B

[解析] Algebra Inequalities
   Determine whether where a, b, c, and d are positive numbers.
   (1) Given that then a=2, b=3, c=6, and d=8 are possible values for a, b, c, and d because and For these values, is true because On the other hand, a=4, b=6, c=2, and d=3 are also possible values of a, b, c, and d because For these values, is false because NOT sufficient.
   (2) Given that then
   
   The correct answer is B;
   statement 2 alone is sufficient.

C

[解析] Arithmetic Sets
   Let a be the number of members in Club X that do not belong to Club Y, let b be the number of members in Club Y that do not belong to Club X, and let c be the number of members that belong to both Club X and to Club Y. Determine whether a+c＞b+c, or equivalently, whether a＞b.
   (1) If a=80, b=79, and c=20, then 20 percent of the members of Club X are also members of Club Y (because c=20 is 20 percent of a+c=100) and a＞b is true. However, if a=80, b=80, and c=20, then 20 percent of the members of Club X are also members of Club Y (because c=20 is 20 percent of a+c=100) and a＞b is false. Therefore, it cannot be determined whether a＞b; NOT sufficient.
   (2) If a=71, b=70, and c=30, then 30 percent of the members of Club Y are also members of Club X (because c=30 is 30 percent of b+c=100) and a＞b is true. However, if a=70, b=70, and c=30, then 30 percent of the members of Club Y are also members of Club X (because c=30 is 30 percent of b+c=100) and a＞b is false. Therefore, it cannot be determined whether a＞b; NOT sufficient.
   Now assume both (1) and (2). From (1) it follows that or 5c=a+c, and so a=4c. From (2) it follows that or 10c=3b+3c, and so 7c=3b and Since (from the statements it can be deduced that c＞0), it follows that a＞b. Therefore, (1) and (2) together are sufficient.
   The correct answer is C;
   both statements together are sufficient.

D

[解析] Arithmetic Properties of numbers
   Since p, q, r, s, and t are consecutive even integers listed in numerical order, the 5 integers can also be given as p, p+2, p+4, p+6, and p+8. Determine the average of these 5 integers, which is the value of
   (1) Given that q+s=24, then (2+2)+(2+6)=24.
   Therefore, 2p+8=24, or p=8, and hence p+4=12; SUFFICIENT.
   (2) Given that then q+r=(2)(11)=22, or (p+2)+(p+4)=22. Therefore, 2p+6=22, or p=8, and hence p+4=12; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

C

[解析] Algebra Functions
   Determine if the least integer greater than or equal to x, is equal to 0, which is the same as determining if x satisfies -1＜x≤0.
   (1) Given that -1＜x＜1, then it is possible that (for example, if x=0, then and it is possible that (for example, if then NOT sufficient.
   (2) Given that x＜0, then it is possible that (for example, if then and it is possible that (for example, if x=-5, then NOT sufficient.
   Taking (1) and (2) together gives -1＜x＜0, which implies that
   The correct answer is C;
   both statements together are sufficient.

C

[解析] Arithmetic Properties of integers
   Determine if the integer x is greater than the integer y.
   (1) It is given that x+y＞0, and so -x＜y. If, for example, x=-3 and y=4, then x+y=-3+4=1＞0 and x＜y. On the other hand, if x=4 and y=-3, then x+y=4-3=1＞0 and x＞y; NOT sufficient.
   (2) It is given that y^x＜0, so y＜0. If, for example, x=3 and y=-2, then (-2)³=-8＜0 and x＞y. On the other hand, if x=-3 and y=-2, then and x＜y; NOT sufficient.
   Taking (1) and (2) together, from (2) y is negative and from (1) -x is less than y. Therefore, -x is negative, and hence x is positive. Since x is positive and y is negative, it follows that x＞y.
   The correct answer is C;
   both statements together are sufficient.

B

[解析] Algebra Second-degree equations
   Determine whether the product of the roots to x²+bx+c=0, where b and c are constants, is negative.
   If r and s are the roots of the given equation, then (x-r)(x-s)=x²+bx+c. This implies that x²-(r+s)x+rs=x²+bx+c, and so rs=c. Therefore, rs is negative if and only if c is negative.
   (1) Given that b＜0, then c could be negative or
   positive. For example, if b=-1 and c=-6, then the given equation would be x²-x-6=(x-3)(x+2)=0, and the product of its roots would be (3)(-2), which is negative. On the other hand, if b=-6 and c=5, then the given equation would be x²-6x+5=(x-5)(x-1)=0, and the product of its roots would be (5)(1), which is positive; NOT sufficient.
   (2) Given that c＜0, it follows from the explanation above that rs＜0; SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

A

[解析] Geometry Polygons
   (1) Given that the area of the garden is 189 ft², and observing that area of the garden can be calculated by imagining the garden as a square with dimensions 15 ft by 15 ft from which its upper-right square corner with dimensions (15-k) ft by (15-k) ft is removed, it follows that 189=(15)²-(15-k)². Therefore, (15-k)²=225-189=36, so 15-k=6 or 15-k=-6. The figure implies 15-k＞0, so it follows that 15-k=6, or k=9; SUFFICIENT.
   (2) Given that the perimeter of the garden is 60 ft, and because 15+15+k+(15-k)+(15-k)+k=60, any value of k between 0 and 15 is possible, since for any such value of k the sum of the lengths of all the sides would be 60 ft; NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

