Sat - MC Grawhill part 85 pdf

830 MCGRAW-HILL’S SAT 10. 104 Quadrilateral ABCD is composed of two identical triangles, each with an area of 240 square inches. Area of ΔABC = 1 ⁄2bh 240 = 1 ⁄2(20)(h) h = 24 inches To solve for side BC, set up the Pythagorean theorem: 10 2 + 24 2 = (BC) 2 Simplify: 100 + 576 = (BC) 2 Combine like terms: 676 = (BC) 2 Take the square root: 26 = BC (Or simply notice that each right triangle is a classic 5-12-13 triangle times 2: 10-24-26.) The perimeter of quadrilateral ABCD = 26 + 26 + 26 + 26 = 104 (Chapter 10, Lesson 5: Areas and Perimeters) 11. 30 If two things are equal, you can substitute either one for the other. Since g(x) = x 2 − 5, f(g(x)) = f(x 2 − 5) Plug x 2 − 5 into f(x) and simplify: f(x) = 7(x 2 − 5) + 2 Distribute: 7x 2 − 35 + 2 Plug in 3 for x: 7(3) 2 − 35 + 2 Simplify: 63 − 35 + 2 = 30 (Chapter 11, Lesson 2: Functions) 12. 3 Start by simplifying the expression: Combine like terms: Simplify: x − 3 This expression is 3 less than x. (Chapter 8, Lesson 1: Solving Equations) 13. 18 Approach this problem logically, but keep the restrictions in mind. If we want the largest possible value of a and a + b < 20, try a = 19. But that is not a possibility, because b is a positive integer and so can be no less than 1, and 19 + 1 is equal to, not less than 20. Therefore, the largest value of a that fits the re- striction is 18. If a = 18 and b = 1, then ab = (18)(1) = 18, an even number. (Chapter 9, Lesson 3: Numerical Reasoning Problems) 515 5 x − 69 5 6 5 69 6 5 xx xx− −= −−−+ 14. 28 Since rectangle U and rectangle V share a side with integer length, this length must be a common factor of 18 and 21. Similarly, the side that rectangle U and rectan- gle W share must be a common factor of 18 and 12. Therefore, the common side between U and V is 3, and the common side between U and W is 6. So U is a 6-by-3 rectangle, V is a 7-by-3 rectangle, and W is a 6-by-2 rectangle, which means rectangle X must have an area of 14. The sum of the areas of those four rectangles is 18 + 21 + 12 + 14 = 65. The area of the entire rectangle is given as 117. Thus, the area of rectangles Y and Z together must be 117 − 65 = 52. Set up an equation: 6x + 7x = 52 Combine like terms: 13x = 52 Divide by 13: x = 4 If x = 4, then the area of rectangle Z is 4 × 7 = 28. (Chapter 10, Lesson 5: Areas and Perimeters) 15. 45 Begin by finding the amount the class would spend on 55 regular-price tickets: 55 × $15 = $825. Then calculate how much 60 discounted tickets cost: 60 × $13 = $780. Then subtract to find the amount saved: $825 − $780 = $45. (Chapter 9, Lesson 3: Numerical Reasoning Problems) 16. 22.5 The length of WY –– , as shown above, is 15. Point X is the midpoint of WY –– , so WX = XY = 7.5. YZ = 2WX = 2(7.5) = 15. So XZ = 22.5. (Chapter 6, Lesson 2: Analyzing Problems) 17. 12 r ▫ Solve for x: 3 ▫ x ▫ 3 = 12 ▫ (Chapter 9, Lesson 1: New Symbol or Term Problems) 3 12 3 12 3 108 9 12 2 = () () − == 2 32 32 12 1 12 2 = () () − ===x s rs rs = − 2 U W Y V X Z 6 6 6 7 7 7 3 2 x W Y 15 WX7.5 7.5 15YZ B D C A 10 10 24 2626 2626 24 CHAPTER 16 / PRACTICE TEST 4 831 18. 37.5 Since DC –– ⊥ AB –– , angle CDB is a right angle, so ΔCDB is a 45°-45°-90° right triangle. Therefore, DB = DC = 5. Since AD = 2DB, AD = (5)(2) = 10. The area of ΔABC = 1 ⁄2(b)(h) = 1 ⁄2(15)(5) = 37.5. (Chapter 10, Lesson 5: Areas and Perimeters) Section 6 1. B The reflex produces an immediate (or instan- taneous) response. transient = short-lived; stagnant = not moving; revitalized = filled with new life and energy 2. D Although the cats are emaciated (excessively thin) and starved for food, summoning energy would help them to fight hard or aggressively for the scraps. humanely = with mercy; vigilantly = in a watchful way; fluently = smoothly; ferociously = fiercely 3. C Jennifer irritated her peers with her supercil- ious (overly proud) and pretentious (haughty) re- marks. These are characteristic of an arrogant or showy demeanor. reticent = reserved, unwilling to speak; belligerent = warlike; lofty = pompous; self- effacing = modest; discomfited =uneasy, uncomfortable 4. A The first part of the sentence indicates that a sushi master’s work is not easily learned. Therefore, much training and studying are required to become a master