac{x^2}{(x+y)^2} \,dy \)
b. \(dx = 0.5\) and \(dy = -2\) so
\( \begin{align*} dz &= f_x(10, 15) \, dx + f_y(10,15)\, dy \\
&= \frac{15^2}{25^2}\,dx + \frac{10^2}{25^2}\, dy \\
&= \frac{225}{625}\, (0.5) + \frac{100}{625} (-2) \\
&= \frac{9}{25}\left(\frac{1}{2}\right) + \frac{4}{25} (-2) \\
&= \frac{18}{100} - \frac{32}{100} \\
&= .18 - .32 = -0.14 \end{align*}\)
33) Let \( z=f(x,y)=xe^y.\) State its total differential. Then compute \( Δz\) from \( P(1,2)\) to \( Q(1.05,2.1)\) and then find the approximate change in \( z\), \(dz\), from point \( P\) to point \( Q\). Recall \( Δz=f(x+Δx,y+Δy)−f(x,y)\), and \( dz\) and \( Δz\) should be approximately equal, if \(dx\) and \(dy\) are both reasonably small.
- Answer
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Total Differential: \(dz = e^y\,dx + xe^y\, dy \)
\( Δz≈1.185422\) and \( dz≈1.108.\) Note that they are relatively close.
34) The volume of a right circular cylinder is given by \( V(r,h)=πr^2h.\) Find the differential \( dV\). Interpret the formula geometrically.
- Answer
- \( dV = 2 \pi r h\, dr + \pi r^2 \,dh \)
35) See the preceding problem. Use differentials to estimate the amount of aluminum in an enclosed aluminum can with diameter \( 8.0cm\) and height \( 12cm\) if the aluminum is \( 0.04\) cm thick.
- Answer
- \( 16\,\text{cm}^3\)
36) Use the differential \( dz\) to approximate the change in \( z=\sqrt{4−x^2−y^2}\) as \( (x,y)\) moves from point \( (1,1)\) to point \( (1.01,0.97).\) Compare this approximation with the actual change in the function.
37) Let \( z=f(x,y)=x^2+3xy−y^2.\) Find the exact change in the function and the approximate change in the function as \( x\) changes from \( 2.00\) to \( 2.05\) and \( y\) changes from \( 3.00\) to \( 2.96\).
- Answer
- \( Δz=\) exact change \( =0.6449\), approximate change is \( dz=0.65\). The two values are close.
38) The centripetal acceleration of a particle moving in a circle is given by \( a(r,v)=\frac{v^2}{r},\) where \( v\) is the velocity and \( r\) is the radius of the circle. Approximate the maximum percent error in measuring the acceleration resulting from errors of \( 3\%\) in \( v\) and \( 2\%\) in \( r\). (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in a is given by \( \frac{da}{a}\).)
39) The radius \( r\) and height \( h\) of a right circular cylinder are measured with possible errors of \( 4\%\) and \( 5\%\), respectively. Approximate the maximum possible percentage error in measuring the volume (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in \( V\) is given by \( \frac{dV}{V}\).)
- Answer
- \( 13\%\) or \( 0.13\)
40) The base radius and height of a right circular cone are measured as \( 10\) in. and \( 25\) in., respectively, with a possible error in measurement of as much as \( 0.1\) in. each. Use differentials to estimate the maximum error in the calculated volume of the cone.
41) The electrical resistance \( R\) produced by wiring resistors \( R_1\) and \( R_2\) in parallel can be calculated from the formula \( \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}\). If \( R_1\) and \( R_2\) are measured to be \( 7Ω\) and \( 6Ω\), respectively, and if these measurements are accurate to within \( 0.05Ω\), estimate the maximum possible error in computing \( R\). (The symbol \( Ω\) represents an ohm, the unit of electrical resistance.)
- Answer
- \( 0.025\)
42) The area of an ellipse with axes of length \( 2a\) and \( 2b\) is given by the formula \( A=πab\). Approximate the percent change in the area when \( a\) increases by \( 2\%\) and \( b\) increases by \( 1.5\%.\)
43) The period \( T\) of a simple pendulum with small oscillations is calculated from the formula \( T=2π\sqrt{\frac{L}{g}}\), where \( L\) is the length of the pendulum and \( g\) is the acceleration resulting from gravity. Suppose that \( L\) and \( g\) have errors of, at most, \( 0.5\%\) and \( 0.1\%\), respectively. Use differentials to approximate the maximum percentage error in the calculated value of \( T\).
- Answer
- \( 0.3\%\)
44) Electrical power \( P\) is given by \( P=\frac{V^2}{R}\), where \( V\) is the voltage and \( R\) is the resistance. Approximate the maximum percentage error in calculating power if \( 120 V\) is applied to a \( 2000−Ω\) resistor and the possible percent errors in measuring \( V\) and \( R\) are \( 3\%\) and \( 4\%\), respectively.
For exercises 45 - 49, find the linear approximation of each function at the indicated point.
45) \( f(x,y)=x\sqrt{y},\quad P(1,4)\)
- Answer
- \( L(x,y) = 2x+\frac{1}{4}y−1\)
46) \( f(x,y)=e^x\cos y;\quad P(0,0)\)
47) \( f(x,y)=\arctan(x+2y),\quad P(1,0)\)
- Answer
- \( L(x,y) = \frac{1}{2}x+y+\frac{1}{4}π−\frac{1}{2}\)
48) \( f(x,y)=\sqrt{20−x^2−7y^2},\quad P(2,1)\)
49) \( f(x,y,z)=\sqrt{x^2+y^2+z^2},\quad P(3,2,6)\)
- Answer
- \( L(x,y,z) = \frac{3}{7}x+\frac{2}{7}y+\frac{6}{7}z\)
50) [T] Find the equation of the tangent plane to the surface \( f(x,y)=x^2+y^2\) at point \( (1,2,5),\) and graph the surface and the tangent plane at the point.
