GMAT Review 11th Ed. (Orange)

Data Sufficiency Page 25, #41 If x + y/z > 0, is x < 0? 1. x < y 2. z < 0

Data Sufficiency, Page 61, (Arithmetic - Inequalities) If x + y/z > 0, then either one of two cases holds true. Either (x + y) > 0 and z > 0 or (x + y) < 0 and z < 0. In other words, in order for the term to be greater than zero, it must be true that either 1) both the numerator and denominator are greater than 0 or 2) both the numerator and denominator are less than 0. 1.) Regardless of whether (x + y) is positive or negative, the positive or negative value of z must be in agreement with the sign of (x + y) in order for x + y/z > 0. However, there is no information about z here; INSUFFICIENT. 2.) If z < 0, then (x + y) must be less than 0. However, this statement gives no information about (x + y); INSUFFICIENT. This can be solved using 1 and 2 together. From 2 it is known that z < 0 and going back to the original analysis, for the term to be greater than zero, (x + y) must also be less than 0. Now if y > 0 then x must be less than y(-1) or x < -y, thereby confirming that x < 0. If instead y < 0, since x < y, x is also less than 0. Finally, if y = 0, then by substitution, x < 0. Answer choice C.

Problem Solving Sample Questions, Page 172, #152 The shaded portion of the rectangular lot shown above represents a flower bed. If the area of the bed is 24 square yards and x = y + 2, then z equals √13 2√13 6 8 10

...

Problem Solving Sample Questions, Page 173, #157 If x + 5y = 16 and x = -3y, then y = -24 -8 -2 2 8

I'm thinking it's B

Problem Solving Sample Questions Page 164, #96 1/2 + [(2/3 x 3/8) ÷ 4] - 9/16 = 29/16 19/16 15/16 9/13 0

Page 217, (Algebra - Second-degree equations) Page 217, (Arithmetic - Operations on rational numbers) Page 217, (Arithmetic - Statistics)

Problem Solving Sample Questions, Page 178, #189 If x > 0, x/50 + x/25 is what percent of x? 6% 25% 37% 60% 75%

Page 247, (Algebra + Arithmetic - Simplifying algebraic expressions + percents) Simplifying the expression by using a common denominator for the fractions and solve for x. If x > 0, then x/50 + x/25 x/50 + 2x/50 3x/50 = 6x/100 or 6% of x Answer choice A

Problem Solving Sample Questions, Page 174, #161 The positive integer n is divisible by 25. If √n is greater than 25, which of the following could be the value of n/25? 22 23 24 25 26

...

Problem Solving Sample Questions, Page 175, #169 Thirty percent of the swim club members have passed the lifesaving test. Among the members who have NOT passed the test, 12 have taken the preparatory course and 30 have not taken the course. How many members are there in the swim club? 60 80 100 120 140

...

Problem Solving Sample Questions, Page 183, #226 In the figure above, if z = 50, then x + y = 230 250 260 270 290

...

Problem Solving Sample Questions Page 167, #114 Which of the following is the product of two integers whose sum is 11? -42 -28 12 26 32

My previous work said 14 x -3 = -42.

Problem Solving Sample Questions Page 167, #115 Mary's income is 60% more than Tim's income, and Tim's income is 40% less than Juan's income. What percent of Juan's income is Mary's income? 124% 120% 96% 80% 64%

My previous work said answer choice C.

Problem Solving Sample Questions Page 159, #56 Which of the following is NOT equal to the square of an integer? √√1 √4 18/2 41-25 36

Page 204, (Algebra - Second-degree equations)

Problem Solving Sample Questions Page 171, #144 If 4-x/2+x = x, what is the value of x^2 + 3x - 4? -4 -1 0 1 2

Page 232, (Algebra - Second degree equations) Work the problem: 4-x/2+x = x 4-x = x(2+x) (multiply both sides by (2+x) 4-x = 2x + x^2 0 = x^2 + 3x -4 Answer choice C.

Problem Solving Sample Questions, Page 174, #162 1 // 1+1/2+1/3 3/10 7/10 6/7 10/7 10/3

Page 238, (Arithmetic - Operations on rational numbers)

Problem Solving Sample Questions Page 161, #69 If y(3x-5/2) = y and y ≠ 0, then x = 2/3 5/3 7/3 1 4

asdf

Problem Solving Sample Questions Page 159, #54 The average of 10, 30, and 50 is 5 more than the average of 20, 40, and: 15 25 35 45 55

Page 204, (Arithmetic - Statistics) Using the formula: sum of n values/n = average, the given information about the first set of numbers can be expressed in the equation: 10+30+50/3 = 30. From the given information then, the average of the second set of numbers is 30-5 = 25. Letting x represent the missing number, set up the equation for calculating the average for the second set of numbers, and solve for x. 20 + 40/3 = 25 60 + x = 75 x = 15 Answer choice A

Problem Solving Sample Questions Page 162, #81 y = 248 - 398x Which of the following values of x gives the greatest value of y in the equation above? 200 100 0.5 0 -1

Page 213, (Algebra - Applied problems) The equation given is linear. Thus, since lines continue in two directions indefinitely, it has no greatest value. So, the solution must be selected from the answer choices offered. The x term is being subtracted; therefore, the larger the x value, the smaller the y value. Conversely, the smaller the x value, the greater the y value. The smallest x value among the answer choices is -1. Answer choice E.

Page 22, #18 If n is the product of the integers 1 to 8, inclusive, how many different prime factors greater than 1 does n have? 4 5 6 7 8

Page 53, Section 3.5 (Arithmetic - Number Properties) If n is the product of integers 1 to 8 then its prime factors will be the prime numbers from 1 to 8. There are four prime numbers between 1 and 8: 2, 3, 5, and 7 Answer choice A

Data Sufficiency Page 24, #34 Of the companies surveyed about the skills they require in prospective employees, 20% required both computer skills and writing skills. What percent of the companies surveyed required neither computer skills nor writing skills? 1. Of those companies surveyed that required computer skills, half required writing skills. 2. 45 percent of the companies surveyed required writing skills but not computer skills.

asdf

Problem Solving Sample Questions Page 161, #70 If x + 5 > 2 and x - 3 < 7, the value of x must be between which of the following pairs of numbers? -3 and 10 -3 and 4 2 and 7 3 and 4 3 and 10

asdf

Problem Solving Sample Questions Page 159, #60 In the figure above (FIGURE) if PQRS is a parallelogram, then y - x = 30 35 40 70 100

Page 204, (Algebra - Second-degree equations)

Problem Solving Sample Questions, Page 176, #175 As x increases from 165 to 166, which of the following must increase? I. 2x - 5 II. 1 - 1/x III. 1/x^2 - x II only III only I and II I and II II and III

ddddddd

Problem Solving Sample Questions Page 171, #140 In a weightlifting competition the total weight of Joe's two lifts was 750 pounds. If twice the weight of his first lift was 300 pounds more than the weight of the second lift, what was the weight, in pounds, of his first lift? 225 275 325 350 400

...

Problem Solving Sample Questions, Page 176, #180 In a nationwide poll, N people were interviewed. if 1/4 of them answered "yes" to question 1, and of those, 1/3 answered "yes" to question 2, which of the following expressions represents the number of people interviewed who did not answer "yes" to both questions? N/7 6N/7 5N/12 7N/12 11N/12

...

Problem Solving Sample Questions, Page 176, #181 The ratio of two quantities is 3 to 4. If each of the quantities is increased by 5, what is the ratio of these new quantities? 3/4 8/9 18/19 23/24 It cannot be determined.

...

Problem Solving Sample Questions, Page 177, #184 If the two-digit integers M and N are positive and have the same digits, but in reverse order, which of the following CANNOT be the sum of M and N? 181 165 121 99 44

...

Problem Solving Sample Questions, Page 185, #238 The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture, in inches? 9√2 3/2 9/√2 15(1-1/√2) 9/2

...

Problem Solving Sample Questions Page 156, #30 (This problem has a circular image) If O is the center of the circle above, what fraction of the circular region is shaded? 1/12 1/9 1/6 1/4 1/3

Page 198, (Geometry - Circles and area) Vertical angles are congruent, so 150˚ + 150˚ = 300˚ of the circle is not shaded. Sine there are 360˚ in a circle, this makes 360˚ - 300˚ = 60˚ of the circle shaded. The fraction of the circular region that is shaded is thus 60/360 = 1/6. Answer Choice C.

Problem Solving Sample Questions Page 157, #41 If x = 1-3t and y = 2t-1, then for what value of t does x = y? 5/2 3/2 2/3 2/5 0

Page 201, (Algebra - Simultaneous equations) Since it is given that x = y, set the expressions for x and y equal to each other and solve for t. 1 - 3t = 2t - 1 2 = 5t 2/5 = t Answer Choice D

Problem Solving Sample Questions Page 158, #45 If 3 and 8 are the lengths of two sides of a triangular region, which of the following can be the length of the third side? I. 5 II. 8 III. 11 II only III only I and II only II and III only I, II, and III only

Page 202, (Geometry - Triangles) In a triangle the length of the longest side must be smaller than the sum of the lengths of the two other sides. Applying this, test each of these lengths with the lengths of the two sides given to determine which can be possible for the third side of the triangle. I. 8 = 3 + 5 CANNOT be a triangle II. 8 < 3 + 8 CAN be a triangle III. 11 = 3 + 8 CANNOT be a triangle Answer choice A

Problem Solving Sample Questions Page 159, #58 If x^2 = 2y^3 and 2y = 4, what is the value of x^2 + y? -14 -2 3 6 18

Page 204, (Algebra - Simplifying algebraic expressions) Solve the given equations for the values of x^2 and y. First, solve for y. Since 2y = 4, y = 2. Then substitute 2 for y and solve for x^2: x^2 = 2y^3 x^2 = 2(2)^3 x^2 = 16 Therefore, by substitution, x^2 = y = 16 + 2 = 18 Answer choice E.

Problem Solving Sample Questions Page 160, #62 Lucy invested $10,000 in a new mutual fund exactly 3 years ago. The value of the account increased by 10% during the first year, increased by 5% during the second year, and decreased by 10% during the third year. What is the value of the account today? $10,350 $10,395 $10,500 $11,500 $12,705

Page 206, (Arithmetic - Percents) The first year's increase of 10% can be expressed as 1.10; the second year's increase of 5% can expressed as 1.05; and the third year's decrease of 10% can be expressed as 0.90. Multiply the original value of the account by each of these yearly changes. 10,000(1.10)(1.05)(0.90) = 10,395 Answer choice B.

Problem Solving Sample Questions Page 163, #87 Machine A produces 100 parts twice as fast as machine B does. Machine B produces 100 parts in 40 minutes. If each machine produces parts at a constant rate, how many parts does machine A produce in 6 minutes? 30 25 20 15 7.5

Page 213, (Arithmetic - Operations on rational numbers) If machine A produces the parts twice as fast as machine B does, then machine A requires half the time that machine B does to produce 100 parts. So, if machine B takes 40 minutes for the job, machine A takes 20 minutes for the job. This is a rate of 100 parts/20 minutes = 5 parts per minute. At this rate, in 6 minutes machine A will produce 5(6) = 30 parts. Answer choice A.

Problem Solving Sample Questions Page 164, #94 In a certain city, 60% of the registered voters are Democrats and the rest are Republicans. In a mayoral race, if 75% of the registered voters who are Democrats and 20% of the registered voters who are Republicans are expected to vote for Candidate A, what percent of the registered voters are expected to vote for Candidate A? 50% 53% 54% 55% 57%

Page 215, (Arithmetic + Algebra - Percents + Applied problems) Letting v be the number of registered voters in the city, then the information that 60% of the registered voters are Democrats can be expressed as 0.60v. From this it can be stated that 1.00v - 0.60v = 0.40v are Republicans. The percentage of Democrats and the percentage of Republicans who are expected to vote for candidate A can then be expressed as (0.75)(0.60v) + (0.20)(0.40v). Simplify the expression to determine the total percentage of voters expected to vote for Candidate A. (0.75)(0.60v) + (0.20)(0.40v) 0.45v + 0.08v 0.53v Answer choice B.

Problem Solving Sample Questions Page 165, #99 On a scale that measures the intensity of a certain phenomenon, a reading of n+ 1 corresponds to an intensity that is 10 times the intensity corresponding to a reading of n. On that scale, the intensity corresponding to a reading of 8 is how many times as great as the intensity corresponding to a reading of 3? 5 50 10^5 5^10 8^10-3^10

Page 217, (Arithmetic - Operations on rational numbers) Since each increase of 1 in the scale creates an intensity increase of a factor of 10, the intensity of the reading of 8 is 10^8/10^3 = 10^8 - 10^3 = 10^5 times the intensity of reading 3. Answer choice C.

Problem Solving Sample Questions Page 166, #109 1+2+3+4+5/3 1/3(1+1+1+1+1) 1/3+1/3+1/3+1/3+1/3 2/3(1/2+1/2+1/2+1/2+1/2) 1/3 + 2/6 + 3/9 + 4/12 +5/15

Page 220, (Arithmetic - Operations on rational numbers) Simplifying each answer choice will show which one has a different value from the rest. Starting with A, 1+2+3+4+5/3 = 15/3 = 5 Notice that by looking over the other four answer choices quickly, it is possible to observe that not one of them can have a value close to 5. Before putting time into doing calculations, it can be beneficial to see whether the problem can be solved simply by observation. In this case the following calculations will determine the one answer choice that has a different value. A. 5 B. 1/3(1+1+1+1+1) = 5/3 C. 1/3+1/3+1/3+1/3+1/3 = 5/3 D. 2/3(1/2+1/2+1/2+1/2+1/2) = 5/3 E. 1/3 + 2/6 + 3/9 + 4/12 +5/15 = 5/3 Answer choice A.

Problem Solving Sample Questions Page 166, #110 If candy bars that regularly sell for $.40 each are on sale for two for $.75, what is the percent reduction in price of two such candy bars pruchased at the sale price? 2 1/2% 6 1/4% 6 2/3% 8% 12 1/2%

Page 220, (Arithmetic - Percents) Two candy bars at the regular price cost 2 x $0.40 = $0.80. The two candy bars at the sale price cost $0.80 - $0.75 = $0.05 less. The percent of the reduction from the regular price can therefore be established as: $0.05/$0.80 = 0.0625 = 6.25% = 6 1/4% Answer choice B.

Problem Solving Sample Questions Page 168, #123 The price of lunch for 15 people was $207.00, including a 15% gratuity for service. What was the average price per person, EXCLUDING the gratuity? $11.73 $12.00 $13.80 $14.00 $15.87

Page 225, (Arithmetic + Algebra - Statistics + Applied problems) Let c be the total price of lunch for everyone excluding the gratuity. Since $207.00 is given as the total price including the 15% gratuity, the total price for the group lunch excluding the gratuity can be expressed as $207 = 1.15c, or $207/1.15 = $180 = c. The average price per person, or sum of v values/v = average, was thus, $180/15 = $12.00 for each of the 12 individuals. Answer choice B.

Problem Solving Sample Questions Page 169, #126 It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective rates, to complete the order? 7/12 1 1/2 1 5/7 3 1/2 7

Page 226, (Arithmetic - Operations on rational numbers) The first machine can complete 1/4 of the production order in one hour, and the second machine can complete the 1/3 of the same order in one hour. Thus, working together they can complete 1/4 + 1/3 = 3/12 + 4/12 = 7/12 of the order in one hour. Therefore it will take 12/7 = 1 5/7 hours for the two machines working simultaneously to complete the production order. Answer choice C. I don't quite understand this. Once you know the method, you can use the shortcut. If one machine takes A hours and the second takes B hours, together they will take: AB ----- A + B A = 3 B = 3 = (3 * 4) / (3 + 4) = 12/7 = 1 5/7 hours

Problem Solving Sample Questions Page 169, #129 On a recent trip, Cindy drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons of gasoline, rounded to the nearest gallon. The actual number of miles per gallon that Cindy's car got on this trip must have been between: 290/12.5 and 290/11.4 295/12 and 284/11.4 284/12 and 295/12 284/12.5 and 295/11.4 295/12.5 and 284/11.4

Page 227, (Arithmetic - Estimations) The lowest number of miles per gallon can be calculated using the lowest possible miles and the highest amount of gasoline. Conversely the highest number of miles per gallon can be calculated using the highest possible miles and the lowest amount of gasoline. Since the miles are rounded to the nearest 10 miles, they can range anywhere from 284 miles to 295 miles. Since the gallons of gasoline are rounded to the nearest gallon, they can range anywhere from 11.4 gallons to 12.5 gallons. Therefore, the following calculations can be set up. Lowest number of miles per gallon: 284/12.5 Highest number of miles per gallon: 295/11.4 Answer choice D.

Problem Solving Sample Questions Page 168, #121 There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played? 15 16 28 56 64

Page 227, (Arithmetic - Operations on rational numbers) Since no team needs to play itself, each team needs to play 7 other teams. In addition, each game needs to be counted only once. Since two teams play each game, 8x7/2 = 28 games are needed. Or, you could look at the Ivy League standings from last year. Answer choice C.

