3.4: Pedigree Analysis
Background Information
Recall that autosomal chromosomes are all the chromosomes in a genome that are not the sex chromosomes. For disorders that are caused by a dominant allele there are some trends that you will see in a population of a family tree that carries that allele. Since you only need 1 copy of a dominant allele to express a dominant trait, autosomal dominant disorders often will occur in every generation of a family. If two individuals who do not have the disorder reproduce, it is impossible for any of their children to be affected by the disorder. Huntington’s disease is an example of a disease caused by an autosomal dominant allele.
If the gene (and its resulting disorder) reduces the chance of surviving or reproducing, its frequency should decrease over time. This is because if the disorder results in death before a person reaches reproductive age, they will never get the chance to pass the deleterious allele to the next generation. This is not always the case however, because some disorders do not become evident until after the individual has reached reproductive age, and already unknowingly passed the disorder to their children. This is the case with Huntington’s disease, a debilitating and deadly disease that leads to a serious degeneration of the nervous system, and does not appear until the age of 30 or later. In other instances, such as achondroplasia, this disorder does not affect the person’s life expectancy significantly or impact normal reproductive events, and will therefore persist generation after generation.
Autosomal recessive inheritance means that a person can be a carrier of the deleterious allele, but not be affected with the disorder. Cystic fibrosis, phenylketonuria (PKU), and albinism are all examples of autosomal recessive traits. This inheritance pattern has characteristic trends in a population. Heterozygotes are typically symptom-free carriers and only homozygous recessive individuals are affected. Although there are exceptions to this rule, such as sickle cell anemia, where heterozygotes display some symptoms, but they are so mild that it generally does not affect their overall life expectancy. Sickle cell anemia is an example of a disorder that displays codominance. Because inheritance of an autosomal recessive disorder requires 2 copies of the allele to be expressed, these types of disorders tend to skip generations. Two heterozygous parents have a 50 percent chance of producing heterozygous children and a 25 percent chance of producing a homozygous recessive child. When both parents are homozygous recessive, then all children will be affected.
Sex-linked traits : Disorders that are carried on the X chromosome have very different patterns in a population.  Males cannot inherit the X chromosome from their father, since the paternal X chromosome would result in a female offspring. Clues that a trait is a recessive X-linked:
- More males than females are affected because males only get one shot at an X chromosome. They cannot be carriers for a disorder, they either have it, or they don’t. Females need two copies, therefore their chances of inheriting both deleterious alleles is greatly diminished compared to males.
- An affected father cannot pass trait to sons, and in order for a female to be affected by a sex-linked disorder, her father MUST express the disorder as well.
- X-linked dominant disorders are often fatal, and do not result in viable offspring that survive outside of the womb and are exceedingly rare. For this reason, we will not address X-linked dominant disorders.
- A note on X-linked annotation. With autosomal alleles, the correct annotation is to use capital letters for dominant and lower case for recessive. Genes found on the X chromosome however, are annotated a bit differently. Capital and lowercase are still used, but most often these are added as a sub- or superscript next to the X. In this example Punnett square for example, the mother who is a carrier for, let’s say color blindness, would be annotated XBXb.
Exercise 3: Pedigree Analysis
Inheritance patterns of traits can often be determined by analyzing a pedigree , a chart that depicts several generations of parents and their children and illustrates the individuals that are affected by a particular trait. In a pedigree males are depicted with squares and females with circles. When an individual is affected by the trait being studied then their square or circle is black. When the trait is absent from an individual then their square or circle is white. Parents are connected by a horizontal line and children are shown beneath the parents, connected to the parents with a vertical line. Each generation is shown on the same horizontal line.
Figure X. How to read pedigrees. “BI-102 General Biology 2” by Linn Benton Virtual College. 7/2020, CC BY. https://www.oercommons.org/courses/b...biology-2/view
Pedigree 1
Things to note visually in this pedigree:
- Both male and females are affected at about the same percentage
- A large percentage of offspring at each generation are affected
- There are no offspring with the disease that don’t have at least one affected parent
- Is this disorder autosomal or sex-linked?
- Is this disorder most likely caused by a dominant or recessive gene?
- Circle the individual(s) that proves your choices.
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What are the genotypes of the individuals labeled in the pedigree?
- Grandfather:
- Son:
- Grandson 1:
- Daughter 1:
- Granddaughter 1:
Pedigree 2
Things to note visually in this pedigree:
- A much higher percentage of males are affected by this disorder.
- Females are only affected if their father is also affected.
- An offspring is affected by the disorder even though both parents are not.
- Is this disorder autosomal or sex-linked?
- Is this disorder most likely caused by a dominant or recessive gene?
- Circle the individual(s) that proves your choices.
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What are the genotypes of the individuals labeled in the pedigree?
- Grandfather:
- Grandmother:
- Granddaughter 1:
- Grandson 1:
- Daughter 1:
Pedigree 3
- Is this disorder autosomal or sex-linked?
- Is this disorder most likely caused by a dominant or recessive gene?
- Circle the individual(s) that proves your choices.
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What are the genotypes of the individuals labeled in the pedigree?
- Grandfather:
- Grandson 1:
- Daughter 1:
- Granddaughter 1:
- Daughter-in-Law:
- Son 1:
Pedigree 4
- Is this disorder autosomal or sex-linked?
- Is this disorder most likely caused by a dominant or recessive gene?
- Circle the individual(s) that proves your choices.
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What are the genotypes of the individuals labeled in the pedigree?
- Grandfather:
- Son 1:
- Daughter-in-Law:
- Granddaughter 1
- Grandson 1
Pedigree 5
- Is this disorder autosomal or sex-linked?
- Is this disorder most likely caused by a dominant or recessive gene?
- Circle the individual(s) that proves your choices.
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What are the genotypes of the individuals labeled in the pedigree?
- Grandfather:
- Grandmother:
- Daughter 1:
- Granddaughter 1:
- Grandson 1:
Questions for Review
- In typical situations, how many chromosomes are inherited from our mother? How many from our father?
- What does it mean if someone is a carrier for a trait?
- Why is it that even though some dominant disorders are fatal, they continue to persist in human populations?
- Explain why disorders caused by a dominant allele cannot have carriers of the disorder.
- Why is it possible for two individuals who express a dominant disorder to have children who are unaffected?
- A son receives his X-chromosome from:
- Explain why men cannot be carriers for an X-linked trait.
Testing Understanding Post-Lab
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Imagine you are a genetic counselor, and a couple who is planning on starting a family comes to you for information. Charles was married once before, and he and his first wife had a child who suffered from cystic fibrosis (an autosomal recessive disorder). His current wife Elaine’s brother died of cystic fibrosis, but both parents did not have the disorder.
- What is Charles’ genotype?
- What is the genotype of Elaine’s parents?
- What is the genotype of Elaine’s brother?
- What is the chance that Elaine is a carrier for the disorder?
- A couple is both phenotypically normal, but their son suffers from hemophilia, an X-linked recessive disorder. What percent of their children are likely to suffer from hemophilia? What percent are likely to be carriers for the disorder?
- Roger and Gayle are both of normal phenotype but have a son with Muscular Dystrophy, a sex-linked recessive disorder. Gayle is pregnant and amniocentesis shows the fetus is male. What is the probability that the fetus will suffer from muscular dystrophy?


4.3.2: Javanese Traditional - The Love Dance of Klana Sewandana
Like sung drama, danced drama can be found around the world. Some of the richest traditions hail from Southeast Asia, where one can find dozens of distinct varieties of danced storytelling. Our example belongs to the wayang wong tradition, which developed in the courts of central Java, an island that is now part of Indonesia. Although wayang wong is unique to this small region, many of its musical and dramatic elements can be found across Java and throughout Indonesia. The most important of these is gamelan music, which is played on pitched gongs and keyed metallophones (marimba and xylophone are Western examples of such instruments).
Indonesia as Kingdom and Colony
Indonesia’s performing arts traditions reflect its diverse cultural heritage. Civilization on the islands dates back thousands of years. Indonesia was at first ruled over by a series of Hindu and Buddhist kingdoms, but the arrival of Islam in the 15th century resulted in the gradual conversion of the population. Today, Indonesia is the largest Muslim nation in the world, although many inhabitants practice a form of the religion that contains traces of the region’s Hindu and Buddhist heritage.
Javanese music is well known in the West due to the island’s colonial history. Dutch traders began to visit the Indonesian islands in the 16th century. The profitability of trade led to an influx of Dutch settlers, who established a local government under the auspices of the United East India Company. When the company failed in 1800, the Dutch government formally annexed the Dutch East Indies—which included most of Java—as a colony. Under Dutch rule, indigenous courts were allowed to remain intact, but they were granted only ceremonial powers. While this denied Indonesians their political autonomy, it contributed to the flourishing of Indonesian art, which the Dutch encouraged. It also facilitated the spread of Javanese music and art to Europe.
During World War II, Indonesia was occupied by the Japanese, who drove out the Dutch in an attempt to claim the islands for themselves. At the conclusion of the war, the Indonesians proclaimed independence. The Netherlands attempted to reassert its claim to the territory by force, and a military conflict ensued. In 1949, however, facing intense pressure from other powers, the Dutch acknowledged Indonesia as an independent nation. The impact of Dutch colonialism on the spread of Javanese cultural products resonates into the present day, and gamelan music in particular can be heard throughout the world. Indeed, there are over one hundred gamelan ensembles in the United States alone.
Wayang Wong
The dance form we will explore here, wayang wong , developed during the colonial era and resulted from the conflict between indigenous and colonial powers. Wayang wong was created at the court of Hamengkubuwono I, the first Sultan of Yogyakarta. Hamengkubuwono was to inherit the throne of the Mataram Sultanate from his brother, Pakubuwono II, but instead led his followers in a civil war after Pakubuwono agreed to cooperate with the United East India Company. The war ended in 1755 with a treaty establishing courts at Yogyakarta and Surakarta. Although this was nominally a victory for Hamengkubuwono, the Dutch ultimately benefited by pitting the courts against one another over the next two centuries. All the same, Yogyakarta remained a stronghold of Javanese resistance, and today serves as the capital of an independent region of the same name.
Wayang wong was initially conceived of as a decadent three-day spectacle celebrating and affirming the power of the newly-established Yogyakarta court. Although the first performance required dozens of dancers and lasted from dawn to dusk on each of the days, it has persisted in scaled-down forms. The term wayang wong literally means “human wayang.” This is in deference to the principal form of Indonesian theater, wayang kulit, which uses shadow puppets to act out traditional epics to the accompaniment of music.