chef. Apprenticeship, tutelage, and cultivation are all good choices for the first word. This training will give someone the autonomy to create his or her own work. apprenticeship = working as a beginner under the assistance of an instructor; autonomy = in- dependence; tutelage = instruction; ineptitude = lack of skill; dormancy = lack of activity; sovereignty = supreme authority; cultivation = the act of improving; boorish = rude, lacking manners; quiescent = not active 5. A The journalist had a reputation for breaking news early, almost as if she were able to see the fu- ture. Later she admitted that she had privileged sources and did not use prophecy at all. prophetic = able to tell the future; prescience = knowledge of fu- ture events; premeditated = planned ahead of time; predilection = preference; dismissive = indifferent; omniscience = total knowledge; preeminent = supe- rior; reluctance = resistance; insolvent = bankrupt; foresight = thinking ahead 6. B Neither passage describes a discovery, but rather the world picture (line 5) of the medieval mind (lines 26–27), that is, medieval theories about the na- ture of the universe. Although Passage 1 provides a counterexample to an often-heard charge (lines 14–15), Passage 2 does not attempt to disprove any assump- tions. Neither passage questions the medieval theories presented. Instead, the passages merely describe those theories. Lastly, neither passage discusses the everyday life in medieval Europe. 7. D The passage states that where our universe is thought to be dark, the other one was presumed to be illuminated, which means that the medieval uni- verse was perceived to be full of light, unlike our modern universe. 8. C Pascal is said to be disturbed by the silence of the vast spaces between the stars (lines 10–11), in con- trast to the medieval thinkers who formerly thought that the universe produced the “music of the spheres” (lines 12–13). 9. E Dante’s theory is described in Passage 1 to counter the charge that medieval thinkers were focused on man’s sense of self-importance (lines 16–17), but Hilde- gard’s theory presented in the final sentence of Passage 2 is clearly anthropocentric, or human-centered. The world views of both Dante and Hildegard are focused on religion and an ordered hierarchy, but neither ad- dresses scientific methods. Lastly, Passage 1 does not discuss the public acceptance of Dante’s theory. 10. E This passage is concerned primarily with de- scribing the relationships among the plants, animals, and climate of the Serengeti. Therefore, it is describing how a particular ecosystem works. Although the passage mentions human intervention tangentially in the last paragraph, where it refers to badly drawn park bound- aries, it is not a central focus of the passage. Although it does mention individual plants, the passage as a whole does not focus on them, but instead shows how they play a role in a larger ecosystem. It does not mention natural disasters, and only mentions the distinction between grazers and browsers as a minor point. 11. C These sentences suggest that the variety in the diet of grazers increases with the length of the grass. When the grass is short, all the animals apparently eat much the same sort of grass, but when it is long, they diversify their diets. 12. A Browsers are said to feed on shrubs or the leaves of trees (lines 15–16), as opposed to the grazers, which eat the abundant grass that springs up like a well-mown lawn (lines 9–11). A D C B 45 ° 45 ° 10 5 5 5 2 832 MCGRAW-HILL’S SAT 13. B The passage states that unlike all the other grazers on the plain, (zebras) have teeth in both jaws (lines 27–28). All the rest (besides the zebras) are var- ious species of antelope (lines 30–31), which have toothless upper palates (lines 32–33). 14. E The second paragraph states that where the grass is all short . . . all the animals apparently eat the same sort of grass . . . but where the grass is of varied lengths . . . each animal copes differently with the avail- able fodder (lines 19–24). This difference is then de- scribed in the third paragraph, where the grazing sequence is specified. 15. C The rains are said to bring on fresh growth (line 53), encouraging the grazers to return to old grazing lands. 