Problem Solving Sample Questions, Page 177, #183 If 1/2 of the air in a tank is removed with each stroke of a vacuum pump what fraction of the original amount of air is removed after 4 strokes? 15/16 7/8 1/4 1/8 1/16

Page 245, (Arithmetic - Operations on rational numbers) With each stroke's removal of 1/2 the tank's air, the amount of air being removed from the tank on that stroke is equal to the amount of air remaining in the tank after the stroke. With the first stroke, 1/2 of the air is removed; With the second stroke, 1/2 x 1/2 = 1/4 of the air is removed, leaving 1/4 of the air. With the third stroke, 1/2 x 1/4 = 1/8 of the air is removed, leaving 1/8 of the air, and With the fourth stroke, 1/2 x 1/8 = 1/16 of the air is removed. Therefore, with four strokes, 1/2 + 1/4 + 1/8 + 1/16 = 8/16 + 4/16 + 2/16 + 1/16 = 15/16 of the air has been removed. Answer choice A.

Problem Solving Sample Questions, Page 183, #228 If a two digit positive integer has its digits reversed, the resulting integer differs from the original by 27. How much do the two digits differ? 3 4 5 6 7

Page 261, (Algebra - Applied problems) Let the one two-digit integer be represented by 10t+s, where s and t are digits and let the other integer with the reversed digits be represented by 10s+t. The information that the difference between the integers is 27 can be expressed in the following equation, which can be solved for the answer (10s+t) - (10t+s) = 27 10s + t - 10t + s = 27 9s - 9t = 27 s - t = 3 Thus the difference between the two digits is 3. Answer choice A.

Data Sufficiency Page 25, #47 In a certain classroom, there are 80 books of which 24 are fiction and 23 are written in Spanish. How many of the fiction books are written in Spanish? 1. Of the fiction books, there are 6 more that are not written in Spanish than are written in Spanish. 2. Of the books written in Spanish, there are 5 more nonfiction books than fiction books.

Step 1: Determine, from the question stem, what kind of information is needed to answer this question? Step 2: Evaluate each statement individually. -- Statement 1: -- Statement 2: Step 3: Combine the statements if necessary. -- Combined statements:

Problem Solving Sample Questions, Page 177, #187 If p is an even integer and q is an odd integer, which of the following must be an odd integer? p/q pq 2p+q 2(p+q) 3p/q

...

Problem Solving Sample Questions, Page 178, #192 (√2+1)(√2-1)(√3+1)(√3-1) = 2 3 2√6 5 6

...

Problem Solving Sample Questions, Page 179, #199 In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB and the x-axis is the perpendicular bisector of segment BC. If the coordinates of Point A are (2,3) what are the coordinates of C? (-3, -2) (-3, 2) (2, -3) (3, -2) (2, 3)

...

Problem Solving Sample Questions, Page 182, #219 What is the 25th digit to the right of the decimal point in the decimal form of 6/11? 3 4 5 6 7

...

Page 23, #20 A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere? √3:1 1:1 1/2:1 √2:1 2:1

Page 54, Section 3.5 (Geometry - Volume) As the diagram on page 54 shows, the height of the cone will be the radius of the hemisphere, so the ratio is 1:1 Answer B

Problem Solving Sample Questions Page 157, #43 (0.3)^5/(0.3)^3 = 0.001 0.01 0.09 0.9 1.0

Page 201, (Arithmetic - Operations on rational numbers) Work the problem. (0.3)^5/(0.3)^3 = (0.3)^5-3 = (0.3)^2 = 0.09 Answer choice C

Problem Solving Sample Questions Page 152, #4 A case contains c cartons. Each carton contains b boxes, and each box contains 100 paper clips. How many paper clips are contained in 2 cases? 100bc 100b/c 200bc 200b/c 200/bc

Problem Solving, Page 191, (Arithmetic - Simplifying algebraic expressions) Each case has bc boxes, each of which has 100 paper clips. The total number of paper clips in 2 cases is thus 2(bc)(100) = 200bc Answer choice C.

Problem Solving Sample Questions Page 154, #21 If x and y are prime numbers, which of the following cannot be the sum of x and y? 5 9 13 16 23

Problem Solving, Page 195, (Arithmetic - Number Properties) Using known prime numbers, attempt to write each answer choice as the sum of two prime numbers. 5 = 2 + 3 (sum of two prime numbers) 9 = 2 + 7 (sum of two prime numbers) 13 = 2 + 11 (sum of two prime numbers) 16 = 3 + 13 (sum of two prime numbers) 23 = 2 + 21 (NOT the sum of two prime numbers) Answer choice E

Problem Solving Sample Questions Page 155, #23 This question has a graph, but the answer is (7, -5)

Problem Solving, Page 195, (Arithmetic - Operations on rational numbers) NOTE THAT THIS PROBLEM HAS A GRAPH The x coordinate V is 7 and the y coordinate V is -5. thus, the coordinates (x, y) of V are (7, -5).

Problem Solving Sample Questions Page 154, #14 Which cannot be the value of 1/x-1? -1 0 2/3 1 2

Problem Solving, Page 193, (Arithmetic - Number properties) I'm guessing it's 0, since a 0 would have to be in the numerator. Since 1 divided by any number can never equal zero, 1/x-1 ≠ 0 Answer choice B (I wuz right)

Problem Solving Sample Questions Page 153, #12 0.1 + (0.1)^2 + (0.1)^3 0.1 0.111 0.1211 0.2341 0.3

Problem Solving, Page 193, (Arithmetic - Operations on rational numbers) Calculate the squared and the cubed term, and then add the three terms. 0.1 + (0.1)^2 + (0.1)^3 0.1 + 0.01 + 0.001 = 0.111 Answer choice B.

Problem Solving Sample Questions Page 158, #44 In a horticultural experiment, 200 seeds were planted in plot I and 300 were planted in plot II. If 57 percent of the seeds in plot I germinated and 42 percent of the seeds in plot II germinated, what percent of the total number of planted seeds germinated? 45.5% 46.5% 48.0% 49.5% 51.0%

Problem Solving, Page 196, (Arithmetic - Percents) The total number of seeds that germinated was 200(0.57) + 300(0.42) = 114 + 126 = 240 Because this was out of 500 seeds planted, the percent of the total planted that germinated was 240/500 = 0.48 or 48% Answer choice C

Problem Solving Sample Questions Page 157, #39 If x = -3, what is the value of -3x^2? -27 -18 18 27 81

Problem Solving, Page 200, (Algebra - Simplifying algebraic expressions) Since x = -3, the value of -3x^2 - -3(-3)^2 (-3)(-3)(-3) = -27 Answer choice A.

Problem Solving Sample Questions Page 166, #108 If x and y are different prime numbers, each greater than 2, which of the following must be true? I. x + y ≠ 91 II. x - y is an even integer III. x/y is not an integer II only I and II only I and III only II and III only I, II, and III

...

Problem Solving Sample Questions Page 167, #116 Each dot in the mileage table above (SEE CHART) represents an entry indicating the distance between a pair of the five cities. If the table were extended to represent the distances between all pairs of 30 cities and each distance were to be represented by only one entry, how many entries would the table then have? 60 435 450 465 900

...

Problem Solving Sample Questions Page 171, #146 If 0 ≤ x ≤ 4 and y < 12, which of the following CANNOT be the value of xy? -2 0 6 24 48

...

Problem Solving Sample Questions, Page 183, #224 For the past n days, the average daily production at a company was 50 units. If today's production of 90 units raises the average to 55 units per day, what is the value of n? 30 18 10 9 7

...

Problem Solving Sample Questions, Page 184, #230 In an electrical circuit, two resistors with resistances x and y are connected in parallel. In this case, if r is the combined resistance of these two resistors, then the reciprocal of r is equal to the sum of the reciprocals of x and y. What is r in terms of x and y? xy x + y 1/x + y xy/x+y x + y/xy

...

Problem Solving Sample Questions, Page 184, #232 If 1/x - 1/x+1 = 1/x+4, then x could be 0 -1 -2 -3 -4

...

Problem Solving Sample Questions, Page 184, #236 If a, b, and c are consecutive positive integers and a < b< c which of the following must be true? I. c - a = 2 II. abc is an integer III. a + b + c/3 is an integer I only II only I and II only II and III only I, II, and III

...

Problem Solving Sample Questions, Page 185, #241 If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have? 4 5 6 8 9

...

Problem Solving Sample Questions, Page 186, #243 A straight pipe 1 yard in length was marked off in fourths and thirds. If the pipe was then cut into separate pieces at each of these markings which of the following gives all the different lengths of the pieces, in fractions of a yard? 1/6 and 1/4 only 1/4 and 1/3 only 1/6, 1/4 and 1/3 1/12, 1/6 and 1/4 1/12, 1/6 and 1/3

...

Problem Solving Sample Questions, Page 186, #244 If 0.0015 x 10^m/0.03 x 10^k = 5 x 10^7, then m-k = 9 8 7 6 5

...

Problem Solving Sample Questions, Page 186, #245 If x + y = a and x - y = b, then 2xy = a^2 - b^2/2 b^2 - a^2/2 a-b/2 ab/2 a^2 + b^2/2

...

Problem Solving Sample Questions Page 161, #72 At least 2/3 of the 40 members of a committee must vote in favor of a resolution for it to pass. What is the greatest number of members who could vote against the resolution and still have it pass? 19 17 16 14 13

Page 209, (Arithmetic - Operations on rational numbers) If at least 2/3 of the members much vote in favor of a resolution, then no more than 1/3 of the members can be voting against it. On this 40 member committee, 1/3(40) = 13 1/3, which means that no more than 13 members can vote against the resolution and still have it pass. Answer choice E.

Problem Solving Sample Questions Page 156, #29 √(16)(20) + (8)(32) = 4√20 24 25 4√20 + 8√2 32

Problem Solving, Page 197, (Arithmetic - Operations on radical expressions) Work the problem. √(16)(20) + (8)(32) = √(320) + (256) = √576 = 24 Answer choice B.

Problem Solving Sample Questions Page 162, #84 90-8(20÷4)//1/2 = 25 50 100 116 170

...

Problem Solving Sample Questions, Page 183, #227 In the coordinate system above, which of the following is the equation of line ∂? 2x - 3y = 6 2x + 3y = 6 3x + 2y = 6 2x - 3y = -6 3x - 2y = -6

...

Problem Solving Sample Questions, Page 183, #229 The circle with center C shown above is tangent to both axes. If the distance from O to C is equal to k, what is the radius of the circle, in terms of k? k k/√2 k/√3 k/2 k/3

...

Problem Solving Sample Questions, Page 184, #233 (1/2)^-3 x (1/4)^-2 x (1/16)^-1 = (1/2)^-48 (1/2)^-11 (1/2)^-6 (1/8)^-11 (1/8)^-6

...

Problem Solving Sample Questions Page 160, #61 If 1 kilometer is approximately 0.6 mile, whic of the following best approximates the number of kilometers in 2 miles? 10/3 3 6/5 1/3 3/10

Page 204, (Algebra - Second-degree equations)

Problem Solving Sample Questions Page 161, #73 In the Johnson's monthly budget the dollar amounts allocated to household expenses, food, and misc. items are in the ratio 5:2:1 respectively. If the total amount allocated to these three categories is $1,800, what is the amount allocated to food? $900 $720 $675 $450 $225

Page 209, (Algebra - Applied problems) Since the ratio is 5:2:1, let 5x be the money allocated to household expenses, let 2x be the money allocated to food, and 1x be the money allocated to misc. items. The given information can then be expressed in the following equation and solved for x: 5x + 2x + 1x = $1,800 8x = $1,800 x = $225 The money allocated to food is: 2x = 2($225) = $450 Answer choice D.

Problem Solving Sample Questions Page 163, #91 If Sam were twice as old as he is, he would be 40 years older than Jim. If Jim is 10 years younger than Sam, how old is Sam? 20 30 40 50 60

Page 214, (Algebra - Applied problems + Simultaneous equations) Let S be Sam's current age, and let J be Jim's current age. The information in the problems can be expressed in the two equations: 2S = J + 40 J = S-10 Substitute this value of J into the first equation, and solve for S: 2S = (S-10) + 40 2S = S + 30 S = 30 Answer choice B.

Problem Solving Sample Questions Page 164, #92 In a certain furniture store, each week Nancy ears a salary of $240 plus 5 percent of the amount of her total sales that exceed $800 for the week. If Nancy earned a total of $450 one week what were her total sales that week? $2,200 $3,450 $4,200 $4,250 $5,000

Page 214, (Algebra - Applied problems) Let x represent Nancy's total sales for the week. Then, 5% of her total sales over $800(x-800) can be expressed as 0.05(x-800). Her earnings for the week can be expressed in the following equation, which can be solved for x. 450 = 240 + 0.05(x-800) 210 = 0.05x - 40 250 = 0.05x 5,000 = x Answer choice E.

Problem Solving Sample Questions Page 164, #98 If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = -3 -1/2 0 1/2 3/2

Page 216, (Algebra - Second-degree equations + Simultaneous equations) Distributing in both given equations gives: 2x^2 + x = 0 2x^2 -2x - 3/2 = 0 Then, subtracting the second equation from the first gives: 3x + 3/2 = 0 3x = -3/2 x = =1/2 Answer choice B

Problem Solving Sample Questions Page 164, #97 Water consists of hydrogen and oxygen, and the approximate ratio, by mass, of hydrogen to oxygen is 2:16. Approximately how many grams of oxygen are there in 144 grams of water? 16 72 112 128 142

Page 217, (Algebra - Applied problems) From this, the ratio of oxygen to water's hydrogen and oxygen combination is known to be 16/2+16. Letting x be the number of grams of oxygen in 144 grams of water, the proportion comparing oxygen to water's hydrogen and oxygen combination can be expressed as 16/2+16 = x/144 and solved for x as follows: 16/2+16 = x/144 16/18 = x/144 8/9 = x/144 9x = 1,152 x = 128 Answer choice D.

Problem Solving Sample Questions Page 166, #107 What is the smallest integer n for which 25^n > 5^12 6 7 8 9 10

Page 219, (Arithmetic - Operations with rational numbers) Since it can be stated that 25 = 5^2 then it can be stated that 5^12 = (5^2)^6 = 25^6. Thus, n must be greater than 6 and the smallest integer for which the inequality holds true is there 7. Answer choice B.

Problem Solving Sample Questions Page 167, #113 If a = -0.3 which of the following is true? a < a^2 < a^3 a < a^3 < a^2 a^2 < a < a^3 a^2 < a^3 < a a^3 < a < a^2

Page 221, (Arithmetic - Operations on rational numbers) First, determine the relative values of a, a^2, and a^3, remembering that (negative)(negative) = positive. If a = -0.3 then a^2 = (-0.3)^2 = (-0.3)(-0.3) = 0.09, and a^3 = (-0.3)^3 = (-0.3)(-0.3)(-0.3) = -0.027. Since -0.3 < -0.027 < 0.09, then a < a^3 < a^2 Answer choice B

Problem Solving Sample Questions Page 170, #132 In a small snack shop, the average revenue was $400 a day per day over a 10 day period. During this period, if the average daily revenue was $360 for the first 6 days, what was the average daily revenue for the last 4 days? $420 $440 $450 $460 $480

Page 228, Answer choice D.

Problem Solving Sample Questions Page 170, #133 A certain country had a total annual expenditure of $1.2 x 10^12 last year. If the population of the country was 240 million last year, what was the per capita expenditure? $500 $1,000 $2,000 $3,000 $5,000

Page 229, (Arithmetic - Operations on rational numbers) In scientific notation, 240 million is 2.4 x 10^8. So, the per capita expenditure was: $1.2 x 10^12/2.4 x 10^8 = ($1.2/2.4) x 10^12-8 = $0.5 x 10^4 $5,000 Answer choice E.

Problem Solving Sample Questions Page 171, #139 A restaurant meal costs $35.50 and there was no tax. If the tip was more than 10% but less than 15% of the total cost of the meal, then the total amount paid must have been between: $40 and $42 $39 and $41 $38 and $40 $37 and $39 $36 and $37

Page 231, (Arithmetic - Estimation and percent) First calculate the actual total amount for the meal with a 10% tip and a 15% tip. To calculate each, multiply the cost of the meal by (1+the percent as a decimal): 10% tip: $35.50(1.10) = $39.05 15% tip: $35.50(1.15) = $40.825 The only answer choice that includes all values between $39.05 and $40.83 is B. Answer choice B.