51) [T] Find the equation for the tangent plane to the surface at the indicated point, and graph the surface and the tangent plane: \( z=\ln(10x^2+2y^2+1),\quad P(0,0,0).\)
- Answer
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\( z=0\)
52) [T] Find the equation of the tangent plane to the surface \( z=f(x,y)=\sin(x+y^2)\) at point \( \left(\frac{π}{4},0,\frac{\sqrt{2}}{2}\right)\), and graph the surface and the tangent plane.
Contributors
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Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .
- Paul Seeburger (Monroe Community College) created all but part e of exercise 31.


9.5: Ending Inventory Budget
Learning Outcomes
- Create an ending inventory budget
As the senior accountant at Hupana, it is your job to oversee the preparation of the master budget. You have been waiting on the manufacturing overhead budget, so you can complete the finished goods inventory budget. Having the balance for finished goods inventory is needed before you can move on to the budgeted financial statements that are part of your job! Luckily, you now have all of the information needed to move forward. Your managers have provided you with the details to fill in all of the blanks in this finished goods inventory budget!
Ok, so we have completed the first five budgets needed to help us figure out the value of our ending inventory among other things!
Calculating the per unit cost of our finished goods inventory is an important figure. We will need it to calculate the cost of goods sold. How much was spent to manufacture the remaining items in inventory will also give us the value of what we still have in our stock. This is an asset on our balance sheet, so getting the calculation right is important.
| Hupana Running Company Ending Finished Goods Inventory Budget | |||||
|---|---|---|---|---|---|
| Item | Quantity | Cost | Total | ||
| Production cost per pair | 5 | unit | $2 | unit | $10 |
| Direct materials (from direct materials budget) | 0.5 | hour | $20 | hour | $10 |
| Direct labor (from direct labor budget) | 0.5 | hour | $11 | hour | $6 |
| Manufacturing overheard (from manufacturing overhead budget) | $26 | ||||
| Unit production | |||||
| Budgeted finished goods inventory | |||||
| Ending finished goods inventory (in pairs)-from production budget | 150 | ||||
| Unit product cost (from above) | $26 | ||||
| Ending finished goods inventory in dollars | $3,825 |
The finished goods inventory budget, references several other budgets we have prepared. Without those budgets, we would not have had a good number to use for our balance sheet that included all of the costs involved in the manufacture of the shoes left in our ending inventory.
So we now have a ton of information about our production schedule, our costs of manufacturing and a really good handle on our inventory. This is a great start, but there is a lot more to the overall business than what happens in manufacturing. There are costs relating to administration and selling expenses that are necessary to get the orders to make the products that we sell.
Let’s get started!
Practice Questions


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Review of Key Points
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Adler helped Freud gain recognition for Freud’s personality theory. This lends credence to Adler’s claim that he was never a student or follower of Freud, but rather a colleague interested in similar psychiatric/psychological questions.
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Striving for superiority takes the form of compensating for our weaknesses, which, unfortunately, can sometimes lead to overcompensation.
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Adler believed that all thought and behavior was tied to some goal. Our overriding goal is the life plan, and we pursue it by living a characteristic style of life.
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According to Adler, social interest is the best way to achieve superiority, and it can most easily be seen in cooperation.
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Adler described three life tasks: work, communal life, and love.
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Each person exists within their own perception of the world, known as the scheme of apperception. This scheme guides all experience to fit into our style of life and our goals.
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Adler was very interested in the effects of birth order and the family constellation. Being an only child is likely to result in pampering, whereas an oldest child must deal with being dethroned.
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The key to Adlerian psychotherapy is understanding the patient’s style of life.
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Once the style of life is understood, Adlerian psychotherapists work to strengthen social interest and reorient the style of life.
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Sullivan began his formal psychiatric career at two prestigious hospitals: St. Elizabeth’s and Sheppard Pratt.
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Sullivan referred to the energy transformation that underlies our personal interactions as dynamisms. Perhaps the most important dynamism is the self-system.
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Before we can truly understand relationships, we develop images in our mind known as personifications. The personified self includes everything we can consciously describe about ourselves.
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Sullivan described seven developmental epochs, which provide a framework for our unfolding abilities to engage in healthy interpersonal relationships.


2.3: Carbometallation and Carbocyclization Reactions
Organometallic compounds add to carbon-carbon multiple bonds to give a new organometallic species, which could be further modified to yield new carbon-carbon bonds. These processes are called as “carbometallation reactions”. It primarily refers to the relationship between the reactants and products (Scheme \(\PageIndex{1}\)). This section covers some examples of the asymmetric carbometallation reactions using Rh, Cu and Pd-based systems.
Rhodium-Catalyzed Reactions
Hydrogen-mediated carbon-carbon bond formation has emerged as powerful industrial process in chemical industries. For example, the hydroformylation and Fischer-Tropsch reactions are well known for the hydrogen-mediated carbon-carbon bond formation reactions. These processes require heterolytic activation of molecular hydrogen to give monohydride species, where the C-H reductive elimination pathway is disabled. Addition of a metal hydride to carbon-carbon multiples bonds (i.e., alkene and alkyne) give organometallic species that could be rapidly captured by an electrophile (i.e., aldehydes and imine) prior to its reaction with molecular hydrogen via oxidative addition or σ-bond metathesis of carbon-metal bond. Scheme \(\PageIndex{2}\) illustrates metal-dihydride route (leading to hydrogenation) and metal-monohydride route (leading to C-C bond formation) with an alkyne.
