430 McGRAW-HILL’S SAT SAT Practice 5: Data Analysis Questions 1 and 2 refer to the following information: 3. According to the graph above, Wacky Water Park experienced its largest increase in park at- tendance between which two consecutive months? (A) March and April (B) April and May (C) May and June (D) June and July (E) July and August Revenue per Unit Cost per Unit Widgets Gadgets Tinkers $w $x $6 $4 $3 $y COST/REVENUE ANALYSIS FOR THE TRINKET FACTORY 1. Which of the following best approximates the slope m (in pounds per week) of the line that best approximates these data? (A) m > 1 (B) 0 < m < 1 (C) m = 1 (D) −1 < m < 0 (E) m < −1 2. If the line of best fit for the data presented above passed through the points (32, 6.5) and (39.5, 8.0), it can be estimated that a baby born at 28 weeks would most nearly weigh how many pounds? (A) 5.3 (B) 5.5 (C) 5.7 (D) 5.9 (E) 6.1 60 45 30 15 0 MONTHLY TICKET SALES AT WACKY WATER PARK Mar Apr May Jun Jul Aug Number of Visitors (in thousands) 4. The table above shows the per unit revenue and cost of three products at the Trinket Factory. If profit equals revenue minus cost, how much profit do they make if they produce and sell two of each item? (A) 2w + 2x − 2y − 2 (B) 2y − 2x − 2w − 2 (C) w + x − y − 1 (D) x + 2w + y − 7 (E) 2x + 2y − 2w + 2 CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 431 Questions 5 and 6 refer to the following tables: 6. If the girls who bought turkey sandwiches have $206 in total to spend on their lunches, what is the greatest number of turkey sandwiches with fries they could buy without exceeding their budget? (A) 23 (B) 24 (C) 25 (D) 26 (E) 27 5. Based on the tables above, if every boy bought a sandwich without fries and every girl bought a sandwich with fries, how much more money did the boys spend at the deli than the girls? Turkey Ham Veggie Boys Girls 75 80 22 40 35 64 Number of sandwiches ordered at a local deli Sandwich Sandwich and Fries Turkey Ham Veggie $ 4.50 $ 5.00 $ 3.75 $ 5.50 $ 6.00 $ 4.75 Cost of food 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 432 McGRAW-HILL’S SAT 4. A The revenue generated from two widgets, two gadgets, and two tinkers is $2w, $2x, and $2(6), respectively. The cost of producing two widgets, two gadgets, and two tinkers is $8, $6, and $2y, respectively. Therefore, the total profit can be found by subtracting the cost from the revenue: (2w + 2x + 12) − (8 + 6 + 2y) = 2w + 2x − 2y − 2. 5. 86 The boys bought only sandwiches and spent ($4.50)(75) + ($5.00)(80) + $3.75(22) = $820.00. If the girls bought only sandwiches with fries, then they spent ($5.50)(40) + ($6.00)(35) + $4.75(64) = $734.00. $820.00 − $734.00 = $86.00 6. D If x = sandwiches: x(4.5) + (40 − x)(5.5) = 206 Distribute: 4.5x + 220 − 5.5x = 206 Combine like terms: −1.0x =−14 Divide by −1: x = 14 There were (40 − x) = 40 − 14 = 26 meals with fries. Concept Review 5 1. A best fit line is a straight line that “hugs” the data most closely on a scatterplot. 2. It can be created by connecting the “outermost” points on the plot or by drawing a line that best “hugs” the points and divides them in half. Try to ignore any outliers that don’t fit with the rest of the data. 3. For the SAT, you just need to be able to tell if a slope is positive or negative, or perhaps greater or less than 1. Positive slopes go up to the right, and negative slopes go down to the right. If the slope is positive and the “rise” is greater than the “run,” the slope is greater than 1; if the rise is less than the run, the slope is less than 1. 4. If you know what percent of the data are in a sec- tor of the pie chart, multiply the percentage by 360° to obtain the degree measure of that sector (e.g., a sector that represents 40% of the circle would be (.40)(360°) = 144°). 5. The biggest percent change occurs between Thursday and Fridays. Percent change = (100%)(10.5 − 5)/(5) = 110%. 6. There were a total of 13 + 7 + 14 + 5 + 5 + 10.5 + 11.5 = 66,000 accidents; 35,000 of them occurred on Friday, Saturday, or Sunday. 35/66 = 53% 7. 25% of the kids said blue was their favorite color. 25% of 4,000 = 0.25 × 4,000 = 1,000. 20% of the kids said yellow was their favorite color. 