16. A This paragraph states that if the migrant herds . . . were confined (lines 63–64), they would so weaken the grass that it would die out (lines 67–68). So maintaining the grasslands requires that the animals not be confined. 17. A The thesis of the passage is that men come greatly to desire that these capricious gifts of Nature (that is, the natural resources that are hard for some and easy for others to find , by luck alone) might be in- tercepted by some agency having the power and the goodwill to distribute them justly. . . . This desire is So- cialism (lines 41–47). 18. C This primitive cultivator is a person who tries to stick a spade into the earth and make wheat and other edible matters spring from it. This is a farmer. Although the author uses figurative and metaphorical language throughout the passage, this particular phrase is being used literally. 19. D The astronomer is said to regard the earth as simply a ball . . . without ulterior motives (lines 3–4), while the foolish spendthrift . . . suddenly realizes that the earth is offering him gold (lines 21–24). Therefore, the astronomer regards the earth as impersonal, while the spendthrift regards it as generous. 20. B The closed hand represents the tendency of the Earth to hide its diamonds and good red wheat (line 31). 21. A The author is discussing how capricious na- ture is in revealing its resources, and suggests that anyone trying to harvest the earth’s resources must become a gambler (line 33), and scoff at theorists who prate (speak inconsequentially) of moral virtues such as industry and honesty and equality. Therefore, the author is suggesting that these virtues are not as valu- able as many people claim they are. 22. C This fate is the fate of the gambler (line 33), who is at the whim of mother earth. 23. E The author states that the Social Democratic State . . . remains to be tried (lines 51–52). 24. C The author states that our own choice (that is, the choice of his society) is shown by our continual as- piration to possess property (lines 59–60). Section 7 1. B 3x + 5x + 8x = 32 Combine like terms: 16x = 32 Divide by 16: x = 2 (Chapter 8, Lesson 1: Solving Equations) 2. D Simplify: 2x = 8 Divide by 2: x = 4 (Chapter 8, Lesson 1: Solving Equations) 3. A 5b − 10 ≥ 15 Add 10: 5b ≥ 25 Divide by 5: b ≥ 5 (Chapter 8, Lesson 6: Inequalities, Absolute Values, and Plugging In) 4. C First find 30% of 50: (.3)(50) = 15 t% of 60 is 15 Set up the equation: Cross-multiply: 60t = 1,500 Divide by 60: t = 25 (Chapter 7, Lesson 5: Percents) 5. C First convert 8 hours into minutes: Cross-multiply: x = 480 minutes Then set up a ratio to answer the question: Cross-multiply: 40y = 480h Divide by 40: y = 12h (Chapter 8, Lesson 7: Word Problems) (Chapter 7, Lesson 4: Ratios and Proportions) 6. A First find the product of −1.5 × 1.25:−1.875. Point A is closest to −1.875 on the number line presented. (Chapter 7, Lesson 1: Numbers and Operations) 480 40minutes cards minutes cardsyh = 81 60 hours minutes hour minutesx = t 100 15 60 = 1 3 68 x x x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ () = CHAPTER 16 / PRACTICE TEST 4 833 7. C If the four people must each have a different positive number of cards, then the least that three may have is one, two, and three cards. This leaves a maximum of 100 − 6 = 94 for the remaining person. (Chapter 9, Lesson 3: Numerical Reasoning Problems) 8. C A quick plot of the data listed in the table will point you to answer choice (C). Since the miles per gallon are de- creasing as the age in- creases, you can eliminate choice (A). There is no point where the data levels out, which eliminates an- swer choice (D). Finally, because it is not de- creasing at a constant rate, you can eliminate choices (B) and (E). (Chapter 11, Lesson 5: Data Analysis) 9. D If the question is at most how many of these integers could be odd, begin by imagining that ALL of them are odd. The integers may be the same, so imag- ine that they are all 1: 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7. But 7 is odd, so try 6 odds and 1 even: 1 + 1 + 1 + 1 + 1 + 1 + 2 = 8. Therefore, the most that could be odd is 6. (Chapter 9, Lesson 3: Numerical Reasoning Problems) 10. D You can write out a quick calendar for your- self to track the days: If you do