D

[解析] Arithmetic Ratio and proportion
   Letting s, d, and j represent, respectively, the numbers of shirts, dresses, and jackets in the closet, then s=9x, d=4x, and j=5x, where x is a positive integer. It is given that 4x＞7, and so 4x≥8 or x≥2 since x is an integer. Determine the value of 9x+4x+5x=18x.
   (1) This indicates that 9x+5x＜30, and so 14x≤28 or x≤2 since x is an integer. It follows from x≥2 and x≤2 that x=2 and 18x=36; SUFFICIENT.
   (2) This indicates that 9x+4x=26 or 13x=26, or x=2. It follows that 18x=36; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

A

[解析] Geometry Circles
   In this circle graph, the expenses of Division R are equal to the value of multiplied by $5,400,000, or $15,000x. Therefore, it is necessary to know the value of x in order to determine the expenses for Division R.
   (1) The value of x is given as 94, so the expenses of Division R can be determined; SUFFICIENT.
   (2) This gives a comparison among the expenses of some of the divisions of Company H, but no information is given about the value of x; NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

A

[解析] Algebra Properties of numbers
   (1) Given that x²＞9, it follows that x＜-3 or x＞3, a result that can be obtained in a variety of ways. For example, consider the equivalent inequality that reduces to |x|＞3, or consider when the two factors of x²-9 are both positive and when the two factors of x²-9 are both negative, or consider where the graph of the parabola y=x²-9 is above the x-axis, etc. Since it is also given that x is negative, it follows that x＜-3; SUFFICIENT.
   (2) Given that x³＜-9, if x=-4, then x³=-64, and so x³＜-9 and it is true that x＜-3. However, if x=-3, then x³=-27, and so x³＜-9, but it is not true that x＜-3; NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

E

[解析] Geometry Rectangular solids and cylinders
   (1) No information about the size of the cans is given; NOT sufficient.
   (2) No information about the size of the carton is given; NOT sufficient.
   Taking (1) and (2) together, there is still not enough information to answer the question. If the carton is a rectangular solid that is 1 inch by 1 inch by 2,304 inches and the cans are cylindrical with the given dimensions, then 0 cans can be packed into the carton. However, if the carton is a rectangular solid that is 16 inches by 12 inches by 12 inches and the cans are cylindrical with the given dimensions, then 1 or more cans can be packed into the carton.
   The correct answer is E;
   both statements together are still not sufficient.

D

[解析] Arithmetic Properties of numbers
   In the following discussion, "row/column convention" means that each of the numbers 1, 2, and 3 appears exactly once in any given row and exactly once in any given column.
   (1) Given that v+z=6, then both v and z are equal to 3, since no other sum of the possible values is equal to 6. Applying the row/column convention to row 2, and then to row 3, it follows that neither u nor x can be 3. Since neither u nor x can be 3, the row/column convention applied to column 1 forces r to be 3; SUFFICIENT.
   (2) If u=3, then s+t+x=3. Hence, s=t=x=1, since the values these variables can have does not permit another possibility. However, this assignment of values would violate the row/column convention for row 1, and thus u cannot be 3. If x=3, then s+t+u=3. Hence, s=t=u=1, since the values these variables can have does not permit another possibility. However, this assignment of values would violate the row/column convention for row 1, and thus x cannot be 3. Since neither u nor x can be 3, the row/column convention applied to column 1 forces r to be 3; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

B

[解析] Algebra Inequalities
   Since x+y=10, the relation x＞y is equivalent to x＞10-x, or x＞5.
   (1) The given information is consistent with x=5.5 and y=4.5, and the given information is also consistent with x=y=5. Therefore, it is possible for x＞y to be true and it is possible for x＞y to be false; NOT sufficient.
   (2) Given that 3x+5y＜40, or 3x+5(10-x)＜40, then 3x-5x＜40-50. It follows that -2x＜-10, or x＞5;
   SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

E

[解析] Arithmetic Applied problems
   Determine the speed of the train when it had completed half the total distance of the trip.
   (1) Given that the train traveled 460 miles in 4 hours, the train could have traveled at the constant rate of 115 miles per hour for 4 hours, and thus it could have been traveling 115 miles per hour when it had completed half the total distance of the trip. However, the train could have traveled 150 miles per hour for the first 2 hours (a distance of 300 miles) and 80 miles per hour for the last 2 hours (a distance of 160 miles), and thus it could have been traveling 150 miles per hour when it had completed half the total distance of the trip; NOT sufficient.
   (2) Given that the train traveled at an average rate of 115 miles per hour, each of the possibilities given in the explanation for (1) could occur, since 460 miles in 4 hours gives an average speed of miles per hour; NOT sufficient.
   Assuming (1) and (2), each of the possibilities given in the explanation for (1) could occur. Therefore, (1) and (2) together are not sufficient.
   The correct answer is E;
   both statements together are still not sufficient.

