Quantitative Solve the problem and indicate the best of the answer choices given. Numbers: All numbers used are real numbers. Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.
1. If n is a positive integer and the product of all the integers from 1 to n, inclusive, is divisible by 990, what is the least possible value of n?
A.8
B.9
C.10
D.11
E.12
A B C D E
D
[解析] Arithmetic Properties of numbers For convenience, let N represent the product of all integers from 1 through n. Then, since N is divisible by 990, every prime factor of 990 must also be a factor of N. The prime factorization of 990 is 2×32×5×11, and therefore, 11 must be a factor of N. Then, the least possible value of N with factors of 2, 5, 32, and 11 is 1×2×3×…×11, and the least possible value of n is 11. The correct answer is D.
2. The probability that event M will not occur is 0.8 and the probability that event R will not occur is 0.6. If events M and R cannot both occur, which of the following is the probability that either event M or event R will occur? A. B. C. D. E.
A B C D E
C
[解析] Arithmetic Probability Let P(M) be the probability that event M will occur, let P(R) be the probability that event R will occur, and let P(M and R) be the probability that events M and R both occur. Then the probability that either event M or event R will occur is P(M)+P(R)-P(M and R). From the given information, it follows that P(M)=1.0-0.8=0.2, P(R)=1.0-0.6=0.4, and P(M and R)=0. Therefore, the probability that either event M or event R will occur is The correct answer is C.
3. The total cost for Company X to produce a batch of tools is $10,000 plus $3 per tool. Each tool sells for $8. The gross profit earned from producing and selling these tools is the total income from sales minus the total production cost. If a batch of 20,000 tools is produced and sold, then Company X's gross profit per tool is
A.$3.00
B.$3.75
C.$4.50
D.$5.00
E.$5.50
A B C D E
C
[解析] Arithmetic Applied problems The total cost to produce 20,000 tools is $10,000+$3(20,000)=$70,000. The revenue resulting from the sale of 20,000 tools is $8(20,000)=$160,000. The gross profit is $160,000-$70,000=$90,000, and the gross profit per tool is The correct answer is C.
4. If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers? A. B. C. D. E.
A B C D E
A
[解析] Arithmetic Statistics For an odd number of data values, the median is the middle number. Thus, 120 is the middle number, and so half of the Q-1 remaining values are at most 120 and the other half of the Q-1 remaining values are at least 120. In particular, data values lie to the right of 120 when the data values are listed in increasing order from left to right, and so the largest data value is . Alternatively, it is evident that (B), (C), or (E) cannot be correct since these expressions do not have an integer value when Q is odd. For the list consisting of the single number 120 (i.e., if Q=1), (D) fails because and (A) does not fail because The correct answer is A.
5. A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach? A. 35 B. 42 C. D. E.
A B C D E
D
[解析] Geometry Triangles
Given the figure above, determine BC Then add 7 to determine how far above the ground the ladder reaches. Triangle △ABC is a 30°-60°-90° triangle with hypotenuse of length 70 feet. Since the lengths of the sides of a 30°-60°-90° triangle are in the ratio and Therefore, the ladder reaches feet above the ground. The correct answer is D.
6.
The window in the figure above consists of a rectangle and a semicircle with dimensions as shown. What is the area, in square feet, of the window?
A.40+8π
B.40+2π
C.32+8π
D.32+4π
E.32+2π
A B C D E
E
[解析] Geometry Area The semicircle has a radius of 2 ft, and thus its area is . The rectangle has dimensions 4 ft by 8 ft, where 8=10-2 is the full height of the window minus the radius of the semicircle, and thus has area (4)(8)=32 ft2. Therefore, in square feet, the area of the window is 32+2π. The correct answer is E.