20% of 4,000 = 0.20 × 4,000 = 800. 1,000 − 800 = 200 kids. 8. The color red represents 10% of the circle, which is 10% of 360°. ° 9. There are currently (.25)(4,000) = 1,000 votes for blue. In order for blue to be 50% of the circle, it would need (.50)(4,000) = 2,000 votes. Therefore, 1,000 votes must change. 10. Hamden High ordered 36 + 28 + 81 = 145 tickets. New Haven Public ordered 64 + 23 + 64 = 151. Waterbury High ordered 53 + 31 + 51 = 135. 11. 81 Hamden High students bought 5-day passes at $200.00/pass, spending (81)(200) = $16,200. 64 students at New Haven Public bought 5-day passes at $200/pass, spending (64)(200) = $12,800. The difference is $16,200 − $12,800 = $3,400. 10 360 100 36 ( ) = Answer Key 5: Data Analysis SAT Practice 5 1. B A line connecting (32, 6.5) and (39.5, 8.0) is a good line of fit, and has a slope of 1.5/7.5 = 0.2, which is between 0 and 1. 2. C If the slope is about 0.2, you can use the slope equation to solve: Plug in values: Simplify: Multiply by 4: (6.5 − y 1 ) = 0.8 Subtract 6.5: −y 1 =−5.7 Divide by −1: y 1 = 5.7 3. C From March to April: 30 − 15 = 15,000 From April to May: 30 − 30 = 0 From May to June: 60 − 30 = 30,000 From June to July: 65 − 60 = 5,000 From July to August: 69 − 65 = 4,000 65 4 02 1 . . − () () = y 65 32 28 02 1 . . − () − () = y yy xx 21 21 02 − () − () = . CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 433 Lesson 6: Negative and Fractional Exponents Exponents Review In Chapter 8, Lesson 3, we discussed the practical de- finition of exponentials: The expression x n means x multiplied by itself n times. This is a useful definition when you need to evaluate something like 4 3 : you simply multiply 4 × 4 × 4 and get 64. But what about expressions like 4 0 or 4 −3 or 4 1/2 ? How do you multiply 4 by itself 0 times, or −3 times, or half of a time? It doesn’t make much sense to think of it that way. So to understand such ex- pressions, you must expand your understanding of exponents. Zero and Negative Exponents Using what you have learned in Lesson 1 of this chap- ter, what are the next three terms of this sequence? 81, 27, 9, 3, _____, _____, _____ The rule seems to be “divide by 3,” so the next three terms are 1, 1 ⁄ 3 , and 1 ⁄ 9 . Now, what are the next three terms of this sequence? 3 4 , 3 3 , 3 2 , 3 1 , _____, _____, _____ Here, the rule seems to be “reduce the power by 1,” so that the next three terms are 3 0 , 3 −1 , and 3 −2 . Notice that the two sequences are exactly the same, that is, 3 4 = 81, 3 3 = 27, 3 2 = 9, and 3 1 = 3. This means that the pattern can help us to understand zero and negative exponents: 3 0 = 1, 3 −1 = 1 ⁄ 3 , and 3 −2 = 1 ⁄ 9 . Now, here’s the million-dollar question: Without a calculator, how do you write 3 −7 without a negative exponent? If you follow the pattern you should see that and, in general: x 0 ϭ 1 and Notice that raising a positive number to a neg- ative power does not produce a negative result. For instance 3 −2 does not equal −9; it equals 1 ⁄ 9 . Fractional Exponents What if a number is raised to a fractional exponent? For instance, what does 8 1/3 mean? To understand expressions like this, you have to use the basic rules of exponents from Chapter 8, Lesson 3. Specifically, you need to remember that x n ϫ x m ϭ x m+n . (8 1/3 ) 3 = 8 1/3 × 8 1/3 × 8 1/3 . Using the rule above, 8 1/3 × 8 1/3 × 8 1/3 = 8 1/3 + 1/3 + 1/3 = 8 1 = 8. In other words, when you raise 8 1/3 to the 3rd power, the result is 8. This means that 8 1/3 is the same as the cube root of 8, and, in general: The expression x 1/n means , or the nth root of x. For example, can be written as a 1/2 . Example: What is the value of 16 3/4 ? The first step is to see that 16 3/4 is the same as (16 1/4 ) 3 (because (16 1/4 ) 3 = 16 1/4 × 16 1/4 × 16 1/4 = 16 3/4 ). Using the definition above, 16 1/4 is the 4th root of 16, which is 2 (because 2 4 = 16). So (16 1/4 ) 3 = 2 3 = 8. The expression x m/n means the nth root of x raised to the mth power. For instance, 4 3/2 means the square root of 4 raised to the third power, or 2 3 = 8. a x n x x n n − = 1 3 1 3 7 7 − = 434 McGRAW-HILL’S SAT Concept Review 6: Negative and Fractional Exponents Evaluate the following expressions without a calculator. 