B

[解析] Arithmetic Statistics
   Let T, J, and S be the purchase prices for Tom's, Jane's, and Sue's new houses. Given that the average purchase price is 120,000, or T+J+S=3(120,000), determine the median purchase price.
   (1) Given T=110,000, the median could be 120,000 (if J=120,000 and S=130,000) or 125,000 (if J=125,000 and S=125,000); NOT sufficient.
   (2) Given J=120,000, the following two cases include every possibility consistent with T+J+S=(3)(120,000), or T+S=(2)(120,000).
   (i) T=S=120,000
   (ii) One of Tor S is less than 120,000 and the other is greater than 120,000.
   In each case, the median is clearly 120,000; SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

B

[解析] Algebra Inequalities
   The inequality x³＜x² is equivalent to x³-x²＜0, or x²(x-1)＜0. Since this inequality is false for x=0, it follows that x≠0, and hence x²＞0. Therefore, x²(x-1)＜0 can only hold if x-1＜0, or if x＜1. Thus, the problem is equivalent to determining the value of x given that x≠0 and x＜1.
   (1) Given that -2＜x＜2, it is not possible to determine the value of x. For example, the value of x could be -1 (note that -1＜1) and the value of x could be 0.5 (note that 0.125＜0.25); NOT sufficient.
   (2) Given that the value of x is an integer greater than -2, then the value of x must be among the integers -1, 0, 1, 2, 3, .... However, from the discussion above, x≠0 and x＜1, so the value of x can only be -1; SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

D

[解析] Arithmetic Properties of numbers
   If w≥10, then (10, w)=10, and if w＜10, then (10, w)=w. Therefore, the value of min(10, w) can be determined if the value of w can be determined.
   (1) Given that w=max(20, z) then w≥20. Hence, w≥10, and so min(10, w)=10; SUFFICIENT.
   (2) Given that w=max(10, w), then w≥10, and so rain(10, w)=10; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

C

[解析] Arithmetic Statistics
   (1) The information given says nothing about the number of books on the lower shelf. If there are fewer than 25 books on the lower shelf, then the median number of pages will be the number of pages in one of the books on the upper shelf or the average number of pages in two books on the upper shelf. Hence, the median will be at most 400. If there are more than 25 books on the lower shelf, then the median number of pages will be the number of pages in one of the books on the lower shelf or the average number of pages in two books on the lower shelf. Hence, the median will be at least 475; NOT sufficient.
   (2) An analysis very similar to that used in (1) shows the information given is not sufficient to determine the median; NOT sufficient.
   Given both (1) and (2), it follows that there is a total of 49 books. Therefore, the median will be the 25th book when the books are ordered by number of pages. Since the 25th book in this ordering is the book on the upper shelf with the greatest number of pages, the median is 400. Therefore, (1) and (2) together are sufficient.
   The correct answer is C;
   both statements together are sufficient.

D

[解析] Geometry Circles
   If R is the radius of the larger circle and r is the radius of the smaller circle, then the desired area is πR²-πr². Thus, if both the values of R and r can be determined, then the desired area can be determined.
   (1) Given that AB=r=3 and BC=2, then AB+BC=R=3+2=5; SUFFICIENT.
   (2) Given that CD=1 and DE=4, then CD+DE=R=1+4=5. Since is a diameter of the larger circle, then AD+DE=2R. Also, since is a diameter of the smaller circle, then AD=2r. Thus, 2r+DE=2R or 2r+4=10, and so r=3; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

E

[解析] Arithmetic Applied problems; Estimating
   (1) Given that Joan's estimate for the distance was within 5 miles of the actual distance, it is not possible to determine whether her estimate for the time was within 0.5 hour without information about her estimated average speed. For example, if her estimated distance was 20 miles and was within 5 miles of the actual distance, then the actual distance would be between 15 and 25 miles. If her estimated speed was 20 miles per hour (mph) and was within, say, 10 mph of her actual speed, then her actual speed would be between 10 and 30 mph. Her estimated time would then be hour and the actual time would be between hour (least distance over greatest speed) and hours (greatest distance over least speed). Since 1.0 hour is between 0.5 hour and 2.5 hours, her estimate for the time could equal the actual time, and thus it is possible that her estimate of the time is within 0.5 hour of the actual time. However, her estimate for the time could be as much as 2.5-1.0=1.5 hours over the actual time, and thus it is possible that her estimate of the time is not within 0.5 hour; NOT sufficient.
   (2) Given that Joan's estimate for her average speed was within 10 miles per hour of her actual average speed, the same examples used in (1) can be used to show that it cannot be determined whether her estimate for the time would be within 0.5 hour of the actual time; NOT sufficient.
   Taking (1) and (2) together is of no more help than either (1) or (2) taken separately because the same examples used to show that (1) is not sufficient also show that (2) is not sufficient.
   The correct answer is E;
   both statements together are still not sufficient.