7. If there are fewer than 8 zeros between the decimal point and the first nonzero digit in the decimal expansion of , which of the following numbers could be the value of t? Ⅰ. 3 Ⅱ. 5 Ⅲ. 9
A.None
B.Ⅰ only
C.Ⅱ only
D.Ⅲ only
E.Ⅱ and Ⅲ
A B C D E
A
[解析] Arithmetic Properties of numbers; Decimals The decimal expansion of has 11 zeros to the right of the decimal point followed by a single digit that is 1. Therefore, in the decimal expansion of the number of zeros between the decimal point and the first nonzero digit is equal to 12 minus the number of digits in the integer t4. Since this number of zeros is fewer than 8, it follows that the number of digits in the integer t4 is greater than 4. The integer t cannot be 3, because the number of digits in 34=81 is not greater than 4. The integer t cannot be 5, because the number of digits in 54=625 is not greater than 4. The integer t cannot be 9, because the number of digits in 94=6,561 is not greater than 4. Therefore, none of 3, 5, or 9 could be the value of t. The correct answer is A.
8. A three-digit code for certain locks uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 according to the following constraints. The first digit cannot be 0 or 1, the second digit must be 0 or 1, and the second and third digits cannot both be 0 in the same code. How many different codes are possible?
A.144
B.152
C.160
D.168
E.176
A B C D E
B
Arithmetic Elementary combinatorics Since the first digit cannot be 0 or 1, there are 8 digits possible for the first digit. Since the second digit must be 0 or 1, there are 2 digits possible for the second digit. If there were no other restrictions, all 10 digits would be possible for the third digit, making the total number of possible codes 8×2×10=160. But, the additional restriction that the second and third digits cannot both be 0 in the same code eliminates the 8 codes 2-0-0, 3-0-0, 4-0-0, 5-0-0, 6-0-0, 7-0-0, 8-0-0, and 9-0-0. Therefore, there are 160-8=152 possible codes. The correct answer is B.
9. Jackie has two solutions that are 2 percent sulfuric acid and 12 percent sulfuric acid by volume, respectively. If these solutions are mixed in appropriate quantities to produce 60 liters of a solution that is 5 percent sulfuric acid, approximately how many liters of the 2 percent solution will be required?
A.18
B.20
C.24
D.36
E.42
A B C D E
E
[解析] Algebra Simultaneous equations Let x represent the quantity of the 2% sulfuric acid solution in the mixture, from which it follows that the 2% sulfuric acid solution contributes 0.02x liters of sulfuric acid to the mixture. Let y represent the quantity of the 12% sulfuric acid solution in the mixture, from which it follows that the 12% sulfuric acid solution contributes 0.12y liters of sulfuric acid to the mixture. Since there are 60 liters of the mixture, x+y=60. The quantity of sulfuric acid in the mixture, which is 5% sulfuric acid, is then (0.05)(60)=3 liters. Therefore, 0.02x+0.12y=3. Substituting 60-x for y gives 0.02x+0.12(60-x)=3. Then, The correct answer is E.
10. If Jake loses 8 pounds, he will weigh twice as much as his sister. Together they now weigh 278 pounds. What is Jake's present weight, in pounds?
A.131
B.135
C.139
D.147
E.188
A B C D E
E
[解析] Algebra Systems of equations Let J represent Jake's weight and S represent his sister's weight. Then J-8=2S and J+S=278. Solve the second equation for S and get S=278-J. Substituting the expression for S into the first equation gives The correct answer is E.
11. For each student in a certain class, a teacher adjusted the student's test score using the formula y=0.8x+20, where x is the student's original test score and y is the student's adjusted test score. If the standard deviation of the original test scores of the students in the class was 20, what was the standard deviation of the adjusted test scores of the students in the class?
A.12
B.16
C.28
D.36
E.40
A B C D E
B
Arithmetic Statistics Let n be the number of students in the class, let be the mean of the students' unadjusted scores. It follows that the standard deviation of the unadjusted scores is To find the standard deviation of the adjusted scores, first find their mean: . Then, subtract the adjusted mean from each adjusted score: (0.8x+20)-(0.8μ+20)=0.8(x-μ). Next, square each difference: (0.8(x-μ))2=0.64(x-μ)2. Next, find the average of the squared differences: Finally, take the nonnegative square root: The correct answer is B.