All forms of wayang tell stories that have a long history in Indonesian culture. The principal sources are two Indian epics, the Mahabharata and the Ramayana . Wayang performances also draw from the Panji stories, which concern the life of a legendary Javanese prince. Each of these epics is much too long and complex to be related in a single performance. Instead, wayang performers select individual stories, which they often elaborate in ways that add novelty without disrupting the traditional narrative. Indonesian audiences are already familiar with the stories, and they appreciate seeing the characters and events presented with originality.
Gamelan
Our example presents an excerpt from the Panji stories, which we will consider in greater detail below. Before that, however, we owe some attention to the music that accompanies wayang wong dance. The instruments used in gamelan music date back to at least the 8th century, although they did not acquire their present form until the 15th or 16th century, when gamelan music became an important component of court life. Gamelan music is ubiquitous in traditional Indonesian culture. It can be performed on its own, but it is also used to accompany dance and theater. It is played for court celebrations and religious ceremonies, pursued by amateurs for their own enjoyment, and consumed by fans as entertainment.
A gamelan is a collection of bronze percussion instruments that are struck using a variety of mallets. The components of a gamelan include vertically-hung gongs of various sizes, smaller gongs suspended horizontally in wooden frames, and melodic instruments consisting of metal bars suspended over resonating chambers. The instruments of a gamelan are built together and they remain together. It is not possible for players to bring their own instruments, or for an instrument to be substituted. This is because there is no fixed pitch system in Indonesian music, so gamelan instruments are tuned to one another by the original builder. Every gamelan therefore plays a distinct set of pitches.
In addition to being communally stored and maintained, gamelans are bestowed with names. This not only recognizes the gamelan’s status as a complete ensemble but also signifies the traditional belief that a gamelan possesses a spirit. This spirit resides primarily in the largest suspended gong, the gong ageng , which might be provided offerings of food, flowers, and incense. The blacksmith who forges the bronze components of the gamelan instruments is traditionally understood to hold special powers, and he is expected to prepare for his task through fasting and self-purification. His carefully-tuned gongs and bars are then set in ornately-carved wooden frames.
Each instrument in a gamelan occupies a specific place in the musical texture. The vertically- and horizontally-suspended gongs are all used to mark the pulses of the rhythmic cycles that underlie gamelan music. These cycles, each of which is named, consist of patterns of timbres provided by the various gongs. The gong ageng always marks the end of the cycle, which repeats throughout a performance and provides the structure.
The other bronze instruments—which include two sizes of bonang , gendèr , slenthem (an instrument that resembles the gendèr but is pitched lower and has fewer keys), and two sizes of saron — play the same melody in heterophonic texture. However, each of these instruments interprets the melody in a markedly different way. Some play ornate versions, some play spare versions, and some play syncopated versions. The result is a complex texture woven out of the brilliant and resonant timbres of the many instruments.
Gamelan music also uses a few additional instruments, although these are not part of the main gamelan ensemble. Such instruments usually provide improvised melodies that follow the contour of the main melody but are otherwise independent. The instruments that might fill this role include the rebab (a bowed fiddle related to the Chinese jinghu ), the gambang (a wood-keyed xylophone), and the suling (a type of flute). Finally, a male choir or female soloist might sing, drawing their texts from a body of poems and riddles that belong to the gamelan tradition. These texts, however, are not associated with any particular musical composition. Instead, they might be heard in a variety of contexts.
A gamelan performance is led by the drummer, who plays three different sizes of a drum known as kendhang . The drummer starts and stops performances, but his most important task is to control the pace and to signal the dramatic shifts in tempo that are characteristic of gamelan music. During performance, the ensemble will periodically slow to half of its former tempo or accelerate to twice the tempo. This is accompanied by changes in how the melody is interpreted by each of the instruments, which will add notes at slow tempos in order to maintain a consistent texture.
The Love Dance of Klana Sewandana
We are now prepared to turn to our example: A wayang wong performance that presents an episode from the Panji legends. Wayang wong is practiced both in masked and unmasked versions. This is obviously an example of the former. The use of masks in Indonesian dance dates back thousands of years. The masks are symbolically significant, and their shape, color, and details carry information about the character.
Many episodes of the Panji legend concern his romantic relationship with Sekartaji, who often finds herself separated from the Prince due to the intervention of malevolent forces. To survive, she frequently disguises herself as a man, at one point becoming the King of Bali and meeting Prince Panji on the battlefield. (There is also a long history of cross dressing in wayang dance itself.) The scene we will examine opens on Sekartaji alone in the forest. She has fled from the unwanted affections of King Klana Sewandana, whose obsession with her has driven him insane. Klana Sewandana attempts to seduce Sekartaji, but she rebuffs his advances. Luckily, Prince Panji has been drawn to the scene by his great love for Sekartaji. He fights and defeats Klana Sewandana, and the lovers are reunited.
| Time | Form |
What to listen for |
|---|---|---|
| 32’14” |
A few of the metallophones play a slow, Sekartaji alone in simple melody; the rebab and solo the forest female singer improvise high melodies; a percussionist taps rhythms on a wooden box |
|
| 35’12” | Klana Sewandana |
The music at first becomes faster and more enters and rhythmically intense; this is followed by attempts to fluctuations in tempo and dynamic seduce Sekartaji |
| 36’08 | A male solo singer enters and the rhythm becomes more regular | |
| 37’52 | The brighter/louder metallophones reenter | Prince Panji arrives and fights Klana the texture Sewandana |
| 38’21” | The male choir enters and most of the instruments drop out | |
| 39’06” | The rebab enters | |
| 39’35” | The solo female singer joins the male choir Instruments and singers continue to [...] leave and reenter the texture as the music fluctuates in intensity | |
| 45’30 | Prince Panji and Sekartaji are The music slows in tempo, then accelerates reunited | |
| 48’19 | The music slows in anticipation of the final note |
The three dancers in the scene move differently, for they represent the three character types of wayang wong . As a female type, Sekartaji keeps her feet close to the ground while executing refined movements with her hands, neck, and head. Klana Sewandana is a strong male, as betrayed by his wide stance, large steps, and aggressive movements. Prince Panji, on the other hand, is a refined male. As such, he keeps his feet close to the ground and moves fluidly.
Throughout the episode, the gamelan provides a constant musical backdrop while accentuating the dramatic contour. Klana Sewandana’s arrival in the forest, for example, is marked by an intensification of the music, which grows in volume and increases in tempo. Musical outbursts punctuate the fight scene, while a musical calm descends on the final scene between Panji and Sekartaji. From time to time we hear the rebab , the suling , and various singers, who can be seen seated behind the dancers. The words they are singing have nothing in particular to do with the drama unfolding onstage, but voices contribute an important aesthetic element to any performance.


5.10: The Endosymbiotic Theory
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Briefly describe what is meant by the endosymbiotic theory.
- Give some evidence supporting the theory that mitochondria and chloroplasts may have arisen from prokaryotic organisms.
It is thought that life arose on earth around four billion years ago. The endosymbiotic theory states that some of the organelles in today's eukaryotic cells were once prokaryotic microbes. In this theory, the first eukaryotic cell was probably an amoeba-like cell that got nutrients by phagocytosis and contained a nucleus that formed when a piece of the cytoplasmic membrane pinched off around the chromosomes. Some of these amoeba-like organisms ingested prokaryotic cells that then survived within the organism and developed a symbiotic relationship. Mitochondria formed when bacteria capable of aerobic respiration were ingested; chloroplasts formed when photosynthetic bacteria were ingested. They eventually lost their cell wall and much of their DNA because they were not of benefit within the host cell. Mitochondria and chloroplasts cannot grow outside their host cell.
Evidence for this is based on the following:
- Chloroplasts are the same size as prokaryotic cells, divide by binary fission, and, like bacteria, have Fts proteins at their division plane. The mitochondria are the same size as prokaryotic cells, divide by binary fission, and the mitochondria of some protists have Fts homologs at their division plane.
- Mitochondria and chloroplasts have their own DNA that is circular, not linear.
- Mitochondria and chloroplasts have their own ribosomes that have 30S and 50S subunits, not 40S and 60S.
- Several more primitive eukaryotic microbes, such as Giardia and Trichomonas have a nuclear membrane but no mitochondria.
Although evidence is less convincing, it is also possible that flagella and cilia may have come from spirochetes.
Example \(\PageIndex{1}\)
- Briefly describe what is meant by the endosymbiotic theory.
- Give three points of evidence supporting the theory that mitochondria and chloroplasts may have arisen from prokaryotic organisms.
Solutions
- The endosymbiotic theory states that some of the organelles in eukaryotic cells were once prokaryotic microbes.
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- Mitochondria and chloroplasts are the same size as prokaryotic cells and divide by binary fission.
- Mitochondria and chloroplasts have their own DNA which is circular, not linear.
- Mitochondria and chloroplasts have their own ribosomes which have 30S and 50S subunits, not 40S and 60S.
Summary
The endosymbiotic theory states that mitochondria and chlopoplasts in today's eukaryotic cells were once separate prokaryotic microbes.


13.8.5: Divide Square Roots
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Learning Objectives
By the end of this section, you will be able to:
- Divide square roots
- Rationalize a one-term denominator
- Rationalize a two-term denominator
Note
Before you get started, take this readiness quiz.
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Find a fraction equivalent to \(\frac{5}{8}\) with denominator 48.
If you missed this problem, review Exercise 1.6.1 . -
Simplify: \((\sqrt{5})^2\).
If you missed this problem, review Example 9.4.13 . -
Multiply: (7+3x)(7−3x).
If you missed this problem, review Exercise 6.4.22 .
Divide Square Roots
We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.
Example \(\PageIndex{1}\)
Simplify: \(\frac{\sqrt{54}}{6}\)
- Answer
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\(\frac{\sqrt{54}}{6}\) Simplify the radical. \(\frac{\sqrt{9}·\sqrt{6}}{6}\) Simplify. \(\frac{3\sqrt{6}}{6}\) Remove the common factors. \(\frac{3\sqrt{6}}{3·2}\) Simplify. \(\frac{\sqrt{6}}{2}\)
Example \(\PageIndex{2}\)
Simplify: \(\frac{\sqrt{32}}{8}\).