1. 5 −2 2. 9 1/2 3. 2 −5 4. 25 −1/2 5. 4 3/2 6. Simplify the following expressions, eliminating any negative or fractional exponents. 7. x 1/3 8. (4g) 1/2 9. 4x −2 10. (4y) −2 11. (9m) 3/2 12. (27b) 1/3 /(9b) −1/2 13. If x 3/4 = 27, what is the value of x? 14. If b −1/2 = 4, what is the value of b? 15. If (2 m ) −6 = 16, what is the value of 2 3m ? 16 25 32 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 435 SAT Practice 6: Negative and Fractional Exponents 1. If 4 n = 20, then what is the value of 4 −n ? 2. If 5 4 × m = 5 2 , then m = (A) −5 2 (B) 5 −2 (C) (D) 5 1/2 (E) 3. If 2 m × 2 m × 2 m × 2 m = 2, then m = 1 2 1 5 4. For all values of n, (A) 3 (B) (C) 3 n (D) 9 2 (E) 9 n 5. If x is a positive number, then (A) x 3/4 (B) x –(1/4) (C) x 3/4 (D) (E) 6. If x = a 5 = b 3 and x is positive, then ab = (A) x 1/5 (B) x 1/8 (C) x 8/15 (D) x 8 (E) x 15 x 5 x 3 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ n 33 9 2 × = n n 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 x 3/2 × x 1/2 x –(1/2) 436 McGRAW-HILL’S SAT 5. E Simplify numerator (add exponents): Simplify quotient (subtract exponents): Simplify exponent: x 5/2 Rewrite as a root: 6. C x = a 5 = b 3 Solve for a (raise to the 1/5): x 1/5 = a Solve for b (raise to the 1/3): x 1/3 = b Multiply a and b: ab = x 1/5 × x 1/3 Simplify (add exponents): ab = x 1/5 + 1/3 Simplify: ab = x 8/15 (Remember the quick way to add fractions: “zip-zap-zup” from Chapter 7, Lesson 3.) x 5 x 42 12−−(/) x x 42 12− xx x 3 2 1 2 1 2 × = − Concept Review 6 1. 5 −2 = 1/(5 2 ) = 1/25 2. 3. 2 −5 = 1/(2 5 ) = 1/32 4. 5. 6. (16/25) −3/2 = 1/(16/25) 3/2 = (25/16) 3/2 = ((25/16) 1/2 ) 3 7. 8. 9. 4x −2 = 4/x 2 10. (4y) −2 = 1/(4y) 2 = 1/(16y 2 ) 11. 99 327 32 12 3 3 mm mmm () = () ⎛ ⎝ ⎞ ⎠ = () = 442 12 ggg () == xx 13 3 = = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 25 16 5 4 125 64 3 3 44 428 32 12 3 3 3 = () = () == 25 1 25 1 25 1 5 12 12− = () == 993 12 == 12. 13. x 3/4 = 27 Raise both sides to the 4/3 power: (x 3/4 ) 4/3 = 27 4/3 Simplify: x 1 = (27 1/3 ) 4 Simplify: x = 3 4 Simplify: x = 81 14. b −1/2 = 4 Raise both sides to the −2 power: (b −1/2 ) −2 = 4 −2 Simplify: b 1 = 1/(4 2 ) Simplify: b = 1/16 15. (2 m ) −6 = 16 Simplify: 2 −6m = 16 Raise to the −1/2 power: 2 3m = 16 −1/2 Simplify: 2 3m = Simplify: 2 3m = 1/4 116 27 9 27 9 3 3 9 13 12 13 12 13 12 1 bb b b bb b ( ) ( ) = ( ) × ( ) =× = − 66 6 9= b Answer Key 6: Negative and Fractional Exponents SAT Practice 6 1. 1/20 or 0.05 4 −n = 1/4 n = 1/20 2. B 5 4 × m = 5 2 Divide by 5 4 : m = 5 2 /5 4 Simplify (subtract exponents): m = 5 −2 3. 1/4 or 0.25 2 m × 2 m × 2 m × 2 m = 2 Simplify (add exponents): 2 4m = 2 Exponents must be equal: 4m = 1 Divide by 4: m = 1/4 4. A Write as powers of 3: Simplify denominator: Divide numerator and denominator by 3 2n :3 1 /1 = 3 Perhaps a simpler method is to simply pick n to be 0 (because n can be any number). This gives (3 × 3 0 )/9 0 = (3 × 1)/1 = 3. The only choice that equals 3 when n = 0 is (A). 33 3 12 2 × n n 33 3 12 2 × () n n 33 9 2 × n n WRITING A GREAT ESSAY 1. Map the SAT Essay Assignment 2. Analyze the Assignment Closely 3. Brainstorm Your Alternatives Creatively 4. Connect to Your Knowledge with “Source Summaries” 5. Write a Strong and Creative Thesis 6. Organize Your Thoughts 7. Write Logically 8. Write Clearly 9. Write Concisely 10. Write Forcefully 11. Write Masterfully 12. Finish with a Bang CHAPTER 12 437 ✓ Copyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use. 438 McGRAW-HILL’S SAT Lesson 1: Map the SAT Essay Assignment Consider carefully the issue discussed in the following passage, and then write an essay that answers the question posed in the assignment. Our leaders love to tell us that only victory will do, as if they are imparting great wisdom. They seek to defeat the enemy, to achieve the goal. Yet many times a loss, particularly one that is hard fought, is more valuable than victory. We cannot live a life full of only victories, nor should we. The quality of our lives depends as much on how we manage our losses as on how we achieve our victories. Quality is much more important than