Problem Solving Sample Questions Page 171, #145 The trapezoid shown in the FIGURE ABOVE represents a cross section of the rudder of a ship. If the distance from A to B is 13 feet, what is the area of the cross section of the rudder in square feet? 39 40 42 45 46.5

Page 232, (Arithmetic - Operations on rational numbers)

Problem Solving Sample Questions Page 171, #142 Of the 3,600 employees of Company X, 1/3 are clerical. If the clerical staff were to be reduced by 1/3, what percent of the total number of the remaining employees would then be clerical? 25% 22.5% 20% 12.5% 11.1%

Page 232, (Arithmetic - Percents) First calculate the size of the clerical staff. Then calculate the changes to the clerical staff and the total company staff because of the reduction. Clerical staff = 3,600(1/3) = 1,200 Clerical staff lost = 1,200(1/3) = 400 Remaining clerical staff = 1,200 - 400 = 800 Remaining company employees = 3,600 - 400 = 3,200 Percent of remaining employees who are clerical staff = 800/3,200 = 1/4 = 25%

Problem Solving Sample Questions, Page 174, #164 Lois has x dollars more than Jim, and together they have a total of y dollars. Which of the following represents the number of dollars Jim has? y-x/2 y-x/2 y/2 - x 2y - x y - 2x

Page 239, (Algebra - Simplifying algebraic expressions) Let J be the number of dollars that Jim has. Then the amount that Lois has can be expressed as J + x dollars. If Lois and Jim together have a total of y dollars, then: y = J + (J+x) or total dollars = Jim's dollars + Lois' dollars y = 2J + x y-x = 2j y-x/2 = j Answer choice A

Problem Solving Sample Questions, Page 174, #165 During a season a team won 80 percent of its first 100 games and 50 percent of its remaining games. If the team won 70% of its games for the entire season, what was the total number of games that the team played? 180 170 156 150 105

Page 239, (Arithmetic + Algebra) - Percents + applied problems) Let G equal the number of games played by the team this season. The given information can be expressed as (0.80)(100) + 0.50(G-100) = 0.70G, that is, 80% of the first 100 games plus 50% of the remaining games equals 70% of the total number of games played. This equation can be solved for G to determine the answer to the problem: (0.80)(100) + (0.50)(G-100) = 0.70G 80 + 0.50G - 50 = 0.70G 30 = 0.20G 150 = G Answer choice D.

Problem Solving Sample Questions, Page 175, #170 In a certain company the ratio of the number of managers to line workers is 5 to 72. If 8 additional line workers were to be hired the ratio of the number of managers to line workers would be 5 to 74. How many managers does the company have? 5 10 15 20 25

Page 240, (Algebra - Applied problems) Letting m represent the number of managers and p represent the number of production line workers, the given information can be expressed as follows: m/p = 5/72 (original proposition) m/p+8 = 5/74 (added workers) Since the product of the means equals the product of the extremes for both equations: 72m = 5p 74m = 5p+40 Subtract the first equation from the second and solve for m. 2m = 40 m = 20 Answer choice D.

Problem Solving Sample Questions, Page 179, #196 The ratio, by volume, of soap to alcohol to water in a solution is 2:50:100. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how many cubic centimeters of water will it contain? 50 200 400 625 800

Page 250, (Arithmetic - Operations on rational numbers) When a ratio is doubled or halved it means the first value of the ratio is doubled or halved. Thus, when the soap to alcohol ratio of 2:50 is doubled the new ratio of soap to alcohol is 4:50. When the soap to water ratio of 2:100 is halved the new ratio of soap to water is 1:100. Originally the ratio of soap to alcohol to water was 2:50:100. Since soap is now represented by 4, it is necessary to change 1:100 to 4:400 to incorporate all the ratios together. The new solution ratio of soap to alcohol to water is thus 4:50:400. Since 100 cubic centimeters represents the 50 parts of alcohol in the new solution, 800 cubic centimeters will represent the 400 parts of water in the solution. Answer choice E.

Problem Solving Sample Questions, Page 179, #197 If 75% of a class answered the first question on a certain test correctly, 55% answered the second question correctly, and 20% answered neither question correctly, what percent answered both correctly? 10% 20% 30% 50% 65%

Page 250, (Arithmetic - Percents) For questions of this type, it is convenient to draw a Venn diagram to represent the conditions in the problem. For example, the given information can be depicted: Q1: 75% Q2: 55% Intersection: 20% In the diagram it can be seen that the 80% of the class answering a question correctly is represented by the two circles. Let x represent the percent of the class that answered both questions correctly, that is, the shaded region of the intersection. Since the sum of the circles minus their overlap equals 80% of the class (20% or .20 - 1), the information given in the problem can then be expressed as 75% to 55% - x = 80% This equation can be solved for x as follows: 75% + 55% - x = 80% 130% - x = 80% -x = -50% x = 50% Answer choice D.

Problem Solving Sample Questions, Page 181, #210 A television manufacturer produces 600 units of a model each month at a cost of $90 per unit and all of the units are sold each month. What is the minimum selling price per unit that will ensure the monthly profit (revenue - cost) on the sales of these units will be at least $42,000? $110 $120 $140 $160 $180

Page 255, (Arithmetic - Inequalities) Letting x be the amount the manufacturer sells each unit for in dollars, the profit per unit can be expressed as x - 90. Then, the information that the profit on 600 units is the least (greater than or equal to) $42,000 can be expressed in the following inequality, which can be solved for x. 600(x-90) ≥ 42,000 x - 90 ≥ 70 x ≥ 160 The correct answer is D.

Problem Solving Sample Questions, Page 181, #214 Of the 50 researchers in a workgroup, 40% will be assigned to Team A and 60% will be in Team B. However, 70% prefer Team A and 30% prefer Team B. What is the lowest number of researchers who will NOT be assigned to the team they prefer? 15 17 20 25 30

Page 256, (Arithmetic - Percents) The number of researchers assigned to Team A will be (0.40)(50) = 20, and so 30 will be assigned to team B. The number of researchers who prefer team A is (0.70)(50) = 35, and the rest, 15, prefer team B. If all 15 who prefer team B are assigned to team B, which is to have 30 researchers, then 15 who prefer team A will need to be assigned to team B. Alternately, since there area only 20 spots on team A, 35-20 = 15 who prefer team A but will have to be placed on team B instead. Answer choice A.

Problem Solving Sample Questions, Page 182, #220 John and Mary were each paid x dollars in advance to do a certain job together. John worked on the job for 10 hours and Mary worked 2 hours less than John. If Mary gave John y dollars of her payment so that they would have received the same hourly wage, what was the dollar amount, in terms of y, that John was paid in advance? 4y 5y 6y 8y 9y

Page 258, (Algebra - Applied problems) Let w be the amount of Mary and John's same hourly wage. To set the hourly pay equal, John, who worked 10 hours, needs to be paid 10w, and Mary, who worked 8 hours, needs to be paid 8w. Since Mary gave John y dollars, Mary now has x-y dollars and John now has x+y dollars. Their pay can thus be expressed as follows: x-y=8w (Mary's pay) x+y=10w (John's pay) Subtract the first equation from the second and solve for w: 2y = 2w y = w Substitute y for w in the second equation and solve for x, the amount each was paid in advance. x + y = 10y x = 9y Answer choice E

Problem Solving Sample Questions, Page 184, #235 If 2//1 + 2/y = 1, then y = -2 -1/2 1/2 2 3

Page 265, (Algebra - First degree equations) Solve for y. 2//1 + 2/y = 1 1 + 2/y = 2 (multiply both sides by 2 - I guess. Actually, they say that the product of the means equals the product of the extremes) 2/y = 1 (subtract 1 from from both sides) y = 2 (solve for y) Answer choice D. The product of the means is equal to the product of the extremes: When you cross multiply to show 2 fractions are equivalent. Ex a/c =b/d so cross multiplying would show a x d = c x b c x b are the means a x d are the extremes Their products are equal in a proportion or equivalent fractions that is the answer and it is correct

Problem Solving Sample Questions, Page 185, #240 Seed mixture X is 40% ryegrass and 60% bluegrass by weight; seed mixture Y is 25% ryegrass and 75% fescue. If a mixture of X and Y contains 30% ryegrass, what percent of the weight of the mixture is X? 10% 33 1/3% 40% 50% 66 2/3%

Page 267, (Algebra - Applied problems) Let X be the amount of seed mixture X in the final mixture, and let Y be the amount of seed mixture Y in the final mixture. The final mixture of X and Y needs to contain 30 percent ryegrass seed, so any other kinds of grass seed are irrelevant to the solution of this problem. The information about the ryegrass percentages for X, Y and the final mixture can be expressed in the following equation and solved for X. 0.40X + 0.25Y = 0.30(X + Y) 0.40X + 0.25Y = 0.30X + 0.30Y 0.10X = 0.05Y X = 0.5Y Using this, the percent of the weight of the combined mixture (X + Y) that is X is X/X+Y = 0.5Y/0.5Y + Y = 0.5Y/1.5Y = 0.5/1.5 = 0.33333 = 33 1/3% Answer choice B

Problem Solving Sample Questions, Page 186, #249 If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors. 2 3 5 6 8

Page 271, (Arithmetic - Properties of numbers) Since 14n/60 = 7n/30 is an integer and n is an integer, it follows that n is divisible by 30. The possible values of n that are less than 200 are therefore 30, 60 90, 120, 150, and 180. Determining the positive prime factors of these 6 numbers will thus identify how many different prime factors n has. 30 = (2)(3)(5) 60 = (2^2)(3)(5) 90 = (2)(3^2)(5) 120 = (2^3)(3)(5) 150 = (2)(3)(5^2) 180 = (2^2)(3^2)(5) The positive prime factors of n are thus 2, 3, and 5 Answer choice B

Page 22, #17 If n = √16/81, what is the value of √n? 1/9 1/4 4/9 2/3 9/2

Page 53, Section 3.5 (Arithmetic - Operations on Radical Expressions) Since n = √16/81 = 4/9, then √n = √4/9 = 2/3 Answer choice D.

Problem Solving Sample Questions Page 153, #8 When 1/10 percent is 5,000 is subtracted from 1/10 of 5,000, the difference is: 0 50 450 495 500

Problem Solving, Page 192, (Arithmetic - Percents) Convert the fractions to decimals and work the problem. 5,000(0.10) - 5,000(0.001) = 500 - 5 = 495 Answer choice D.

Problem Solving Sample Questions Page 157, #42 In the circular region with center O, shown above, the two unshaded sections constitute 3/7 and 1/3 of the area of the circular region. The shaded section constitutes what fractional part of the area of the circular region? 3/5 6/7 2/21 5/21 16/21

Problem Solving, Page 201, (Algebra - Operations on Rational Numbers) The two unshaded sections constitute: 3/7 + 1/3 = 9/21 + 7/21 = 16/21 of the area of the circular region. Thus, the shaded section constitutes: 1 - 16/21 = 5/21 = of the shaded region Answer choice D.

Problem Solving Sample Questions Page 168, #119 The mean salary of 15 people in the shipping department at a certain firm is $20,000. The salary of 5 of the employees is $25,000 each and the salary of 4 of the employees is $16,000. What is the average salary of the remaining employees? $19,250 $18,500 $18,000 $15,850 $12,300

...

Problem Solving Sample Questions, Page 185, #242 If n is a positive integer, then n(n+1)(n+2) is even only when n is even even only when n is odd odd whenever n is odd divisible by 3 only when n is odd divisible by 4 whenever n is even

...

Problem Solving Sample Questions Page 162, #76 Which of the following ratios is most nearly equal to the ratio 1+√5 to 2? 8 to 5 6 to 5 5 to 4 2 to 1 1 to 1

...

Problem Solving Sample Questions Page 162, #77 7//1/5 + 5//1/7 = 35/74 74/35 35 70 74

...

Problem Solving Sample Questions Page 162, #83 What is the decimal equivalent of (1/5)^5? 0.00032 0.0016 0.00625 0.008 0.003125

...

Problem Solving Sample Questions Page 163, #85 A dealer originally bought 100 identical batteries at a total cost of q dollars. If each battery was sold at 50% above the original cost per battery, then, in terms of q, for how many dollars was each battery sold? 3q/200 3q/2 150q q/100 + 50 150/q

...

Problem Solving Sample Questions Page 163, #90 At the rate of m meters per s seconds, how many meters does a cyclist travel in x minutes? m/sx mx/s 60m/sx 60ms/x 60mx/s

...

Problem Solving Sample Questions Page 165, #102 If T = 5/9(K - 32), and if T - 290, then K = 1,738/9 322 490 544 2,898/5

...

Problem Solving Sample Questions Page 165, #103 The water from one outlet, flowing at a constant rate, can fill a swimming pool in 9 hours. The water from a second outlet, flowing at a constant rate, can fill the same pool in 5 hours. If both outlets are used at the same time, approximately what is the number of hours required to fill the pool? 0.22 0.31 2.50 3.21 4.56

...

Problem Solving Sample Questions Page 166, #106 The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers? 5 8 10 12 15

...

Problem Solving Sample Questions Page 167, #117 Which of the following has a value of less than 1? 2(7/13) √10/2 2/√2 1//1/2 (9/10)^2

...

Problem Solving Sample Questions, Page 166, #112 The front of a 6 foot by 8 foot rectangular door has brass rectangular trim as indicated by the shading in THE DIAGRAM. If the trim is uniformly 1 foot wide, what fraction of the door's front surface is covered by the trim? 13/48 5/12 1/2 7/12 5/8

I sketched it out on a post-it note and came up with 7/12. (sketched it out making 48 squares and counting) Page 221, (Geometry - Area) To determine the area of the trim, find the area of the unshaded portions of the door and subtract. The width of each unshaded rectangle is the width of the door minus two trim strips, or 6-2 = 4 feet. The amount of height available for both unshaded rectangles is the height of the door minus three trim strips or 8-3 = 5 feet. Thus, the area of the unshaded portions is 4x5 = 20 square feet. The area of the entire door is 6x8 = 48 feet, so the area of the trim is 48-20 = 28 feet. Therefore, the fraction of the door's front surface is 28/48 = 7/12 Answer choice D.

Problem Solving Sample Questions Page 161, #74 There are 4 more women than men on Centerville's board of education. If there are 10 members on the board, how many are women? 3 4 6 7 8

...

Problem Solving Sample Questions Page 161, #75 Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an interest rate of 8% compounded semi-annually. What was the total amount of interest paid on this certificate at maturity? $10,464 $864 $816 $800 $480

...

Problem Solving Sample Questions Page 163, #89 In the xy-coordinate system, if (a,b) and (a+3, b+k) are two points on the line defined by the equation x = 3y-7, then k = 9 3 7/3 1 1/3

...

Problem Solving Sample Questions Page 168, #120 David has d books, which is 3 times as many as Jeff and 1/2 as many as Paula. How many books do the three of them have altogether, in terms of d? 5/6d 7/3d 10/3d 7/2d 9/2d

...

Problem Solving Sample Questions Page 168, #122 An operation ø is defined by the equation a ø b = a-b/a+b, for all numbers a and b such that a ≠ -b. If a ≠ -c and a ø c = 0, then c = -a -1/a 0 1/a a

...

Problem Solving Sample Questions Page 169, #127 To mail a package, the rate is x cents for the first pound and y cents for each additional pound, where x > y. Two packages weighing 3 pounds and 5 pounds respectively can be mailed separately or combined as one package. Which method is cheaper, and how much money is saved? Combined, with a saving of x - y cents Combined, with a saving of y - x cents Combined, with a saving of x cents Separately, with a saving of x - y cents Separately, with a saving of y cents

...

Problem Solving Sample Questions Page 169, #130 O . . . . . O ---------- -5 . . . . . 3 Which of the following inequalities is an algebraic expression for the shaded part of the number line above? |x| ≤ 3 |x| ≤ 5 |x - 2| ≤ 3 |x - 1| ≤ 4 |x = 1| ≤ 4

...

Problem Solving Sample Questions Page 170, #131 A factory has 500 workers, 15% of whom are women. If 50 additional workers are to be hired and all of the present workers remain, how many additional workers must be women in order to raise the percentage of women employees to 20%? 3 10 25 30 35

...

Problem Solving Sample Questions, Page 173, #159 If the product of the integers w, w, y, and z is 770, and if 1 < w < x < y < z, what is the value of x + z? 10 13 16 18 21

...

Problem Solving Sample Questions Page 170, #135 The DIAGRAM above shows the various paths along which a mouse can travel from point x, where it is released, to point Y, where it is rewarded with a food pellet. How many different paths from x to y can the mouse take if it goes directly from x to y without retracing any point along a path? 6 7 12 14 17

I'm guessing 7.

Problem Solving Sample Questions Page 155, #27 The dots on the graph indicate the weights and fuel efficiencies for 20 cars . . .

Looks to me like 5 Yup. Page 197 says it's B, or 5 Problem Solving, Page 193, (Arithmetic - Interpretation of Graphs and Tables) Answer choice B.

Problem Solving Sample Questions, Page 172, #153 Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in 5 years? 9 19 21 23 33

J = B + 14 10 + J = 2(B + 10) 10 + B + 14 = 2(B + 10) 24 + B = 2B + 20 4 = B Bill is 4 years old; Jack is 14 years older than him, so Jack is 18. How old will Jack be in 5 years? In 5 years Jack will be 23. Answer Choice D.