- Answer
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\(\frac{\sqrt{2}}{2}\)
Example \(\PageIndex{3}\)
Simplify: \(\frac{\sqrt{75}}{15}\).
- Answer
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\(\frac{\sqrt{3}}{3}\)
Example \(\PageIndex{4}\)
Simplify: \(\frac{6−\sqrt{24}}{12}\).
- Answer
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\(\frac{6−\sqrt{24}}{12}\) Simplify the radical. \(\frac{6−\sqrt{4}·\sqrt{6}}{12}\) Simplify. \(\frac{6−2\sqrt{6}}{12}\) Factor the common factor from the numerator. \(\frac{2(3−\sqrt{6})}{12}\) Remove the common factors. \(\frac{2(3−\sqrt{6})}{2·6}\) Simplify. \(\frac{3−\sqrt{6}}{6}\)
Example \(\PageIndex{5}\)
Simplify: \(\frac{8−\sqrt{40}}{10}\).
- Answer
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\(\frac{4−\sqrt{10}}{5}\)
Example \(\PageIndex{6}\)
Simplify: \(\frac{10−\sqrt{75}}{20}\).
- Answer
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\(\frac{5−\sqrt{3}}{4}\)
We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says
\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\), \(b \ne 0\).
Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.
\(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\), \(b \ne 0\).
We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.
Definition: QUOTIENT PROPERTY OF SQUARE ROOTS
If a , b are non-negative real numbers and \(b \ne 0\), then
\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) and \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\)
We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.
Example \(\PageIndex{7}\)
Simplify: \(\frac{\sqrt{27}}{\sqrt{75}}\)
- Answer
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\(\frac{\sqrt{27}}{\sqrt{75}}\) Neither radicand is a perfect square, so rewrite using the quotient property of square root. \(\sqrt{\frac{27}{75}}\) Remove common factors in the numerator and denominator. \(\sqrt{\frac{9}{25}}\) Simplify. \(\frac{3}{5}\)
Example \(\PageIndex{8}\)
Simplify: \(\frac{\sqrt{48}}{\sqrt{108}}\)
- Answer
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\(\frac{2}{3}\)
Example \(\PageIndex{9}\)
Simplify: \(\frac{\sqrt{96}}{\sqrt{54}}\)
- Answer
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\(\frac{4}{3}\)
We will use the Quotient Property for Exponents, \(\frac{a^m}{a^n}=a^{m−n}\), when we have variables with exponents in the radicands.
Example \(\PageIndex{10}\)
Simplify: \(\frac{\sqrt{6y^5}}{\sqrt{2y}}\)
- Answer
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\(\frac{\sqrt{6y^5}}{\sqrt{2y}}\) Neither radicand is a perfect square, so rewrite using the quotient property of square root. \(\sqrt{\frac{6y^5}{2y}}\) Remove common factors in the numerator and denominator. \(\sqrt{3y^4}\) Simplify. \(y^2\sqrt{3}\)
Example \(\PageIndex{11}\)
Simplify: \(\frac{\sqrt{12r^3}}{\sqrt{6r}}\).
- Answer
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\(r\sqrt{2}\)
Example \(\PageIndex{12}\)
Simplify: \(\frac{\sqrt{14p^9}}{\sqrt{2p^5}}\)
- Answer
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\(p^2\sqrt{7}\)
Example \(\PageIndex{13}\)
Simplify: \(\frac{\sqrt{72x^3}}{\sqrt{162x}}\)
- Answer
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\(\frac{\sqrt{72x^3}}{\sqrt{162x}}\) Rewrite using the quotient property of square roots. \(\sqrt{\frac{72x^3}{162x}}\) Remove common factors. \(\sqrt{\frac{18·4·x^2·x}{18·9·x}}\) Simplify. \(\sqrt{\frac{4x^2}{9}}\) Simplify the radical. \(\frac{2x}{3}\)
ExAMPLe \(\PageIndex{14}\)
Simplify: \(\frac{\sqrt{50s^3}}{\sqrt{128s}}\).
- Answer
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\(\frac{5s}{8}\)
Example \(\PageIndex{15}\)
Simplify: \(\frac{\sqrt{75q^5}}{\sqrt{108q}}\).
- Answer
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\(\frac{5q^2}{6}\)
Example \(\PageIndex{16}\)
Simplify: \(\frac{\sqrt{147ab^8}}{\sqrt{3a^3b^4}}\).
- Answer
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\(\frac{\sqrt{147ab^8}}{\sqrt{3a^{3}b^{4}}}\) Rewrite using the quotient property of square roots. \(\sqrt{\frac{147ab^8}{3a^{3}b^{4}}}\) Remove common factors. \(\sqrt{\frac{49b^4}{a^2}}\ Simplify the radical. \(\frac{7b^2}{a}\)
Example \(\PageIndex{17}\)
Simplify: \(\frac{\sqrt{162x^{10}y^{2}}}{\sqrt{2x^6y^6}}\).
- Answer
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\(\frac{9x^2}{y^2}\)
Example \(\PageIndex{18}\)
Simplify: \(\frac{\sqrt{300m^{3}n^{7}}}{\sqrt{3m^{5}n}}\).
- Answer
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\(\frac{10n^3}{m}\)
Rationalize a One Term Denominator
Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. Figure shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.
If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.
For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.
Definition: RATIONALIZING THE DENOMINATOR
The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator .
Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator , we write an equivalent fraction with a rational number in the denominator.
Let’s look at a numerical example.
\[\begin{array}{ll} {\text{Suppose we need an approximate value for the fraction.}}&{\frac{1}{\sqrt{2}}}\\ {\text{A five decimal place approximation to} \sqrt{2} \text{is} 1.41421}&{\frac{1}{1.41421}}\\ {\text{Without a calculator, would you want to do this division?}}&{1.41421) \overline{1.0}}\\ \nonumber \end{array}\]
But we can find a fraction equivalent to \(\frac{1}{\sqrt{2}}\) by multiplying the numerator and denominator by \(\sqrt{2}\).
Now if we need an approximate value, we divide \(2) \overline{1.41421}\). This is much easier.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.
Similarly, a square root is not considered simplified if the radicand contains a fraction.
Definition: SIMPLIFIED SQUARE ROOTS
A square root is considered simplified if there are
- no perfect-square factors in the radicand
- no fractions in the radicand
- no square roots in the denominator of a fraction
To rationalize a denominator, we use the property that \((\sqrt{a})^2=a\). If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
Example \(\PageIndex{19}\)
Simplify: \(\frac{4}{\sqrt{3}}\).
- Answer
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To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by the same factor.
\(\frac{4}{\sqrt{3}}\)
Multiply both the numerator and denominator by \(\sqrt{3}\)\(\frac{4·\sqrt{3}}{\sqrt{3}·\sqrt{3}}\) Simplify. \(\frac{4\sqrt{3}}{3}\)
Example \(\PageIndex{20}\)
Simplify: \(\frac{5}{\sqrt{3}}\).
- Answer
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\(\frac{5\sqrt{3}}{3}\)
Example \(\PageIndex{21}\)
Simplify: \(\frac{6}{\sqrt{5}}\).
- Answer
-
\(\frac{6\sqrt{5}}{5}\)
Example \(\PageIndex{22}\)
Simplify: \(−\frac{8}{3\sqrt{6}}\)
- Answer
-
To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by \(\sqrt{6}\).
Multiply both the numerator and the denominator by \(\sqrt{6}\). Simplify. Remove common factors. Simplify.
Example \(\PageIndex{23}\)
Simplify: \(\frac{5}{2\sqrt{5}}\).
- Answer
-
\(\frac{\sqrt{5}}{2}\)
Example \(\PageIndex{24}\)
Simplify: \(−\frac{9}{4\sqrt{3}}\).
- Answer
-
\(−\frac{3\sqrt{3}}{4}\)
Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.
Example \(\PageIndex{25}\)
Simplify: \(\sqrt{\frac{5}{12}}\).
- Answer
-
The fraction is not a perfect square, so rewrite using the
Quotient Property.Simplify the denominator. Rationalize the denominator. Simplify. Simplify.
Example \(\PageIndex{26}\)
Simplify: \(\sqrt{\frac{7}{18}}\).
- Answer
-
\(\frac{\sqrt{14}}{6}\)
Example \(\PageIndex{27}\)
Simplify: \(\sqrt{\frac{3}{32}}\).
- Answer
-
\(\frac{\sqrt{6}}{8}\)
Example \(\PageIndex{28}\)
Simplify: \(\sqrt{\frac{11}{28}}\)
- Answer
-
Rewrite using the Quotient Property. Simplify the denominator. Rationalize the denominator. Simplify. Simplify.
Example \(\PageIndex{29}\)
Simplify: \(\sqrt{\frac{3}{27}}\).
- Answer
-
\(\frac{1}{3}\)
Example \(\PageIndex{30}\)
Simplify: \(\sqrt{\frac{10}{50}}\)
- Answer
-
\(\frac{\sqrt{5}}{5}\)
Rationalize a Two-Term Denominator
When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.
\[\begin{array}{ll} {(a−b)(a+b)}&{(2−\sqrt{5})(2+\sqrt{5})}\\ {a^2−b^2}&{2^2−(\sqrt{5})^2}\\ {}&{4−5}\\ {}&{−1}\\ \nonumber \end{array}\]
When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.
Example \(\PageIndex{31}\)
Simplify: \(\frac{4}{4+\sqrt{2}}\).
- Answer
-
Multiply the numerator and denominator by the conjugate of the denominator. Multiply the conjugates in the denominator. Simplify the denominator. Simplify the denominator. Remove common factors from the numerator and denominator. We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.
Example \(\PageIndex{32}\)
Simplify: \(\frac{2}{2+\sqrt{3}}\).
- Answer
-
\(\frac{2(2−\sqrt{3})}{1}\)
Example \(\PageIndex{33}\)
Simplify: \(\frac{5}{5+\sqrt{3}}\).
- Answer
-
\(\frac{5(5−\sqrt{3})}{22}\)
Example \(\PageIndex{34}\)
Simplify: \(\frac{5}{2−\sqrt{3}}\).
- Answer
-
Multiply the numerator and denominator by the conjugate of the denominator. Multiply the conjugates in the denominator. Simplify the denominator. Simplify the denominator. Simplify.
Example \(\PageIndex{35}\)
Simplify: \(\frac{3}{1−\sqrt{5}}\).
