Answers and Explanations—5.2 The Golden Ratio

P1	Paragraph 1	Comments
S1	In math, simple relationships can often take on critical roles.
2	One such relationship, the Golden Ratio, has captivated the imagination and appealed to mathematicians, architects, astronomers, and philosophers alike.	Golden Ratio = captivating.
3	The Golden Ratio has perhaps had more of an effect on civilization than any other well-known mathematical constant.
4	To best understand the concept, start with a line and cut it into two pieces (as seen in the figure).	How to find = cut line in 2.
5	If the pieces are cut according to the Golden Ratio, then the ratio of the length of the longer piece to the length of the shorter piece (A : B) would be the same as the ratio of the length of the entire line to the length of the longer piece (A+B : A).
6	Rounded to the nearest thousandth, both of these ratios will equal 1.618 : 1.	Ratio = 1.618 : 1.

	According to paragraph 1, which of the following is true about the Golden Ratio?	Fact. S2–3 introduce the Golden Ratio and discusses the ratio’s broad appeal and applications. S4–6 define it mathematically.
✗	A It was invented by mathematicians.	The paragraph does not discuss the invention of the ratio. If anything, the ratio would be discovered rather than invented.
✓	B It has significantly impacted society in general.	Correct. S3: “The Golden Ratio has perhaps had more of an effect on civilization than any other well-known mathematical constant.”
✗	C It is most useful to astronomers and philosophers.	S2 states that the ratio has appeal to astronomers and philosophers. But S2 does not argue that the ratio is more useful to these groups than it is to mathematicians, philosophers, or others.
✗	D It is used to accurately calculate a length.	S4–6 explain the Golden Ratio using the example of a line cut into pieces, but the Golden Ratio is not used to accurately calculate a length. It is used to talk about a relationship.

	The phrase “appealed to” in the passage is closest in meaning to	Vocabulary. In this context, “appealed” means to attract or to be interesting to someone or something.
✓	A interested	Correct. The Golden Ratio “appealed to” mathematicians = it attracted or interested them.
✗	B defended	Unrelated. “Defended” can either mean to protect or guard, or to support or justify.
✗	C requested	“Appealed to” can also mean “made a request to,” but that does not make sense in context. The Golden Ratio isn’t asking anything of mathematicians. Nor is it asking for mathematicians (which is what “requested mathematicians” means).
✗	D repulsed Opposite. “Repulsed” = drive away or reject.

P2	Paragraph 2	Comments
S1	The first recorded exploration of the Golden Ratio comes from the Greek mathematician Euclid in his 13-volume treatise on mathematics, Elements, published in approximately 300 BC.	First recorded discussion of the ratio.
2	Many other mathematicians since Euclid have studied the ratio.
3	It appears in various elements related to certain “regular” geometric figures, which are geometric figures with all side lengths equal to each other and all internal angles equal to each other.	Examples of the ratio in mathematics.
4	Other regular or nearly regular figures that feature the ratio include the pentagram (a five-sided star formed by five crossing line segments, the center of which is a regular pentagon) and three-dimensional solids such as the dodecahedron (whose 12 faces are all regular pentagons).

	According to paragraph 2, all of the following are true of the Golden Ratio EXCEPT:	Negative Fact. Three answers are true and supported by the passage. One answer is false or just unsupported. Facts about the Golden Ratio could be anywhere in P2.
✗	A Its first known description occurred in approximately 300 BC.	S1 notes that the ratio was first recorded in a book published around 300 BC.
✗	B It was studied by mathematicians after Euclid.	S2: “Many other mathematicians since [= after] Euclid have studied the ratio.”
✓	C It only occurs in regular geometric figures.	Correct. S3: the ratio shows up in “certain ‘regular’ geometric figures.” It doesn’t say that the ratio only occurs in them. In fact, S4 mentions that the Golden Ratio is featured in “nearly regular” figures.
✗	D It appears in some three-dimensional geometric figures.	S4 mentions that the Golden Ratio is featured in “three-dimensional solids such as the dodecahedron.”