Problem Solving Sample Questions Page 154, #20 ___________________ -2 -1 0 1 2 Of the five coordinates associated with points A, B, C, D, and E on the number line above has the greatest absolute value? A B C D E

On page 195 the book says this is A but I don't know why. Well, after a search of the internet, I see that there was an error (one of several) in the book. Answer choice A

Problem Solving Sample Questions Page 156, #34 What percent of 30 is 12? 2.5 3.6 25 40 250

Page 199 First of all, let me say that at 11:15pm on 2 December after a couple of glasses of Montes cabernet I misread the problem as "what is 30% of 12" and came up with 3.6 or answer B. Nope. 12/30 = 2/5 = 40/100 = 40% Answer D

Problem Solving Sample Questions, Page 182, #221 1//1 + 1 + 1/1+1/3 = 4/7 4/3 11/8 11/7 7/4

Page 258, (Arithmetic - Operations on rational numbers) 1//1 + 1 + 1/1+1/3 = 1//1 + 1 + 1/4/3 = 1//1 + 4/3 = 1 + 1//7/4 = 1 + 4/7 = 11/7 Answer choice D.

Prime numbers are numbers that are divisible exactly by only themselves and one. The prime numbers from 1-100 are:

Prime numbers are numbers that are divisible exactly by only themselves and one. The prime numbers from 1-100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. 1 is not a prime number because 1 is divisible by only itself. 0 is not a prime number because it doesn't have any factors. Both 1 and 0 are called ''Special.'' Also 1 and 0 are neither prime nor composite.

Problem Solving Sample Questions Page 156, #33 What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive? 420 840 1,260 2,520 5,040

Problem Solving, Page 198, (Arithmetic - Operations on rational numbers) A number that is divisible by the integers from 1 through 7 inclusive must have 2, 3, 4, 5, 6, and 7 as factors. The lowest positive integer will have no duplication of factors. The lowest multiple of 2, 3, 4, and 6 is 12, and 5 and 7 are prime, so the lowest positive integer that is divisible by each of the integers 1 through 7 inclusive is 12(5)(7) = 420 Answer choice A

Data Sufficiency Page 24, #35 What is the value of w + q? 1. 3w = 3 - 3q 2. 5w + 5q = 5

Step 1: Determine, from the question stem, what kind of information is needed to answer this question? Step 2: Evaluate each statement individually. -- Statement 1: -- Statement 2: Step 3: Combine the statements if necessary. -- Combined statements:

Data Sufficiency Page 25, #44 If m and n are positive integers, is (√m)^n an integer? 1. (√m) is an integer 2. (√n) is an integer

Step 1: Determine, from the question stem, what kind of information is needed to answer this question? Step 2: Evaluate each statement individually. -- Statement 1: -- Statement 2: Step 3: Combine the statements if necessary. -- Combined statements:

Data Sufficiency Page 25, #48 If p is the perimeter of rectangle Q, what is the value of p? 1. Each diagonal of rectangle Q has length of 10. 2. The area of rectangle Q is 48.

Step 1: Determine, from the question stem, what kind of information is needed to answer this question? Step 2: Evaluate each statement individually. -- Statement 1: -- Statement 2: Step 3: Combine the statements if necessary. -- Combined statements:

Problem Solving Sample Questions Page 170, #136 If the operation © is defined by x © y = √xy for all positive numbers x and y, then (5 © 45) © 60 = 30 60 90 30√15 60√15

xx

Problem Solving Sample Questions Page 159, #53 Of the following, which is the closest approximation of 50.2 x 0.49/199.8? 1/10 1/8 1/4 5/4 25/2

Page 204, (Arithmetic - Estimation) Simplify the expression using approximations. 50.2 x 0.49/199.8 50 x 0.5/200 25/200 = 1/8 Answer Choice B.

Problem Solving Sample Questions Page 168, #124 According to a car dealers sales report, 1/3 of the cars sold during a certain period were sedans and 1/5 of the other cars sold were station wagons. If N station wagons were sold during that period, how many sedans, in terms of N, were sold? (2/15)N (3/5)N (5/3)N (5/2)N (15/2)N

...

Problem Solving Sample Questions Page 168, #125 If p/q < 1, and p and q are positive integers, which of the following must be greater than 1? √p/q p/q^2 p/2q q/p^2 q/p

...

Problem Solving Sample Questions Page 170, #134 A rectangular window is twice as long as it wide. If its perimeter is 10 feet, then its dimension in feet are: 3/2 by 7/2 5/3 by 10/3 2 by 4 2 by 6 10/3 by 20

...

Problem Solving Sample Questions Page 170, #137 A bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitesly. What is the value of (10^4 - 10^2)(0.00121212)? 0 0.121212 1.2 10 12

...

Problem Solving Sample Questions Page 170, #138 At the loading dock, each worker on the night crew loaded 3/4 as many boxes as each worker on the day crew. If the night crew has 4/5 as many workers as the day crew, what fraction of all the boxes loaded by the two crews did the day crew load? 1/2 2/5 3/5 4/5 5/8

...

Problem Solving Sample Questions Page 171, #141 A club collected exactly $599 from its members. If each member contributed at least $12, what is the greatest number of members the club could have? 43 44 49 50 51

...

Problem Solving Sample Questions Page 172, #148 What is the lowest integer that is the sum of the three different primes each greater than 20? 69 73 75 79 83

...

Problem Solving Sample Questions Page 172, #149 The mean of 6, 8, and 10 equals the average of 7, 9, and 5 7 8 9 11

...

Problem Solving Sample Questions, Page 172, #150 If x = -1, then x^4 - x^3 + x^4/x - 1 = -3/2 -1/2 0 1/2 3/2

...

Problem Solving Sample Questions, Page 172, #151 A toy store regularly sells all stock at a discount of 20% to 40%. If an additional 25% were deducted from the discount price during a special sale, what would be the lowest possible price of a toy costing $16 before any discount? $5.60 $7.20 $8.80 $9.60 $15.20

...

Problem Solving Sample Questions, Page 173, #154 An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. How much more time will it take to finish filling the pool? 5 hrs. 30 mins. 5 hrs. 20 mins. 4 hrs. 48 mins. 3 hrs. 12 mins. 2 hrs. 40 mins.

...

Problem Solving Sample Questions, Page 173, #155 A positive number x is multiplied by 2, and this product is then divided by 3. If the positive square root of the result of these two operations equals x, what is the value of x? 9/4 3/2 4/3 2/3 1/2

...

Problem Solving Sample Questions, Page 173, #156 A tank containing 10,000 gallons of a solution that is 5 percent sodium chloride by volume. If 2,500 gallons of water evaporate from the tank, the remaining solution will be approximately what percent of sodium chloride? 1.25% 3.75% 6.25% 6.67% 11.7%

...

Problem Solving Sample Questions, Page 173, #158 A committee is composed of w women and m men. If 3 women and 2 men are added to the committee, and if one person is selected at random from the enlarged committee, the the probability that a woman is selected can be represented by w/m w/m+w w+3/m+2 w+3/w+m+3 w+3/w+m+5

...

Problem Solving Sample Questions, Page 173, #160 The FIGURE above shows a circular flower bed, with its center at 0, surrounded by a circular path that is 3 feet wide. What is the area of the path, in square feet? 25π 38π 55π 57π 64π

...

Problem Solving Sample Questions, Page 174, #163 A fruit salad mixture consists of apples, peaches, and grapes in the ratio 6:5:2, respectively, by weight. If 39 pounds of the mixture is prepared, the mixture includes how many more pounds of apples than grapes? 15 12 9 6 4

...

Problem Solving Sample Questions, Page 174, #166 Of 30 applicants for a job 14 had at least 4 years experience, 18 had degrees, and 3 had less than 4 years experience and did not have a degree. How many of the applicants had at least 4 years experience and a degree? 14 13 9 7 5

...

Problem Solving Sample Questions, Page 174, #167 If 1 + 1/x = 2 - 2/x, then x = -1 1/3 2/3 2 3

...

Problem Solving Sample Questions, Page 175, #168 Last year, for every 100 million vehicles that travelled on a certain highway, 96 vehicles were involved in accidents. If 3 billion vehicles traveled on a highway last year, how many of those vehicles were involved in accidents? (1 billion = 1,000,000,000). 288 320 2,880 3,200 28,800

...

Problem Solving Sample Questions, Page 175, #171 If (x-1)^2 = 400 which of the following could be the value of x -5? 15 14 -24 -25 -26

...

Problem Solving Sample Questions, Page 175, #172 Which of the following describes all values of x for which 1-x^2≥0? x ≥ 1 x ≤ -1 0 ≤ x 1 x ≤ 1 or x ≥ 1 -1 ≤ x ≤ 1

...

Problem Solving Sample Questions, Page 176, #176 A rectangular box is 10 inches wide, 10 inches long, and 5 inches high. What is the greatest possible distance, in inches, between any two points on the box? 15 20 25 10√2 10√3

...

Problem Solving Sample Questions, Page 177, #182 If the average of x and y is 60 and the average of y and z is 80, what is the value of z - x? 70 40 20 10 Cannot determine

...

Problem Solving Sample Questions, Page 177, #185 Car X and Car Y traveled the same 80 mile route. If Car x took 2 hours and Car Y traveled at an average speed that was 50% faster than he average speed of Car X, how many hours did it take Car Y to travel the route? 2/3 1 1 1/3 1 3/5 3

...

Problem Solving Sample Questions, Page 177, #186 If the average of the four numbers K, 2K + 3, 3K - 5 and 5K +1 is 63 what is the value of K? 11 15 3/4 22 23 25 3/10

...

Problem Solving Sample Questions, Page 178, #188 Drum X is 1/2 full of oil and drum Y, which has twice the capacity of drum X, is 2/3 full of oil. If all the oil in drum X is poured into drum Y, then drum Y will be filled to what fraction of its capacity? 3/4 5/6 11/12 7/6 11/6

...

Problem Solving Sample Questions, Page 178, #191 The inside dimensions of a rectangular wooden box are 6 inches x 8 inches x 10 inches. A cylindrical canister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such canisters that could be used, what is the radius, in inches, of the one that has maximum volume? 3 4 5 6 8a

...

Problem Solving Sample Questions, Page 178, #193 In a certain calculus class the ratio of the number of math majors to non-math majors is 2 to 5. If 2 more math majors enter the class the ratio would be 1 to 2. How many students are in the class? 10 12 21 28 35

...

Problem Solving Sample Questions, Page 178, #194 What is the units digit of (13)^4 x (17)^2 x (29)^3? 9 7 5 3 1

...

Problem Solving Sample Questions, Page 178, #195 __|_____|_____|___ __|_____|_____|__ __|_____|_____|___ Pat will walk from intersection X to intersection Y along a route that is confined to the square grid of four streets and three avenues shown in the map above. How many routes from X to Y can Pat take that have the minimum possible length? 6 8 10 14 16

...

Problem Solving Sample Questions, Page 179, #200 A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel? 1 2 3 4 12

...

Problem Solving Sample Questions, Page 179, #201 If the sum of n consecutive integers is 0, which of the following must be true? I. n is an even number II. n is an odd number III. The average of the n integers is 0 I only II only III only I and III only II and III only

...

Problem Solving Sample Questions, Page 180, #202 In the formula V = 1/(2r)^3, if r is halved, then V is multiplied by 64 8 1 1/8 1/64

...

Problem Solving Sample Questions, Page 180, #203 A certain bakery has 6 employees. It pays annual salaries of $14,000 to each of 2 employees, $16,000 to 1 employee, and $17,000 to each of the remaining 3 employees. The average annual salary of these employees is closest to which of the following? $15,200 $15,500 $15,800 $16,000 $16,400

...

Problem Solving Sample Questions, Page 180, #204 If x is equal to the sum of the even integers from 40 to 60, inclusive, and y is the number of even integers from 40 to 60, inclusive, what is the value of x + y? 550 551 560 561 572

...

Problem Solving Sample Questions, Page 180, #205 Jar Red Grn Red+Green P x y 80 Q y z 120 R x z 160 In the table above, what is the number of green marbles in jar R 70 80 90 100 110

...

Problem Solving Sample Questions, Page 180, #206 In the circle above, PQ is parallel to the diameter OR, and OR has length 18. What is the length of minor arc PQ? 2π 9π/4 7π/2 9π/2 3π

...

Problem Solving Sample Questions, Page 180, #207 If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n? 2 3 4 6 8

...

Problem Solving Sample Questions, Page 180, #208 S is a set containing 9 different numbers. T is a set containing 8 different numbers, all of which are members of S. Which of the following statements CANNOT be true? The mean of S is equal to the mean of T The median of S is equal to the median of T The range of S is equal to the range of T The mean of S is greater than the mean of T The range of S is less than the range of T

...

Problem Solving Sample Questions, Page 181, #209 How many different positive integers are factors of 441? 4 6 7 9 11

...

Problem Solving Sample Questions, Page 181, #211 If 4x + 3y = -2 and 3x + 6 = 0 what is the value of y? -3 1/3 -2 -2/3 2/3 2

...

Problem Solving Sample Questions, Page 181, #213 Which of the following is the lowest positive integer that is divisible by 2, 3, 4, 5, 6, 7, 8, and 9? 15,120 3,024 2,520 1,890 1,680

...

Problem Solving Sample Questions, Page 181, #215 If m is the average of the first 10 positive multiples of 5 and if M is the median of the first 10 positive multiples of 5, what is the value of M-m? -5 0 5 25 27.5

...

Problem Solving Sample Questions, Page 181, #216 If m > 0 and x is m percent of y, then, in terms of m, y is what percent of x? 100m 1/100m 1/m 10/m 10,000/m

...

Problem Solving Sample Questions, Page 182, #217 A certain junior class has 1,000 students and a certain senior class has 800 students. Among these students there are 60 sibling pairs, each consisting of 1 junior and 1 senior. If 1 student is to be selected at random from each class, what is the probability that the the 2 students selected will be a sibling pair? 3/40,000 1/3,600 9/2,000 1/60 1/15

...

Problem Solving Sample Questions, Page 182, #218 Which of the following CANNOT be the median of the three positive integers x, y, and z? x z x + z x + z/2 x + z/3

...

Problem Solving Sample Questions, Page 182, #222 In the rectangular coordinate system above, if point R (not shown) lies on the positive y-axis and the area of ORP is 12, what is the coordinate point R? 3 6 9 12 24

...

Problem Solving Sample Questions, Page 182, #223 Car A is 20 miles behind Car B, which is traveling in the same direction along the same route as Car A. Car A is traveling at a constant speed of 58 miles per hour and Car B is traveling at a constant speed of 50 miles per hour. How many hours will it take for Car A to overtake and drive 8 miles ahead of Car B? 1.5 2.0 2.5 3.0 3.5

...

Problem Solving Sample Questions, Page 183, #225 (x+1/x-1)^2 If x ≠ 0 and x ≠ 1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to: (x+1/x-1)^2 (x-1/x+1)^2 x^2+1/1-x^2 x^2-1/x+1)^2 -(x-1/x+1)^2

...

Problem Solving Sample Questions, Page 175, #173 The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails? 1/8 1/2 3/4 7/8 15/16

It's 12:17am on Tuesday morning, 20 December (and I just got some bad news the day before), but I think it's: 1/2 x 1/2 x 1/2 = 1/8 - 1 = 7/8 Answer choice D Page 242, (Arithmetic - Probability) Edit: it's 3:10am and I got up because Isabelle is sick and keeping me awake. My answer above is correct.

Problem Solving Sample Questions Page 159, #57 Fermat primes are prime numbers that can be written in the form 2^k + 1, where K is an integer and a power of 2. Which of the following is nOT a Fermat prime? 3 5 17 31 257

Page 204, (Algebra - Second-degree equations) F0 = 21 + 1 = 3 is prime F1 = 22 + 1 = 5 is prime F2 = 24 + 1 = 17 is prime F3 = 28 + 1 = 257 is prime

Problem Solving Sample Questions Page 159, #55 If y= 4 + (x -3)^2, then y is lowest when x = 14 13 0 3 4

Page 204, (Algebra - Second-degree equations) The value of y is lowest when (x - 3)^2 is least, and that is when (x - 3)^2 = 0. Solving this equation for x yields: (x - 3)^2 = 0 √(x - 3)^2 = 0 x - 3 = 0 x = 3 Answer Choice D

Problem Solving Sample Questions Page 159, #59 A glucose solution contains 15 grams of glucose per 100 cubic centimeters of solution. If 45 cubic centimeters of the solution were poured into an empty container, how many grams of glucose would be in the container? 3.00 5.00 5.50 6.50 6.75

Page 205, (Algebra - Applied problems) Let x be the number of grams of glucose in the 45 cubic centimeters of solution. The proportion comparing the glucose in the 45 cubic centimeters to the given information about the 15 grams of glucose in the entire 100 cubic centimeters of the solution can be expressed as x/45 = 15/100, and thus 100x = 675 x = 6.75 Answer choice E.

Problem Solving Sample Questions Page 160, #63 A computer chip manufacturer expects the ratio of the number of defective chips to the number of chips in all future shipments to equal the corresponding ratio for shipments S1, S2, S3, and S4 combined, as shown in the table above. What is the expected number of defective chips in a shipment of 60,000 chips? 14 20 22 24 25

Page 206, (Arithmetic + Algebra - Interpretation of tables + Applied problems) Let n be the expected number of defective chips in a shipment of 60,000 chips. The proportion comparing the number of defective chips to the total number of chips shipped can be expressed in the following equation and solved for n: 2 + 5 + 6 + 4 ------------------------------ 5,000 + 12,000 + 18,000 + 16,000 n/60,000 17/51,000 = n/60,000 51,000n = 1,020,000 n - 20 Answer choice B.