- Answer
-
\(−\frac{3(1+\sqrt{5})}{4}\)
Example \(\PageIndex{36}\)
Simplify: \(\frac{2}{4−\sqrt{6}}\).
- Answer
-
\(\frac{4+\sqrt{6}}{5}\)
Example \(\PageIndex{37}\)
Simplify: \(\frac{\sqrt{3}}{\sqrt{u}−\sqrt{6}}\).
- Answer
-
Multiply the numerator and denominator by the conjugate of the denominator. Multiply the conjugates in the denominator. Simplify the denominator.
Example \(\PageIndex{38}\)
Simplify: \(\frac{\sqrt{5}}{\sqrt{x}+\sqrt{2}}\).
- Answer
-
\(\frac{\sqrt{5}(\sqrt{x}−\sqrt{2})}{x−2}\)
Example \(\PageIndex{39}\)
Simplify: \(\frac{\sqrt{10}}{\sqrt{y}−\sqrt{3}}\).
- Answer
-
\(\frac{\sqrt{10}(\sqrt{y}+\sqrt{3})}{y−3}\)
Example \(\PageIndex{40}\)
Simplify:\(\frac{\sqrt{x}+\sqrt{7}}{\sqrt{x}−\sqrt{7}}\).
- Answer
-
Multiply the numerator and denominator by the conjugate of the denominator. Multiply the conjugates in the denominator. Simplify the denominator. We do not square the numerator. In factored form, we can see there are no common factors to remove from the numerator and denominator.
Example \(\PageIndex{41}\)
Simplify: \(\frac{\sqrt{p}+\sqrt{2}}{\sqrt{p}−\sqrt{2}}\).
- Answer
-
\(\frac{(\sqrt{p}+\sqrt{2})^2}{p−2}\)
Example \(\PageIndex{42}\)
Simplify: \(\frac{\sqrt{q}−\sqrt{10}}{\sqrt{q}+\sqrt{10}}\).
- Answer
-
\(\frac{(\sqrt{q}−\sqrt{10})^2}{q−10}\)
Access this online resource for additional instruction and practice with dividing and rationalizing.
- Dividing and Rationalizing
Key Concepts
-
Quotient Property of Square Roots
-
If
a
,
b
are non-negative real numbers and \(b \ne 0\), then
\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) and \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\)
-
If
a
,
b
are non-negative real numbers and \(b \ne 0\), then
-
Simplified Square Roots
A square root is considered simplified if there are- no perfect square factors in the radicand
- no fractions in the radicand
- no square roots in the denominator of a fraction
Glossary
- rationalizing the denominator
- The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.


17.2: Sound Waves
- Explain the difference between sound and hearing
- Describe sound as a wave
- List the equations used to model sound waves
- Describe compression and rarefactions as they relate to sound
The physical phenomenon of sound is a disturbance of matter that is transmitted from its source outward. Hearing is the perception of sound, just as seeing is the perception of visible light. On the atomic scale, sound is a disturbance of atoms that is far more ordered than their thermal motions. In many instances, sound is a periodic wave, and the atoms undergo simple harmonic motion. Thus, sound waves can induce oscillations and resonance effects (Figure \(\PageIndex{2}\)).
This video shows waves on the surface of a wine glass, being driven by sound waves from a speaker. As the frequency of the sound wave approaches the resonant frequency of the wine glass, the amplitude and frequency of the waves on the wine glass increase. When the resonant frequency is reached, the glass shatters.
Video \(\PageIndex{1}\): Breaking a wine glass using resonance by playing sound at its resonance frequency. https://www.youtube.com/watch?v=17tqXgvCN0E
A speaker produces a sound wave by oscillating a cone, causing vibrations of air molecules. In Figure \(\PageIndex{2}\), a speaker vibrates at a constant frequency and amplitude, producing vibrations in the surrounding air molecules. As the speaker oscillates back and forth, it transfers energy to the air, mostly as thermal energy. But a small part of the speaker’s energy goes into compressing and expanding the surrounding air, creating slightly higher and lower local pressures. These compressions (high-pressure regions) and rarefactions (low-pressure regions) move out as longitudinal pressure waves having the same frequency as the speaker—they are the disturbance that is a sound wave. (Sound waves in air and most fluids are longitudinal, because fluids have almost no shear strength. In solids, sound waves can be both transverse and longitudinal.)
Figure \(\PageIndex{2a}\) shows the compressions and rarefactions, and also shows a graph of gauge pressure versus distance from a speaker. As the speaker moves in the positive x-direction, it pushes air molecules, displacing them from their equilibrium positions. As the speaker moves in the negative x-direction, the air molecules move back toward their equilibrium positions due to a restoring force. The air molecules oscillate in simple harmonic motion about their equilibrium positions, as shown Figure \(\PageIndex{2b}\). Note that sound waves in air are longitudinal, and in the figure, the wave propagates in the positive x-direction and the molecules oscillate parallel to the direction in which the wave propagates.
Sound can be modeled as a pressure wave by considering the change in pressure from average pressure,
\[\Delta P = \Delta P_{max} \sin (kx \mp \omega t + \phi) \ldotp \label{17.1}\]
This equation is similar to the periodic wave equations seen in Waves, where \(\Delta\)P is the change in pressure, \(\Delta P_{max}\) is the maximum change in pressure, \(k = \frac{2 \pi}{\lambda}\) is the wave number, \(\omega = \frac{2 \pi}{T} = 2 \pi f\) is the angular frequency, and \(\phi\) is the initial phase. The wave speed can be determined from
\[v = \frac{\omega}{k} = \frac{\lambda}{T}.\]
Sound waves can also be modeled in terms of the displacement of the air molecules. The displacement of the air molecules can be modeled using a cosine function:
\[s(x, t) = s_{max} \cos(kx \mp \omega t + \phi) \ldotp \label{17.2}\]
In this equation, \(s\) is the displacement and \(s_{max}\) is the maximum displacement.
Not shown in the figure is the amplitude of a sound wave as it decreases with distance from its source, because the energy of the wave is spread over a larger and larger area. The intensity decreases as it moves away from the speaker, as discussed in Waves . The energy is also absorbed by objects and converted into thermal energy by the viscosity of the air. In addition, during each compression, a little heat transfers to the air; during each rarefaction, even less heat transfers from the air, and these heat transfers reduce the organized disturbance into random thermal motions. Whether the heat transfer from compression to rarefaction is significant depends on how far apart they are—that is, it depends on wavelength. Wavelength, frequency, amplitude, and speed of propagation are important characteristics for sound, as they are for all waves.


1.2: Chapter 2 – Opportunity Recognition and Design Thinking
Entrepreneurs see ways to put resources and information together in new combinations. They not only see the system as it is, but as it might be. They have a knack for looking at the usual and seeing the unusual, at the ordinary and seeing the extraordinary. Consequently, they can spot opportunities that turn the commonplace into the unique and unexpected. – Mitton (1989, p. 12)
In my opinion, all previous advances in the various lines of invention will appear totally insignificant when compared with those which the present century will witness. I almost wish that I might live my life over again to see the wonders which are at the threshold. – Charles H. Duell, Commissioner, U.S. Office of Patents, 1898-1901 (who has been incorrectly quoted as having said “Everything that can be invented has been invented”)
The significant problems we face cannot be solved at the same level of thinking we were at when we created them. – Albert Einstein
I think there is a world market for maybe five computers. – Thomas Watson, Chairman of IBM, 1943
Learning Objectives
After completing this chapter you will be able to
- Discuss opportunity recognition concepts and methods as developed and/or advocated by leading thinkers like Drucker, Mitchell, Schumpeter, and Vesper
- Describe what design thinking is
- Apply design thinking to develop and assess new venture ideas
Overview
This chapter introduces a sample of perspectives and tools designed to help individuals recognize potential business opportunities. The concept of design thinking is also examined in some detail.
The objective is to help you improve your ability to apply inspiration , ideation , and implementation as part of the design thinking process.
Opportunity Recognition
The following is a discussion of entrepreneurship theorists and practitioners who have developed the concept of opportunity recognition . While the tools introduced in the next sections can be applied for a variety of purposes, they are particularly useful for recognizing new venture opportunities.
Baron
Opportunity recognition is
the active, cognitive process (or processes) through which individuals conclude that they have identified the potential to create something new that has the potential to generate economic value and that is not currently being exploited or developed, and is viewed as desirable in the society in which it occurs (i.e. its development is consistent with existing legal and moral conditions). (Baron, 2004b, p. 52)
Because opportunity recognition is a cognitive process, according to Baron (2004b), people can learn to be more effective at recognizing opportunities by changing the way they think about opportunities and how to recognize them.
Drucker
Systematic innovation involves “monitoring seven sources for innovative opportunity” (Drucker, 1985, p. 35). The first four are internally focused within the business or industry, in that they may be visible to those involved in that organization or sector. The last three involve changes outside the business or industry.
-
Internally Focused
- The unexpected (unexpected success, failure, or outside events)
- The incongruity between reality as it actually is and reality as it is assumed to be or as it ought to be
- Innovation based on process need
- Changes in industry structure or market structure that catch everyone unawares
-
Externally Focused
- Demographics (population changes)
- Changes in perception, mood, and meaning
- New knowledge , both scientific and nonscientific
Mitchell
One of the components of Mitchell’s (2000) New Venture Template TM asks whether the venture being examined represents a new combination. To determine this, he suggests considering two categories of entrepreneurial discovery: scientific discovery and circumstance .
-
Scientific Discovery
- Physical/technological insight
- New and valuable way
-
Circumstantial Discovery
- Specific knowledge of time, place, or circumstance
- When and what you know
The second set of variables to consider are the market imperfections that can create profit opportunities: excess demand and excess supply . This gives rise to the following four types of entrepreneurial discovery.