Formation of the monohydride organometallic species depends on the choice of the catalytic system. For example, the heterolytic activation of molecular hydrogen is observed with cationic rhodium complexes in the presence of base. The reaction takes place via the oxidation addition of the molecular hydrogen with metal species followed by a base induced reductive elimination of HX (Scheme \(\PageIndex{3}\)).
For example, Scheme \(\PageIndex{4}\) presents enantioselective reductive cyclization of 1,6-enynes using Rh(COD) 2 OTf and ( R )-BINAP in the presence of molecular hydrogen. This carbocyclization reaction is compatible with various functional groups, however, the yield and enantioselectivity of the product depends on the structure of 1,6-enynes and the ligands.
A possible mechanism has been proposed for this reaction based on deuterium labeling control experiments (Scheme \(\PageIndex{5}\)). The catalytic cycle starts with cycloaddition of RhL n and 1,6-enyne forming rhodacyclopentene. Homolytic hydrogen activation via oxidative addition of molecular hydrogen or σ-bond metathesis may lead to the formation of vinyl-rhodium vinyl species that could afford cyclization product by reductive elimination to complete the catalytic cycle.
1,4-Conjugate addition of organometallic reagents to α,β -unsaturated carbonyl compounds afford effective method for carbon-carbon bond formation. Much effort has been on the development of asymmetric version of the reaction using a series of catalytic systems. The first reductive aldol cyclization of keto-enone with phenylboronic acid has been shown utilizing Rh[(COD)Cl] 2 and ( R )-BINAP with yield and enantioselectivity (Scheme \(\PageIndex{6}\)).
The mechanism of the reaction is presented in Scheme 7. The observed stereochemistry has been rationalized by assuming Z -enolate formation.
Copper-Catalyzed Reactions
CuH is found to be highly efficient catalyst for the asymmetric reductive aldol cyclization of keto-enones to give the target product as a singly diastereoisomer with high enantiopuritiy (Scheme \(\PageIndex{8}\)). These reactions use ferrocenylphosphines, ( S , R )-PPE-P( t -Bu) 2 , as effective chiral ligands in the presence of silane as a hydride source (Scheme \(\PageIndex{8}\)). These reactions can also be carried out under heterogeneous as well as aqueous conditions with surfactant.
Palladium-Catalyzed Reactions
The palladium-catalyzed cross-coupling reactions of aryl or alkenyl halides with alkenes in the presence of base are among the powerful reactions in organic synthesis to construct carbon-carbon bonds. The asymmetric version of the reaction is also well explored. Scheme \(\PageIndex{9}\)-\(\PageIndex{12}\) illustrates some examples for the intramolecular and intermolecular Heck reactions. In 1970, the Heck reaction was discovered and, in 1989, the first example of asymmetric intramolecular Heck reactions appeared using Pd(OAc) 2 with ( R )-BINAP with moderate enantioselectivity (Scheme \(\PageIndex{9}\)).
The intramolecular Heck reaction finds wide applications in organic synthesis. Among those applications the synthesis of optically active oxindoles having a quaternary asymmetric center has been considerably explored. Because the oxindole moiety serves as useful synthetic intermediate in the synthesis of numerous natural products. For example, ( E )- α,β -unsaturated-2-iodoanilide undergoes cyclization in the presence of Pd 2 (dba) 3 -CHCl 3 and ( R )-BINAP to give oxindoles with ( S ) or ( R ) configuration under cationic and neutral conditions, respectively. It is noteworthy that a dramatic switching in the direction of asymmetric induction has been observed between the two conditions even though the same chiral ligand ( R )-BIANP is employed. In these reactions, Ag 3 PO 4 and PMP act as HI scavenger.
The use of TADDOL-based monophosphoramide has been demonstrated instead of BINAP in the reaction of intramolecular cyclization of cyclohexadienone derivatives (Scheme \(\PageIndex{11}\)). This reaction can be performed in the absence of silver salt.
Intermolecular Heck reaction is also well studied. For example, dihydrofuran reacts with phenyl triflate to give 2-phenyl-2,3-dihydrofuran along with small amount 2-phenyl-2,5-dihydrofuran in the presence of Pd-BINAP with excellent enantioselectivity (Scheme \(\PageIndex{12}\)). A mechanism has been proposed to explain the high enantioselectivity of the major product and inversion configuration of the minor product (Scheme \(\PageIndex{13}\)). It involves a kinetic resolution process that enhances the enantioselectivity of the major product.
Scheme \(\PageIndex{14}\) exemplifies the reactions of 2,3-dihydrofuran and 2,2-dimethyl-2,3-dihydrofuran with phenyl triflate, 2-carbethoxy cyclohexenyl triflate and cyclohexenyl trilfate using palladium complexes with oxazoline based aminophosphine and (D-glucosamine)phosphiteoxazoline as the ligands. The reactions are effective affording the products with excellent enantioselectivity.