Problem Solving Sample Questions Page 160, #64 A = {2, 3, 4, 5} B = {4, 5, 6, 7, 8} Two integers will be randomly selected from the sets above, one integer from set A and one integer from set B. What is the probability that the sum of the two integers will equal 9? 0.15 0.20 0.25 0.30 0.33

Page 207, (Arithmetic + Algebra - Probability + Concepts of sets) The total number of different pairs of numbers, one from set A and one from a set B is (4)(5) = 20. Of these 20 pairs of numbers, there are 4 possible pairs that sum to 9: 2 and 7, 3 and 6, 4 and 5, and 5 and 4. Thus the probability that the sum of the two integers will be 9 is equal to 4/20 = 0.20. Answer choice B.

Problem Solving Sample Questions Page 160, #66 4 u 7 n 2 3 + 1 6 2 ------ 1,222 If n and u represent single digits in the correctly worked computation above, what is the value of n + u? 7 9 10 11 13

Page 207, (Arithmetic - Operations on rational numbers) Since the sum of the units digits is 7 + 3 + 2 = 12, the sum of the tens digits must be 1 + u + 2 + 6 = 12 because 1 is carried from the sum of the units digits. Solving the equation for u gives u = 3, and 1 is then carried to the hundreds column, making 1 + 4 + n + 1 = 12. Solving this equation for n gives n = 6. Thus n + u = 6 + 3 + 9. Answer choice B.

Problem Solving Sample Questions Page 160, #65 2, 4, 6, 8, n, 3, 5, 7, 9 In the list above, if n is an integer between 1 and 10, inclusive, the the median must be either 4 or 5 either 5 or 6 either 6 or 7 n 5.50

Page 207, (Arithmetic - Statistics) Since the list has an odd number of values, the median will be the middle number when the values are written in ascending order. The given numbers in ascending order are 2, 3, 4, 5, 6, 7, 8, and 9. If n is 1, 2, 3, 4, or 5, the median will be 5 because that will be the middle number in the list. If n is 6, 7, 8, 9, or 10, the median will be 6 because that will become the middle number in the list. Answer choice B.

Problem Solving Sample Questions Page 161, #71 A gym class can be divided into 8 teams with an equal number of players on each team or into 12 teams with an equal number of players on each team. What is the lowest possible number of students in the class? 20 24 36 48 96

Page 209, (Arithmetic - Properties of numbers) The lowest value that can be divided evenly by 8 and 12 is their least common multiple (LCM). Since 8 = 2^3 and 12 = 2^2(3), the LCM is 2^3(3) = 24. Answer choice B.

Problem Solving Sample Questions Page 162, #79 R is the set of positive odd integers less than 50, and S is the set of the squares of the integers in R. How many elements does the intersection of R and S contain? None Two Four Five Seven

Page 211, (Algebra - Concept of Sets) There are 25 positive odd integers less than 50. Of these 25 integers in set R, only the four integers, 1, 9, 25, and 49, (1^2 = 1, 3^2 = 9, 5^2 = 25, and 7^2 = 49) are also the square of integers. Answer choice C.

Problem Solving Sample Questions Page 162, #78 From January 1, 1991 to January 1 1993 the number of people enrolled in health maintenance organizations increased by 15%. The enrollment on January 1, 1993 was 45 million. How many million people, to the nearest million, were enrolled in HMOs on 1 January 1991. 38 39 40 41 42

Page 211, (Arithmetic + Algebra - Percents + Applied problems) Let x be the number of people, in millions, enrolled in HMOs on January 1, 1991. The information that this enrollment had increased by 15% to 45 million can be expressed as follows, and solved for x. x(1.15) = 45 x = 39.13 Thus, to the nearest million, 39 million people were enrolled in HMOs on January 1st, 1991. Answer choice B.

Problem Solving Sample Questions Page 162, #80 A retail appliance store priced a video recorder at 20% above wholesale coast of $200. If a store employee applied the 10% employee discount to the retail price to buy the recorder, how much did the employee pay for the recorder? $198 $216 $220 $230 $240

Page 211, (Arithmetic - Percents) From the given information it can be stated that the retail price of the video recorder in 1.20($200) = $240. The employee received a 10% discount from this price, which means the employee paid 100 - 10 = 90% of the retail price or (0.90)$240 = $216. Answer choice B.

Problem Solving Sample Questions Page 162, #82 Machine A produces bolts at a uniform rate of 120 every 40 seconds, and machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts? 22 25 28 32 56

Page 212, (Algebra - Applied problems) Determine the production rates for each machine separately and then calculate their production rate together. Rate of Machine A = 120/40 = 3 bolts per second Rate of Machine B = 100/20 = 5 bolts per second Combined rate = 3 + 5 = 8 bolts per second Build an equation with s = the number of seconds it takes to produce 200 bolts. 8s = 200 (rate)(time) = amount produced s = 25 (solve for s) Answer choice B.

Problem Solving Sample Questions Page 163, #88 A necklace is made by stringing N individual beads together in the repeating pattern red bead, green bead, blue bead, and yellow bead. If the necklace design begins with a red bead and ends with a white bead, then N could equal: 16 32 41 54 68

Page 213, (Algebra - Applied problems) The bead pattern repeats after every fifth bead. Since the first bead in this design (or the first in the pattern) is red and the last bead in this design (or third in the pattern) in white, the number of beads in this design is 3 more than some multiple of 5. This can be expressed as 5n+3, where n is an integer. Test each of the answer choices to determine which is a multiple of 5 plus a value of 3. Of the options, only 68 = 5(13) + 3 can be written in the form 5n+3. Answer choice E.

Problem Solving Sample Questions Page 163, #86 In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence? 585 580 575 570 565

Page 213, (Algebra - First degree equations) Let the first five consecutive integers be represented by x, x+1, x+2, x+3, and x+4. Then, since the sum of the integers is 560, then: x+ x + 1 + x + 2 + x + 3 + x + 4 = 560 Simplify: 5x + 10 = 560 5x = 550 x = 110 The first integer in the sequence is 110, so the next integers are 111, 112, 113, 114. From this, the last five integers in the sequence, and thus their sum, can be determined. The sum of the 6th, 7th, 8th, 9th, and 10th integers is 115 + 116 + 117 + 118 + 119 = 585 Answer choice A. Damn, what a weird question.

Problem Solving Sample Questions Page 164, #95 A certain company retirement plan has a "Rule of 70" provision that allows an employee to retire when the employee's age plus years of employment with the company totals at least 70. In what year could a female employee hired in 1986 on her 32nd birthday first be eligible to retire under this provision? 2003 2004 2005 2006 2007

Page 215, (Algebra - Applied problems) Construct a table with the data for the first few years after the employee was hired; the pattern of the plan can then be identified from the table. Year Years after hire Age Rule of 70 Value 1986 0 32 32+0 = 32 1987 1 33 33+1 = 34 1988 2 34 34+2 = 36 1989 3 35 35+3 = 37 It becomes clear from the table that for each year of service the employee makes 2 years of progress toward her "Rule of 70" eligibility. Letting y be the number of years after the employee's hire, the following equation can be set up to determine how many years after her hiring this employee will have the 70 combined years needed to be eligible to retire under the "Rule of 70" provision 32+2y = 70 2y = 38 y = 19 From this, the employee will reach "Rule of 70" eligibility 19 years after she is hired, or 1986 + 19 = 2005. Answer choice C.

Problem Solving Sample Questions Page 164, #93 List I: 3, 6, 8, 19 List II: x, 3, 6, 8, 19 If the median of the numbers in List I above is equal to the median of the numbers in List II above, what is the value of x? 6 7 8 9 10

Page 215, (Arithmetic - Statistics) Since List I has an even number of numbers, the median of List I is the average of the middle two numbers or 6+8/2 = 7 is the median of List I. Since List II has an odd number of numbers, the median of List II will be the middle number when the five numbers are put in ascending order. Since the median of List II must be 7 (the median of List I) and since 7 is not in List II, then x = 7. Answer choice B.

Problem Solving Sample Questions Page 165, #100 Gasoline tax: 12% Truck tax: (x + 4)% Highway trust fund: 72% Tax on tires: x% According to the graph above, what percent of the funds for highway maintenance came from the tax on tires? 3% 6% 8% 10% 16%

Page 217, (Arithmetic - Interpretation of graphs) Since all the percentages have to add up to 100%, the combined sources of funding can be expressed in the following equation, which can be solved for x, the tax on tires. 72 + x + 12 + x + 4 = 100 88 + 2x = 100 2x = 12 x = 6 Answer choice B

Problem Solving Sample Questions Page 165, #101 A poll reveals that the average (mean) income of 10 households is $25,000. If 6 of the households have incomes of $30,000 each what is the income of the other 4 households? $21,500 $20,000 $17,500 $7,500 $7,000

Page 217, (Arithmetic - Statistics) Using the formula: sum of values/number of values = average, and letting x be the average income of the other 4 households, this information can be expressed as shown and solved for x. 6(30,000) + 4x/10 = 25,000 180,000 + 4x = 250,000 4x = 70,000 x = 17,500 Answer choice C

Problem Solving Sample Questions, Page 186, #248 Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and y coordinates of P, Q, and R are to be integers that satisfy the inequalities -4 ≤ x ≤ 5 and 6 ≤ y ≤ 16. How many different triangles with these properties could be constructed? 110 1,100 9,900 10,000 12,100

Page 217, (Arithmetic - Statistics) Page 271, (Algebra - Second-degree equations)

Problem Solving Sample Questions Page 165, #104 Diana bought a stereo for $530, which was the retail price plus a 6% sales tax. How much money could she have saved if she had bought the stereo at the same price in a neighboring state where she would have paid 5% sales tax? $1.00 $2.65 $4.30 $5.00 $5.30

Page 218, (Algebra - Applied problems) Letting r be the retail price of the stereo, the information about the total purchase price given the 6% sales tax can be expressed as 530 = 10.6r, and solved for r. 530 = 1.06r 500 = r In the neighboring state the stereo would then have cost $500(1.05) = $525. Thus, the amount saved by reducing the sales tax would have been $530 - $525 = $5. Answer choice D.

Problem Solving Sample Questions Page 165, #105 If a square mirror has a 20" diagonal, what is the approximate perimeter of the mirror in inches? 40 60 80 100 120

Page 219, (Geometry - Perimeter + Pythagorean theorem) Let x be the length of one of the sides of the square mirror. The triangles created by the diagonal are isosceles right triangles for which the Pythagorean theorem yields the following equation that can be solved for x: x^2 + x^2 = 20^2 2 x^2 = 400 x^2 = 200 x = 14.14 Thus, the perimeter of the square mirror would be 4(14.14) = 56.56 ~ 60. Answer choice B.

Problem Solving Sample Questions Page 167, #118 The ratio of the length to the width of a rectangular advertising display is approximately 3.3 to 2. If the width of the display is 8 meters, what is the approximate length of the display, in meters? 7 11 13 16 26

Page 220, (Algebra - Applied problems) Letting l be the length of the advertising display, the proportion for the ratio of the length to the width can be expressed in the following equation, which can be solved for l: 3.3/2 = l/8 13.2 = l (multiply both sides by 8) Answer choice C.

Problem Solving Sample Questions Page 166, #111 If s > 0 and √r/s = s, what is r in terms of s? 1/s √s s√s s^3 s^2 - s

Page 221, (Algebra - Equations) Solve the equation for r as follows: √r/s = s r/s = s^2 (square both sides) r = s^3 (multiply both sides of the equation by s) Answer choice D.

Problem Solving Sample Questions Page 169, #128 If money is invested at r percent interest, compounded annually the amount of the investment will double in approximately 70/r years. If Pat's parents invested $5,000 in a long term bond that pays 8% interest, compounded annually, what will be the approximate total amount of the investment 18 years later, when Pat is ready for college? $20,000 $15,000 $12,000 $10,000 $9,000

Page 227, (Algebra - Applied problems) Since the investment will double in approximately 70/r = 70/8 = 8.75 ~ 9 years, it will double every 9 years. The value of the investment over the 18 years will thus be doubled twice. Therefore, its approximate value will be $5,000(2)(2) = $20,000. Answer choice A.

Problem Solving Sample Questions Page 171, #143 3.003/2.002 = 1.05 1.50015 1.501 1.5015 1.5

Page 232, (Arithmetic - Operations on rational numbers) Simplify the expression as follows: 3.003/2.002 = 3(1.001)/2(1.001) = 3/2 = 1.5 Answer choice E.

Problem Solving Sample Questions, Page 177, #174 Of the final grades received by the students in a math course, 1/5 are A's, 1/4 are B's, 1/2 are C's, and the remaining 10 grades are D's. What is the number of students in the course? 80 110 160 200 400

Page 242, (Algebra - Applied problems) Let x be the number of students in the course. Then: (1/5+1/4+1/2)x or (4/20+5/20+10/20)x or (19/20)x of the students received grades of A, B, or C. This means that the 10 remaining grades represent 1/20th of the students in the course. Thus, 1/20x = 10, and x = 200 Answer choice D.

Problem Solving Sample Questions, Page 176, #178 If x * y = xy - 2(x+y) for all integers x and y, then 2 * (-3) = -16 -11 -4 4 16

Page 243, (Arithmetic - Operations on rational numbers) Substitute 2 * (-3) = xy in the given equation and work the problem. x * y = xy - 2(x+y) 2 * (-3) = 2 * (-3) - 2(2 * (-3)) 2 * (-3) = -6 - 2(-1) 2 * (-3) = -6 + 2 2 * (-3) = -4 Answer choice C.

Problem Solving Sample Questions, Page 176, #179 Club # Chess 40 Drama 30 Math 25 The table above shows the number of students in three clubs at McAuliffe School. Although no student is in all three clubs, 10 students are in both Chess and Drama, 5 students are in both Chess and Math, and 6 students are in both Drama and Math. How many different students are in the 3 clubs? 68 69 74 79 84

Page 244, (Arithmetic - Interpretation of graphs) A good way to solve this problem is to create a Venn diagram. To determine how many students to put in each section begin by putting the given shared-student data in the overlapping sections. Put 0 in the intersection of all three clubs, 10 in the Chess and Drama intersection, 5 in the Chess and Math intersection, and 6 in the Drama and Math intersection. (SEE DIAGRAM) Subtracting the shared students from the totals in each club that are listed in the table establishes the members who belong only to that club. Through this process, it can be determined that the Chess club has 25 such members (40-10-5=25), the Drama club has 14 such members (30-10-6 =14), and the Math club has 14 such members (25-5-6=14). Putting the number of unshared club members into the Venn diagram and then adding up all the sections of the diagram gives 25+14+14+10+5+6=74 students. Answer choice C.

Problem Solving Sample Questions, Page 178, #190 If the operation © is defined for all a and b they the equation a © b = a^2(b)/3, then 2©(3©-1) = 4 2 -4/3 -2 -4

Page 247, (Arithmetic - Operations on rational numbers) The results of the operation a © b can be used whenever the operation a © b occurs. Solve the equation 2 © (3 © -1) by calculating each occurrence of the operation © separately.

Problem Solving Sample Questions, Page 176, #177 A company accountant estimates that airfares next year for business trips of a thousand miles or less will increase by 20% and all other business trips will increase by 10%. This year total airfares for trips of 1,000 miles or less were $9,900 and other trips were $13,000. With the same number of trips next year as this year, how much will be spent on airfare next year? $22,930 $26,180 $26,330 $26,490 $29,770

Page 247, (Arithmetic - Percents) Since the airfare for business trips of a thousand miles or less will increase by 20% next year, the amount spent will be (1.20)($9,900) = $11,880. Since the airfares for all other business trips will increase by 10% next year, the amount spent will be (1.10)($13,000) = $14,300. Thus, according to the accountant's estimate, the total amount spent for airfare next year will be $11,880 + $14,300 = $26,180. Answer choice B.

Problem Solving Sample Questions Page 172, #147 In the FIGURE above, V represents an observation point at one end of a pool. From V, an object that is actually located on the bottom of the pool at point R appears to be at point S. If VR = 10 feet, what is the distance RS, in feet, between the actual position and the perceived position of the object? 10-5√3 10-5√2 2 2 1/2 4

Page 249, (Arithmetic - Percents)

Problem Solving Sample Questions, Page 179, #198 If 1/2 + 1/3 + 1/4 = 13/x, which of the following must be an integer? I. x/8 II. x/12 III. x/24 I only II only I and III only II and III only I, II, and III

Page 250, (Algebra - First degree equations) First, using the common denominator of the fractions, solve the equation for x: 1/2 + 1/3 + 1/4 = 13/x 6/12 + 4/12 + 3/12 = 13/x 13/12 = 13/x x = 12 Then, consider this value of x in each of the answer choices: I. 12/8 NOT an integer II. 12/12 INTEGER III. 12/24 NOT an integer Answer choice B.