D

[解析] Arithmetic Statistics
   Let the numbers be x, y, and z so that x＜y＜z.
   Determine whether or equivalently, whether 3y=x+y+z, or equivalently, whether 2y=x+z.
   (1) Given that the range is equal to twice the difference between the greatest number and the median, it follows that z-x=2(z-y), or z-x=2z-2y, or 2y=x+z; SUFFICIENT.
   (2) Given that the sum of the 3 numbers equals 3 times one of the numbers, it follows that x+y+z=3x or x+y+z=3y or x+y+z=3z. If x+y+z=3x, then y+z=2x, or (y-x)+(z-x)=0. Also, if x+y+z=3z, then x+y=2z, or 0=(z-x)+(z-y). In each of these cases, the sum of two positive numbers is zero, which is impossible. Therefore, it must be true that x+y+z=3y, from which it follows that x+z=2y, and hence by the initial comments, the median of the 3 numbers equals the average of the 3 numbers; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

A

[解析] Geometry Coordinate geometry
   Since the line does not pass through the origin, the line is either vertical and given by the equation x=c for some constant c such that c≠0, or the line is not vertical and given by the equation y=mx+b for some constants m and b such that b≠0. Determine whether the line is not vertical, and if so, determine the slope of the line, which is the value of m.
   (1) Given that the x-intercept of the line is twice the y-intercept of the line, it follows that the line is not vertical, since a vertical line that does not pass through the origin will not have ay-intercept. Thus, the line is given by the equation y=mx+b. The x-intercept of the line is the solution to 0=mx+b, or mx=-b, which has solution and the y-intercept of the line is b. Therefore, Since b≠0, both sides of the last equation can be divided by b to get or m=-1/2; SUFFICIENT.
   (2) Given that the x- and y-intercepts of the line are both positive, it is not possible to determine the slope of the line. For example, if the line is given by y=-x+1, then the x-intercept is 1 (solve -x+1=0), the y-intercept is 1 (b=1), and the slope is -1. However, if the line is given by y=-2x+2, then the x-intercept is 1 (solve -2x+2=0), the y-intercept is 2 (b=2), and the slope is -2; NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

D

[解析]
   Algebra Coordinate geometry
   Since (p, r) is on each of the lines, each of the following three equations is true:
   (i) r=ap-5
   (ii) r=p+6
   (iii) r=3p+b
   Determine the value of b.
   (1) Given that a=2, Equations (i) and (ii) become r=2p-5 and r=p+6. Subtracting equations gives 0=p-11, or p=11. Now using r=p+6, it follows that r=11+6=17. Finally, using p=11 and r=17 in Equation (iii) gives 17=3(11)+b, or b=17-33=-16; SUFFICIENT.
   (2) Given that r=17, Equation (ii) becomes 17=p+6, and so p=17-6=11. Using r=17 and p=11, Equation (iii) becomes 17=3(11)+b, or b=17-33=-16; SUFFICIENT.
   the correct answer is D;
   each statement alone is sufficient.

A

[解析] Geometry Triangles; Perimeter
   (1) Given that triangle SWV has perimeter 9, the perimeter of polygon PQWTUVR can be determined, since the perimeter of polygon PQWTUVR equals the sum of the perimeters of the two equilateral triangles, which is 3(6)+3(6)=36, minus the perimeter of triangle SWV; SUFFICIENT.
   
   (2) Given that VW=3.5, and using the fact that the perimeter of polygon PQWTUVR equals the sum of the perimeters of the two equilateral triangles, which is 3(6)+3(6)=36, minus the perimeter of triangle SWV, it is not possible to determine the perimeter of polygon PQWTUVR. For example, the perimeter of polygon PQWTUVR could be 36-3(3.5)=25.5 (arrange triangles PQR and STU so that VW=WS=SV=3.5, as shown in Figure 1), and the perimeter of polygon PQWTUVR could be greater than 25.5 (arrange triangles PQR and STU so that VW=3.5, WS is slightly greater than 3.5, and SV is dose to 0, as shown in Figure 2); NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

E

[解析] Arithmetic Statistics
   Set S has a range of x, set T has a range of y, and T is a subset of S. Determine if x is greater than y.
   (1) It is given that S contains exactly 7 numbers, but nothing additional is known about T. Thus, if S={1, 2, 3, 4, 5, 6, 7} and T={1,2, 3,4,5,6}, then x=7-1=6, y=6-1=5, and x is greater than y. On the other hand, if S={1, 2, 3, 4, 5, 6, 7} and T={1, 3, 4, 5, 6, 7}, then x=7-1=6, y=7-1=6, and x is not greater than y; NOT sufficient.
   (2) It is given that T contains exactly 6 numbers, but nothing additional is known about T. Since the same examples given in (1) can also be used in (2), it cannot be determined if x is greater than y; NOT sufficient.
   Taking (1) and (2) together, the examples used in (1) can be used to show that it cannot be determined if x is greater than y.
   The correct answer is E;
   both statements together are still not sufficient.