1.1.4: Activity of Aqueous Solutions
A given aqueous solution is prepared using 1 kg of water(\(\lambda\), molar mass \(\mathrm{M}_{1}\), and \(m_{j}\) moles of solute at temperature \(\mathrm{T}\) and pressure \(p\) (which is close to the standard pressure \(p_{0}\)) At fixed \(\mathrm{T}\) and \(p\), the activity of water \(a_{1}(\mathrm{aq})\) is related to the chemical potential of water in the aqueous solution using equation (a) where \(\mu_{1}^{*}(\lambda)\) is the chemical potential of water(\(\lambda\)) at the same \(\mathrm{T}\) and \(p\).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right) \nonumber \]
Further equation (b) relates \(\mu_{1}(\mathrm{aq})\) to the molality of a simple solute, \(\mathrm{m}_{j}\) (e.g. urea) where \(\mathrm{R}\) is the gas constant (\(=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\)).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Here \(\mathrm{m}_{j}\) is the molality of solute \(j\) and \(\phi\) is the practical osmotic coefficient. If the thermodynamic properties of the aqueous solution are ideal (i.e. no solute-solute interactions) the practical osmotic coefficient is unity.
\[\text { At fixed } \mathrm{T} \text { and } \mathrm{p}, \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \nonumber \]
\[\text { Hence, } \mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Therefore in the case of an ideal solution, addition of a solute, molality \(\mathrm{m}_{j}\), stabilises the solvent; i.e. lowers the chemical potential of the solvent. In the event that solute \(j\) is a salt which forms with complete dissociation \(ν\) ions for each mole of salt in solution, \(\mu_{1}(\mathrm{aq})\) is given by equation (e).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
We consider an aqueous solution containing a simple neutral solute \(j\). In order to understand the properties of this solution, we need to consider water-solute interactions and solute \(j\) – solute \(j\) interactions. Solute-solute interactions determine the extent to which the properties of a given solution differ from those of the corresponding solution having thermodynamic properties which are ideal [1].
The extent to which the thermodynamic properties of solutions are not ideal also reflects in part the role of water-solute interactions. For example the extent to which urea-urea interactions differ from ethanol-ethanol interactions in aqueous solutions reflects the different hydration characteristics of urea and ethanol.
Comparison of equations (a) and (b) yields the following important equation relating activity of solvent, water, and the molality of simple neutral solute \(j\).
\[\text { Thus, } \quad \ln \left(\mathrm{a}_{1}\right)=-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
The minus signs in equations (b) and (f) are extremely significant. If the thermodynamic properties of the solutions are ideal, \(\phi\) is unity.
\[\text { Then, } \ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=-\mathrm{M}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}=-\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \nonumber \]
Here \(n_{1}{}^{0}\) is the amount of solvent, water, molar mass \(\mathrm{M}_{1}\); \(n_{j}\) is the amount of solute \(j\). Therefore a plot of \(\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}\) against molality \(\mathrm{m}_{j}\) is linear with slope ‘\(-\mathrm{M}){1}\)’. Furthermore the plots for a range of neutral solutes will be super-imposable. In other words\(\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}\) is related to the ratio of amounts of solute to solvent. By adding a solute to a fixed amount of (solvent) water(\(\lambda\)) we lower the activity of water(\(\lambda\)) , (i.e. the chemical potential of water, µ(aq) in an aqueous solution) and stabilise the solvent.
The chemical potential of solute \(j\) in an aqueous solution \(\mu_{\mathrm{j}}(\mathrm{aq})\) is related to the molality of solute mj using equation (h) where \(\mu_{j}^{0}(\mathrm{aq})\) is the chemical potential of solute \(j\) in an aqueous solution, molality \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) and \(\gamma_{j}=1\) at all \(\mathrm{T}\) and \(p\), (taken as close to the standard pressure \(p^{0}\) ).
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
\[\text { By definition, at all T and } p \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}} \rightarrow 1 \nonumber \]
The role of water activity in determining enzyme activity is an important consideration [2]. .
Footnotes
[1] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd edn. Revised, 1965.
[2] G. Bell, A. E. M. Janssen and P. J. Halling, Enzyme and Microbial Technology, 1997,20,471.


11.2: What Copyright Covers
Several types of works can be protected including:
- Literary works
- Musical works, including any accompanying words
- Dramatic works, including any accompanying music
- Pantomimes and choreography
- Pictorial, graphic, and sculptural works
- Motion pictures and other audiovisual works
- Sound recordings
- Architectural works
What Isn’t Covered by Copyright?
Not all works are covered by copyright. Those not covered include:
Works already in the public domain
- Moby Dick
- Shakespeare’s plays
- Beethoven’s works
Works not fixed in a tangible medium
- A song in your head, but not recorded or written down
Ideas
- Boy meets girl, they fall in love and live happily ever after
- Hero protagonist saves the world with the help of wacky sidekick
Facts
- 1+1=2
- King George IV died in 1830
- Copenhagen is the capital of Denmark
Works of the U.S. government produced by government employees
- Federal government reports
- Acts/Bills of Congress
- The White House website
Copyright in Cases of a Work Made for Hire
If you create something as part of your job duties, it is likely a work made for hire. In these cases, your employer is considered the author and rights holder of a work made for hire rather than the employee. Read the United States Copyright Office’s Works Made for Hire circular for a more nuanced discussion.
Exercises: Copyrightable?
An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://minnstate.pressbooks.pub/ctar/?p=211#h5p-31


3.5: Discussion- Mission Accomplished?
Post 1: Initial Post
Instructions
This posting should be a minimum of one short paragraph and a maximum of two paragraphs. Word totals for this post should be in the 100–200-word range. Whether you agree or disagree, explain why with supporting evidence and concepts from the readings or a related experience. Include a reference, link, or citation when appropriate.