this problem too quickly, you might assume that since the fourth Wednesday is the 22nd, the fourth Monday would be the 20th. But the first Monday comes after the first Wednesday, which makes the fourth Monday the 27th. (Chapter 9, Lesson 3: Numerical Reasoning Problems) 11. D Be careful with this question. Make sure you understand the chart before choosing an answer. The question asks about teachers who use gas heat. There are 60 + 13 = 73 teachers who use gas heat, and 60 of these live in a house. (Chapter 11, Lesson 5: Data Analysis) (Chapter 7, Lesson 3: Fractions) 12. D “Stack” the equations: 5x + 7y = 18 2x − 4y = 6 Add straight down: 7x + 3y = 24 (Chapter 8, Lesson 2: Systems) 13. A (1,36) (3,31) (5,20) Miles per gallon Age (years) y ° z ° x ° x ° x ° l m Su M T W Th F Sa 1234 8910567 11 15 16 1712 13 14 18 22 23 2419 20 21 29 30 3126 27 28 25 Starting with the angle marked x° in the original fig- ure, you can mark its vertical angle x° as well. Since line l is parallel to line m, the corresponding angle in the top triangle is also x°. Set up an equation for the triangle: x + z + (180 − y) = 180 Subtract 180: x + z − y = 0 Add y: x + z = y (You might also simply notice that the angle marked y° is an “exterior” angle to the triangle, so its measure is equal to the sum of the two “remote interior” an- gles: y = x + z.) Subtract z: x = y − z (Chapter 10, Lesson 1: Lines and Angles) (Chapter 10, Lesson 2: Triangles) 14. B This problem involves rates, so it helps to re- call the rate equation: d = rt. Because she travels home along the same route, you can use d for the distance both to and from work. Be- cause she spends a total of 2 hours in the car, if she spends t hours on the way to work, she will spend 2 − t hours on the way home from work. Set up rate equations for both legs of the trip: To work: d = 40(t) From work: d = 24(2 − t) Set the expressions equal: 40t = 24(2 − t) Distribute: 40t = 48 − 24t Add 24t: 64t = 48 Divide by 64: t = .75 Plug 0.75 in for t and solve for d: d = 40(.75) = 30 Check by confirming that plugging t = .75 into the other rate equation gives the same distance from home to work. (Chapter 9, Lesson 4: Rate Problems) 15. B The graph of the original function will be shifted up 4 and right 2. Answer choice (B) shows the proper representation of the new graph. (Chapter 11, Lesson 3: Transformations) 834 MCGRAW-HILL’S SAT 16. A Line segment BD –– is tangent to the circle at point A, so angles BAO and DAO are right angles. This means that both ΔDAO and ΔBAO are 30°-60°-90° triangles. 2. B The sentence states that one feature of Joyce’s work gave way to another, suggesting that the first missing word is a noun that directly contrasts the ad- jective in the second blank. The second word is paired with arcane, which means secret or little-understood. This word must also describe works that feature ne- ologisms (invented words) and obscure literary tricks. Therefore the first word should mean something like clarity and the second phrase should include an ad- jective like hard to understand. lucidity = clarity; opaque = very difficult to understand or translate; con- cise = brief and to the point; anachronism = quality of being out of place in time; derivative = copied from others 3. B Churchill was known to choose his strategies arbitrarily (without logical reason), so he was whim- sical or impulsive. diligent = working with great effort; impulsive = acting without thought; vicious = evil, harsh; malevolent = wishing harm, malicious 4. D If the king executed those who acted irrever- ently (without respect), he must have demanded ut- most respect. insolence = brazen rudeness; impudence = disrespect; truculence = inclination to pick fights; deference = respect; ignominy = humiliation 5. D Since Galileo contradicted church teachings, he was a heretic. ostracized = cut off from society; hermit = one who seeks solitude; venerated = wor- shipped; demagogue = powerful leader; hallowed = respected as holy; revisionist = one who rethinks or reshapes a commonly accepted view; denounced = accused or condemned for being a villain; heretic = one who holds controversial opinions; reviled = at- tacked with harsh language; luminary = one who in- spires others 6. A The Senator is known for her