12. Last year 26 members of a certain club traveled to England, 26 members traveled to France, and 32 members traveled to Italy. Last year no members of the club traveled to both England and France, 6 members traveled to both England and Italy, and 11 members traveled to both France and Italy. How many members of the club traveled to at least one of these three countries last year?
A.52
B.67
C.71
D.73
E.79
A B C D E
B
[解析] Arithmetic Applied problems The numbers in the following diagram represent the numbers of members of the club who traveled to the indicated countries, and these numbers can be determined as follows. Since no members traveled to both England and France, both regions that form the overlap of England and France are labeled with 0. It follows that none of the 6 members who traveled to both England and Italy traveled to France, and so the region corresponding to England and Italy only is labeled with 6. It also follows that none of the 11 members who traveled to both France and Italy traveled to England, and so the region corresponding to France and Italy only is labeled with 11. At this point it can be determined that 26-6=20 members traveled to England only, 26-11=15 members traveled to France only, and 32-(6+11)=15 members traveled to Italy only.
From the diagram it follows that 20+15+6+11+15=67 members traveled to at least one of these three countries. The correct answer is B.
13. A store reported total sales of $385 million for February of this year. If the total sales for the same month last year was $320 million, approximately what was the percent increase in sales?
A.2%
B.17%
C.20%
D.65%
E.83%
A B C D E
C
[解析] Arithmetic Percents The percent increase in sales from last year to this year is 100 times the quotient of the difference in sales for the two years divided by the sales last year. Thus, the percent increase is The correct answer is C.
14. When positive integer x is divided by positive integer y, the remainder is 9. , what is the value of y?
A.96
B.75
C.48
D.25
E.12
A B C D E
B
[解析] Arithmetic Properties of numbers The remainder is 9 when x is divided by y, so x=yq+9 for some positive integer q. Dividing both sides by y gives But, . Equating the two expressions for Thus, q=96 and 9=0.12y y=75 The correct answer is B.
15. A. -3 B. C. 0 D. E.
A B C D E
B
[解析] Algebra Second-degree equations; Simultaneous equations Setting each factor equal to O, it can be seen that the solution set to the first equation is and the solution set to the second equation is Therefore, is the solution to both equations. The correct answer is B.
16.
Figures X and Y above show how eight identical triangular pieces of cardboard were used to form a square and a rectangle, respectively. What is the ratio of the perimeter of X to the perimeter of Y? A. 2:3 B. C. D. 1:1 E.
A B C D E
C
[解析] Geometry Perimeter Because Figure X is a square and the diagonals of a square are the same length, are perpendicular, and bisect each other, it follows that each triangular piece is a 45°-45°-90° triangle. Thus, the length of each side of the square is , and the perimeter is . The perimeter of the rectangle is 2(a+2a)=6a. It follows that the ratio of the perimeter of the square to the perimeter of the rectangle is The correct answer is C.
17. A certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are
A.0 and 13
B.0 and 14
C.1 and 10
D.1 and 9
E.2 and 8
A B C D E
A
[解析] Algebra Simultaneous equations; Inequalities It is given that 2(32)=64 classes are taught by 37 teachers. Let k, m, and n be the number of teachers who taught, respectively, 1, 2, and 3 of the classes. Then k+m+n=37 and k+2m+3n=64. Subtracting these two equations gives m+2n=64-37=27, or 2n=27-m, and therefore 2n≤27. Because n is an integer, it follows that n≤13 and B cannot be the answer. Since n=0 is possible, which can be seen by using m=27 and k=10 (obtained by solving 2n=27-m with n=0, then by solving k+m+n=37 with n=0 and m=27), the answer must be A. It is not necessary to ensure that n=13 is possible to answer the question. However, it is not difficult to see that k=23, m=1, and n=13 satisfy the given conditions. The correct answer is A.
18. For the positive numbers, n, n+1, n+2, n+4, and n+8, the mean is how much greater than the median?
A.0
B.1
C.n+1
D.n+2
E.n+3
A B C D E
B
Algebra Statistics Since the five positive numbers n, n+1, n+2, n+4, and n+8 are in ascending order, the median is the third number, which is n+2. The mean of the five numbers is Since (n+3)-(n+2)=1, the mean is 1 greater than the median. The correct answer is B.