D

[解析] Geometry Triangles
   If x and y are the lengths of the legs of the triangle, then it is given that x²+y²=100. Determining the value of x+y+10, the perimeter of the triangle, is equivalent to determining the value of x+y.
   (1) Given that the area is 25, then or xy=50. Since (x+y)²=x²+y²+2xy, it follows that (x+y)²=100+2(50), or
   (2) Given that x=y, since x²+y²=100, it follows that 2x²=100, or Hence, SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

B

[解析] Arithmetic Operations on rational numbers
   Given that the shipments on the first truck had a value greater than of the total value of the 6 shipments, determine if S3 was shipped on the first truck.
   To avoid dealing with fractions, it will be convenient to create scaled values of the shipments by multiplying each fractional value by 60, which is the least common denominator of the fractions. Thus, the scaled values associated with S1, S2, S3, S4, S5, and S6 are 15,12, 10, 9, 8, and 6, respectively. The given information is that the scaled value of the shipments on the first truck is greater than
   (1) Given that the first truck includes shipments with scaled values 12 and 9, it may or may not be the case that S3 (the shipment with scaled value 10) is on the first truck. For example, the first truck could contain only S2, S3, and S4, for a total scaled value 12+10+9=31＞30. Or, the first truck could contain only S1, S2, and S4, for a total scaled value 15+12+9=36＞30; NOT sufficient.
   (2) Given that the second truck includes shipments with scaled values 15 and 6, the second truck cannot contain S3. Otherwise, the second truck would contain shipments with scaled values 15, 6, and 10, for a total scaled value 15+6+10=31, leaving at most a total scaled value 29 (which is not greater than 30) for the first truck; SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

A

[解析] Arithmetic Place value
   Letting x=100a+10b+c, y=100p+10q+r, and z=100t+10u+v, where a, b, c, p, q, r, t, u, and v are digits, determine if a=p+t.
   (1) It is given that b=q+u (which implies that c+v≤9 because if c+v＞9, then in the addition process a ten would need to be carried over to the tens column and b would be q+u+1). Since b is a digit, 0≤b≤9. Hence, 0≤q+u≤9, and so 0≤10(q+u)≤90. Therefore, in the addition process, there are no hundreds to carry over from the tens column to the hundreds column, so a=p+SUFFICIENT.
   (2) It is given that c=r+v. If x=687, y=231, and z=456, then, y+z=231+456=687=x, r+v=1+6=7=c, and p+t=2+4=6=a. On the other hand, if x=637, y=392, and z=245, then y+z=392+245=637=x, r+v=2+5=7=c, and p+t=3+2=5≠6=a; NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

A

[解析] Arithmetic Sets
   If x is the number of voters who responded "Favorable" for both candidates, then it follows from the table that the number of voters who responded "Favorable" to at least one candidate is 40+30-x=70-x. This is because 40+30 represents the number of voters who responded "Favorable" for Candidate M added to the number of voters who responded "Favorable" for Candidate N, a calculation that counts twice each of the x voters who responded "Favorable" for both candidates.
   (1) Given that there were 40 voters who did not respond "Favorable" for either candidate and there were 100 voters surveyed, the number of voters who responded "Favorable" to at least one candidate is 100-40=60. Therefore, from the comments above, it follows that 70-x=60, and hence x=10; SUFFICIENT.
   (2) The information given affects only the numbers of voters in the categories "Unfavorable" for Candidate M only, "Unfavorable" for Candidate N only, and "Unfavorable" for both candidates. Thus, the numbers of voters in the categories "Favorable" for Candidate M only, "Favorable" for Candidate N only, and "Favorable" for both candidates are not affected. Since these latter categories are only constrained to have certain integer values that have a total sum of 70-x, more than one possibility exists for the value of x. For example, the numbers of voters in the categories "Favorable" for Candidate M only, "Favorable" for Candidate N only, and "Favorable" for both candidates could be 25, 15, and 15, respectively, which gives 70-x=25+15+15, or x=15. However, the numbers of voters in the categories "Favorable" for Candidate M only, "Favorable" for Candidate N only, and "Favorable" for both candidates could be 30, 20, and 10, respectively, which gives 70-x=30+20+10, or x=10; NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

B

[解析] Arithmetic Properties of numbers
   Determine if n is divisible by m.
   (1) Given that 3n is divisible by m, then n is divisible by m if m=9 and n=27 (note that 3＜m＜13＜n, 3n=81, and m=9, so 3n is divisible by m) and n is not divisible by m if m=9 and n=30 (note that 3＜m＜13＜n, 3n=90, and m=9, so 3n is divisible by m); NOT sufficient.
   (2) Given that 13n is divisible by m, then for some integer q.
   Since 13 is a prime number that divides qm (because 13n=qm) and 13 does not divide m (because m＜13), it follows that 13 divides q.
   Therefore, is an integer, and since
    is an integer. Thus, n is divisible by m; SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