For Discussion
Choose ONE of the following questions to respond to in your initial post.
- Think of a company whose product or service you feel strongly about (positive or negative). Research that company’s mission statement. Do you think the company is achieving its mission? Be sure to include the company, the mission statement, and reasons why you think the company is or isn’t achieving its mission.
- In your reading on forecasting, you read examples of botched forecasts by business leaders, including one from Microsoft CEO Steve Ballmer predicting in 2012 that the iPhone wouldn’t get any significant market share. Why do you think it’s difficult for businesses to accurately forecast the future when planning?
-
In your reading, we identified six steps of the planning process:
- Define objectives
- Develop premises
- Evaluate alternatives
- Identify resources
- Plan and implement tasks
- Determine tracking and evaluation methods
Which step do you think businesses most often overlook or fail to adequately address when making plans? Why?
Posts 2 and 3: Respond to Classmates’ Posts
Instructions
After you have created your initial post, look over the discussion posts of your classmates and give at least two thoughtful responses to two different classmates (one per classmate).
A response posting should be a minimum of one short paragraph. Word totals for these posts should be in the 75–100-word range. Whether you agree or disagree, explain why with supporting evidence and concepts from the readings or a related experience. Include a reference, link, or citation when appropriate. The goal of your response posts is to extend discussions already taking place or pose new possibilities or opinions not previously voiced. Your goal should be to motivate the group discussion and present a creative approach to the topic.
Contributors and Attributions
- Discussion: Mission Accomplished?. Authored by : Lumen Learning. License : CC BY: Attribution
- General Discussion Instructions. Authored by : Linda Williams and Lumen Learning. Located at : https://courses.lumenlearning.com/wmopen-introbusiness/chapter/discussion-external-factors-internal-pressure/ . License : CC BY: Attribution


8.5: Mole-Mole Relationships in Chemical Reactions
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Learning Objectives
- To use a balanced chemical reaction to determine molar relationships between the substances.
Previously, you learned to balance chemical equations by comparing the numbers of each type of atom in the reactants and products. The coefficients in front of the chemical formulas represent the numbers of molecules or formula units (depending on the type of substance). Here, we will extend the meaning of the coefficients in a chemical equation.
Consider the simple chemical equation
\[\ce{2H_2 + O_2 → 2H_2O}\nonumber \]
The convention for writing balanced chemical equations is to use the lowest whole-number ratio for the coefficients. However, the equation is balanced as long as the coefficients are in a 2:1:2 ratio. For example, this equation is also balanced if we write it as
\[\ce{4H_2 + 2O_2 → 4H_2O}\nonumber \]
The ratio of the coefficients is 4:2:4, which reduces to 2:1:2. The equation is also balanced if we were to write it as
\[\ce{22H_2 + 11O_2 → 22H_2O}\nonumber \]
because 22:11:22 also reduces to 2:1:2.
Suppose we want to use larger numbers. Consider the following coefficients:
\[12.044 \times 10^{23}\; H_2 + 6.022 \times 10^{23}\; O_2 → 12.044 \times 10^{23}\; H_2O\nonumber \]
These coefficients also have the ratio 2:1:2 (check it and see), so this equation is balanced. But 6.022 × 10 23 is 1 mol, while 12.044 × 10 23 is 2 mol (and the number is written that way to make this more obvious), so we can simplify this version of the equation by writing it as
\[\ce{2 \;mol\; H_2 + 1\; mol\; O_2 → 2 \;mol\; H_2O}\nonumber \]
We can leave out the word mol and not write the 1 coefficient (as is our habit), so the final form of the equation, still balanced, is
\[\ce{2H_2 + O_2 → 2H_2O}\nonumber \]
Now we interpret the coefficients as referring to molar amounts, not individual molecules. The lesson? Balanced chemical equations are balanced not only at the molecular level but also in terms of molar amounts of reactants and products. Thus, we can read this reaction as “two moles of hydrogen react with one mole of oxygen to produce two moles of water.”
2 molecules H 2 1 molecule O 2 2 molecules H 2 O
2 moles H 2 1 mole O 2 2 moles H 2 O
2 x 2.02 g=4.04 g H 2 32.0 g O 2 2 x 18.02 g=36.04 g H 2 O
Figure \(\PageIndex{1}\): This representation of the production of water from oxygen and hydrogen show several ways to interpret the quantitative information of a chemical reaction.
By the same token, the ratios we constructed to describe molecules reaction can also be constructed in terms of moles rather than molecules. For the reaction in which hydrogen and oxygen combine to make water, for example, we can construct the following ratios:
\[\mathrm{\dfrac{2\: mol\: H_2}{1\: mol\: O_2}\: or\: \dfrac{1\: mol\: O_2}{2\: mol\: H_2}}\nonumber \]
\[\mathrm{\dfrac{2\: mol\: H_2O}{1\: mol\: O_2}\: or\: \dfrac{1\: mol\: O_2}{2\: mol\: H_2O}}\nonumber \]
\[\mathrm{\dfrac{2\: mol\: H_2}{2\: mol\: H_2O}\: or\: \dfrac{2\: mol\: H_2O}{2\: mol\: H_2}}\nonumber \]
We can use these ratios to determine what amount of a substance, in moles, will react with or produce a given number of moles of a different substance. The study of the numerical relationships between the reactants and the products in balanced chemical reactions is called stoichiometry . The ratio of coefficients in a balanced chemical equation, used in computations relating amounts of reactants and products is called the stoichiometric factor.