-
Invention I
- Uses science to exploit excess demand (a market imperfection)
- Becomes an opportunity to discover and apply the laws of nature to satisfy excess demand
- Inventions in one industry have ripple effects in others
- Example: invention of airplane
-
Observation
- Circumstances reveal opportunity to exploit excess demand (a market imperfection)
- Not necessarily science-oriented
- Example: airline industry = need for food service for passengers
-
Invention II
- Uses science to exploit excess supply (a market imperfection)
- Example: Second most abundant element on earth after oxygen = silicon microchips
-
Coordination
- Circumstances reveals opportunity to exploit excess supply (a market imperfection)
- Example: Producer’s capacities to lower prices = Wal-Mart
Schumpeter
Schumpeter’s (1934) five kinds of new combinations (see page 13) can occur within each of the four kinds of entrepreneurial discovery (Mitchell, 2000):
-
New or improved good/service
- Distinction between true advances and promotional differences
-
New method of production
- Example: assembly line method to automobile production, robotics, agricultural processing
-
Opening of a new market
- Global context: Culture, laws, local buyer preferences, business practices, customs, communication, transportation all set up new distribution channels
- Example: Honda created a new market for smaller modestly powered motorbikes
-
Conquest of a new source of supply of raw materials
- Enhance availability of products by providing at lower cost
- Enhance availability by making more available without compromising quality
- Reorganization of an industry
Murphy
Murphy (2011) claimed that there was a single-dimensional logic that oversimplified the approach taken to understand entrepreneurial discovery. He was bothered by the notion that entrepreneurs either deliberately searched for entrepreneurial opportunities or they serendipitously discovered them. Murphy’s (2011) multidimensional model of entrepreneurial discovery suggests that opportunities may be identified (a) through a purposeful search; (b) because others provide the opportunity to the entrepreneur; (c) through prior knowledge, entrepreneurial alertness, and means other than a purposeful search; and, (d) through a combination of lucky happenstance and deliberate searching for opportunities.
Vesper
According to experimentation research, entrepreneurial creativity is not correlated with IQ (people with high IQs can be unsuccessful in business and those with lower IQs can be successful as an entrepreneur). Research has also shown that those who practice idea generation techniques can become more creative. The best ideas sometimes come later in the idea-generation process—often in the days and weeks following the application of the idea-generating processes (Vesper, 1996).
Vesper (1996) identified several ways in which entrepreneurs found ideas:
- Prior job
- Recreation
- Chance event
- Answering discovery questions
Although would-be entrepreneurs usually don’t discover ideas by a deliberate searching strategy (except when pursuing acquisitions of ongoing firms), it is nevertheless possible to impute to their discoveries some implicit searching patterns. (Vesper, 1996, p. 60)
Vesper (1996) categorized discovery questions as follows:
-
Search questions
, which might prompt venture ideas by placing one’s mind into a mode where the subconscious will work to push ideas into the conscious mind
- What is bothering me and what might relieve that bother?
- How could this be made or done differently that it is now?
- What else might I like to have?
- How can I fall the family tradition?
-
Questions based on encounters
with a potential customer request, someone else’s idea, or another event
- Can I play some role in providing this product or service to a broader market?
- Could there be a way to do this better for the customer?
-
Questions based on evaluative reactions
to ideas
- Could I do this job on my own instead of as an employee?
- If people elsewhere went for this idea, might they want it here too?
 He suggested that generating innovative ideas involved two tasks: to depart from what is usual or customary and to apply an effective way to direct this departure. The mental blocks in the way of departure include the following:
-
Perceptual blocks
- difficulty viewing things from different perspectives
- seeing only what you expect to see or think what others expect you to see
-
Emotional blocks
- intolerance of ambiguity
- preference for judging rather than seeking ideas
- tunnel vision
- insufficient patience
-
Cultural blocks
- a belief that reason and logic are superior to feeling, intuition, and other such approaches
- thinking that tradition is preferable to change
- disdain for fantasy, reflection, idea playfulness, humor
-
Imagination blocks
- fear of subconscious thinking
- inhibition about some areas of imagination
-
Environmental blocks
- distrust of others who might be able to help
- distractions
- discouraging responses from other people
-
Intellectual blocks
- lack of information
- incorrect information
- weak technical skills in areas such as financial analysis
-
Expressive blocks
- poor writing skills
- inability to construct prototypes
Understanding these mental blocks to departure is a first step in figuring out how to cope with them. Some tactics for departure include the following (Vesper, 1996):
- Trying different ways of looking at and thinking about venture opportunities
- Trying to continually generate ideas about opportunities and how to exploit them
- Seeking clues from business and personal contacts, trade shows, technology licensing offices, and other sources
- Not being discouraged by others’ negative views because many successful innovations were first thought to be impossible to make
- Generating possible solutions to obstacles before stating negative views about them
-
Using idea generating tricks like
- Brainstorming
- Considering multiple consequences of possible future events or changes
- Rearranging, reversing, expanding, shrinking, combining, or altering ideas
- Developing scenarios
Design Thinking
Design thinking is a human-centered approach to innovation that draws from the designer’s toolkit to integrate the needs of people, the possibilities of technology, and the requirements for business success – Tim Brown, president and CEO (IDEO, 2015, para. 5)
The Hasso Plattner Institute of Design at Stanford University, called the d.school ( http://dschool.stanford.edu/ ), is an acknowledged leader at promoting design thinking. You can download the Bootcamp Bootleg manual from the d.school website at https://dschool.stanford.edu/resources/the-bootcamp-bootleg . The following description of design thinking is from the IDEO website:
Design thinking is a deeply human process that taps into abilities we all have but get overlooked by more conventional problem-solving practices. It relies on our ability to be intuitive, to recognize patterns, to construct ideas that are emotionally meaningful as well as functional, and to express ourselves through means beyond words or symbols. Nobody wants to run an organization on feeling, intuition, and inspiration, but an over-reliance on the rational and the analytical can be just as risky. Design thinking provides an integrated third way.
The design thinking process is best thought of as a system of overlapping spaces rather than a sequence of orderly steps. There are three spaces to keep in mind: inspiration , ideation , and implementation . Inspiration is the problem or opportunity that motivates the search for solutions. Ideation is the process of generating, developing, and testing ideas. Implementation is the path that leads from the project stage into people’s lives (IDEO, 2015, para. 7-8).


9.1: Introduction
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The 1930s and 1940s were trying times. A global economic crisis gave way to a global war that became the deadliest and most destructive in human history. Perhaps eighty million individuals lost their lives during World War II. The war saw industrialized genocide and nearly threatened the eradication of an entire people. It also unleashed the most fearsome technology ever used in war. And when it ended, the United States found itself alone as the world’s greatest superpower. Armed with the world’s greatest economy, it looked forward to the fruits of a prosperous consumers’ economy. But the war raised as many questions as it would settle and unleashed new social forces at home and abroad that confronted generations of Americans to come.


1.3: The Scientific Process
By the end of this section, you will be able to do the following:
- Understand the process of scientific inquiry
- Formulate a good hypothesis for a hypothetical scenario
- Compare independent and dependent variables
- Explain the purpose of a control group and distinguish a positive control from a negative control
Biologists study the living world by posing questions about it and seeking science-based responses. This approach is common to other sciences as well and is often referred to as the scientific method. The scientific process was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626) (Figure \(\PageIndex{1}\)), who set up inductive methods for scientific inquiry. The scientific method is not exclusively used by biologists but can be applied to almost anything as a logical problem solving method.
Question
The scientific process typically starts with an observation (often a problem to be solved) that leads to a question . Remember that science is very good at answering questions having to do with observations about the natural world, but is very bad at answering questions having to do with morals, ethics, or personal opinions.
| Questions that can be answered using science | Questions that cannot be answered using science |
|---|---|
| • What is the optimum temperature for the growth of E. coli bacteria? | • How tall is Santa Claus? |
| • Do birds prefer bird feeders of a specific color? | • Do angels exist? |
| • What is the cause of this disease? | • Which is better: classical music or rock and roll? |
| • How effective is this drug in treating this disease? | • What are the ethical implications of human cloning? |
Let’s think about a simple problem that starts with an observation and apply the scientific method to solve the problem. Imagine that one morning when you wake up and flip the switch to turn on your bedside lamp, the light won’t turn on. That is an observation that also describes a problem: the lights won’t turn on. Of course, you would next ask the question: “Why won’t the light turn on?”
Hypothesis
Recall that a hypothesis is a suggested explanation that can be tested. A hypothesis is NOT the question you are trying to answer – it is what you think the answer to the question will be and why. To solve a problem, several hypotheses may be proposed. For example, one hypothesis might be, “The light won’t turn on because the bulb is burned out.” But there could be other answers to the question, and therefore other hypotheses may be proposed. A second hypothesis might be, “The light won’t turn on because the lamp is unplugged” or “The light won’t turn on because the power is out.” A hypothesis should be based on credible background information. A hypothesis is NOT just a guess (not even an educated one), although it can be based on your prior experience (such as in the example where the light won’t turn on). In general, hypotheses in biology should be based on a credible, referenced source of information.
A hypothesis must be testable to ensure that it is valid.  It should also be falsifiable, meaning that it can be disproven by experimental results. An example of an unfalsifiable hypothesis is “Red is a better color than blue.” There is no experiment that might show this statement to be false. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. This is important: a hypothesis can be disproven, or eliminated, but it can never be proven. Science does not deal in proofs like mathematics. If an experiment fails to disprove a hypothesis, then that explanation (the hypothesis) is supported as the answer to the question. However, that doesn’t mean that later on, we won’t find a better explanation or design a better experiment that will be found to falsify the first hypothesis and lead to a better one.
Variables
A variable is any part of the experiment that can vary or change during the experiment. Typically, an experiment only tests one variable and all the other conditions in the experiment are held constant.
- The variable that is tested is known as the independent variable.
- The dependent variable is the thing (or things) that you are measuring as the outcome of your experiment.
- A constant is a condition that is the same between all of the tested groups.
- A confounding variable is a condition that is not held constant that could affect the experimental results.
A hypothesis often has the format “If [I change the independent variable in this way] then [I will observe that the dependent variable does this] because [of some reason].” For example, the first hypothesis might be, “If you change the light bulb, then the light will turn on because the bulb is burned out.” In this experiment, the independent variable (the thing that you are testing) would be changing the light bulb and the dependent variable is whether or not the light turns on. It would be important to hold all the other aspects of the environment constant, for example not messing with the lamp cord or trying to turn the lamp on using a different light switch. If the entire house had lost power during the experiment because a car hit the power pole, that would be a confounding variable.
You may have learned that a hypothesis can be phrased as an “If..then…” statement. Simple hypotheses can be phrased that way (but they must also include a “because”), but more complicated hypotheses may require several sentences. It is also very easy to get confused by trying to put your hypothesis into this format. Hypotheses do not have to be phrased as “if..then..” statements, it is just sometimes a useful format.