A

[解析] Algebra Absolute value
   (1) It is given that t-q=|t-s|+|s-q|, which can be rewritten without absolute values in four mutually exclusive and collectively exhaustive cases by making use of the algebraic definition of absolute value. Recall that |x|=x if x＞0, and |x|=-x if x＜0. Thus, for example, if t-s＜0, then |t-s|=-(t-s).
   Case 1: t＞s and s＞q. In this case, t-s＞0 and s-q＞0, and so t-q=|t-s|+|s-q| is equivalent to t-q=(t-s)+(s-q), which is an identity. Therefore, the case for which t＞s and s＞q is consistent with the given information and the assumption t-q=|t-s|+|s-q|.
   Case 2: t＞s and s＜q. In this case, t-s＞0 and s-q＜0, and so t-q=|t-s|+|s-q| is equivalent to t-q=(t-s)-(s-q), or s=q, which is not consistent with the assumption that q, s, and t are all different numbers. Therefore, the case for which t＞s and s＜q is not consistent with the given information and the assumption t-q=|t-s|+|s-q|.
   Case 3: t＜s and s＞q. In this case, t-s＜0 and s-q＞0, and so t-q=|t-s|+|s-q| is equivalent to t-q=-(t-s)+(s-q), or t=s, which is not consistent with the assumption that q, s, and t are all different numbers. Therefore, the case for which t＜s and s＞q is not consistent with the given information and the assumption t-q=|t-s|+|s-q|.
   Case 4: t＜s and s＜q. In this case, t-s＜0 and s-q＜0, and so t-q=|t-s|+|s-q| is equivalent to t-q=-(t-s)-(s-q), or t=9, which is not consistent with the assumption that q, s, and t are all different numbers. Therefore, the case for which t＜s and s＜q is not consistent with the given information and the assumption t-q=|t-s|+Is-q|. The only case that is consistent with the given information and the assumption t-q=|t-s|+|s-q| is Case 1. Therefore, it follows that t＞s and s＞q, and this implies q＜s＜t; SUFFICIENT.
   (2) Given that t＞q, it is possible that q＜s＜t is true (for example, when s is between t and q) and it is possible that q＜s＜t is false (for example, when s is greater than t); NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

C

[解析] Arithmetic Statistics
   (1) Although 25 percent of the projects have 4 or more employees, there is essentially no information about the middle values of the numbers of employees per project. For example, if there were a total of 100 projects, then the median could be 2 (75 projects that have exactly 2 employees each and 25 projects that have exactly 4 employees each) or the median could be 3 (75 projects that have exactly 3 employees each and 25 projects that have exactly 4 employees each); NOT sufficient.
   (2) Although 35 percent of the projects have 2 or fewer employees, there is essentially no information about the middle values of the numbers of employees per project. For example, if there were a total of 100 projects, then the median could be 3 (35 projects that have exactly 2 employees each and 65 projects that have exactly 3 employees each) or the median could be 4 (35 projects that have exactly 2 employees each and 65 projects that have exactly 4 employees each); NOT sufficient.
   Given both (1) and (2), 100-(25+35) percent=40 percent of the projects have exactly 3 employees. Therefore, when the numbers of employees per project are listed from least to greatest, 35 percent of the numbers are 2 or less and (35+40) percent=75 percent are 3 or less, and hence the median is 3.
   The correct answer is C;
   both statements together are sufficient.

B

[解析] Algebra Applied problems
   For a company that made a profit last year from selling product X and had total expenses
    for product X of $100,000+0.05R, where R is the total revenue for selling product X, determine whether the company sold more than 21,000 units of product X last year. Note that since the company made a profit, revenue-cost, which is given by R-($100,000+0.05R)=0.95R-$100,000, must be positive.
   (1) It is given that R＞$110,000. It is possible to vary the unit price and the number of units sold so that R＞$110,000 and more than 21,000 units were sold, and also so that R＞$110,000 and less than 21,000 units were sold. For example, if 25,000 units were sold for $10 per unit, then R=25,000($10)=$250,000＞$110,000 and 25,000＞21,000. On the other hand, if 20,000 units were sold for $10 per unit, then R=20,000($10)=$200,000＞$110,000 and 20,000＜21,000; NOT sufficient.
   (2) It is given that the company's revenue for each unit of product X was $5. If the company manufactured and sold x units of product X, then its revenue was $5x. Because the company made a profit, 0.95($5x)-$100,000＞0, and so
   
   To avoid long division in the last step, note that 4.75(21,000)=99,750, and thus from 4.75x＞100,000, it follows that x＞21,000.
   The correct answer is B;
   statement 2 alone is sufficient.