Problem Solving Sample Questions, Page 181, #212 I. 72, 73, 74, 75, 76 II. 74, 74, 74, 74, 74 III. 62, 74, 74, 74, 89 The data sets above are ordered from the greatest standard deviation to the least standard deviation in which of the following? I, II, III I, III, II II, III, I III, I, II III, II, I

Page 256, (Algebra - Statistics) To have a large standard deviation means the values are spread far apart with significant variation. Data set II has no variation, so the standard deviation is 0. Data set I has only small variations, so it will have some standard deviation. Data set III has the extreme values 62 and 89, making its standard deviation the largest. The order of the data sets from largest to smallest standard deviation is therefore III, I, II. Answer choice D.

Problem Solving Sample Questions, Page 184, #231 Xavier, Yvonne, and Zelda each try independently to solve a problem. If their individual probabilities for success are 1/4, 1/2, and 5/8 respectively, what is the probability that Xavier and Yvonne, but not Zelda, will solve the problem? 11/8 7/8 9/64 5/64 3/64

Page 263, (Arithmetic - Probability) Since the individual's probabilities are independent, they can be multiplied to figure out the combined probability. The probability of Xavier's success is given as 1/4 and the probability of Yvonne's success is given as 1/2. Since the probability of Zelda's success is given as 5/8, then the probability of her NOT solving the problem is 1-5/8 = 3/8. Thus, the combined probability is: (1/4)(1/2)(3/8) = 3/64 Answer choice E.

Problem Solving Sample Questions, Page 184, #234 In a certain game a large container is filled with red, yellow, green, and blue beads worth 7, 5, 3, and 2 points respectively. A number of beads are then removed from the container. If the product of the point values of the removed beads is 147,000, how many red beads were removed? 5 4 3 2 0

Page 264, (Arithmetic - Number properties) From this, the red beads represent factors of 7 in the total point value of 147,000. Since 147,000 = 147(1,000), and 1,000 = 10^3, then 147 is all that needs to be factored to determine the factors of 7. Factoring 147 yields 147 (3)(49) = (3)(7^2). This means there are 2 factors of 7, or 2 red beads. Answer choice D.

Problem Solving Sample Questions, Page 185, #237 A part-time employee whose hourly wage was increased by 25% decided to reduce the number of hours worked per week so that the employee's total weekly income would remain unchanged. By what percent should the number of hours worked be reduced? 12.5% 20% 25% 50% 75%

Page 265, (Algebra - Applied problems) Let w represent the original hourly wage. Letting h be the original number of hours the employee worked per week, the original weekly income can be expressed as wh. Given a 25% increase in hourly wage, the employee's new wage is thus 1.25w. Letting H be the reduced number of hours, the problem can then be expressed as: 1.25wH = wh (new wage)(new hours) = (old wage)(old hours) By dividing both sides by w, this equation can be solved for H: 1.25H = h H = 0.8h Since the new hours should be 0.8 = 80% of the original hours, the number of hours worked should be reduced by 20%. Answer choice B.

Problem Solving Sample Questions, Page 185, #239 Of the 200 students at College H majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in BOTH chemistry or biology could be any number from 20 to 50 40 to 70 50 to 130 110 to 130 110 to 150

Page 266, (Arithmetic - Operations on rational numbers) A Venn diagram will help with this problem. There are two extremes that need to be considered: (1) having the least number of students majoring in both chemistry and biology, and (2) having the greatest number of students majoring in both chemistry and biology. (1) If at least 30 science majors are not majoring in either chemistry or biology, then at most 200-30=170 students can be majoring in either or both. Since there are 130 + 150 = 280 biology and chemistry majors (some of whom are individual students majoring in both areas), then there are at least 280 - 170 = 110 majoring in both. The diagram following shows this relationship: 170 TOTAL STUDENTS FOR CHEMISTRY AND BIOLOGY MAJORS Chemistry: 20 Biology: 40 Intersection: 110 (2) The maximum number of students who can be majoring in both chemistry and biology is 130, since 130 is the number given as majoring in chemistry, the smaller of the two subject areas. Logically, there cannot be more double majors than there are majors in the smaller field. The diagram below shows this relationship in terms of the given numbers of majors in each subject area. Chemistry: 0 Biology: 20 Intersection: 130 Additionally, from this diagram it can be seen that the total number of students who are majoring in chemistry, or in biology, or in both = 130 + 20 = 150. Thus, there are 200 - 150 = 50 students who are neither chemistry or biology majors. This number is not in conflict with the condition that 30 is the minimum number of non-chemistry or non-biology majors. Thus, the number of students majoring in both chemistry and biology could be any number from a minimum of 110 to a maximum of 130. Answer choice D.

Problem Solving Sample Questions, Page 186, #246 A rectangular circuit board is designed to have width w inches, perimeter p inches, and area k square inches. Which of the following equations must be true? w^2 + pw + k = 0 w^2 - pw = 2k = 0 2w^2 + pw + 2k = 0 2w^2 - pw - 2k = 0 2w^2 - pw + 2k = 0

Page 270, (Algebra - Applied problems) Since there are squared items in all the choices, the solution will most likely come from the area formula for a rectangle. The area, k, of a rectangle equals its length, l, times its width, w. First, the unknown length of the rectangle needs to be determined from the perimeter. The formula for a perimeter p is 2L + 2w = p, and thus 2L = p - 2w, and L = p -2w/2. Substituting this into the formula area = (length)(width) or k = lw gives: k = (p-2w/2)w k = pw-2w^2/2 (distribute the w) 2k = pw - 2w^2 (multiply both sides by 2) 2w^2 -pw + 2k = 0 (set equal to 0 by moving all terms to the left side) Answer choice E.

Problem Solving Sample Questions, Page 186, #247 p, r, s, t, u An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters shown above is an arithmetic sequence, which of the following must also be an arithmetic sequence? I. 2p, 2r, 2s, 2t, 2u II. p-3, r-3, s-3, t-3, u-3 III. p^2, r^2, s^2, t^2, u^2 I only II only III only I and II II and III

Page 270, (Algebra - Concepts of sets + Functions) It follows from the definition of arithmetic sequence given in the first sentence that there is a constant c such that r - p = s - r = t - s = u - t = c. To test a sequence to determine whether it is arithmetic, calculate the difference of each pair of consecutive terms in that sequence to see if a constant difference is found. I. 2r - 2p = 2(r-p) = 2c finish this later.

Data Sufficiency Page 24, #32 Is the range of the integers 6, 3, y, 4, 5 and x greater than 9? 1. y > 3x 2. y > x > 3

Page 58, Data Sufficiency, (Arithmetic - Statistics) The range of a set of integers is equal to the difference between the largest integer and the smallest integer. The range of the set of integers 3, 4, 5, and 6 is 3. 1). Although

Problem Solving Sample Questions Page 155, #24 A rope 40' long is cut into two pieces. If one piece is 18' longer than the other, what is the length, in feet, of the shorter piece? 9 11 18 22 29

Problem Solving, Page 196, (Algebra - First Degree Equations) Build an equation to express the given information and solve for the answer. Let x = length of the shortest piece of rope in feet. Then x + 18 = length of the longer piece of rope in feet. Thus x + (x+18) = length of the longer piece of rope in feet. 2x + 18 = 40 2x = 22 x = 11 Answer choice B.

Problem Solving Sample Questions Page 155, #25 The earth travels around the sun at a speed of 18.5 miles per second. This is how many miles per hour? 1,080 1,160 64,800 66,600 3,996,00

Problem Solving, Page 196, (Arithmetic - Operations on rational numbers) Calculate the equivalent per hour speed, given that there are 60 seconds in one minute and 60 minutes in one hour. 18.5 miles/1 second x 60 seconds/1 minute x 60 minutes/1 hour = 66,600/1 hour 66,600 miles/1 hour Answer choice D.

Problem Solving Sample Questions Page 160, #67 r = 400(D+S-P/P) If stock is sold three months after it is purchased, the formula above relates P, D, S, and r, where P is the purchase price of the stock, D is the amount of any dividend received, S is the selling price of the stock, and r is the yield of the investment as a percent. If Rose purchased $400 worth of stock, received a dividend of $5 and sold the stock for $420 three months after purchasing it, what was the yield of her investment according to the formula? 1.25% 5% 6.25% 20% 25%

asdf

Problem Solving Sample Questions Page 161, #68 The temperatures in degrees Celsius recorded at 6 in the morning in various parts of a certain country were 10˚, 2˚, -1˚, -5˚, and 15˚. What is the median of these temperatures? -2˚C -2˚C 2˚C 3˚C 5˚C

asdf

Data Sufficiency Page 25, #38 If Paula drove the distance from her home to her college at an average speed that was greater than 70 khm, did it take her less than 3 hours to drive this distance? 1. The distance that Paula drove from her home to college was greater than 200 km. 2. The distance that Paula drove from her home to her college was less than 205 km.

Data Sufficiency, Page 60, (Arithmetic - Distance problem) A distance problem uses the formula distance = rate x time. To find the time, the formula would be rearranged as time distance/rate. To solve this problem, it is necessary to know the rate (given here as 70 kilometers per hour) and the distance. 1. If the distance was less than 210 kilometers, then Paula drove for less than 3 hours (209/70). However, if the distance was 210 kilometers or greater, Paula drove 3 hours or more (210/70). However, there is no way to know if the distance was more or less than 210 kilometers. INSUFFICIENT. 2. From this, the most time that Paula drove would have been 205/70 or approximately 2.93 hours. SUFFICIENT. Answer choice B.

Problem Solving Sample Questions Page 155, #28 How many minutes does it take John to type y words if he types at the rate of x words per minute? x/y x/y xy 60x/y y/60x

Page 197, (Algebra - First degree equation) Let m represent the number of minutes it takes John to type y words. In this rate problem, the number of words typed = (typing rate)(time). Thus, y = xm, or m = y/x Answer choice B.

Problem Solving Sample Questions Page 156, #31 If Juan takes 11 seconds to run y yards, how many seconds will it take him to run x yards at the same rate? 11x/y 11y/x x/11y 11/xy xy/11

Page 198, (Algebra - Applied Problems) Juan's running rate can be expressed as y/11 yards per second. Use this value in the formula distance = (rate)(time), letting t equal the time in seconds that it will take Juan to run the distance of x yards: x = y/11(t) 11(x)/y = t Answer choice A.

Problem Solving Sample Questions Page 156, #36 In the figure above, the point on segment PQ that is twice as far from P as from Q is: (refer to drawing) (3,1) (2,1) (2,-1) (1.5,0.5) (1,0)

Page 199, (Geometry - Coordinate geometry) On a segment, a point that is twice as far from one end as the other is 1/3 the distance from one end. The points (0, -1), (1,0), (2,1), and (3,2) are on segment PQ and they divide the segment into three intervals of equal length as shown in the figure below. FIGURE ON PAGE 199 Note that the point (2,1) is twice as far from P(0,-1) as from Q(3,2) and also that it is 1/3 the distance from Q. Answer choice B

Problem Solving Sample Questions Page 157, #38 If 4 is one solution of the equation x^2 + 3x + k = 10, where k is a constant, what is the other solution? -7 -4 -3 1 6

Page 200, (Algebra - Second-degree equations) If 4 is one solution of the equation, then substitute 4 for x and solve for k. x^2 + 3x + k = 10 4^2 + 3(4) + k = 10 16 + 12 + k = 10 28 + k = 10 k = -18 Then, substitute -18 for k and solve for x. x^2 + 3x - 18 = 10 x^2 + 3x - 28 = 0 (x + 7)(x - 4) - 28 = 0 x = -7, x = 4 Answer choice A

Problem Solving Sample Questions Page 158, #47 A car dealer sold x used cars and y new cars during May. If the number of used cars sold was 10 greater than the number of new cars sold, which of the following expresses this relationship? x > 10y x > y + 10 x > y - 10 x = y + 10 x = y - 10

Page 202, (Algebra - Applied problems) According to the given information, if x is 10 more than y, then x = y + 10.

Problem Solving Sample Questions Page 158, #46 How many integers n are there such that 1 < 5n + 5 < 25? 5 4 3 2 1

Page 202, (Algebra - Inequalities) Isolate the variable in the inequalities to determine the range within which n lies. 1 < 5n + 5 < 25 -4 < 5n < 20 -4/5 < n < 4 There are four integers between -4/5 and 4, namely 0, 1, 2, and 3 Answer Choice B

Problem Solving Sample Questions Page 158, #48 If a 10% deposit that has been paid toward the purchase of a certain product is $110, how much more remains to be paid? 880 990 1,000 1,100 1,210

Page 202, (Arithmetic - Percents) Let x be the purchase price of the product. The information that the $110 already paid is 10 percent of the purchase price can be expressed as $110 = (0.10)x. From this $110/0.10 = x or $1,100 = x. Subtracting the deposit from this purchase price yields $1,100 - $110 = $990 still remaining to be paid.

Problem Solving Sample Questions Page 158, #50 In a certain population, there are 3 times as many people aged 21 or under as there are people over 21. The ratio of those 21 or under to the total population is: 1:2 1:3 1:4 2:3 3:4

Page 203, (Applied Problems) Let x represent the people over 21. Then 3x represents the number of people 21 or under and x + 3x = 4x represents the total population. Thus the ratio of those 21 or under to the total population is 3x/4x = 3/4 or 3 to 4 Answer Choice E.

Problem Solving Sample Questions Page 158, #49 (√7 + √7)^2 98 49 28 21 14

Page 203, (Arithmetic - Operations with radical expressions) Simplify the expression: (√7 + √7)^2 = (2√7)^2 = (2)^2 x (√7)^2 = 4 x 7 = 28 Answer Choice C.

Problem Solving Sample Questions Page 159, #52 Kelly and Chris packed several boxes with books. If Chris packed 60 percent of the total number of boxes, what was the ratio of the number of boxes Kelly packed to the number of boxes Chris packed? 1:6 1:4 2:5 3:5 2:3

Page 203, (Arithmetic - Percent) If Chris packed 60 percent of the boxes, then Kelly packed 100 - 60 = 40 percent of the boxes. The ratio of the number of boxes Kelly packed to the number Chris packed is 40%/60% = 2/3 Answer Choice E.

Problem Solving Sample Questions Page 158, #51 (Note that this has a figure) In the figure above, the value of y is: 6 12 24 36 42

Page 203, (Geometry - Angle Measure in Degrees) The sum of the measures of angles that form a straight line equals 180. From this, 2x + 3x = 180 so 5x = 180 and thus x = 36. Then, because the vertical angles are congruent, the measure in degrees of angle 2x equals the measure in degrees of angle (y+30). This can be expressed in the following equation and solved for y. 2x = y + 30 2(36) = y + 30 72 = y + 30 42 = y Answer Choice E.

Page 21, #10 is a geometry problem that requires images. Go to page 21 to solve.

Page 21, #10 is a geometry problem that requires images. Go to page 21 to solve.

Page 20, #1 Last month a certain music club offered a discount to preferred customers. After the first CD was purchased, preferred customers paid $3.99 for each additional CD they bought. If one bought 6 CDs and paid $15.95 for the first disc, then the dollar amount the customer paid for the 6 CDs is: 5(4.00) + 15.90 5(4.00) + 15.95 5(4.00) + 16.00 5(4.00 - 0.01) + 15.90 5(4.00 - 0.05) + 15.95

Page 46, Section 3.5 (Arithmetic - Operations on Rational Numbers) A. 5(4.00) + 15.90

Page 20, #2 The average of the integers from 200 to 400, inclusive, is how much greater than the average of the integers from 50 to 100 inclusive? 150 175 200 225 300

Page 46, Section 3.5 (Arithmetic - Statistics) D. The middle value between 200 and 400 inclusive is 300. The middle value between 50 and 100 is 75. The difference is 300 - 75 = 225

Page 20, #4 Among a group of 2,500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from the 2,500 people, what is the probability that the person selected will be one who invests in municipal bonds but not in oil stocks? 9/50 7/25 7/20 21/50 27/50

Page 47, Section 3.5 (Arithmetic - Probability) B. The probability of an event is: Number of desired events/ Total of number of events that could occur. 700/2,500 = 7/25

Page 20, #5 A closed cylindrical tank contains 36π cubic feet of water and is filled to half its capacity. When the tank is placed upright on its circular base on level ground, what is the height, in feet, of the surface of the water above the ground? 2 3 4 6 9

Page 47, Section 3.5 (Geometry - Volume) Since the cylinder is half full, it will be filled to half its height, whether it is upright or on its side. When the cylinder is on its side half its height is equal to its radius. V = πr^2 x height (volume = π x radius^2 x height) 36π = πr^2 x height The known volume of water is 36π 36 = r^2(4) Substitute 4 for "height" and divide both sides by π 9 = r^2 Solve for radius, 3 = r "r" or radius = height of the water in the cylinder on its side.