P3	Paragraph 3	Comments
S1	The Fibonacci sequence, described by Leonardo Fibonacci, demonstrates one application of the Golden Ratio.	The Fibonacci sequence demonstrates the ratio.
2	The Fibonacci sequence is defined such that each term in the sequence is the sum of the previous two terms, where the first two terms are 0 and 1.
3	The next term would be 0 + 1 = 1, followed by 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, etc.
4	This sequence continues: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.
5	As the sequence progresses, the ratio of any number in the sequence to the previous number gets closer to the Golden Ratio.
6	This sequence appears repeatedly in various applications in mathematics.

	According to paragraph 3, which of the following is true about the Fibonacci sequence?	Fact. S1 introduces the Fibonacci sequence. S2–4 define the sequence. S5 explains how this sequence relates to the Golden Ratio. S6 mentions that the sequence is important in mathematics.
✓	A It can be used to estimate the Golden Ratio.	Correct. S5: “As the sequence progresses, the ratio of any number in the sequence to the previous number gets closer to the Golden Ratio.” So you can estimate the Golden Ratio by computing many items in the sequence, then calculating the ratio of two consecutive terms in the sequence.
✗	B It cannot be computed without the use of the Golden Ratio.	S2–4 describe how to calculate the Fibonacci sequence without using the Golden Ratio.
✗	C It was discovered by Euclid in approximately 300 BC.	S1 notes that Fibonacci described the sequence. As discussed in P2, Euclid explored the Golden Ratio, not the Fibonacci sequence.
✗	D No two terms in the sequence are equal to one another.	S4 mentions the beginning of the sequence: 0, 1, 1, 2, 3… The second and third terms are equal to one another.

	The word “progresses” in the passage is closest in meaning to	Vocabulary. “Progress” is about forward movement. In terms of a sequence or series of events, it means to advance, continue, or count up.
✗	A calculates	The sequence itself does not do any calculating. The person trying to determine the numbers in the sequence must calculate them (or look them up).
✗	B declines	Opposite in this case, because the sequence actually keeps getting bigger.
✓	C continues	Correct. The sequence “progresses” = the sequence continues.
✗	D disintegrates	Opposite. “Disintegrate” = to break up or fall apart

P4	Paragraph 4	Comments
S1	The allure of the Golden Ratio is not limited to mathematics, however.	The ratio goes beyond math.
2	Many experts believe that its aesthetic appeal may have been appreciated before it was ever described mathematically.	Aesthetic/artistic appeal may have come first.
3	In fact, ample evidence suggests that many design elements of the Parthenon building in ancient Greece bear a relationship to the Golden Ratio.	The ratio is in the Parthenon in ancient Greece.
4	Regular pentagons, pentagrams, and decagons were all used as design elements in its construction.
5	In addition, several elements of the façade of the building incorporate the Golden Rectangle, whose length and width are in proportion to the Golden Ratio.
6	Since the Parthenon was built over a century before Elements was published, the visual attractiveness of the ratio, at least for the building’s designers, may have played a role in the building’s engineering and construction.

	According to paragraph 4, which of the following is true about the construction of the Parthenon?	Fact. S3 notes that the Parthenon may be related to the Golden Ratio. S4–5 give examples. S6 provides a hypothesis about why this might have been the case.
✗	A It was based upon the writings of Euclid.	P4 does not mention Euclid’s writings, other than to state that the Parthenon was constructed more than 100 years before Euclid’s publication of Elements.
✓	B Aesthetics may have played a role in the Parthenon’s use of elements that exhibit the Golden Ratio.	Correct. S2 states that the ratio’s “aesthetic appeal may have been appreciated before it was ever described mathematically.” S3: “many design elements of the Parthenon… bear a relationship to the Golden Ratio.” S6: “the visual attractiveness of the ratio… may have played a role in the building’s engineering and construction.”
✗	C It is an example of mathematics being prioritized over aesthetics.	S2 argues the opposite: aesthetics may have come before mathematics. P4 goes on to discuss the Parthenon as an example of this point.
✗	D The designer of the Parthenon is unknown.	The paragraph does not mention the designer or designers of the Parthenon by name. But you can’t conclude that the designer or designers are unknown.