C

[解析] Arithmetic Statistics
   (1) If Carl began making $50 withdrawals on or before May 15, his account balance on April 16 would be at least $50 greater than it was on the last day of May. Thus, his account balance on April 16 would be at least $2,600+$50=$2,650, which is contrary to the information given in (1). Therefore, Carl did not begin making $50 withdrawals until June 15 or later. These observations can be used to give at least two possible ranges. Carl could have had an account balance of $2,000 on January 1, made $120 deposits in each of the first 11 months of the year, and then made a $50 withdrawal on December 15, which gives a range of monthly closing balances of (120)(10). Also, Carl could have had an account balance of $2,000 on January 1, made $120 deposits in each of the first 10 months of the year, and then made $50 withdrawals on November 15 and on December 15, which gives a range of monthly closing balances of (120)(9); NOT sufficient.
   (2) On June 1, Carl's account balance was the same as its closing balance was for May, namely $2,600. Depending on whether Carl made a $120 deposit or a $50 withdrawal on June 15, Carl's account balance on June 16 was either $2,720 or $2,550. It follows from the information given in (2) that Carl's balance on June 16 was $2,550. Therefore, Carl began making $50 withdrawals on or before June 15. These observations can be used to give at least two possible ranges. Carl could have had an account balance of $2,680 on January 1, made one $120 deposit on January 15, and then made a $50 withdrawal in each of the remaining 11 months of the year (this gives a closing balance of $2,600 for May), which gives a range of monthly closing balances of (50)(11). Also, Carl could have had an account balance of $2,510 on January 1, made $120 deposits on January 15 and on February 15, and then made a $50 withdrawal in each of the remaining 10 months of the year (this gives a closing balance of $2,600 for May), which gives a range of monthly closing balances of (50)(10); NOT sufficient. Given both (1) and (2), it follows from the remarks above that Carl began making $50 withdrawals on June 15. Therefore, the changes to Carl's account balance for each month of last year are known. Since the closing balance for May is given, it follows that the closing balances for each month of last year are known, and hence the range of these 12 known values can be determined.
   The correct answer is C;
   both statements together are sufficient.

B

[解析] Arithmetic Properties of numbers
   (1) If there are 15 occurrences of the number 4 in the list, then the sum of the numbers in the list is 60 and all the numbers in the list are equal. If there are 13 occurrences of the number 4 in the list, 1 occurrence of the number 3 in the list, and 1 occurrence of the number 5 in the list, then the sum of the numbers in the list is 60 and not all the numbers in the list are equal; NOT sufficient.
   (2) Given that the sum of any 3 numbers in the list is 12, arrange the numbers in the list in numerical order, from least to greatest:
   a₁≤a₂≤a₃≤...≤a₁₅.
   If a₁＜4, then a₁+a₂+a₃＜4+a₂+a₃. Therefore, from (2), 12＜4+a₂+a₃, or 8＜a₂+a₃, and so at least one of the values a₂ and a₃ must be greater than 4. Because a₂≤a₃, it follows that a₃＞4. Since the numbers are arranged from least to greatest, it follows that a₄＞4 and a₅＞4. But then, a₃+a₄+a₅＞4+4+4=12, contrary to (2), and so a₁＜4 is not true. Therefore, a₁≥4. Since a₁ is the least of the 15 numbers, a_n≥4 for n=1, 2, 3,..., 15.
   If a₁₅＞4, then a₁₃+a₁₄+a₁₅＞a₁₃+a₁₄+4. Therefore, from (2), 12＞a₁₃+a₁₄+4, or 8＞a₁₃+a₁₄, and so at least one of the values a₁₃ and a₁₄ must be less than 4. Because a₁₃≤a₁₄, it follows that a₁₃＜4. Since the numbers are arranged from least to greatest, it follows that a₁₁＜4 and a₁₂＜4. But then a₁₁+a₁₂+a₁₃＜4+4+4=12, contrary to (2). Therefore, a₁₅≤4. Since a₁₅ is the greatest of the 15 numbers, a_n≤4 for n=1, 2, 3, ..., 15.
   It has been shown that, for n=1, 2, 3,..., 15, each of a_n≥4 and a_n≤4 is true. Therefore, a_n=4 for n=1, 2, 3, ..., 15; SUFFICIENT.
   The correct answer is B;
   statement 2 alone is sufficient.

D

[解析] Arithmetic Statistics
   If the average of six numbers is 75, then of the sum of the numbers is 75. Therefore, the sum of the numbers is (6)(75).
   (1) If one of the numbers is greater than 75, then we can write that number as 75+x for some positive number x. Consequently, the sum of the 6 numbers must be at least (5)(75)+(75+x)=(6)(75)+x, which is greater than (6)(75), contrary to the fact that the sum is equal to (6)(75). Hence, none of the numbers can be greater than 75. Since none of the numbers can be less than 75 (given information) and none of the numbers can be greater than 75, it follows that each of the numbers is equal to 75; SUFFICIENT.
   (2) If one of the numbers is less than 75, then we can write that number as 75-x for some positive number x. Consequently, the sum of the 6 numbers must be at most (5)(75)+(75-x)=(6)(75)-x, which is less than (6)(75), contrary to the fact that the sum is equal to (6)(75). Hence, none of the numbers can be less than 75. Since none of the numbers can be less than 75 and none of the numbers can be greater than 75 (given information), it follows that each of the numbers is equal to 75; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