Results
The results of your experiment are the data that you collect as the outcome. In the light experiment, your results are either that the light turns on or the light doesn’t turn on. Based on your results, you can make a conclusion. Your conclusion uses the results to answer your original question.
We can put the experiment with the light that won’t go on into the figure above:
- Observation: the light won’t turn on.
- Question: why won’t the light turn on?
- Hypothesis: the lightbulb is burned out.
- Prediction: if I change the lightbulb (independent variable), then the light will turn on (dependent variable).
- Experiment: change the lightbulb while leaving all other variables the same.
- Analyze the results: the light didn’t turn on.
- Conclusion: The lightbulb isn’t burned out. The results do not support the hypothesis, time to develop a new one!
- Hypothesis 2: the lamp is unplugged.
- Prediction 2: if I plug in the lamp, then the light will turn on.
- Experiment: plug in the lamp
- Analyze the results: the light turned on!
- Conclusion: The light wouldn’t turn on because the lamp was unplugged. The results support the hypothesis, it’s time to move on to the next experiment!
In practice, the scientific method is not as rigid and structured as it might at first appear. Sometimes an experiment leads to conclusions that favor a change in approach; often, an experiment brings entirely new scientific questions to the puzzle. Many times, science does not operate in a linear fashion; instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests (Figure \(\PageIndex{3}\)).
Control Groups
Another important aspect of designing an experiment is the presence of one or more control groups. A control group allows you to make a comparison that is important for interpreting your results. Control groups are samples that help you to determine that differences between your experimental groups are due to your treatment rather than a different variable – they eliminate alternate explanations for your results (including experimental error and experimenter bias). They increase reliability, often through the comparison of control measurements and measurements of the experimental groups. Often, the control group is a sample that is not treated with the independent variable, but is otherwise treated the same way as your experimental sample. This type of control group contains every feature of the experimental group except it is not given the manipulation that is hypothesized about (it does not get treated with the independent variable). Therefore, if the results of the experimental group differ from the control group, the difference must be due to the hypothesized manipulation, rather than some outside factor. It is common in complex experiments (such as those published in scientific journals) to have more control groups than experimental groups.
Example \(\PageIndex{1}\)
Question: Which fertilizer will produce the greatest number of tomatoes when applied to the plants?
Prediction and Hypothesis: If I apply different brands of fertilizer to tomato plants, the most tomatoes will be produced from plants watered with Brand A because Brand A advertises that it produces twice as many tomatoes as other leading brands.
Experiment: Purchase 10 tomato plants of the same type from the same nursery. Pick plants that are similar in size and age. Divide the plants into two groups of 5. Apply Brand A to the first group and Brand B to the second group according to the instructions on the packages. After 10 weeks, count the number of tomatoes on each plant.
Independent Variable: Brand of fertilizer.
Dependent Variable: Number of tomatoes.
The number of tomatoes produced depends on the brand of fertilizer applied to the plants.
Constants: amount of water, type of soil, size of pot, amount of light, type of tomato plant, length of time plants were grown.
Confounding variables: any of the above that are not held constant, plant health, diseases present in the soil or plant before it was purchased.
Results: Tomatoes fertilized with Brand A produced an average of 20 tomatoes per plant, while tomatoes fertilized with Brand B produced an average of 10 tomatoes per plant.
You’d want to use Brand A next time you grow tomatoes, right? But what if I told you that plants grown without fertilizer produced an average of 30 tomatoes per plant! Now what will you use on your tomatoes?
Results including control group: Tomatoes which received no fertilizer produced more tomatoes than either brand of fertilizer.
Conclusion: Although Brand A fertilizer produced more tomatoes than Brand B, neither fertilizer should be used because plants grown without fertilizer produced the most tomatoes!
Positive control groups are often used to show that the experiment is valid and that everything has worked correctly. You can think of a positive control group as being a group where you should be able to observe the thing that you are measuring (“the thing” should happen). The conditions in a positive control group should guarantee a positive result. If the positive control group doesn’t work, there may be something wrong with the experimental procedure.
Negative control groups are used to show whether a treatment had any effect. If your treated sample is the same as your negative control group, your treatment had no effect. You can also think of a negative control group as being a group where you should NOT be able to observe the thing that you are measuring (“the thing” shouldn’t happen), or where you should not observe any change in the thing that you are measuring (there is no difference between the treated and control group). The conditions in a negative control group should guarantee a negative result. A placebo group is an example of a negative control group.
As a general rule, you need a positive control to validate a negative result, and a negative control to validate a positive result.
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You read an article in the NY Times that says some spinach is contaminated with Salmonella.
You want to test the spinach you have at home in your fridge, so you wet a sterile swab and wipe it on the spinach, then wipe the swab on a nutrient plate (petri plate).
- You observe growth . Does this mean that your spinach is really contaminated? Consider an alternate explanation for growth: the swab, the water, or the plate is contaminated with bacteria. You could use a negative control to determine which explanation is true. If a swab is wet and wiped on a nutrient plate, do bacteria grow?
- You don’t observe growth. Does this mean that your spinach is really safe? Consider an alternate explanation for no growth: Salmonella isn’t able to grow on the type of nutrient you used in your plates. You could use a positive control to determine which explanation is true. If you wipe a known sample of Salmonella bacteria on the plate, do bacteria grow?
- In a drug trial, one group of subjects are given a new drug, while a second group is given a placebo drug (a sugar pill; something which appears like the drug, but doesn’t contain the active ingredient). Reduction in disease symptoms are measured. The second group receiving the placebo is a negative control group. You might expect a reduction in disease symptoms purely because the person knows they are taking a drug so they should be getting better. If the group treated with the real drug does not show more a reduction in disease symptoms than the placebo group, the drug doesn’t really work. The placebo group sets a baseline against which the experimental group (treated with the drug) can be compared. A positive control group is not required for this experiment.
- In an experiment measuring the preference of birds for various types of food , a negative control group would be a “placebo feeder”. This would be the same type of feeder, but with no food in it. Birds might visit a feeder just because they are interested in it; an empty feeder would give a baseline level for bird visits. A positive control group might be a food that squirrels are known to like. This would be useful because if no squirrels visited any of the feeders, you couldn’t tell if this was because there were no squirrels around or because they didn’t like any of your food offerings!
- To test the effect of pH on the function of an enzyme , you would want a positive control group where you knew the enzyme would function (pH not changed) and a negative control group where you knew the enzyme would not function (no enzyme added). You need the positive control group so you know your enzyme is working: if you didn’t see a reaction in any of the tubes with the pH adjusted, you wouldn’t know if it was because the enzyme wasn’t working at all or because the enzyme just didn’t work at any of your tested pH. You need the negative control group so you can ensure that there is no reaction taking place in the absence of enzyme: if the reaction proceeds without the enzyme, your results are meaningless.
Contributors and Attributions
Remixed and/or curated from the following works:
Bartee, L., Shriner, W., & Creech, C. (2017). 1.2 The Scientific Process . In Principles of Biology . Open Oregon Educational Resources (CC BY 4.0; Access for free at https://openoregon.pressbooks.pub/mhccmajorsbio/ ).


8.4: Induction in a Motionless Loop
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In this section, we consider the problem depicted in Figure \(\PageIndex{1}\), which is a single motionless loop of wire in the presence of a spatially-uniform but time-varying magnetic field. A small gap is introduced in the loop, allowing us to measure the induced potential \(V_T\). Additionally, a resistance \(R\) is connected across \(V_T\) in order to allow a current to flow. This problem was considered in Section 8.3 as an introduction to Faraday’s Law; in this section, we shall actually work the problem and calculate some values. This is intended to serve as an example of the application of Faraday’s Law, a demonstration of transformer emf, and will serve as a first step toward an understanding of transformers as devices.
Figure \(\PageIndex{1}\): A single loop of wire in the presence of an impressed spatially-uniform but time-varying magnetic field.In the present problem, the loop is centered in the \(z=0\) plane. The magnetic flux density is \({\bf B} = \hat{\bf b}B(t)\); i.e., time-varying magnitude \(B(t)\) and a constant direction \(\hat{\bf b}\). Because this magnetic field is spatially uniform (i.e., the same everywhere), we will find that only the area of the loop is important, and not it’s specific shape. For this reason, it will not be necessary to specify the radius of the loop or even require that it be a circular loop. Our task is to find expressions for \(V_T\) and \(I\).
To begin, remember that Faraday’s Law is a calculation of electric potential and not current. So, the approach is to first find \(V_T\), and then find the current \(I\) that flows through the gap resistance in response.
The sign convention for \(V_T\) is arbitrary; here, we have selected “\(+\)” and “\(-\)” terminals as indicated in Figure \(\PageIndex{1}\). 1 Following the standard convention for the reference direction of current through a passive device, \(I\) should be directed as shown in Figure \(\PageIndex{1}\). It is worth repeating that these conventions for the signs of \(V_T\) and \(I\) are merely references ; for example, we may well find that \(I\) is negative, which means that current flows in a clockwise direction in the loop.
We now invoke Faraday’s Law: \[V_T = - N \frac{\partial}{\partial t} \Phi \nonumber \] The number of windings \(N\) in the loop is 1, and \(\Phi\) is the magnetic flux through the loop. Thus:
\[V_T = - \frac{\partial}{\partial t} \int_{\mathcal{S}} { {\bf B} \cdot d{\bf s} } \nonumber \]
where \(\mathcal{S}\) is any open surface that intersects all of the magnetic field lines that pass through the loop. The simplest such surface is simply the planar surface defined by the perimeter of the loop. Then \(d{\bf s}=\hat{\bf n}ds\), where \(ds\) is the differential surface element and \(\hat{\bf n}\) is the normal to the plane of the loop. Which of the two possible normals to the loop? This is determined by the right-hand rule of Stokes’ Theorem. From the “\(-\)” terminal, we point the thumb of the right hand in the direction that leads to the “\(+\)” terminal by traversing the perimeter of the loop. When we do this, the curled fingers of the right hand intersect \(\mathcal{S}\) in the same direction as \(\hat{\bf n}\). To maintain the generality of results derived below, we shall not make the substitution \(\hat{\bf n}=+\hat{\bf z}\); nevertheless we see this is the case for a loop parallel to the \(z=0\) plane with the polarity of \(V_T\) indicated in Figure \(\PageIndex{1}\).