	Paragraph 4 supports the idea that the designers of the Parthenon	Inference. The designers of the Parthenon are mentioned only in S6. But the whole paragraph is concerned with design aspects of the Parthenon—in particular, the aesthetics of the Golden Ratio and its appearance in the Parthenon’s design.
✗	A were able to derive the Golden Ratio mathematically before it was formally recorded by Euclid	S4–5 describe Parthenon design elements that have a relationship to the Golden Ratio. However, you don’t know that the designers were able to define the ratio formally or mathematically. The sentences only imply that they were aware of the ratio on some level.
✓	B were aware of the Golden Ratio on some level, even if they could not formally define it	Correct. S4–5 describe Parthenon design elements that have a relationship to the ratio. So the designers were probably aware of the ratio in some way. S6 mentions that the Parthenon was constructed well before the formal description of the ratio was published (in Elements, mentioned in P2). So the Parthenon designers may not have been able to define the ratio formally.
✗	C were mathematicians	The paragraph does not provide any support for the idea that the designers were mathematicians.
✗	D were more interested in aesthetic concerns than sound architectural principles	The paragraph suggests that the aesthetic appeal of the Golden Ratio played a part in various design elements of the Parthenon. But nothing suggests that these aesthetic considerations were more important than “sound architectural principles.”

	The word “elements” in the passage is closest in meaning to	Vocabulary. The “elements” of something are its parts or components.
✗	A origins	In some contexts, “elements” can mean “basics” or “fundamentals,” which is similar to “origins.” But here, the author means “parts” or “components.”
✗	B substances	In the context of chemistry, an “element” is a certain kind of basic substance or chemical. But here, the author means “parts” or “components.”
✗	C drawings	Sketches or pictures are unrelated here.
✓	D components	Correct. “Many design elements of the Parthenon… bear a relationship to the Golden Ratio” = many components or parts of the building’s design are related to the Golden Ratio.

P5	Paragraph 5	Comments
S1	Numerous studies indicate that many pieces of art now considered masterpieces may also have incorporated the Golden Ratio in some way.	The Golden Ratio also seems to appear in great art.
2	Leonardo da Vinci created drawings illustrating the Golden Ratio in numerous forms to supplement the writing of De Divina Proportione.	Various examples.
3	This book on mathematics, written by Luca Pacioli, explored the application of various ratios, especially the Golden Ratio, in geometry and art.
4	Analysts believe that the Golden Ratio influenced proportions in some of da Vinci’s other works, including his Mona Lisa and Annunciation paintings.
5	The ratio is also evident in certain elements of paintings by Raphael and Michelangelo.
6	Swiss painter and architect Le Corbusier used the Golden Ratio in many design components of his paintings and buildings.
7	Finally, Salvador Dalí intentionally made the dimensions of his work Sacrament of the Last Supper exactly equal to the Golden Ratio, and incorporated a large dodecahedron as a design element in the painting’s background.

	Why does the author mention that Dalí “incorporated a large dodecahedron as a design element in the painting’s background”?	Purpose. S7 discusses Dalí’s painting Sacrament of the Last Supper. The sentence mentions two elements that have a relationship with the Golden Ratio. One of these elements is the dodecahedron mentioned in the highlighted text. Recall that P2 lists the regular dodecahedron as a geometric figure with a relationship to the Golden Ratio.
✗	A To demonstrate Dalí’s frequent use of geometric shapes	S7 is the only sentence that discusses Dalí’s work. You do not know that Dalí made “frequent use of geometric shapes,” because you only have one example (the dodecahedron).
✓	B To illustrate the extent to which the Golden Ratio has influenced some works of art	Correct. P5 focuses on how the Golden Ratio is used in famous artwork. The Dalí painting discussed in S7 is an example of this use. The author mentions the dodecahedron in order to make a connection to the Golden Ratio. So the highlighted text illustrates the extent to which the Golden Ratio has influenced art.
✗	C To argue that certain style elements in art are more effective than others	The highlighted text does not argue or offer an opinion on anything. Also, the paragraph does not discuss the effectiveness of style elements in any way.
✗	D To refer to works by other artists such as da Vinci and Le Corbusier	The highlighted text does not refer to any aspects of the work of other artists. The only common thread among the works described in P5 is that all of them seem to have incorporated the Golden Ratio somehow.