E

[解析] Arithmetic Applied problems
   Let A be the amount of Jean's sales in the first half of 1988. Determine the value of A.
   (1) If the amount of Jean's sales in the first half of 1988 was $10,000, then her commission in the first half of 1988 would have been (5%)($10,000)=$500. On the other hand, if the amount of Jean's sales in the first half of 1988 was $100,000, then her commission in the first half of 1988 would have been (5%)($100,000)=$5,000; NOT sufficient.
   (2) No information is given that relates the amount of Jean's sales to the amount of Jean's commission; NOT sufficient.
   Given (1) and (2), from (1) the amount of Jean's commission in the first half of 1988 is (5%)A. From (2) the amount of Jean's sales in the second half of 1988 is A+$60,000. Both statements together do not give information to determine the value of A.
   The correct answer is E;
   both statements together are still not sufficient.

D

[解析] Geometry Area
   The area of the triangular region D can be represented by where b is the base of the triangle (and is equal to the length of a side of the square region C) and h is the height of the triangle (and is equal to the length of a side of the square region B). The area of any square is equal to the length of a side squared. The Pythagorean theorem is used to find the length of a side of a right triangle, when the length of the other 2 sides of the triangle are known and is represented by a²+b²=c², where a and b are the lengths of the 2 perpendicular sides of the triangle and c is the length of the hypotenuse.
   Although completed calculations are provided in what follows, keep in mind that completed calculations are not needed to solve this problem.
   (1) If the area of B is 9, then the length of each side is 3. Therefore, h=3. Then, b can be determined, since the area of the triangle is, by substitution, Once b is known, the Pythagorean theorem can be used: or The length of a side of A is thus SUFFICIENT.
   (2) If the area of C is then the length of each side is Therefore, The area of a triangle is and 3=h. Once h is known, the Pythagorean theorem can be used as above; SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

D

[解析] Arithmetic Properties of numbers
   Given that k=5.1×10ⁿ, where n is a positive integer, then the value of k must follow the pattern shown in the following table:
   
   (1) Given that 6,000＜k＜500,000, then k must have the value 51,000, and so n=4; SUFFICIENT.
   (2) Given that k²=2.601×10⁹, then SUFFICIENT.
   The correct answer is D;
   each statement alone is sufficient.

C

[解析] Geometry Angles
   
   In the figure above, a, b, c, and d are the degree measures of the interior angles of the quadrilateral formed by the four lines and a+b+c+d=360. Then
   w+x+y+z
   =(180-a)+(180-d)+(180-c)+(180-b)
   =720-(a+b+c+d)
   =720-360
   =360.
   Determine the value of x+y.
   (1) Given that w=95, then 95+x+y+z=360 and x+y+z=265. If z=65, for example, then x+y=200. On the other hand, if z=100, then x+y=165; NOT sufficient.
   (2) Given that z=125, then w+x+y+125=360 and w+x+y=235. If w=35, for example, then x+y=200. On the other hand, if w=100, then x+y=135; NOT sufficient.
   Taking (1) and (2) together, 95+x+y+125=360, and so x+y=140.
   The correct answer is C;
   both statements together are sufficient.

A

[解析] Algebra Inequalities
   Determine if Since each side is positive, squaring each side preserves the inequality, so is equivalent to which in turn is equivalent to n+k＞4n, or to k＞3n.
   (1) Given that k＞3n, then SUFFICIENT.
   (2) Given that n+k＞3n, then k＞2n. However, it is possible for k＞2n to be true and k＞3n to be false (for example, k=3 and n=1) and it is possible for k＞2n to be true and k＞3n to be true (for example, k=4 and n=1); NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

B

[解析] Arithmetic Proportions
   (1) This gives only values for e and i, and, while p is directly proportional to e, the nature of this proportion is unknown. Therefore, p cannot be determined; NOT sufficient.
   (2) Since p is directly proportional to e, which is directly proportional to i, then p is directly proportional to i. Therefore, the following proportion can be set up: If i=70, then Through cross multiplying, this equation yields 50p=140, or p=2.8; SUFFICIENT.
   The preceding approach is one method that can be used. Another approach is as follows: It is given that p=Ke=K(Li)=(KL)i, where K and L are the proportionality constants, and the value of 70KL is to be determined. Statement (1) allows us to determine the value of L, but gives nothing about K, and thus (1) is not sufficient. Statement (2) allows us to determine the value of KL, and thus (2) is sufficient.
   The correct answer is B;
   statement 2 alone is sufficient.

A

[解析] Arithmetic Properties of numbers
   (1) Given that the hundreds digit of 10n is 6, the tens digit of n is 6, since the hundreds digit of 10n is always equal to the tens digit of n; SUFFICIENT.
   (2) Given that the tens digit of n+1 is 7, it is possible that the tens digit of n is 7 (for example, n=70) and it is possible that the tens digit of n is 6 (for example, n=69); NOT sufficient.
   The correct answer is A;
   statement 1 alone is sufficient.

A

[解析] Algebra Simplifying algebraic expressions
   Determine the value of