19. The interior of a rectangular carton is designed by a certain manufacturer to have a volume of x cubic feet and a ratio of length to width to height of 3:2:2. In terms of x, which of the following equals the height of the carton, in feet? A. B. C. D. E.
A B C D E
B
[解析] Geometry; Arithmetic Volume; Ratio and proportion Letting c represent the constant of proportionality, the length, width, and height, in feet, of the carton can be expressed as 3c, 2c, and 2c, respectively, The volume of the carton is then (3c)(2c)(2c)=12c3 cubic feet. But, the volume is x cubic feet, and so 12c3=x. Then, . The height of the carton is 2c and in terms of x, and , the height of the carton can be expressed as The correct answer is B.
20. 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?
A.5
B.8
C.10
D.12
E.15
A B C D E
E
[解析] Algebra Applied problems Let s be the present number of students, and let t be the present number of teachers. According to the problem, the following two equations apply: Solving the first equation for s gives s=30t. Substitute this value of s into the second equation, and solve for t. The correct answer is E.
21. What is the smallest integer n for which 25n＞512?
A.6
B.7
C.8
D.9
E.10
A B C D E
B
[解析] Arithmetic Operations with rational numbers Because 52=25, a common base is 5. Rewrite the left side with 5 as a base: 25n=(52)n=52n. It follows that the desired integer is the least integer n for which 52n＞512. This will be the least integer n for which 2n＞12, or the least integer n for which n＞6, which is 7. The correct answer is B.
22. Sixty percent of the members of a study group are women, and 45 percent of those women are lawyers. If one member of the study group is to be selected at random, what is the probability that the member selected is a woman lawyer?
A.0.10
B.0.15
C.0.27
D.0.33
E.0.45
A B C D E
C
[解析] Arithmetic Probability For simplicity, suppose there are 100 members in the study group. Since 60 percent of the members are women, there are 60 women in the group. Also, 45 percent of the women are lawyers so there are 0.45(60)=27 women lawyers in the study group. Therefore the probability of selecting a woman lawyer is . The correct answer is C.
23. Each year for 4 years, a farmer increased the number of trees in a certain orchard by of the number of trees in the orchard the preceding year. If all of the trees thrived and there were 6,250 trees in the orchard at the end of the 4-year period, how many trees were in the orchard at the beginning of the 4-year period?
A.1,250
B.1,563
C.2,250
D.2,560
E.2,752
A B C D E
D
[解析] Arithmetic Operations on rational numbers Increasing the number of trees each year by of the number of trees in the orchard the preceding year is equivalent to making the number of trees increase 25% per year, compounded yearly. If there were n trees at the beginning of the 4-year period, then there will be 1.25n trees at the end of the first year, 1.25(1.25n)=(1.25)2n trees at the end of the second year, 1.25[(1.25)2n]=(1.25)3n trees at the end of the third year, and 1.25[(1.25)3n]=(1.25)4n trees at the end of the fourth year. Hence, 6,250=(1.25)4n and . The arithmetic can be greatly simplified by rewriting and 6,250 as (625)(10)=(54)(10). Then The correct answer is D.
24.
According to the chart shown, which of the following is closest to the median annual number of shipments of manufactured homes in the United States for the years from 1990 to 2000, inclusive?
A.250,000
B.280,000
C.310,000
D.325,000
E.340,000
A B C D E
C
[解析] Arithmetic Interpretation of graphs and tables; Statistics From the chart, the approximate numbers of shipments are as follows:
Since there are 11 entries in the table and 11 is an odd number, the median of the numbers of shipments is the 6th entry when the numbers of shipments are arranged in order from least to greatest. In order, from least to greatest, the first 6 entries are:
The 6th entry is 310,000. The correct answer is C.
25. For the positive integers a, b, and k, ak||b means that ak is a divisor of b, but ak+1 is not a divisor of b. If k is a positive integer and 2k||72, then k is equal to
A.2
B.3
C.4
D.8
E.18
A B C D E
B
Arithmetic Property of numbers Since 72=(23)(32), it follows that 23 is a divisor of 72 and 24 is not a divisor of 72. Therefore, 23||72, and hence k=3. The correct answer is B.