	According to paragraph 5, da Vinci’s illustrations in De Divina Proportione and two of his paintings, the Mona Lisa and Annunciation,	Fact. S2–4 discuss these works.
✓	A exhibit evidence that da Vinci’s work was influenced by the Golden Ratio	Correct. S2 notes that da Vinci “created drawings illustrating the Golden Ratio” for De Divina Proportione. S4: “Analysts believe that the Golden Ratio influenced proportions in some of da Vinci’s other works,” including the paintings mentioned.
✗	B illustrate that he had a higher commitment to the Golden Ratio than other artists	Nothing in the paragraph suggests that da Vinci was more or less committed to the Golden Ratio than other artists discussed in P5.
✗	C provide examples showing that Renaissance art was more influenced by the Golden Ratio than modern art	Nothing in the passage tells you the historical category of art (Renaissance or modern) of the examples given. Moreover, you aren’t told whether any artist is more or less influenced by the Golden Ratio than any other.
✗	D demonstrate that da Vinci’s work was at least as influential as the work of mathematicians or architects	P5 mentions a mathematician (Pacioli) and an architect (Le Corbusier). But nothing states or implies that da Vinci’s work was any more or less influential than work done by these or any other mathematicians or architects.

P6	Paragraph 6	Comments
S1	The Golden Ratio even appears in numerous aspects of nature.	The ratio even appears in nature.
2	Philosopher Adolf Zeising observed that it was a frequently occurring relation in the geometry of natural crystal shapes.	Various examples.
3	He also discovered a common recurrence of the ratio in the arrangement of branches and leaves on the stems of many forms of plant life.
4	Indeed, the Golden Spiral, formed by drawing a smooth curve connecting the corners of Golden Rectangles repeatedly inscribed inside one another, approximates the arrangement or growth of many plant leaves and seeds, mollusk shells, and spiral galaxies.

	By including the text “formed by drawing a smooth curve connecting the corners of Golden Rectangles repeatedly inscribed inside one another,” the author is	Purpose. S4 discusses elements in nature that exhibit the Golden Spiral. The highlighted text defines the Golden Spiral in relation to the Golden Ratio (which is part of the Golden Rectangle).
✗	A emphasizing the importance of the Golden Spiral	The highlighted text makes no mention of the Golden Spiral’s importance. It does not even mention any applications or examples.
✗	B providing the reader with instructions for creating a Golden Spiral	While the reader may be tempted to create a Golden Spiral based on the given description, that is not the purpose of the highlighted text. Rather, the author is just defining the Golden Spiral, so that the reader can loosely visualize it and understand how it relates to the Golden Ratio.
✓	C defining the Golden Spiral in relation to the Golden Ratio	Correct. The highlighted text defines the Golden Spiral in terms of Golden Rectangles, which themselves are related to the Golden Ratio (as defined in P4).
✗	D providing illustrations of elements in nature that exhibit the Golden Ratio	The highlighted text does not provide “illustrations” or examples of elements in nature. Instead, it defines the Golden Spiral mathematically.

	According to paragraph 6, which of the following is an example from nature that demonstrates the Golden Spiral?	Fact. S4 is the only sentence that discusses the Golden Spiral. The correct answer must be found there.
✗	A The arrangement of branches in some forms of plant life	S3 discusses how the Golden Ratio occurs “in the arrangement of branches… on the stems of many forms of plant life.” But this is before the Golden Spiral is introduced. S4 specifically focuses on “plant leaves and seeds,” not branches.
✗	B The common recurrence of the spiral throughout nature	The Golden Spiral may commonly recur throughout nature, according to S4, but just saying that it does so is not itself an example of the spiral.
✗	C The geometry of natural crystal shapes	S2 discusses how the Golden Ratio occurs “in the geometry of natural crystal shapes.” But this is before the Golden Spiral is introduced.
✓	D The arrangement or growth of some galaxies	Correct. S4: “the Golden Spiral… approximates the arrangement or growth of many… spiral galaxies.”