Example \(\PageIndex{1}\)
How many moles of oxygen react with hydrogen to produce 27.6 mol of H 2 O? The balanced equation is as follows:
\[\ce{2H2 + O2 -> 2H2O} \nonumber \nonumber \]
Solution
Because we are dealing with quantities of H 2 O and O 2 , we will use the stoichiometric ratio that relates those two substances. Because we are given an amount of H 2 O and want to determine an amount of O 2 , we will use the ratio that has H 2 O in the denominator (so it cancels) and O 2 in the numerator (so it is introduced in the answer). Thus,
\(\mathrm{27.6\: mol\: H_2O\times\dfrac{1\: mol\: O_2}{2\: mol\: H_2O}=13.8\: mol\: O_2}\)
To produce 27.6 mol of H 2 O, 13.8 mol of O 2 react.
Exercise \(\PageIndex{1}\)
Using 2H 2 + O 2 → 2H 2 O, how many moles of hydrogen react with 3.07 mol of oxygen to produce H 2 O?
- Answer
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\(\mathrm{3.07\: mol\: O_2\times\dfrac{2\: mol\: H_2}{1\: mol\: O_2}=6.14\: mol\: H_2}\)
Key Takeaway
- The balanced chemical reaction can be used to determine molar relationships between substances.


31.5: A New Way to Interpret a over b
Lesson
Let's investigate what a fraction means when the numerator and denominator are not whole numbers.
Exercise \(\PageIndex{1}\): Recalling Ways of Solving
Solve each equation. Be prepared to explain your reasoning.
\(0.07=10m\qquad 10.1=t+7.2\)
Exercise \(\PageIndex{2}\): Interpreting \(\frac{a}{b}\)
Solve each equation.
- \(35=7x\)
- \(35=11x\)
- \(7x=7.7\)
- \(0.3x=2.1\)
- \(\frac{2}{5}=\frac{1}{2}x\)
Are you ready for more?
Solve the equation. Try to find some shortcuts.
\(\frac{1}{6}\cdot\frac{3}{20}\cdot\frac{5}{42}\cdot\frac{7}{72}\cdot x=\frac{1}{384}\)
Exercise \(\PageIndex{3}\): Storytime Again
Take turns with your partner telling a story that might be represented by each equation. Then, for each equation, choose one story, state what quantity \(x\) describes, and solve the equation. If you get stuck, consider drawing a diagram.
\(0.7+x=12\qquad \frac{1}{4}x=\frac{3}{2}\)
Summary
In the past, you learned that a fraction such as \(\frac{4}{5}\) can be thought of in a few ways.
- \(\frac{4}{5}\) is a number you can locate on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0.
- \(\frac{4}{5}\) is the share that each person would have if 4 wholes were shared equally among 5 people. This means that \(\frac{4}{5}\) is the result of dividing 4 by 5.
We can extend this meaning of a fraction as a quotient to fractions whose numerators and denominators are not whole numbers. For example, we can represent 4.5 pounds of rice divided into portions that each weigh 1.5 pounds as: \(\frac{4.5}{1.5}=4.5\div 1.5=3\). In other words, \(\frac{4.5}{1.5}=3\) because the quotient of 4.5 and 1.5 is 3.
Fractions that involve non-whole numbers can also be used when we solve equations.
Suppose a road under construction is \(\frac{3}{8}\) finished and the length of the completed part is \(\frac{4}{3}\) miles. How long will the road be when completed?
We can write the equation \(\frac{3}{8}x=\frac{4}{3}\) to represent the situation and solve the equation.
The completed road will be \(3\frac{5}{9}\) or about 3.6 miles long.
\(\begin{aligned} \frac{3}{8}x&=\frac{4}{3} \\ x&=\frac{\frac{4}{3}}{\frac{3}{8}} \\ x&=\frac{4}{3}\cdot\frac{8}{3} \\ x&=\frac{32}{9}=3\frac{5}{9}\end{aligned}\)
Glossary Entries
Definition: Coefficient
A coefficient is a number that is multiplied by a variable.
For example, in the expression \(3x+5\), the coefficient of \(x\) is \(3\). In the expression \(y+5\), the coefficient of \(y\) is \(1\), because \(y=1\cdot y\).
Definition: Solution to an Equation
A solution to an equation is a number that can be used in place of the variable to make the equation true.
For example, 7 is the solution to the equation \(m+1=8\), because it is true that \(7+1=8\). The solution to \(m+1=8\) is not \(9\), because \(9+1\neq 8\).
Definition: Variable
A variable is a letter that represents a number. You can choose different numbers for the value of the variable.
For example, in the expression \(10-x\), the variable is \(x\). If the value of \(x\) is 3, then \(10-x=7\), because \(10-3=7\). If the value of \(x\) is \(6\), then \(10-x=4\), because \(10-6=4\).
Practice
Exercise \(\PageIndex{4}\)
Select all the expressions that equal \(\frac{3.15}{0.45}\).
- \((3.15)\cdot (0.45)\)
- \((3.15)\div (0.45)\)
- \((3.15)\cdot\frac{1}{0.45}\)
- \((3.15)\div\frac{45}{100}\)
- \((3.15)\cdot\frac{100}{45}\)
- \((\frac{0.45}{3.15}\)
Exercise \(\PageIndex{5}\)
Which expressions are solutions to the equation \(\frac{3}{4}x=15\)? Select all that apply.