26. A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than m+d?
A.16%
B.32%
C.48%
D.84%
E.92%
A B C D E
D
[解析] Arithmetic Statistics Since 68% lies between m-d and m+d, a total of (100-68)%=32% lies to the left of m-d and to the right of m+d. Because the distribution is symmetric about m, half of the 32% lies to the right of m+d. Therefore, 16% lies to the right of m+d, and hence (100-16)%=84% lies to the left of m+d. The correct answer is D.
27. Four extra-large sandwiches of exactly the same size were ordered for m students, where m＞4. Three of the sandwiches were evenly divided among the students. Since 4 students did not want any of the fourth sandwich, it was evenly divided among the remaining students. If Carol ate one piece from each of the four sandwiches, the amount of sandwich that she ate would be what fraction of a whole extra-large sandwich? A. B. C. D. E.
A B C D E
E
[解析] Algebra Applied problems Since each of 3 of the sandwiches was evenly divided among m students, each piece was of a sandwich. Since the fourth sandwich was evenly divided among m-4 students, each piece was of the fourth sandwich. Carol ate 1 piece from each of the four sandwiches, so she ate a total of The correct answer is E.
28. Which of the following equations has as one of its roots?
A.x2+2x-1=0
B.x2-2x+1=0
C.x2+2x+1=0
D.x2-2x-1=0
E.x2-x-1=0
A B C D E
D
[解析] Algebra Second-degree equations This problem can be solved by working backwards to construct a quadratic equation with as a root that does not involve radicals. The correct answer is D.
29. In Country C, the unemployment rate among construction workers dropped from 16 percent on September 1, 1992, to 9 percent on September 1, 1996. If the number of construction workers was 20 percent greater on September 1, 1996, than on September 1, 1992, what was the approximate percent change in the number of unemployed construction workers over this period?
A.50% decrease
B.30% decrease
C.15% decrease
D.30% increase
E.55% increase
A B C D E
B
[解析] Arithmetic Percents Let U1 and U2 be the numbers of unemployed construction workers on September 1, 1992, and September 1, 1996, respectively, and let N be the number of construction workers on September 1, 1992. Then, from the given information, 1.2N is the number of construction workers on September 1, 1996, U1=0.16N, and U2=0.09(1.2N). Therefore, the percent change from September 1, 1992, to September 1, 1996, of unemployed construction workers is given by The correct answer is B.
30. In a box of 12 pens, a total of 3 are defective. If a customer buys 2 pens selected at random from the box, what is the probability that neither pen will be defective? A. B. C. D. E.
A B C D E
C
[解析] Arithmetic Probability Let A represent the event that the first pen purchased is not defective and let B represent the event that the second pen purchased is not defective, where A and B are dependent events. Since there are 3 defective pens in the box of 12 pens, there are 9 pens in the box that are not defective. Using standard probability notation, and P(B, given that A has occurred)=. (Event A has occurred so there are 11 pens left in the box and 8 of them are not defective.) Then, using the multiplication rule for dependent events, P(neither pen is defective) =P(A and B)=P(A)×P(B, given that A has occurred) Alternately, the probability of selecting 2 pens, neither of which is defective, from a box containing 12 pens, 3 of which are defective, and therefore, 9 of which are non-defective is the number of ways to select 2 non-defective pens from 9 non-defective pens over the number of ways to select 2 pens from 12 pens The correct answer is C.
31. At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is 60 cents. Mary selects a total of 10 apples and oranges from the fruit stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 56 cents. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 52 cents?
A.1
B.2
C.3
D.4
E.5
A B C D E
E
[解析] Algebra Statistics If Mary selected x apples, then she selected (10-x) oranges. The average price of the 10 pieces of fruit is . From this, Thus, Mary selected 2 apples and8 oranges. Next, let y be the number of oranges Mary needs to put back, so that the average price of the [2+(8 -y)] pieces of fruit Mary keeps is 52 cents. Then, Therefore, Mary must put back 5 oranges, so that the average price of the fruit she keeps (that is, the average price of 2 apples and 3 oranges) is 52 cents. The correct answer is E.