P3	Paragraph 3	Comments
S1	The Fibonacci sequence, described by Leonardo Fibonacci, demonstrates one application of the Golden Ratio.
2–4	A The Fibonacci sequence is defined such that each term in the sequence is the sum of the previous two terms, where the first two terms are 0 and 1. The next term would be 0 + 1 = 1, followed by 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, etc. This sequence continues: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.	Placement here would suggest that Pascal’s triangle is an application of the Golden Ratio. But it is not—it is an application of the Fibonacci sequence.
5	B As the sequence progresses, the ratio of any number in the sequence to the previous number gets closer to the Golden Ratio.	“For example” doesn’t work here. Pascal’s triangle is not an example of the way the Fibonacci sequence is built. Placement here interrupts the description of the Fibonacci sequence.
6	C This sequence appears repeatedly in various applications in mathematics.	“For example” doesn’t work here. Pascal’s triangle is not an example of how the ratio of consecutive terms in the Fibonacci sequence gets closer to the Golden Ratio.
End	D	Correct. “For example” works well. Pascal’s triangle is an example of one of the “various applications in mathematics” (S6) of the Fibonacci sequence.

	For example, it can be shown that the sum of the entries in consecutive diagonals of Pascal’s triangle, useful in calculating binomial coefficients and probabilities, correspond to consecutive terms in the Fibonacci sequence. Where would the sentence best fit?	Insert Text. “For example” tells you that this sentence must be an example of the idea in the previous sentence. The paragraph is about the Fibonacci sequence. This sentence describes something “new” that is related to that sequence (namely, some property of Pascal’s triangle).
✗	A Choice A
✗	B Choice B
✗	C Choice C
✓	D Choice D	Correct.

	Whole Passage	Comments
P1	In math, simple relationships can often…	Golden Ratio = captivating. How to find = cut line in 2. Ratio = 1.618 : 1.
P2	The first recorded exploration of the Golden Ratio…	First recorded discussion of the ratio. Examples of the ratio in mathematics.
P3	The Fibonacci sequence, described by…	The Fibonacci sequence demonstrates the ratio.
P4	The allure of the Golden Ratio is not limited to…	The ratio goes beyond math. Aesthetic/artistic appeal may have come first. The ratio is in the Parthenon in ancient Greece.
P5	Numerous studies indicate that many pieces of art…	The Golden Ratio also seems to appear in great art. Various examples.
P6	The Golden Ratio even appears in…	The ratio even appears in nature. Various examples.

	The Golden Ratio, a mathematical relationship that has been known for centuries, has numerous applications and associated examples in the fields of mathematics, architecture, art, and nature.	Summary. Correct answers must be clearly expressed in the passage. They must also be among the major points of the passage. They should tie as directly as possible to the summary given.
✗	a The Fibonacci sequence has numerous applications in the field of mathematics.	P3 is devoted to the Fibonacci sequence as it relates to the Golden Ratio. But the fact that the Fibonacci sequence has applications in the field of mathematics does not fit in directly with the main idea. That main idea focuses on the breadth of areas related to the Golden Ratio. So this fact is a minor detail in the context of the passage as a whole.
✓	b The Golden Ratio appears in mathematical sequences and regular geometrical figures.	Correct. P2–P3 are devoted to discussing various examples of the appearance of the Golden Ratio in mathematics, including the Fibonacci sequence and various geometrical figures.
✗	c Some crystals have configurations that exhibit a relationship to the Golden Ratio.	Mentioned in P6. But this only one minor example of the Golden Ratio as found in nature.
✓	d The Golden Ratio influenced the work of many famous artists, including Da Vinci, Michelangelo, Le Corbusier, and Dalí.	Correct. P5 discusses the impact on the works of many famous artists.
✓	e The Golden Ratio appears in many examples from nature, ranging from crystal formations and plant structures to the shape of mollusk shells and spiral galaxies.	Correct. P6 discusses in detail these examples that illustrate the extent to which the Golden Ratio appears in nature.
✗	f Although described by Euclid in approximately 300 BC, the Golden Ratio was formally defined at a much earlier date.	Not supported by the passage. The author does not argue that the Golden Ratio was formally defined before Euclid. Rather, the author cites opinions that the ratio may have been appreciated aesthetically at an earlier date, before it was mathematically defined.