- \(\frac{15}{\frac{3}{4}}\)
- \(\frac{15}{\frac{4}{3}}\)
- \(\frac{4}{3}\cdot 15\)
- \(\frac{3}{4}\cdot 15\)
- \(15\div\frac{3}{4}\)
Exercise \(\PageIndex{6}\)
Solve each equation.
\(4a=32\qquad 4=32b\qquad 10c=26\qquad 26=100d\)
Exercise \(\PageIndex{7}\)
For each equation, write a story problem represented by the equation. For each equation, state what quantity \(x\) represents. If you get stuck, consider drawing a diagram.
- \(\frac{3}{4}+x=2\)
- \(1.5x=6\)
Exercise \(\PageIndex{8}\)
Write as many mathematical expressions or equations as you can about the image. Include a fraction, a decimal number, or a percentage in each.
(From Unit 3.4.4)
Exercise \(\PageIndex{9}\)
In a lilac paint mixture, 40% of the mixture is white paint, 20% is blue, and the rest is red. There are 4 cups of blue paint used in a batch of lilac paint.
- How many cups of white paint are used?
- How many cups of red paint are used?
- How many cups of lilac paint will this batch yield?
If you get stuck, consider using a tape diagram.
(From Unit 3.4.3)
Exercise \(\PageIndex{10}\)
Triangle P has a base of 12 inches and a corresponding height of 8 inches. Triangle Q has a base of 15 inches and a corresponding height of 6.5 inches. Which triangle has a greater area? Show your reasoning.
(From Unit 1.3.3)


2.4: Generating Sets
In this section, we explore the concept of a generating set for a group.
Let \(G\) be a group and let \(S\) be a subset of \(G\) . A finite product (under the operation of \(G\) ) consisting of elements from \(S\) or their inverses is called a word in \(S\) . That is, a word in \(S\) is of the form \[s_{x_1}s_{x_2}\cdots s_{x_n},\] where each \(s_{x_i}\) is either an element from \(S\) or the inverse of an element from \(S\) . Each \(s_{x_i}\) is called a letter and the set \(S\) is called the alphabet . By convention, the identity of \(G\) can be represented by the empty word , which is the word having no letters. The set of elements of \(G\) that can be written as words in \(S\) is denoted by \(\langle S\rangle\) and is called the group generated by \(S\) .
It is important to pay close attention to our notation. While \(S\) and \(\langle S\rangle\) are both sets, the latter set is the set of elements we can build using letters and their inverses from \(S\) . It turns out that if \(S\) is itself a group, then \(S=\langle S\rangle\) . Otherwise, \(S\) is a proper subset of \(\langle S\rangle\) .
Suppose \(G\) is a group such that \(a,b,c\in G\) and let \(S=\{a,b,c\}\) . Then \(ab\) , \(c^{-1}acc\) , and \(ab^{-1}caa^{-1}bc^{-1}\) are words in \(\langle S\rangle\) . If any one of these words is not equal to \(a\) , \(b\) , or \(c\) , then \(\langle S\rangle\) is strictly larger than \(S\) .
It is worth mentioning that we are slightly abusing notation here. For nonempty \(S\subseteq G\) , we can form infinitely many words in \(\langle S\rangle\) , but often there are many words that represent the same group element. We can partition the collection of words in the alphabet \(S\) into equivalence classes based on which group element a word represents. Strictly speaking, each group element is an equivalence class of words. When we say two words are equal in the group, what we really mean is that both words are in the same equivalence class.
If \(G\) is a group under \(*\) and \(S\) is a subset of \(G\) , then \(\langle S\rangle\) is also a group under \(*\) .
If \(G\) is a group and \(S\) is a subset of \(G\) such that \(G=\langle S\rangle\) , then \(S\) is called a generating set of \(G\) . In other words, \(S\) is a generating set of \(G\) if every element of \(G\) can be expressed as a word in \(S\) . In this case, we say \(S\) generates \(G\) . A generating set \(S\) for \(G\) is a minimal generating set if \(S\setminus\{x\}\) is no longer a generating set for \(G\) for all \(x\in S\) .
A generating set for a group is analogous to a spanning set for a vector space and a minimal generating set for a group is analogous to a basis for a vector space.
If we know what the elements of \(S\) actually are, then we will list them inside the angle brackets without the set braces. For example, if \(S=\{a,b,c\}\) , then we will write \(\langle a, b, c\rangle\) instead of \(\langle \{a,b,c\}\rangle\) . In the special case when the generating set \(S\) consists of a single element, say \(g\) , we have \[G=\langle g\rangle =\{g^k\mid k\in\mathbb{Z}\}\] and say that \(G\) is a cyclic group . As we shall see, \(\langle g\rangle\) may be finite or infinite.
In Section 2.1, we discovered that the set of spins is a non-minimal generating set for \(\text{Spin}_{3\times 3}\) while the set \(T=\{s_{11}, s_{12}, s_{23}, s_{36}, s_{56}, s_{45}, s_{47}, s_{78}, s_{89}\}\) is a minimal generating set.
Consider the rotation group \(R_4\) that we introduced in Problem 2.2.4 . Let \(r\) be the element of \(R_4\) that rotates the square by \(90^\circ\) clockwise.
- Describe the action of \(r^{-1}\) on the square and express \(r^{-1}\) as a word using \(r\) only.
- Prove that \(R_4=\langle r\rangle\) by writing every element of \(R_4\) as a word using \(r\) only.
- Is \(\{r\}\) a minimal generating set for \(R_4\) ?
- Is \(R_4\) a cyclic group?