32. A pharmaceutical company received S3 million in royalties on the first $20 million in sales of the generic equivalent of one of its products and then $9 million in royalties on the next $108 million in sales. By approximately what percent did the ratio of royalties to sales decrease from the first $20 million in sales to the next $108 million in sales?
A.8%
B.15%
C.45%
D.52%
E.56%
A B C D E
C
[解析] Arithmetic Percents The ratio of royalties to sales for the first $20 million in sales is , and the ratio of royalties to sales for the next $108 million in sales is . The percent decrease in the royalties to sales ratios is 100 times the quotient of the difference in the ratios divided by the ratio of royalties to sales for the first $20 million in sales or The correct answer is C.
33.
The light in a restroom operates with a 15-minute timer that is reset every time the door opens as a person goes in or out of the room. Thus, after someone enters or exits the room, the light remains on for only 15 minutes unless the door opens again and resets the timer for another 15 minutes. If the times listed above are the times at which the door opened from 8:00 to 10:00, approximately how many minutes during this two-hour period was the light off?
A.10
B.25
C.35
D.40
E.70
A B C D E
B
[解析] Arithmetic Operations with integers Look for two consecutive times that are more than 15 minutes apart 8:03-8:00=3 minutes 8:04-8:03=1 minute 8:04-8:04=0 minutes 8:06-8:04=2 minutes 8:10-8:06=4 minutes 8:18-8:10=8 minutes 8:19-8:18=1 minute 8:30-8:19=11 minutes 8:31-8:30=1 minute 8:54-8:31=23 minutes, so the light is off for 23-15=8 minutes 8:57-8:54=3 minutes 9:05-8:57=8 minutes 9:11-9:05=6 minutes 9:29-9:11=18 minutes, so the light is off for 18-15=3 minutes 9:31-9:29=2 minutes 10:00-9:31=29 minutes, so the light is off for 29-15=14 minutes Thus, the light is off for a total of 8+3+14=25 minutes during the two-hour period. Alternatively, the light comes on at 8:00 when the door is opened and is scheduled to go off at 8:15. So, when the door is opened at 8:03, twice at 8:04, at 8:06, and 8:10, the light is still on, but by the door being opened at these times, the timer has been reset to turn the light off at 8:18, 8:19, 8:21, and 8:25, respectively. Therefore, the light is still on when the door is opened at 8:18, 8:19, 8:30, and 8:31, but the timer has been reset to turn the light off at 8:33, 8:34, 8:45, and 8:46, respectively. The light is therefore off from 8:46 until the door is opened again at 8:54, which is an interval of 8 minutes. The light is still on when the door is opened at 8:57, 9:05, and 9:11, but the timer has been reset to turn the light off at 9:12, 9:20, and 9:26, respectively. The light is off from 9:26 until the door is opened again at 9:29, which is an interval of 3 minutes. The light is still on when the door is opened again at 9:31, and the timer has been reset to turn the light off at 9:46. Since, according to the chart, the door is not opened again before 10:00, the light remains off from 9:46 until 10:00, which is an interval of 14 minutes. Thus, the light is off a total of 8+3+14=25 minutes during the two-hour period. The correct answer is B.
34.
The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal? A. B. C. D. E.
A B C D E
D
[解析] Geometry Quadrilaterals; Triangles
First, opposite angles of a parallelogram have equal measure, and the sum of the measures of adjacent angles is 180°. This means that each of the angles adjacent to the angle labeled 60° has measure 120°. Since all four sides of the parallelogram have equal length, say x units, the shorter diagonal divides the parallelogram into two isosceles triangles. An isosceles triangle with one angle measuring 60° is equilateral, and so the shorter diagonal has length x units. The longer diagonal divides the parallelogram into two isosceles triangles with one angle measuring 120° and each of the other angles measuring , as shown in the figure above on the right.