Page 21, #6 A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap? 15 20 30 40 45

Page 48, Section 3.5 (Arithmetic - Operations on Rational Numbers) Answer Choice "A": Since it is given that 80 households use neither Brand A nor Brand B, then 200 - 80 = 120 must use Brand A, Brand B, or both. It is also given that 60 households use only Brand A and that three times as many households use Brand B exclusively as use both brands. VENN DIAGRAM ON PAGE 48 If x is the number of households that use both Brand A and Brand B, then 3x use Brand B alone. All the sections in the circles can be added up and set equal to 120, and then the equations can be solved for x: 60 + x + 3x = 120 60 + 4x = 120 4x = 60 x = 15

Page 21, #7 A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member to be chosen as the treasurer? 1/720 1/80 1/10 1/9 1/5

Page 49, Section 3.5 (Arithmetic - Probability) Two probabilities have to be calculated: 1) the probability of Harry's being chosen for secretary and 2) the probability of Harry's being chosen for treasurer. Remember: number of ways desired outcomes can occur/total number of ways the outcome can occur 1) If Harry is to be secretary, he first CANNOT have been chosen for president, and then he must be chosen secretary. Since the probability of being chosen president is 1/10, the probability of NOT being chosen is 9/10. The probability of being chosen secretary is 1/9. Once he is chosen, the probability that he will be chosen for treasurer is 0, so the probability that he will NOT be selected for treasurer is 1 - 0 = 1. Thus, the probability that Harry will be chosen secretary is: 9/10 x 1/9 x 1 = 1/10 2) If Harry is to be treasurer, he needs to NOT chosen president and secretary. For president the odds are 1 - 1/10 = 9/10. For secretary the odds are 1 - 1/9 = 8/9. The probability of being chosen for treasurer is 1/8, so 9/10 x 8/9 x 1/8 = 1/10 Finally: 1/10 + 1/10 = 2/10 = 1/5 or answer choice E

Page 21, #8 If a certain toy store's revenue in November was 2/5 of its revenue in December and its revenue in January was 1/4 of its revenue in November, then the store's revenue in December was how many times the average of its revenue in November and January? 1/4 1/2 2/3 2 4

Page 49, Section 3.5 (Arithmetic - Statistics) Let n be the store's revenue in November, d be the store's revenue in December, and j be the store's revenue in January. The information from the problem can be expressed as n = 2/5 x d and j = 1/4 x n. Substituting 2/5 x d for n in the second equation gives j = 1/4(2/5 x d) = 1/10 x d. Then the average the revenues in November and January can be found by using the formula: average = sum of values/number of values, as follows: 2/5 x d + 1/10 x d/2 4/10d + 1/10d/2 5/10d/2 1/2 x d x (1/2) = 1/4 x d = 4 average = d Thus, the store's revenue in December was 4 times the average in November and January. Answer E.

Page 22, #12 Positive integer y is 50% of 50 percent of positive integer x, and y percent of x equals 100. What is the value of x? 50 100 200 1,000 2,000

Page 50, Section 3.5 (Arithmetic + Algebra Percents - Simultaneous equations) Because y is a positive integer, y percent is notated as y/100. According to the problem: y = 0.50(0.50x) (y/100)x = 100 The first equation simplifies to y = 0.25x and multiplying the second equation by 100 gives xy = 10,000. Substituting the simplified first equation into this second equation gives: x(0.25x) = 10,000 0.25x^2 = 10,000 x^2 = 40,000 x = 200 Answer Choice C.

Page 21, #9 A researcher computed the mean, the median, and the standard deviation for a set of performance scores. If 5 were to be added to each score, which of these three statistics would change? Mean only Median only Standard Deviation only Mean and median Mean and standard deviation

Page 50, Section 3.5 (Arithmetic - Statistics) If 5 were added to each score, the mean would rise by 5, as would the median. However, the spread of the values would remain the same, simply centered around a new value. So, the standard deviation would NOT change. Answer Choice D

Page 22, #11 Of the three digit numbers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two? 90 82 80 45 36

Page 51, Section 3.5 (Arithmetic - Number Properties) In three-digit integers there are three pairs of digits that can be the same while the other digit is different: Tens and ones, Hundreds and tens, and Hundreds and ones. In each of these pairs there are 9 options for having the third digit be different from the other two. The single exception is in the 700-799 set, where the number 700 cannot be included because the problem calls for integers "greater than 700." So, in the 700-799 set there are only 8 options for when the tens and ones are the same. (see table on page 51) Thus, of the three-digit integers greater than 700 there are 9(9) - 1 = 80 numbers that have two digits that are equal to each other when the remaining digit is different from these two. Answer choice C

Page 22, #13 If s and t are positive integers such that s/t = 64.12, which of the following could be the remainder when s is divided by t? 2 4 8 20 45

Page 51, Section 3.5 (Arithmetic - Operations on Rational Numbers) By using a long division model it can be seen that the remainder after dividing s by t is s - 64t. Then, the given equation can be written as 64.12t = s. By splitting proportions of t into its integer multiple and its decimal multiple, this becomes 64t + 0.12t = s, or 0.12t = s - 64t, which is the remainder. So, 0.12t is the remainder; test the choices to find the situation in which t is an integer. 0.12t = 2 or t = 16.67 (NOT an integer) 0.12t = 4 or t = 33.33 (NOT an integer) 0.12t = 8 or t = 66.67 (NOT an integer) 0.12t = 20 or t = 166.67 (NOT an integer) 0.12t = 45 or t = 375 (integer) Answer choice E

Page 22, #15 The product of all the prime numbers less than 20 is closest to which of all of the following powers of 10? 10^9 10^8 10^7 10^6 10^5

Page 52, Section 3.5 (Arithmetic - Number Properties) The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Their product is 9,699,690, or 2 x 3 x 5 x 7 x 11 x 13 x 17 x 19. This is closest to 10,000,000 or 10^7 (10 x 10 x 10 x 10 x 10 x 10 x 10). Answer choice C.

Page 22, #14 Of the 84 parents who attended a meeting a the school, 35 volunteered to supervise children during the school picnic and 11 volunteered both to supervise children during the picnic and bring refreshments to the picnic. If the number of parents who volunteered to bring refreshments was 1.5 times the number who neither volunteered to supervise children during the picnic nor volunteered to bring refreshments, how many of the parents volunteered to bring refreshments? 25 36 38 42 45

Page 52, Section 3.5 (Arithmetic - Operations on Rational Numbers) Out of the 35 parents who agreed to supervise children during the school picnic, 11 parents are also bringing refreshments, so 35 - 11 = 24 parents are only supervising children. NOTE: VENN DIAGRAM ON PAGE 52. Let x be the number of parents who volunteered to bring refreshments and let y be the number of parents who declined to supervise or bring refreshments. That fact that the number of parents who volunteered to bring refreshments is 1.5 times the number who did not volunteer at all can be expressed as x = 1.5y. y + 24 + x = 84 y + x = 60 y = 60 - x (now substitute) x = 1.5(60 - x) x = 90 - 1.5x 2.5x = 90 x = 36 Answer choice B

Page 22, #16 If √3-2x = √2x + 1, 4x^2 1 4 2-2x 4x-2 6x-1

Page 53, Section 3.5 (Algebra - Second Degree Equations) Work with the equation to isolate 4x^2 on one side. √3-2x = √2x + 1 (√3-2x)^2 = (√2x + 1)^2 3-2x = 2x + 2√2x + 1 2 - 4x = 2√2x 1 - 2x = √2x (1 - 2x)^2 = (√2x)^2 1 - 4x + 4x^2 = 2x 4x^2 = 6x - 1 Answer choice E

Page 22, #19 If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k? 1 2 3 4 5

Page 53, Section 3.5 (Geometry - Triangles) In a triangle the sums of the two smaller sides must be larger than the largest side. For k values 3, 4, 5, and 6 the only triangle possible is 2, 7, and k = 6 because only 2 + 6 > 7. For k values 3, 4, and 5 the sum of the smaller two sides is not larger than the third side. Thus 6 is the only possible value of k that satisfies the conditions. Answer choice A.

Page 21, #21 John deposited $10,000 in a savings account paying 4%. How much would he have in 6 months? 10,100 10,101 10,200 10,201 10,400

Page 54, Section 3.5 (Arithmetic - Number Properties) Since John's account is compounded quarterly, he receives 1/4 of his annual interest or 1% every 3 months. This is added to the amount already in the account to accrue interest for the next quarter. After 6 months this process will have occurred twice, so the amount in John's account will then be: ($10,000)(1.01)(1.01) = ($10,000)(1.01)^2 = $10,201 Answer Choice D

Page 23, #22 A container in the shape of a right circular cylinder is 1/2 full of water. If the volume of water in the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of the base of the cylinder in inches? 16/9π 4/√4 12/√π √2/π 4√2/π

Page 54, Section 3.5 (Geometry - Volume) For a right cylinder volume = π(radius)^2(height). Since the volume of water is 36 cubic inches and since this represents 1/2 the container, the water is occupying 1/2 the container's height or 9(1/2) = 4.5 inches. Let r be the radius of the cylinder: 36 = πr^2 (4.5) 8 = πr^2 8/π = r^2 √8/π = r 2√2/√π = Since the diameter is twice the length of the radius, the diameter equals: 2(2√2/√π) = (4)(√2/√π) = (4)2/√π Answer choice E

Page 23, #24 Aaron will jog from home at x miles per hour and then walk back home by the same route at y miles per hour. How many miles from home can Aaron jog so that he spends a total of t hours jogging and walking? xt/y x+t/y xyt/x+y x + y + t/xy x+y/x - t/y

Page 55, Section 3.5 (Algebra - Simplifying algebraic expressions) Let j be the number of hours that Aaron spends jogging; then let t-j be the total number of hours he spends walking. It can be stated that Aaron jogs a distance of xj miles and walks a distance of y(t - j) miles. Because Aaron travels the same route, the miles jogged just equal the miles walked, and they can be set equal. xj = y(t-j) xj = yt - yj xj + yt = yj j(x + y) = yt j = yt/x +y so the number of hours Aaron spends jogging is: j = yt/x +y The number of miles he can jog is xj or, by substitution of this value of j, x(yt/x+y) = xyt/x+t Answer choice C

Page 23, #23 If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 6, then xy must be a multiple of which of the following? I. 8 II. 12 III. 18 II only I and II only I and III only II and III only I, II, and III

Page 55, Section 3.5 (Arithmetic - Number Properties) The product xy must be a multiple of 4(6) = 24 and any of its factors. Test each alternative: I. 24/8 = 3 (8 is a factor of 24; must be a multiple of 8) II. 24/8 = 3 (12 is a factor of 24; must be a multiple of 12). III. 24/8 = 1 1/3 18 is NOT a factor of 24; NEED NOT be a multiple of 18) Answer Choice B

Data Sufficiency Page 24, #26 What is the value of the integer p? 1. Each of the integers 2, 3, and 5 is a factor of p. 2. Each of the integers 2, 5, and 7 is a factor of p.

Page 56, Data Sufficiency, Arithmetic - Number Properties 1) These are factors of p, but it is not clear that they are the only factors of p, NOT sufficient. 2) These are factors of p, but it is not clear that they are the only factors of p, NOT sufficient. Taken together the two answers overlap, but it is not clear they are the only factors of p. Answer choice E.

Problem Solving Sample Questions Page 154, #15 A bakery opened yesterday with its daily supply of 40 dozen rolls. Half of the rolls were sold by noon and 80 percent of the remaining rolls were sold between noon and closing time. How many dozen rolls had not been sold when the bakery closed yesterday? 1 2 3 4 5

Problem Solving, Page 193, (Arithmetic - Operations on rational numbers + Percents) Since half of the 40 dozen rolls were sold by noon, then 1/2(40) = 20 dozen rolls were left to be sold after noon. Because 80 percent of those 20 were sold, 100 - 80 = 20 percent of them or 20(0.20) = 4 dozen rolls had not been sold when the bakery closed. Answer choice D.

Problem Solving Sample Questions Page 153, #11 On Monday, a person mailed 8 packages weighing an average of 12 3/8 lbs and on Tuesday, 4 packages weighing an average of 15 1/4 lbs. What is the average weight in pounds of all the packages the person mailed on both days? 13 1/3 13 13/16 15 1/2 15 15/16 16 1/2

Problem Solving, Page 193, (Arithmetic - Statistics) Since average = sum of values/number of values, the information about the two shipments of packages can be expressed as: 8(12 3/8) + 4(15 1/4)/12 8(99/8) + 4(61/4)/12 99 + 61/12 160/12 13 1/3 Answer choice A

Problem Solving Sample Questions Page 153, #13 A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as high, and twice as wide as the first sandbox, that would be its capacity in cubic feet? 20 40 60 80 100

Problem Solving, Page 193, (Geometry - Volume) When all the dimensions of a 3d object are changed by a factor of 2, the capacity, or volume, changes by a factor of (2)(2)(2) or 2^3 = 8. Thus the capacity of the second sandbox is 10(8) = 80 cubic feet Answer choice D.

Problem Solving Sample Questions Page 154, #19 Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottle could 10 such machines produce in 4 minutes? 648 1,800 2,700 10,800 64,800

Problem Solving, Page 194, (Arithmetic - Operations on rational numbers) Since there are 6 machines, each machine does 1/6 of the work. Each machine can produce 270(1/6) = 45 bottles per minute, so 10 machines can produce 45(10) = 450 bottles per minute. Therefore, the 10 machines can produce 450(4) = 1,800 bottles in 4 minutes. Answer choice B

Problem Solving Sample Questions Page 154, #18 The ratio 2 to 1/3 is equal to the ratio: 6 to 1 5 to 1 3 to 2 2 to 3 1 to 6

Problem Solving, Page 194, (Arithmetic - Operations on rational numbers) The ratio 2 to 1/3 is the same as 2//1/3 = 2(3/1) = 6, which is the same ratio as 6 to 1 Answer choice A

Problem Solving Sample Questions Page 154, #17 150 is what percent of 30? 5% 20% 50% 200% 500%

Problem Solving, Page 194, (Arithmetic - Percents) Let x be the percent missing in the problem. The given information can be expressed in the following equation, which can be solved for x. 150 = (x)(30) 5 = x Then, 5 expressed as a percent is 500% Answer choice E.

Problem Solving Sample Questions Page 154, #16 What is the combined area, in square inches, of the front and back of a rectangular sheet of paper that's 8.5 x 11? 38 44 88 176 187

Problem Solving, Page 194, (Geometry - Area) Since for a rectangle (width)(length) = area, the combined area of the two sides of the sheet is 2(8.5)(11) = 17(11) = 187 square inches. Answer choice E.

Problem Solving Sample Questions Page 155, #22 If each of the following fractions were written as a repeating decimal, which would have the longest sequence of different digits? 2/11 1/3 41/99 2/3 23/37

Problem Solving, Page 195, (Arithmetic - Number Properties - Operations on rational numbers) Compute each fraction's equivalent decimal to determine which one has the longest string of digits. 2/11 = 0.181818 (2 digit sequence) 1/3 = 0.333 (single digit) 41/99 = 0.414141 (2 digit sequence) 2/3 = 0.6666 (single digit) 23/37 = 0.621621 (3 digit sequence) Answer choice E.

Problem Solving Sample Questions Page 155, #26 If the quotient a/b is positive, which of the following must be true? a > 0 b > 0 ab > 0 a-b > 0 a+b > 0

Problem Solving, Page 196, (Arithmetic - Number Properties) If the quotient a/b is positive, then either a and b are both positive or a and b are both negative. A. a can be negative as long as b is negative NEED NOT BE TRUE that a > 0. B. b can be negative as long as a is negative NEED NOT BE TRUE that b > 0. C. (positive)(positive) = positive, and (negative)(negative) = positive MUST BE TRUE When deciding whether something must be true, test the case with values known to be true. It takes only one counterexample to prove it false. D. If a = 4 and b = 8, a - b = -4 NEED NOT BE TRUE that a - b >0 E. If a = -4 and b = -8, a + b = -12 NEED NOT BE TRUE that a + b > 0 Answer choice C.

Problem Solving Sample Questions Page 156, #32 John has 10 pairs of matched socks. If he loses 7 individual socks, what is the greatest number of pairs of matched socks he can have left? 7 6 5 4 3

Problem Solving, Page 198, (Arithmetic - Operations on rational numbers) Determine first the lowest number of pairs of matched socks that can be made from the 7 individual socks. The lowest number of pairs that 7 individual socks can come from is 3 pairs and a single odd sock from a fourth pair. The greatest number of pairs of matched socks John can have left is therefore 10 - 4 = 6 fully matched pairs. Answer choice B

Problem Solving Sample Questions Page 156, #35 If 1.5/0.2 + x = 5, then x = -3.7 0.1 0.3 0.5 2.8

Problem Solving, Page 199, (Algebra - First degree equations) Work the problem to solve for x. 1.5/0.2 + x = 5 1.5 = 5 + 1 0.5 = 5x 0.1 = x Answer choice B

Problem Solving Sample Questions Page 157, #40 29^2 + 29/29 = 870 841 58 31 20

Problem Solving, Page 200, (Arithmetic - Operations on rational numbers) Work the problem. 29^2 + 29/29 29(29 + 1)/29 29(30)/29 = 30 Answer Choice E

Problem Solving Sample Questions Page 157, #37 If a positive integer n is divisible by both 5 and 7, then n must also be divisible by which of the following? I. 12 II. 35 III. 70 None I only II only I and II II and III

Problem Solving, Page 200, (Arithmetic - Properties of numbers) Since 5 and 7 are prime numbers, if n is divisible by both, then n must also be divisible by 5(7) = 35 I. 12 is not a factor of 35 (Need NOT be divisible) II. 35 = 35(1) (MUST be divisible) III. 70 is a multiple of 35, not a factor (Need NOT be divisible) Answer choice C.