quantity, but it’s hard to get a great score with fewer than four paragraphs. This is so because the readers are looking for structure and development, which require good use of paragraphs. Think of your paragraphs as “stepping-stones” on a journey. Only two or three stepping-stones don’t make for much of a journey. Plan to write four well-defined paragraphs—five if you have enough time. The essay assignment asks you to formulate a point of view regarding a particular aspect of human values or behavior. It does not require you to recall any specific knowledge from your studies, although you should try to connect your thesis with your studies. There is never a “right” or “wrong” answer to the question; that is, your actual position does not affect your score. More important (contrary to what a lot of SAT-prep folks claim), the graders are not looking for essays that fit a particular formula. You can use narration, exposition, persuasion, or argument as long as it is focused on devel- oping an interesting point of view that answers the question. Assignment: Can a loss ever be more valuable than a victory? Write an essay in which you answer this question and discuss your point of view on this issue. Support your position logically with examples from literature, the arts, history, politics, science and technology, current events, or your experience or observation. Know What They’re Looking For Two English teachers who have been trained by the Educational Testing Service (ETS) will read and score your essay from 1 (poor) to 6 (outstanding). They are trained to look for five things: Interesting, relevant, and consistent point of view. Do you take a thoughtful and interesting position on the issue? Do you answer the ques- tion as it is presented? Do you maintain a con- sistent point of view? Good reasoning. Do you define any necessary terms to make your reasoning clear? Do you explain the reasons for and implications of your thesis? Do you acknowledge and address possible objections to your thesis without sac- rificing its integrity? Solid support. Do you give relevant and specific examples to support your thesis? Do you explain how these examples support your thesis? Logical organization. Does every paragraph re- late clearly to your thesis? Do you provide logi- cal transitions between paragraphs? Do you have a clear introduction and conclusion? Does the conclusion provide thoughtful commentary rather than mere repetition of the thesis? Effective use of language. Do you use effective and appropriate vocabulary? Do you vary sen- tence length and structure effectively? Do you avoid needless repetition? Do you use paral- lelism, metaphor, personification, or other rhetorical devices to good effect? Do you use strong verbs? Do you avoid needlessly abstract language? Do you avoid cliché? The readers will not mark you down for minor spelling or grammar mistakes, and they won’t mark you up just for using big words. Focus on good reasoning. If you can take an interesting position, explore its implications, discuss relevant examples that support it, and maintain your focus, you will get a very good score. How Long Should It Be? The Assignment Your essay assignment will look something like this: CHAPTER 12 / WRITING A GREAT ESSAY 439 Practice 1: Map the SAT Essay Assignment SAT Essay Grading Review 1. What does it mean for an essay to have good substance? 2. What does it mean for an essay to have strong organization? 3. What does it mean for an essay to be clear? 4. What does it mean for an essay to have an effective and interesting style? 5. How long should your SAT essay be? Check your answers with the answer key at the end of the chapter. . Bang CHAPTER 12 437 ✓ Copyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use. 438 McGRAW-HILL’S SAT Lesson 1: Map the SAT Essay Assignment Consider carefully the issue. 430 McGRAW-HILL’S SAT SAT Practice 5: Data Analysis Questions 1 and 2 refer to the following information: 3 lot of SAT- prep folks claim), the graders are not looking for essays that fit a particular formula. You can use narration, exposition, persuasion, or argument as long as it is focused on devel- oping