Then, because the diagonals of a parallelogram are perpendicular and bisect each other, the two diagonals divide the parallelogram into four 30°-60°-90° triangles, each with hypotenuse x units long. The sides of a 30°-60°-90° triangle are in the ratio of and so, if y represents the length of the side opposite the 60° angle, then . But, y is half the length of the longer diagonal, so the longer diagonal has length units. Therefore, the ratio of the length of the shorter diagonal to the length of the longer diagonal is The correct answer is D.
35. If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?
A.10
B.12
C.14
D.16
E.18
A B C D E
C
[解析] Arithmetic Properties of numbers The table below shows the numbers from 1 to 30, inclusive, that have at least one factor of" 3 and how many factors of 3 each has.
The sum of the numbers in the right column is 14. Therefore, 314 is the greatest power of 3 that is a factor of the product of the first 30 positive integers. The correct answer is C.
36. If n=38-28, which of the following is NOT a factor of n?
A.97
B.65
C.35
D.13
E.5
A B C D E
C
[解析] Arithmetic Properties of numbers Since 38-28 is the difference of the perfect squares (34)2 and (24)2, then 38-28=(34+24)(34-24). But 34-24 is also the difference of the perfect squares (32)2 and (22)2 so 34-24=(32+22)(32-22) and therefore 38-28=(34+24)(32+22)(32-22). It follows that 38-28 can be factored as (81+16)(9+4)(9-4)=(97)(13)(5). Therefore, 7 is not a factor of 38-28, and hence 35=5×7 is not a factor of 38-28. It is easy to see that each of 97, 13, and 5 is a factor of 38-28, and so is 65, since 65=5×13, although this additional analysis is not needed to arrive at the correct answer. The correct answer is C.
37.
In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle? A. 4 B. 3 C. 2 D. E.
A B C D E
C
[解析] Geometry Circles Let R represent the radius of the larger circle and r represent the radius of the smaller circle. Then the area of the shaded region is the area of the larger circular region minus the area of the smaller circular region, or πR2-πr2. It is given that the area of the shaded region is three times the area of the smaller circular region, and so πR2-πr2=3πr2. Then R2-r2=3r2, and so R2=4r2 and R=2r. The circumference of the larger circle is 2πR=2π(2r)=2(2πr), which is 2 times the circumference of the smaller circle. The correct answer is C.
38. Club X has more than 10 but fewer than 40 members. Sometimes the members sit at tables with 3 members at one table and 4 members at each of the other tables, and sometimes they sit at tables with 3 members at one table and 5 members at each of the other tables. If they sit at tables with 6 members at each table except one and fewer than 6 members at that one table, how many members will be at the table that has fewer than 6 members?
A.1
B.2
C.3
D.4
E.5
A B C D E
E
[解析] Arithmetic Properties of numbers Let n be the number of members that Club X has. Since the members can be equally divided into groups of 4 each with 3 left over, and the members can be equally divided into groups of 5 each with 3 left over, it follows that n-3 is divisible by both 4 and 5. Therefore, n-3 must be a multiple of (4)(5)=20. Also, because the only multiple of 20 that is greater than 10 and less than 40 is 20, it follows that n-3=20, or n=23. Finally, when these 23 members are divided into the greatest number of groups of 6 each, there will be 5 members left over, since 23=(3)(6)+5. The correct answer is E.
39. In order to complete a reading assignment on time, Terry planned to read 90 pages per day. However, she read only 75 pages per day at first, leaving 690 pages to be read during the last 6 days before the assignment was to be completed. How many days in all did Terry have to complete the assignment on time?
A.15
B.16
C.25
D.40
E.46
A B C D E
B
[解析] Algebra Applied problems Let n be the number of days that Terry read at the slower rate of 75 pages per day. Then 75n is the number of pages Terry read at this slower rate, and 75n+690 is the total number of pages Terry needs to read. Also, n+6 is the total number of days that Terry will spend on the reading assignment. The requirement that Terry average 90 pages per day is equivalent to Then Therefore, the total number of days that Terry has to complete the assignment on time is n+6=10+6=16. The correct answer is B.