iconoclastic views, which means that she goes against the party line. Because of this, she would have a tough time get- ting traditional party members to support her. con- tentious = quarrelsome; orthodox = traditional; litigious = prone to bringing lawsuits; disingenuous = insincere; vituperative = using harsh censure or con- demnation; dissident = disagreeing; heretical = going against standard beliefs; polemical = pertaining to a highly controversial political or intellectual position 7. D This is an address to the Atlanta Exposition (as the introduction indicates), and the author is clearly addressing those in the commercial world (lines 26–27) and entreating ex-slaves and Southern whites to work together for their mutual benefit. B A D O 30 ° 30 ° 60 ° 60 ° 12 12 6 6 363 Using the 30°-60°-90° reference information at the be- ginning of this section, you can find the values of the remaining sides of the triangles. Find the area of ΔBOD: Plug in values: The radius of the circle is 6, so the area of the entire circle can be found using the equation Area =πr 2 = π(6) 2 = 36π. The shaded region of the circle makes up 120°, or 1 ⁄3 of the circle. Therefore, the area of the shaded region is equal to 1 ⁄3(πr 2 ) = 1 ⁄3(36π) = 12π. The area of the unshaded region of the triangle can be found by subtracting the area of the shaded region from the total area of the triangle: 36 − 12π. (Chapter 10, Lesson 2: Triangles) (Chapter 10, Lesson 8: Circles) Section 8 1. E If businesses are having a hard time staying current, their equipment must be old or outdated be- cause technology is advancing at a fast rate. urgency = pressing importance; progressive = advancing for- ward; conventional = standard; torpidity = lethargy; antiquated = outdated; lassitude = lack of energy; in- novative = inventive, novel 3 Area 12 3 6= () () = 1 2 36 3 Area base height= () () 1 2 x x 2x 3 6 3 60 ° 30 ° 6 12 60 ° 30 ° CHAPTER 16 / PRACTICE TEST 4 835 8. A The author does not directly address those in the antislavery movement but does address (B) those of the white race who look to the incoming of those of foreign birth . . . for the prosperity of the South (lines 45–48), (C) those of my race who depend upon bettering their condition (lines 12–14), (D) those for whom African Americans have tilled your fields, cleared your forests . . . (lines 56–57), and (E) those African Americans who underestimate the impor- tance of cultivating friendly relations with the South- ern white man (lines 14–16). 9. C The captain did not heed the first, second, or third call but heeded the fourth call at last (line 9), suggesting that he did not believe the responses were helpful at first. 10. C The phrase this chance refers to the man’s chance (that African Americans can have) in the com- mercial world (lines 26–27). 11. A In saying that there is as much dignity in tilling a field as in writing a poem (lines 40–41), the author is saying that such manual labor is valuable work. 12. D These indicate the thoughts of a Negro who has given earnest thought to the situation of his people in America (lines 68–70). 13. C In saying that because of incessant self- questioning (line 83) . . . the best energy of the Negro people cannot be marshalled to do the bidding of the race (lines 89–91), the author means that introspec- tion keeps African Americans from organizing them- selves to meet the needs of their race. 14. D Such people are said to make room for every rascal and demagogue who chooses to cloak his selfish deviltry under the veil of race pride (lines 91–93); that is, they allow themselves to be influenced by selfish and evil people. 15. D That point refers to the point farther than (which) our Americanism does not go (lines 121–122). In other words, this is the point up to which African Americans share much in common with all Ameri- cans but beyond which they are a unique people. 16. C This broader humanity is that which freely rec- ognizes differences in men, but sternly deprecates (dis- approves of) inequality in their opportunities of development (lines 139–141). In other words, its mem- bers value equal opportunity for all races. 