Taking this all into account, we have
\begin{aligned}
V_{T} &=-\frac{\partial}{\partial t} \int_{S}(\hat{\mathbf{b}} B(t)) \cdot(\hat{\mathbf{n}} d s) \\
&=-(\hat{\mathbf{b}} \cdot \hat{\mathbf{n}}) \frac{\partial}{\partial t} \int_{\mathcal{S}} B(t) d s
\end{aligned}
Since the magnetic field is uniform, \(B(t)\) may be extracted from the integral. Furthermore, the shape and the orientation of the loop are time-invariant, so the remaining integral may be extracted from the time derivative operation. This leaves:
\[V_{T}=-(\hat{\mathbf{b}} \cdot \hat{\mathbf{n}})\left(\frac{\partial}{\partial t} B(t)\right) \int_{\mathcal{S}} d s \nonumber \]
The integral in this expression is simply the area of the loop, which is a constant; let the symbol \(A\) represent this area. We obtain
\[\boxed{ V_T = - \left(\hat{\bf b}\cdot\hat{\bf n}A\right) \frac{\partial}{\partial t} B(t) } \label{m0056_eVT} \]
which is the expression we seek. Note that the quantity \(\hat{\bf b}\cdot\hat{\bf n}A\) is the projected area of the loop. The projected area is equal to \(A\) when the the magnetic field lines are perpendicular to the loop (i.e., \(\hat{\bf b}=\hat{\bf n}\)), and decreases to zero as \(\hat{\bf b}\cdot\hat{\bf n}\to 0\). Summarizing:
The magnitude of the transformer emf induced by a spatially-uniform magnetic field is equal to the projected area times the time rate of change of the magnetic flux density, with a change of sign. (Equation \ref{m0056_eVT}).
A few observations about this result:
- As promised earlier, we have found that the shape of the loop is irrelevant; i.e., a square loop having the same area and planar orientation would result in the same \(V_T\). This is because the magnetic field is spatially uniform, and because it is the magnetic flux (\(\Phi\)) and not the magnetic field or shape of the loop alone that determines the induced potential.
- The induced potential is proportional to \(A\); i.e., \(V_T\) can be increased by increasing the area of the loop.
- The peak magnitude of the induced potential is maximized when the plane of the loop is perpendicular to the magnetic field lines.
- The induced potential goes to zero when the plane of the loop is parallel to the magnetic field lines. Said another way, there is no induction unless magnetic field lines pass through the loop.
- The induced potential is proportional to the rate of change of \({\bf B}\). If \({\bf B}\) is constant in time, then there is no induction.
Finally, the current in the loop is simply \[I = \frac{V_T}{R} \nonumber \] Again, electromagnetic induction induces potential, and the current flows only in response to the induced potential as determined by Ohm’s Law. In particular, if the resistor is removed, then \(R\to\infty\) and \(I\to0\), but \(V_T\) is unchanged.
One final comment is that even though the current \(I\) is not a direct result of electromagnetic induction, we can use \(I\) as a check of the result using Lenz’s Law (Section 8.2). We’ll demonstrate this in the example below.
Example \(\PageIndex{1}\): Induction in a motionless circular loop by a linearly-increasing magnetic field
Let the loop be planar in the \(z=0\) plane and circular with radius \(a=10\) cm. Let the magnetic field be \(\hat{\bf z}B(t)\) where
\begin{aligned}
B(t) &=0 \quad, t<0 \\
&=B_{0} t / t_{0}, 0 \leq t \leq t_{0} \\
&=B_{0} \quad, t>t_{0}
\end{aligned}
i.e., \(B(t)\) begins at zero and increases linearly to \(B_0\) at time \(t_0\), after which it remains constant at \(B_0\). Let \(B_0=0.2\) T, \(t_0=3\) s, and let the loop be closed by a resistor \(R=1~\mbox{k}\Omega\). What current \(I\) flows in the loop?
Solution
Adopting the sign conventions of Figure \(\PageIndex{1}\) we first note that \(\hat{\bf n}=+\hat{\bf z}\); this is determined by the right-hand rule with respect to the indicated polarity of \(V_T\). Thus, Equation \ref{m0056_eVT} becomes
\[V_T = - \left(\hat{\bf b}\cdot\hat{\bf z}A\right) \frac{\partial}{\partial t} B(t) \nonumber \]
Note \(\hat{\bf b}\cdot\hat{\bf z}A=A\) since \(\hat{\bf b}=\hat{\bf z}\); i.e., because the plane of the loop is perpendicular to the magnetic field lines. Since the loop is circular, \(A = \pi a^2\). Also
\begin{aligned}
\frac{\partial}{\partial t} B(t) &=0 \quad, t<0 \\
&=B_{0} / t_{0}, 0 \leq t \leq t_{0} \\
&=0 \quad, t>t_{0}
\end{aligned}
Putting this all together:
\[V_T = -\pi a^2 \frac{B_0}{t_0} = -2.09~\mbox{mV} ~\mbox{,}~0\le t \le t_0 \nonumber \]
and \(V_T=0\) before and after this time interval, since \({\bf B}\) is constant during those times. Subsequently the induced current is \[I = \frac{V_T}{R} = -2.09~\mu\mbox{A} ~\mbox{,}~0\le t \le t_0 \nonumber \] and \(I=0\) before and after this time interval. We have found that the induced current is a constant clockwise flow that exists only while \({\bf B}\) is increasing.
Finally, let’s see if the result is consistent with Lenz’s Law. The current induced while \({\bf B}\) is changing gives rise to an induced magnetic field \({\bf B}_{ind}\). From the right-hand rule that relates the direction of \(I\) to the direction of \({\bf B}_{ind}\) (Section 7.5), the direction of \({\bf B}_{ind}\) is generally \(-\hat{\bf z}\) inside the loop. In other words, the magnetic field associated with the induced current opposes the increasing impressed magnetic field that induced the current, in accordance with Lenz’s Law.
Example \(\PageIndex{3}\): Induction in a motionless circular loop by a sinusoidally-varying magnetic field.
Let us repeat the previous example, but now with \[B(t) = B_0\sin{2\pi f_0 t} \nonumber \] with \(f_0=1\) kHz.
Solution
Now \[\frac{\partial}{\partial t}B(t) = 2\pi f_0 B_0\cos{2\pi f_0 t} \nonumber \] So \[V_T = -2\pi^2 f_0 a^2 B_0 \cos{2\pi f_0 t} \nonumber \] Subsequently \[I = \frac{V_T}{R} = -\frac{2\pi^2 f_0 a^2 B_0 }{R} \cos{2\pi f_0 t} \nonumber \] Substituting values, we have: \[I = -(39.5~\mbox{mA}) \cos{ \left[ (6.28~\mbox{krad/s}) t \right] } \nonumber \]
It should be no surprise that \(V_T\) and \(I\) vary sinusoidally, since the source (\({\bf B}\)) varies sinusoidally. A bit of useful trivia here is that \(V_T\) and \(I\) are 90\(^{\circ}\) out of phase with the source. It is also worth noting what happens when \({\bf B}(t)=0\). This occurs twice per period, at \(t=n/2f_0\) where \(n\) is any integer, including \(t=0\). At these times \({\bf B}(t)\) is zero, but \(V_T\) and hence \(I_R\) are decidedly non-zero; in fact, they are at their maximum magnitude. Again, it is the change in \({\bf B}\) that induces voltage and subsequently current, not \({\bf B}\) itself.
- A good exercise for the student is to repeat this problem with the terminal polarity reversed; one should obtain the same answer.↩


10.1: Data Tables and Ranges
By the end of this section, you will be able to:
- Differentiate between a spreadsheet and a Data Table
- Explain the advantages of a Data Table
- Create/insert a Data Table from data in a spreadsheet
- Define a cell range
- Select and name a cell range
- Differentiate between a cell range and an array
WorldCorp has sufficient savings for capital investment for the Fairfax project, yet the chief financial officer (CFO) wants to borrow from a bank about 50 percent of the amount of the estimate for the new addition. There are many banks in Fairfax, each offering different interest rates and years to pay back the loan. The CFO is not sure which offer is best, as the longer the loan period is, the more interest WorldCorp will end up paying, yet the shorter the loan period, the higher the payment will be. The CFO is also analyzing the different interest rates and how they affect the installments and subsequently the total interest charge over the life of the loan.
This case shows why it’s important to learn the intricacies of using Microsoft Excel’s Data Tables. Data Tables can be helpful to assist management in determining the best course of action by evaluating interest charges and monthly payments. In this section, you will create a data table from existing data and will discover the advantages of using the Data Table tool in the What-If Analysis, plus related concepts and definitions, such as data ranges and arrays. The Data Table’s powerful analysis options will be further explored in Organizing and Presenting Data.
Differentiating Between a Spreadsheet and a Data Table
Spreadsheets have a wide variety of applications and can be used for accounting, budgeting, data collection and analysis, and operations efficiency. Spreadsheets are used in all departments of corporations, such as marketing or operations. These departments can use spreadsheets in a variety of ways, such as compiling information from market studies, or performing statistical analysis on operations from the manufacturing plant.
A Data Table is a scenario-building tool. It lets users choose a set of cells on the spreadsheet, and then outputs different solutions or forecasts depending on different scenarios. The difference between a Data Table and a spreadsheet is that a spreadsheet is a file with blank cells that can be formatted using the various tools and commands contained in the application. A Data Table can be part of (embedded in) a spreadsheet, but a spreadsheet can also include a graph, some text for information, or the listing of the inputs for the Data Table. The Data Table feature of Excel automatically builds a table of data based on certain inputs that is part of the spreadsheet. Figure 10.2 shows an example of an Excel-generated Data Table, as well as the data used in the scenarios. You will build a Data Table after learning more about its uses.
The Advantages of a Data Table
The advantages of Data Tables are many. One of their most important advantages is their ability to automate scenario-building. Another advantage is increased accuracy due to less manual entry of data. If you were to create a scenario from a blank workbook, you would need to build out the formulas initially and then copy them through the desired levels of the inputs. This increases the possibility of user error caused by inputting values or formulas incorrectly. By using the Data Table feature, all that work is done automatically—and quickly. Data Tables allow users to view all the levels of inputs in a single table without inputting multiple formulas.