40. , what is r in terms of s? A. B. C. D. s3 E. s3-s
A B C D E
D
[解析] Algebra Equations Solve the equation for r as follows: The correct answer is D.
data values lie to the right of 120 when the data values are listed in increasing order from left to right, and so the largest data value is 
. Alternatively, it is evident that (B), (C), or (E) cannot be correct since these expressions do not have an integer value when Q is odd. For the list consisting of the single number 120 (i.e., if Q=1), (D) fails because 
and (A) does not fail because 
of length 70 feet. Since the lengths of the sides of a 30°-60°-90° triangle are in the ratio 
and 
feet above the ground.
. The rectangle has dimensions 4 ft by 8 ft, where 8=10-2 is the full height of the window minus the radius of the semicircle, and thus has area (4)(8)=32 ft2. Therefore, in square feet, the area of the window is 32+2π.
, which of the following numbers could be the value of t?
has 11 zeros to the right of the decimal point followed by a single digit that is 1. Therefore, in the decimal expansion of 
the number of zeros between the decimal point and the first nonzero digit is equal to 12 minus the number of digits in the integer t4. Since this number of zeros is fewer than 8, it follows that the number of digits in the integer t4 is greater than 4. The integer t cannot be 3, because the number of digits in 34=81 is not greater than 4. The integer t cannot be 5, because the number of digits in 54=625 is not greater than 4. The integer t cannot be 9, because the number of digits in 94=6,561 is not greater than 4. Therefore, none of 3, 5, or 9 could be the value of t.
be the mean of the students' unadjusted scores. It follows that the standard deviation of the unadjusted scores is 
To find the standard deviation of the adjusted scores, first find their mean: 
. Then, subtract the adjusted mean from each adjusted score: (0.8x+20)-(0.8μ+20)=0.8(x-μ). Next, square each difference: (0.8(x-μ))2=0.64(x-μ)2. Next, find the average of the squared differences: 
, what is the value of y?
But, 
. Equating the two expressions for 
Thus, q=96 and 
and the solution set to the second equation is 
Therefore, 
is the solution to both equations.
, and the perimeter is 
. The perimeter of the rectangle is 2(a+2a)=6a. It follows that the ratio of the perimeter of the square to the perimeter of the rectangle is 
. The height of the carton is 2c and in terms of x, 
and 
, the height of the carton can be expressed as 
.
of the number of trees in the orchard the preceding year. If all of the trees thrived and there were 6,250 trees in the orchard at the end of the 4-year period, how many trees were in the orchard at the beginning of the 4-year period?
of the number of trees in the orchard the preceding year is equivalent to making the number of trees increase 25% per year, compounded yearly. If there were n trees at the beginning of the 4-year period, then there will be 1.25n trees at the end of the first year, 1.25(1.25n)=(1.25)2n trees at the end of the second year, 1.25[(1.25)2n]=(1.25)3n trees at the end of the third year, and 1.25[(1.25)3n]=(1.25)4n trees at the end of the fourth year. Hence, 6,250=(1.25)4n and 
. The arithmetic can be greatly simplified by rewriting 
and 6,250 as (625)(10)=(54)(10). Then 
of a sandwich. Since the fourth sandwich was evenly divided among m-4 students, each piece was 
of the fourth sandwich. Carol ate 1 piece from each of the four sandwiches, so she ate a total of 
as one of its roots?
as a root that does not involve radicals.
and P(B, given that A has occurred)=
. (Event A has occurred so there are 11 pens left in the box and 8 of them are not defective.) Then, using the multiplication rule for dependent events, P(neither pen is defective) =P(A and B)=P(A)×P(B, given that A has occurred)
. From this,
, and the ratio of royalties to sales for the next $108 million in sales is 
. The percent decrease in the royalties to sales ratios is 100 times the quotient of the difference in the ratios divided by the ratio of royalties to sales for the first $20 million in sales or
, as shown in the figure above on the right.
and so, if y represents the length of the side opposite the 60° angle, then 
. But, y is half the length of the longer diagonal, so the longer diagonal has length 
units. Therefore, the ratio of the length of the shorter diagonal to the length of the longer diagonal is 
Then
, what is r in terms of s?
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