Consider the dihedral group \(D_3\) introduced in Problem 2.2.5 . To give us a common starting point, let’s assume the triangle and hole are positioned so that one of the tips of the triangle is pointed up. Let \(r\) be rotation by \(120^\circ\) in the clockwise direction and let \(s\) be the reflection in \(D_3\) that fixes the top of the triangle.
- Describe the action of \(r^{-1}\) on the triangle and express \(r^{-1}\) as a word using \(r\) only.
- Describe the action of \(s^{-1}\) on the triangle and express \(s^{-1}\) as a word using \(s\) only.
- Prove that \(D_3=\langle r,s\rangle\) by writing every element of \(D_3\) as a word in \(r\) or \(s\) .
- Is \(\{r,s\}\) a minimal generating set for \(D_3\) ?
- Explain why there is no single generating set for \(D_3\) consisting of a single element. This proves that \(D_3\) is not cyclic.
It is important to point out that the fact that \(\{r,s\}\) is a minimal generating set for \(D_3\) does not imply that \(D_3\) is not a cyclic group. There are examples of cyclic groups that have minimal generating sets consisting of more than one element (see Problem 2.6.4 ).
Let’s consider the group \(D_3\) again. Let \(s\) be the same reflection as in Problem \(\PageIndex{2}\) and let \(s'\) be the reflection in \(D_3\) that fixes the bottom right corner of the triangle.
- Express \(r\) as a word in \(s\) and \(s'\) .
- Use part (a) together with Problem \(\PageIndex{2}\) to prove that \(\langle s,s'\rangle=D_3\) .
Consider the dihedral group \(D_4\) introduced in Problem 2.2.6 . Let \(r\) be clockwise rotation by \(90^\circ\) and let \(s\) be the reflection over the vertical midline of the square.
- Describe the action of \(r^{-1}\) on the square and express \(r^{-1}\) as a word using \(r\) only.
- Describe the action of \(s^{-1}\) on the square and express \(s^{-1}\) as a word using \(s\) only.
- Prove that \(\{r,s\}\) is generating set for \(D_4\) .
- Is \(\{r,s\}\) a minimal generating set for \(D_4\) ?
- Find a different generating set for \(D_4\) .
- Is \(D_4\) a cyclic group?
Consider the symmetric group \(S_3\) that was introduced in Problem 2.2.7 . Let \(s_1\) be the action that swaps the positions of the first and second coins and let \(s_2\) be the action that swaps the positions of the second and third coins. Prove that \(S_3=\langle s_1, s_2\rangle\) .
Find a minimal generating set for \((\mathbb{Z},+)\) . Is \(\mathbb{Z}\) a cyclic group under addition?


1.5: There’s More to Research Methods Than Statistics
So far, most of what I’ve talked about is statistics, and so you’d be forgiven for thinking that statistics is all I care about in life. To be fair, you wouldn’t be far wrong, but research methodology is a broader concept than statistics. So most research methods courses will cover a lot of topics that relate much more to the pragmatics of research design, and in particular the issues that you encounter when trying to do research with humans. However, about 99% of student fears relate to the statistics part of the course, so I’ve focused on the stats in this discussion, and hopefully I’ve convinced you that statistics matters, and more importantly, that it’s not to be feared. That being said, it’s pretty typical for introductory research methods classes to be very stats-heavy. This is not (usually) because the lecturers are evil people. Quite the contrary, in fact. Introductory classes focus a lot on the statistics because you almost always find yourself needing statistics before you need the other research methods training. Why? Because almost all of your assignments in other classes will rely on statistical training, to a much greater extent than they rely on other methodological tools. It’s not common for undergraduate assignments to require you to design your own study from the ground up (in which case you would need to know a lot about research design), but it is common for assignments to ask you to analyse and interpret data that were collected in a study that someone else designed (in which case you need statistics). In that sense, from the perspective of allowing you to do well in all your other classes, the statistics is more urgent.
But note that “urgent” is different from “important” – they both matter. I really do want to stress that research design is just as important as data analysis, and this book does spend a fair amount of time on it. However, while statistics has a kind of universality, and provides a set of core tools that are useful for most types of psychological research, the research methods side isn’t quite so universal. There are some general principles that everyone should think about, but a lot of research design is very idiosyncratic, and is specific to the area of research that you want to engage in. To the extent that it’s the details that matter, those details don’t usually show up in an introductory stats and research methods class.
References
Evans, J. St. B. T., J. L. Barston, and P. Pollard. 1983. “On the Conflict Between Logic and Belief in Syllogistic Reasoning.” Memory and Cognition 11: 295–306.
Bickel, P. J., E. A. Hammel, and J. W. O’Connell. 1975. “Sex Bias in Graduate Admissions: Data from Berkeley.” Science 187: 398–404.
- The quote comes from Auden’s 1946 poem Under Which Lyre: A Reactionary Tract for the Times , delivered as part of a commencement address at Harvard University. The history of the poem is kind of interesting: http://harvardmagazine.com/2007/11/a-poets-warning.html
- Including the suggestion that common sense is in short supply among scientists.
- In my more cynical moments I feel like this fact alone explains 95% of what I read on the internet.
- Earlier versions of these notes incorrectly suggested that they actually were sued – apparently that’s not true. There’s a nice commentary on this here: https://www.refsmmat.com/posts/2016-05-08-simpsons-paradox-berkeley.html . A big thank you to Wilfried Van Hirtum for pointing this out to me!
- Which might explain why physics is just a teensy bit further advanced as a science than we are.