Consider the example of expanding Fairfax’s WorldCorp location. You have gathered the needed information to determine the amount of the loan you will need to finance the added capacity. You have the loan terms from banks in the area and you know that the loan payment period will be ten years. WorldCorp is working on its annual budget, so the CFO wants you to investigate the annual payments, not the monthly payments. The CFO also wants to examine the variability of the annual installment if the interest rate is increased by 0.5 percent increments per year over the life of the loan. The Data Table in the What-If Analysis will automatically perform the annual payment calculations for each increment. The CFO will then be able to look at the results and determine which loan from which bank will best meet the company’s needs.
The first step in this analysis is to set up your data. Since you’re looking at how the interest rate affects the annual payment, you can use Excel’s Payment (PMT) function, which uses the present value (the value of the loan), the number of payments, and the rate. The PMT function is in the Financial library of functions found on the Formulas tab. The present value (PV) is principal to be borrowed; in this case, we will use $750,000, which is half of the total cost of the expansion. This was determined by the initial research by the CFO as the amount needed to finance the company growth. The time period of the loan has been set at ten years. Most loans are paid monthly, but since you are analyzing the annual payments, you will use 10 (not 120) for N (shown as Nper in the function builder) to define the number of payments. After gathering all the interest rates offered by the banks, it was determined that the average interest rate (I) of all is 6.5 percent. Using these input values, the yearly payment can be determined using the financial PMT function. Figure 10.3 shows the building of the PMT function in Excel.
The result is an annual payment on the loan of $104,328.52 (Figure 10.4). The output for Excel with the PMT function is negative by default, which is displayed in parentheses and in red. This is to indicate that the loan total will decrease by that amount each year after the payment is made. Keep this in mind as you examine the output.
The next step in the analysis is to compare the payments with different loan terms. You can calculate the payment manually using the PMT function many times—that is, every time you change the interest rate. You can play around with it by changing the interest rate and seeing how it affects the payment. If the rate is lower, the payment will be lower, and if the rate is higher, the payment will be higher. For example, in Figure 10.3, if you change the rate to 5 percent, the payment automatically updates to $97,128.43. However, this approach does not retain all of the options easily. Unless you make a separate table to capture the impact of changes in the interest rate on the payment, that information is lost each time you make a change to the inputs. If you do create a table, you would need to format it in a manner suitable for additional analysis and decision making. It would be a time-consuming exercise to record all of the data for each interest rate, and then you would need to take an extra step to summarize it in an easy-to-follow table. By creating a Data Table, you can use various levels of the interest rate to calculate the payment and summarize it in one concise table.
PMT is a useful tool for comparing different interest rates and their impact on monthly payment. It can also be used to determine a mortgage payoff schedule. Paying down a mortgage at a faster pace than agreed upon can lead to significant savings in interest payments. Even one additional payment a year can have a large impact on the amount of interest paid. You can find any number of amortization templates available online, either where you input the information and the site calculates it for you, or through an Excel template. Do some online research to find an Excel template to help you calculate your mortgage payoff schedule. Open the template and review the formulas. Notice the use of IF and PMT functions throughout. Using the template, locate a home to purchase and determine the payoff schedule for the home using today’s interest rates. If you own a home already, determine what one additional payment per year will do to your final home cost upon payoff.
Creating a Data Table from Data in a Spreadsheet
To facilitate the process of comparing various interest rates on the annual payment, you can create a Data Table . To do so, the information must be arranged on the spreadsheet in a specific way. First, make a list in a column of the interest rates being examined. The order doesn’t matter to Excel, but it makes the data easier to analyze if you put them in numerical order, either lowest rate to highest or vice versa. In the next column to the right and one row above, use the PMT function to calculate the payment at the current rate (recall, you used 6.5 percent), as shown in Figure 10.5.
Next, select the area of the spreadsheet where the changing input values are contained. In this example, that is cells E1:F8. On the Data tab, you will see the What-If Analysis menu group on the right side. Select the drop-down arrow and choose Data Table (Figure 10.6).
When the Data Table tool is selected, a dialog box appears for inputting data. Because the interest rates are listed in columns, use the “Column input cell” box. In this cell, place the cell reference for the original interest rate used in the PMT function. Notice the dollar sign ($) in front of the letter and number in the cell reference. This will set the column input cell to always refer to the value in cell B2. This is referred to as an absolute reference . You will learn more about absolute references in Absolute Cell References. For this example, the cell is B2 (Figure 10.7).
The Data Table will automatically calculate the payment at each of the interest rates listed in the column below the PMT, as shown in Figure 10.8. You will need to format the data to be Accounting or Currency.
This process will save a lot of time, as it eliminates manually creating a formula for each cell. Furthermore, with the same Data Table, any of the values in the list can be changed and the information will automatically update to the correct value. For example, to see the payments for a 6.4 percent interest rate instead of 6.5 percent, simply change that one value, and the row’s value will be autofilled to calculate the new installment payment. The rest of the data table stays the same, as you can see in Figure 10.9.
Suppose you have taken this Data Table to the WorldCorp CFO, who tells you that the company will not pay high interest rates and will only negotiate and meet with the executives of banks that offer rates at or below 7 percent. There is no need to create a new Data Table that only includes the data for that range of rates because you may need the information at a later point. Instead, you can temporarily hide the values that exceed the 7 percent maximum using Excel’s filtering tools.
To use a filter within the Data Table, select the rate column, and then go to the Data tab. In the Sort & Filter command group, select Filter (the funnel icon). When Filter is selected, a small arrow will appear at the top of the column, as shown in Figure 10.10a. Select that arrow and go to Number Filters, then select Less Than Or Equal To.... On the dialog box that appears next, select 7.00% from the choices, and then click OK. The result is that the Data Table will only show values that are less than or equal to 7 percent, as shown in Figure 10.10b.
Now that you have created a Data Table with one variable, you will follow a similar process to analyze two variables. With the added capacity in Fairfax, WorldCorp can sell more products to help the company reach its strategic goals. Increasing unit sales could also allow the company to reduce its price per unit. You would like to better understand the impact of the increased sales and price changes on the annual revenue for WorldCorp. Let’s assume that in 2020, WorldCorp sold 5,000 RX-500 digital scales in the state of Virginia, at the retail price of $17.99 per unit. That would mean an annual revenue of $89,950 for this item. What would happen if the price was decreased to $16.99, in hopes that the unit sales would increase? Or, conversely, what would happen if the price increased to $18.99? WorldCorp is not sure how the public will react to either option, but you can estimate the revenue it will produce. Let’s assume that either sales will stay the same (i.e., 5,000 per year), decrease to 3,000, or increase to 7,000. With the Data Table tool located in the What-If Analysis , all the revenues will be autofilled for the different unit sales and the different prices (Figure 10.11). To create this table, follow the steps for the loan interest rate example. However, instead of only including the column input, also include the row input.
Defining, Selecting, and Naming a Cell Range
A cell range is a selection of rows or columns of cells that is given a name to make it easier for the worksheet designer to find that specific group of cells later. It can include a group of cells that are next to each other or separated across the worksheet or workbook . Defining a cell range is especially useful when designing a large Excel worksheet with a substantial amount of data, or when interconnecting datasets or different spreadsheets. Naming a range of cells is referred to in Excel as defining a range. You can define a range in three different ways.
Method 1: Define a Range
First, select the range of cells you want to name. Then, click on the leftmost box below the ribbon (Excel calls this the Name Box ). After the cursor on the Name Box begins blinking, type the name you want to use for the data range, and press Enter. Subsequently, you may refer to this range of cells as the name you typed in the Name Box. Type “digital scale” in the Name Box, as shown in Figure 10.12.
Method 2: Define a Range
Begin by selecting the desired range of cells, then go to the Formulas tab, and select Define Name. A new dialog box will appear; it has three fields: Name, Scope, and Comment. Next to Name, simply type the name you want to use for the data range; under Scope, select Workbook; the Comment field is optional. Click OK to complete the process (Figure 10.13).
Method 3: Define a Range
The third way to define a range is more comprehensive, as it allows you to select various columns or a whole Data Table. You need to select the entire table, or the set of columns (or rows), that you want to define. After making your selection, go to the Formulas tab in the ribbon and choose Create from Selection. A dialog box will appear and where you select the headers. (Many tables have column headings only, and other tables have row headings.) In the dialog box, you will select if the table has the column headings on the bottom or on the top, or if the row headings are on the right or on the left.
Using the WorldCorp example of the loan interest rates, choose Left column, and click OK, as shown in Figure 10.14. Because you didn’t type anything, as this method is more geared for tables, Excel automatically gave defined names to the data ranges (rows). Excel uses the text in the left column of the table to name (define) the data ranges. Spaces are automatically converted to “_” by Excel. Consequently, the data ranges are Years_N, Rate_I, Principal_PV, and Payment_PMT.
Differentiating Between a Cell Range and an Array
An array is similar to a cell range in that it is a collection of data in ranges of cells, but arrays do not have to be contiguous (right next to each other). Arrays can contain text or numbers and can span multiple rows and columns if needed. The array can be a collection of cells that consists of unrelated elements, whereas cell ranges have a start and finish and are made of elements related to each other. Arrays also are different from data ranges in that they can work creatively to simultaneously calculate different computations at the same time. An array can replace many standard formulas and make simple computations more efficient. The array formula performs multiple calculations on several values, so using arrays and array formulas can save users time.
Let’s use the WorldCorp digital scale example and recalculate it using arrays. The unit sales of each model are added to the previous data in column C. Then, using arrays, you can determine the total revenue without calculating the revenue for each model separately. Figure 10.15a shows how you can add a column for the revenue of each model and then add them up to get the total revenue of the digital scales. Figure 10.15b shows the total revenue for sales of all models in cell C7, without any of the individual calculations, using an array.
To use an array to calculate the total revenue in Figure 10.15b, click on a blank cell (here, cell C7), and type the function = sum(b2:b6*c2:c6) and hold down Ctrl+Shift as you press Enter. This function calculates the sum of the revenue for each item, as shown in the ranges in the parentheses. As visible in the Formula Bar, Excel recognizes b2:b6 from the defined data ranges, and renames it “Price” in the formula. Excel also adds a curly brace (“{ }”) before and after the whole function (Figure 10.15b) because it recognizes it as an array. Excel then does all of the calculations we did manually in Figure 10.15a in one step and provides the total revenue. You’ll notice that the total revenue values match, regardless of whether you do the calculations manually or use an array. When a selection is defined as an array, then any formula is calculated as an array formula, which allows for multiple calculations to happen simultaneously.