17. E Passage 1 focuses on the manual labor that African Americans have performed in tilling fields, clearing forests, building railroads and cities, etc., while the author of Passage 2 emphasizes contribu- tions like the subtle sense of song that has given Amer- ica its only American music, its only American fairy tales, its only touch of pathos and humor . . . (lines 130–133). 18. A The black tomorrow in Passage 2 is the influ- ence of African Americans in softening the whiteness of the Teutonic today (line 129), which suggests a change in the dominant culture. Passage 1, on the other hand, envisions a future in which African Amer- icans make friends in every manly way of the people of all races by whom we are surrounded (lines 19–20) and incorporate themselves into the existing dominant in- dustries of agriculture, mechanics, . . . commerce, . . . (and) domestic service (lines 21–22). 19. E Passage 1 indicates that the dominant culture can give the African American a man’s chance in the commercial world (lines 26–27) and contains many opportunities (line 44). Passage 2 is more assertive in suggesting that African Americans have changed and will continue to change the dominant culture: We are the first fruits of this new nation, the harbinger of that black tomorrow which is yet destined to soften the whiteness of the Teutonic today (lines 126–129). Section 9 1. C The parallel idiom neither nor requires that the phrase following neither and the phrase fol- lowing nor have the same grammatical form. The only choice that maintains proper idiom and paral- lelism is (C). (Chapter 15, Lesson 3: Parallelism) 2. E This sentence contains three clauses, each of which has the same subject, Georgia. The underlined clause, however, is in the passive voice, unlike the other two. It should be changed to the active voice like the others. (Chapter 15, Lesson 3: Parallelism) 3. D Since any rehearsal would have been com- pleted before the performance, the participle in the underlined phrase should be in the perfect form hav- ing rehearsed. Choice (B) also uses the nonstandard phrase being that, and choices (C) and (E) create run-on sentences. (Chapter 15, Lesson 9: Tricky Tenses) 836 MCGRAW-HILL’S SAT 4. B In the original sentence, the subject outlets does not agree with the verb has acknowledged. In choice (C), the subject scope disagrees with the verb have been acknowledged. Choice (D) is awkward and choice (E) uses an illogical verb tense. Choice (B) con- veys the idea clearly and grammatically. (Chapter 15, Lesson 1: Subject Verb Disagreement) 5. E The original verb have had is in the imperative mood, but should be in the subjunctive mood because it conveys a hypothetical condition. Choice (E) conveys the mood correctly. (Chapter 15, Lesson 14: The Subjunctive Mood) 6. C The statement made in the main clause is not against popular opinion, but rather is contrary to it. Although choice (E) uses the proper modifier, it illog- ically suggests that an opinion can say something. (Chapter 15, Lesson 15: Coordinating Ideas) 7. A The original sentence is the most logical and effective option. 8. B The original phrase misuses the semicolon, because the phrase preceding it is not an independent clause. Similarly, choice (E) uses the conjunction so to join two clauses, but the first is not independent, so the sentence is ungrammatical. Choices (C) and (D) use the unidiomatic phrases struggled for finding and struggled finding. Choice (B) avoids these problems, and is clear and effective. (Chapter 15, Lesson 15: Coordinating Ideas) 9. A The original phrasing is best. It provides the parallel form required by the comparative idiom not so much by . . . as by . . . , whereas the others violate parallel form. 10. D The original phrasing is unidiomatic. The cor- rect idiom is A appears to be B. (Chapter 15, Lesson 10: Idiom Errors) 11. C The comparison is logically between the re- sponse to the revised proposal and the response to the original proposal. Choice (C) is the only one that makes the correct logical and parallel comparison. (Chapter 15, Lesson 4: Comparison Problems) 12. C The original phrasing breaks the idea into two independent clauses. But since it conveys one central idea, it is more effectively phrased with a single inde- pendent clause and a modifying phrase. Choice (C) does this effectively, idiomatically, and concisely. (Chapter 15, Lesson 