Data Sufficiency Page 25, #36 If x and y are points in a plane and x lies inside the circle C with center 0 and radius 2, does y lie inside circle C? 1. The length of line segment xy is 3. 2. The length of line segment 0y is 1.5.

Step 1: Determine, from the question stem, what kind of information is needed to answer this question? Step 2: Evaluate each statement individually. -- Statement 1: -- Statement 2: Step 3: Combine the statements if necessary. -- Combined statements:

Data Sufficiency Page 25, #37 Is x > y? 1. x = y + 2 2. x/2 = y - 1

Step 1: Determine, from the question stem, what kind of information is needed to answer this question? Step 2: Evaluate each statement individually. -- Statement 1: -- Statement 2: Step 3: Combine the statements if necessary. -- Combined statements:

Data Sufficiency Page 25, #45 Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January? 1. More of the people in the auditorium have their birthday in February than in March. 2. Five of the people in the auditorium have their birthday in March

Data Sufficiency, Page 61, (Algebra - Sets and functions) Because it is given that 6 is the greatest number of individuals who can have birthdays in any particular month, these 66 people could be evenly distributed across 11 of the 12 months of the year. That is to say, it could be possible for the distribution to be 11 x 6 = 66, and thus any given month, such as January, would not have a person with a birthday. Assume that January has no people with birthdays, and see if this assumptions is disproved. 1.) the information that more people have February birthdays than March birthdays indicates that the distribution is not even. Therefore, March is underrepresented and must thus have fewer than 6 birthdays. Since no month can have more than 6 people with birthdays, and every month but January already has as many people with birthdays as it can have, January has to have at least 1 person with a birthday. SUFFICIENT. 2.) Again, March is underrepresented with only 5 birthdays and none of the other months can have more than 6 birthdays. Therefore, the extra birthday (from March) must occur in January. SUFFICIENT Answer choice D.

Data Sufficiency, Page 25, #40 If $5,000 invested for one year at p percent simple interest yields $500, what amount must be invested at k percent simple interest for one year to yield the same number of dollars? 1. k = 0.8k 2. k = 8

Data Sufficiency, Page 61, (Arithmetic - interest problem) With simple interest the formula to use is interest = principal x rate x time. It is given that $500 = $5,000 x rate x 1 (year), where the rate p = 10 percent interest. 1. If p is 10 percent, then k = 0.8p is 0.08. Using the same formula, the time is again 1 year; the interest the same amount; and the rate is 0.08 or 8 percent. Thus, $500 = principal x 0.08 x 1 or principal = $6,250. SUFFICIENT 2. If k = 8, then the rate is 8 percent, and the same formula and procedure as above are employed again. SUFFICIENT. Answer choice D.

Data Sufficiency, Page 25, #39 In the x-y plane, if line K has negative slope and passes through the point (-5, r) is the x-intercept of line k positive? 1. The slope of line k is -5 2. r > 0

Data Sufficiency, Page 61, (Geometry - Coordinate geometry) The x-intercept is the x-coordinate of the point in which the line k crosses the x-axis and would have the coordinates (x,0). 1. Knowing the slope of the line does not help in determining the x-intercept, since the point (-5, r) the line k extends in both directions. Without knowing the value of r, the x-intercept could be -5 if r were 0 or it could be other numbers, both positive and negative, depending on the value of r. INSUFFICIENT. 2. Knowing that r > 0 suggests that the x-intercept is not -5; the point (-5, r) where r is a positive number, does lie in quadrant II. It could, however, be any point with an x-coordinate of -5 in that quadrant and line k could have a negative slope, and so the line k would vary with the value of k. Therefore, the x-intercept of line k cannot be determined. INSUFFICIENT. Using 1 and 2 together does not help in the determination of the x-intercept since the point (-5, r) could have any positive y coordinate and thus line k could cross the x-axis at many different places. Answer choice E.

Data Sufficiency Page 25, #42 Does the integer k have at least three different positive prime factors? 1. k/15 is an integer 2. k/10 is an integer

Data Sufficiency, Page 62, (Arithmetic - Properties of numbers) 1. The prime numbers of 15 are 5 and 3. So in this case, k has at least 2 different prime factors, but it is unknown if there are more positive prime factors, INSUFFICIENT. 2. The prime factors of 10 are 2 and 5, showing that k has at least these 2 different positive prime factors, but k might also have more, INSUFFICIENT. Taking both 1 and 2 together, since k is divisible by both 10 and 15, it must be divisible by their different positive prime factors of 2, 3, and 5. Thus k has at least three positive prime factors Answer choice C.

Data Sufficiency Page 25, #43 In City X last April was the average daily temperature greater than the median daily high temperature? 1. In City X last April the sum of the 30 daily high temperatures was 2,160˚. 2. In City X last April 60% of the daily high temperatures were less than the average daily high temperature.

Data Sufficiency, Page 62, (Arithmetic - Statistics) The formula for calculating the arithmetic mean, or the average, is as follows: Average = Sum of v values/v 1. These data will produce an average of 2160/30 = 72˚ for last April in City X. However, there is no information regarding the median for comparison; INSUFFICIENT. 2. The median is the middle temperature of the data. As such, 50% of the daily high temperatures will be above the median, and 50 will be below the median. If 60% of the daily high temperatures were less than the average daily high temperature, then the avearage of the daily highs must be greater than the median. SUFFICIENT! Answer choice B.

Data Sufficiency Page 25, #46 Last year the average salary of the 10 employees of Company X was $42,800. What is the average salary of the same 10 employees this year? 1. For 8 of the 10 employees, this year's salary is 15 greater than last year's salary. 2. For 2 of the 10 employees, this year's salary is the same as last year's salary.

Data Sufficiency, Page 63, (Algebra - Statistics) 1. Since all 10 employees did not receive the same 15 percent increase, it cannot be assumed that the mean this year is 15% higher than last year. It remains unknown whether these 8 salaries were the top 8 salaries, the bottom 8 salaries, or somewhere in-between. Without this type of information from last year, the mean for this year cannot be determined; INSUFFICIENT. 2. If 2 salaries remained the same as last year, then 8 salaries changed. Without further information about the changes, the mean for this year cannot be determined. INSUFFICIENT. Even taking 1 and 2 together it remains impossible to tell the mean salary for this year without additional data. Answer choice E.

Data Sufficiency Page 24, #27 If the length of Wanda's telephone call was rounded up to the nearest whole number by her telephone company, then Wanda was charged for how many minutes for her telephone call? 1. The total charge for Wanda's telephone call was $6.50. 2. Wanda was charged $.50 more for the first minute of the telephone call than for each minute after the first.

Page 56, Data Sufficiency, Arithmetic - Operations. 1) This does not give any information as to the call per minute, NOT sufficient. 2) From this it can be determined only that the call was longer than one minute and that the charge for the first minute was $.50 more than the charge for each succeeding minute, NOT sufficient. Taking 1 and 2 together, the number of minutes cannot be determined as long as the cost of each minute after the first is unknown. For example if the cost of each minute after the first is $.40 then the cost of the first minute is .90. Then the total cost of the other minutes would be $.6.50 -.90 = $5.60, and $5.60 / $.40 would yield 14. In this case the time of the call would be 1 + 14 = 15 minutes. If, however, the cost of each minute after the first minute were $.15, then the cost of the first minute would be $.65. Then $6.50 - $.65 would be $5.85 and this in turn would yield 39 minutes, for a total call time of 40 minutes. More information on the cost of each minute after the first minute is still needed. Answer choice E.

Data Sufficiency Page 24, #25 If the units digit of integer n is greater than 2, what is the units digit of n? 1. The units digit of n is the same as the units digit of n^2. 2. The units digit of n is the same as the units digit of n^3.

Page 56, Data Sufficiency, Arithmetic Operations If the units digit of n is greater than 2, then it can only be the digits 3, 4,5,6, 7,8,9. 1) To solve this problem, it is necessary to find a digit that is the same as the units digit of its square. For example, both 43 squared (1,849) and 303 squared (91,809) have a units digit of 9, which is different from a units digit of 43 and 303. However, 25 squared (625) and 385 squared (148,225) both have a units digit of 5, and 16 and 226 both have a units digit of 6 and their squares (256 and 51,076 respectively) do, too. However, there is no further information to choose between 5 and 6; NOT sufficient. 2) Once again, 5 and 6 are the only numbers which, when cubed, will both have a 5 or 6 respectively in their units digits. However, the information given does not distinguish between them, NOT sufficient. Answer Choice E

Data Sufficiency Page 24, #30 The only gift certificates that a certain store sold yesterday were worth either $100 each or $10 each. If the store sold a total of 20 gift certificates yesterday, how many gift certificates worth $10 each did the store sell yesterday? 1. The gift certificates sold by the store yesterday were worth a total of between $1,650 and $1,800. 2. Yesterday the store sold more than 15 gift certificates worth $100 each.

Page 57, Data Sufficiency, (Algebra - Applied problems + Simultaneous Equations + Inequalities) Let x represent the number of $100 certificates sold and let y represent the number of $10 gift certificates sold. Then the given information can be expressed as: x + y = 20 and thus as y = 20 - x. The value of the $100 certificates sold is $100x and the value of the $10 certificates sold is 10y. 1. From this it is known that 100x + 10y > 1,650. Since y = 20 - x this value can be substituted for y and the inequality can be solved for x: 100x + 10y > 1,650 100x + 10(20-x) > 1,650 100x + 200-10x > 1,650 90x + 200 > 1,650 90x > 1,450 x > 16.1 Thus, more than 16 of the $100 certificates were sold. If 17 $100 certificates were sold then it must be that 3 $10 were also sold for a total of $1,730, which satisfies the condition of being between $1,650 and $1,800. If, however, 18 $100 certificates were sold then it must be that 2 $10 certificates were sold and this totals $1,820, which is more than $1,800 and fails to satisfy the condition. Therefore, 3 of the $10 certificates were sold; SUFFICIENT. 2) From this it can be known only that the number of $10 certificates sold was 4 or fewer; NOT sufficient. Answer Choice A

Data Sufficiency Page 24, #29 In a survey of retailers, what percent had purchased computers for business purposes? 1. 85% of retailers surveyed who owned their own store had purchased computers for business purposes. 2. 40% of the retailers surveyed owned their own store.

Page 57, Data Sufficiency, (Arithmetic - Percents) 1) With only this, it cannot be known what percent of the retailers not owning their own store had purchased computers, and so it cannot be known how many retailers purchased computers overall; NOT sufficient. 2) While this permits the percent of owners and non-owners in the survey to be deducted, the overall percent of retailers who had purchased computers cannot be determined; NOT sufficient. Using 1 and 2 the percent of surveyed owner-retailers who had purchased computers can be deducted and the percent of non-owner retailers can also be deducted. However, the information that would permit a determination of either the percent of non-owner retailers who had purchased computers or the overall percent of all retailers (both owners and non-owners) who had purchased computers is still not provided. Answer Choice E.

Data Sufficiency Page 24, #28 What is the perimeter of isosceles triangle MNP? 1. MN = 16 2. NP = 20

Page 57, Data Sufficiency, (Geometry - Triangles) The perimeter of a triangle is the sum of all three sides. In the case of an isosceles triangle, two of the sides are equal. To determine the perimeter of this triangle, it is necessary to know both the length of an equal side and the length of the base of the triangle. 1) Only gives the length of one side; NOT sufficient. 2) Only gives the length of one side; NOT sufficient Since it is unclear whether MN or NP is one of the equal sides, it is not possible to determine the length of the third side or the perimeter of the triangle. The triangle could either be ((2)(16)) + 20 = 52 or it could be ((2)(20)) + 16 = 56 Answer choice E

Data Sufficiency Page 24, #31 Is the standard deviation of the set of measurements x1, x2 . . . x20 less than 3? 1. The variance for the set of measurements is 4. 2. For each measurement, the difference between the mean and that measurement is 2.

Page 58, Data Sufficiency, (Arithmetic - Statistics) 1) If each variance is 4, then the sum of all the variances is 4 x 20 or 80. Then √80/20 = √4 = 2, which is less than 3. SUFFICIENT. 2) For each measurement the difference between the mean and that measurement is 2. Therefore, the square of each difference is 4 and the sum of all the squares is 4 x 20 = 80. The calculations then proceed as above; SUFFICIENT Answer choice D

Problem Solving Sample Questions Page 152, #6 A rainstorm increased the amount of water stored in State J reservoirs from 124 billion gallons to 138 billion gallons. If the storm increased the amount of water in the reservoirs to 82% of capacity, how many billion gallons were the reservoirs short of total capacity before the storms? 9 14 25 30 44

Problem Solving Sample Questions, Page 191, #7, (Algebra - Applied problems) Letting t be the total capacity of the reservoirs in billions of gallons, the information that the post-storm water amount of 138 billion gallons represented 82% of the total capacity can be expressed as: 0.82t = 138 and thus, 138/0.82 = 168.3 Thus the amount the reservoirs were short of total capacity prior to the storm was 168.3 - 124 = 44.3 billion gallons. Answer choice E.

Problem Solving Sample Questions Page 153, #7 Ont the graph above, when x = 1/2, y=2; and when x =1, y = 1. The graph is symmetrical with respect to the vertical line at x = 2. According to the graph, when x = 3, y = -1 -1/2 0 1/2 1 NOTE: THIS HAS A GRAPH ON PAGE 153

Problem Solving Sample Questions, Page 191, #7, (Arithmetic + Algebra - Interpretation of graphs + second degree equations) Since the graph is symmetric with respect to x = 2, the y value when x = 3 will be the same as the y value when x = 1, which is 1. Answer choice E.

Problem Solving Sample Questions Page 152, #2 For which of the following values of n is 100+n/n NOT an integer? 1 2 3 4 5

Problem Solving, Page 190, (Arithmetic - Number properties) Substitute the value for n given in each answer choice into the expression, and then simplify to determine whether that value results in an integer: 100 + 1/1 = 101/1 = 101 Integer 100 + 2/2 = 102/2 = 51 Integer 100 + 3/3 = 103/3 = 34.33 NOT an Integer 100 + 4/4 = 104/4 = 26 Integer 100 + 5/5 = 105/5 = 21 Integer Answer choice C.

Problem Solving Sample Questions Page 152, #1 A project scheduled to be carried out over a single fiscal year has a budget of $12,600 divided into 12 equal monthly allocations. At the end of the fourth month the total actually spent was $4,580. By what amount was the project over budget? 380 540 1,050 1380 1430

Problem Solving, Page 190, (Arithmetic - Operations with rational numbers) The budget for four months is: $12,600/12 x 4 = $4,200 Thus, the project was $4,580 - $4,200 = $380 over budget for the first four months. Answer choice A.

Problem Solving Sample Questions Page 152, #3 Rectangular floors X and Y have equal area. If floor X is 12 feet x 18 feet and floor y is 9 feet wide, what is the length of floor Y, in feet? 13 1/2 18 18 3/4 21 24

Problem Solving, Page 190, (Geometry - Area) Since for a rectangle, area = (width)(length), the area of floor X = 12(18) = 216. It is given that this is also the area of floor Y, so the length of floor Y can be determined by using the same area in the formula solved for length, or area/width = length. Thus, 216/9 = 24 = length of floor Y. Answer choice E.

Problem Solving Sample Questions Page 152, #5 The sum of prime numbers that are greater than 60 but less than 70 is 67 128 191 197 260

Problem Solving, Page 191, (Arithmetic - Number properties) A prime number is a positive integer divisible by exactly two different positive divisors, 1 and the number itself. Note that: 62, 64, and 66 are also divisible by 2; 63, 66 and 69 are also divisible by 3; 65 is divisible by 5. The only prime numbers between 60 and 70 are 61 and 67, and 61 + 67 = 128 Answer choice B.

Problem Solving Sample Questions Page 153, #9 Which of the following is the value of √3√0.000064? 0.004 0.008 0.02 0.04 0.2

Problem Solving, Page 192, (Arithmetic - Operations on radical expressions) Calculate the value of the expression by first finding the cube root and then finding its square root. √3√0.000064 √0.04 0.2 Answer choice E

Problem Solving Sample Questions Page 153, #10 Raffle tickets numbered consecutively from 101 through 350 are placed in a box. What is the probability that a ticket selected at random will have a number with a hundreds digit of 2? 2/5 2/7 33/83 99/250 100/249

Problem Solving, Page 192, (Arithmetic - Probability) There are 250 integers from 101 to 350 inclusive, 100 of which (that is, 200 to 299) have a hundreds digit of 2. Therefore, the probability that a ticket selected from the box at random will have a hundreds digit of 2 can be expressed as 100/250 = 2/5 Answer choice A.

Data Sufficiency Page 24, #33 Is 5^x+2/25 , 1 1. 5^x < 1 2. x < 0

asdf