15: Coordinating Ideas) 13. E The original phrasing is unnecessarily wordy and does not effectively coordinate the ideas in the sentence. Choice (C) has the same problem. Choices (B) and (D) create clauses with uncoordinated verbs. Only choice (E) conveys the idea concisely and effectively. (Chapter 15, Lesson 15: Coordinating Ideas) 14. C The phrase extra superfluous is redundant, and the phrase to edit for eliminating is unidiomatic. Choice (C) is clear and concise. (Chapter 15, Lesson 12: Other Modifier Problems) Math Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Math A right triangle has a leg of length 3 and a hypotenuse of length 4. What is the length of the other side? The average of 3 consecutive even integers is 80. What is the least of these integers? If 5 – 2(x – 3) = 9, then what is the value of x? If n is a positive real number, what is the simplest way to express n 2 × n 3 ? Stephanie bought a sweater for $42.40, including a 6% sales tax. What was the price before tax? 93 ° 92 ° 105 ° a ° b ° Note: Figure not drawn to scale In the figure above, what is the value of a + b? Math Math Math Math Math Math Math Math Math Math Math Math Formula/Concept: When you multiply exponentials with the same base, you add the exponents. Correct answer: n 5 (n 2 )(n 3 ) = Add the exponents: (n 2+3 ) = n 5 Common mistake: n 6 This is the result if you mistakenly multiply the exponents. Formula/Concept: To find the price before a 6% tax, divide the final price by 1.06. Correct answer: $40 $42.40 = (1.06)(x) Divide by 1.06: $40.00 = x Common mistake: $39.86 This is the result if you mistakenly deduct 6% of $42.40 (which is $2.54), from $42.40. Formula/Concept: (n – 2)180° = the sum of the angles in an n-sided figure. Correct answer: 250 The sum of the angles is (5 – 2)(180°) = 540°, so 105° + 93° + 92° + a + b = 540°. Therefore, a + b = 250°. Common mistake: 70 This is the result if you mistakenly think the sum is 360° instead of 540°. Formula/Concept: The Pythagorean Theorem Correct answer: 3 2 + x 2 = 4 2 Subtract 9: x 2 = 7 Take square root: x = Common mistake: 5 Don’t assume it is a 3-4-5 triangle. In such a triangle, the 3-4 sides must both be legs. 7 7 Formula/Concept: If a set of numbers is “evenly” spaced, the average is the same as the middle number. Correct answer: 78 If the average of consecutive even numbers is 80, then 80 must be the “middle” number in the set, so the numbers are 78, 80, 82. Common mistake: 79 Don’t overlook the fact that the numbers are even. Formula/Concept: Distributing with negative numbers Correct answer: 1 5 – 2(x – 3) = 9 Distribute: 5 – 2x + 6 = 9 Combine like terms: 11 – 2x = 9 Subtract 11: –2x = –2 Common mistake: –5 This results from improperly distributing the –2. 3 4 Math Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Math At the beginning of 1999, stock in ABC company cost $100 per share. It in- creased by 25% in 1999, decreased by 20% in 2000, decreased by 20% in 2001, and increased by 15% in 2002. What was the price at the end of 2002? If the average of x, x + 2, and 2x + 8 is 6, what is the value of x? A set consists of the integers from –12 to n. If the sum of the members of that set is 42, how many integers are in the set? Shaquille O’Neal made 4 of his first 12 free throws. How many consecutive shots x must he hit for his free-throw percentage to reach 60%? If a triangle has two sides of length 8 and 12, then what is the largest possi- ble integer length of the third side? If 10 students in a class of 16 have an average score of 82 on a physics test and the remaining students have an average score of 90, what is the average score of the entire class? . rectan- gle W share must be a common factor of 18 and 12. Therefore, the common side between U and V is 3, and the common side between U and W is 6. So U is a 6-by-3 rectangle, V is a 7-by-3 rectangle,. Transformations) 834 MCGRAW-HILL’S SAT 16. A Line segment BD –– is tangent to the circle at point A, so angles BAO and DAO are right angles. This means that both ΔDAO and ΔBAO are 30 -6 0 -9 0° triangles. 2 grazers, which eat the abundant grass that springs up like a well-mown lawn (lines 9–11). A D C B 45 ° 45 ° 10 5 5 5 2 832 MCGRAW-HILL’S SAT 13. B The passage states that unlike all the other grazers