GMAT Quantitative Review
5.0 Data Sufficiency
5.5 Answer Explanations
The following discussion of data sufficiency is intended to familiarize you with the most efficient and effective approaches to the kinds of problems common to data sufficiency. The particular questions in this chapter are generally representative of the kinds of data sufficiency questions you will encounter on the GMAT. Remember that it is the problem solving strategy that is important, not the specific details of a particular question.
1. What is the tenths digit of the number d when it is written as a decimal?
1. (1) 
2. (2) 1,000d = 2,160
Arithmetic Place value
3. (1) Given that d = 
, it follows that d = 
= 2.16 and the tenths digit is 1; SUFFICIENT.
4. (2) Given that 1,000d = 2,160, it follows that 
and the tenths digit is 1; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
2. A framed picture is shown above. The frame, shown shaded, is 6 inches wide and forms a border of uniform width around the picture. What are the dimensions of the viewable portion of the picture?
1. (1) The area of the shaded region is 24 square inches.
2. (2) The frame is 8 inches tall.
Geometry Area
Let the outer dimensions of the frame be 6 inches by B inches, and let the dimensions of the viewable portion of the picture be a inches by b inches. Then the area of the frame is the area of the viewable portion and the frame combined minus the area of the viewable portion, which equals (6B − ab) square inches. Determine the values of a and b.
3. (1) Given that 6B − ab = 24, then it is not possible to determine the values of a and b. For example, if B = 8, a = 4, and b = 6, then 6B − ab = 6(8) − (4)(6) = 24. However, if B = 7, a = 3, and b = 6, then 6B − ab = 6(7) − (3)(6) = 24; NOT sufficient.
4. (2) Given that B = 8, then 6B − ab = 48 − ab, but it is still not possible to determine the values of a and b; NOT sufficient.
Taking (1) and (2) together, it follows that 6B − ab = 24 and B = 8, and therefore 48 − ab = 24 and ab = 24. Also, letting the uniform width of the border be x inches, the outer dimensions of the frame are (a + 2x) inches = 6 inches and (b + 2x) inches = 8 inches, from which it follows by subtracting the last two equations that b − a = 2. Thus, b = a + 2, and so ab = 24 becomes a(a + 2) = 24, or a2 + 2a − 24 = 0. Factoring gives (a + 6)(a − 4) = 0, so a = −6 or a = 4. Because no dimension of the viewable portion can be negative, it follows that a = 4 and b = a + 2 = 4 + 2 = 6.
The correct answer is C; both statements together are sufficient.
3. What is the value of the integer x?
1. (1) x rounded to the nearest hundred is 7,200.
2. (2) The hundreds digit of x is 2.
Arithmetic Rounding
3. (1) Given that x rounded to the nearest hundred is 7,200, the value of x cannot be determined. For example, x could be 7,200 or x could be 7,201; NOT sufficient.
4. (2) Given that the hundreds digit of x is 2, the value of x cannot be determined. For example, x could be 7,200 or x could be 7,201; NOT sufficient.
Taking (1) and (2) together is of no more help than either (1) or (2) taken separately because the same examples were used in both (1) and (2).
The correct answer is E; both statements together are still not sufficient.
4. Is 2x > 2y?
1. (1) x > y
2. (2) 3x > 3y
Algebra Inequalities
3. (1) It is given that x > y. Thus, multiplying both sides by the positive number 2, it follows that 2x > 2y; SUFFICIENT.
4. (2) It is given that 3x > 3y. Thus, multiplying both sides by the positive number 
, it follows that 2x > 2y; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
5. If p and q are positive, is pq less than 1?
1. (1) p is less than 4.
2. (2) q is less than 4.
Arithmetic Properties of numbers
3. (1) Given that p is less than 4, then it is not possible to determine whether 
is less than 1. For example, if p = 1 and q = 2, then 
= 
and is less than 1. However, if p = 2 and q = 1, then = 2 and 2 is not less than 1; NOT sufficient.
4. (2) Given that q is less than 4, then it is not possible to determine whether is less than 1. For example, if p = 1 and q = 2, then = and is less than 1. However, if p = 2 and q = 1, then = 2 and 2 is not less than 1; NOT sufficient.
Taking (1) and (2) together is of no more help than either (1) or (2) taken separately because the same examples were used in both (1) and (2).
The correct answer is E; both statements together are still not sufficient.
6. In a certain factory, hours worked by each employee in excess of 40 hours per week are overtime hours and are paid for at 1
times the employee”s regular hourly pay rate. If an employee worked a total of 42 hours last week, how much was the employee”s gross pay for the hours worked last week?
1. (1) The employee”s gross pay for overtime hours worked last week was $30.
2. (2) The employee”s gross pay for all hours worked last week was $30 more than for the previous week.
Arithmetic Applied problems
If an employee”s regular hourly rate was $R and the employee worked 42 hours last week, then the employee”s gross pay for hours worked last week was 40R + 2(1.5R). Determine the value of 40R + 2(1.5R) = 43R, or equivalently, the value of R.
3. (1) Given that the employee”s gross pay for overtime hours worked last week was $30, it follows that 2(1.5R) = 30 and R = 10; SUFFICIENT.
4. (2) Given that the employee”s gross pay for all hours worked last week was $30 more than for the previous week, the value of R cannot be determined because nothing specific is known about the value of the employee”s pay for all hours worked the previous week; NOT sufficient.
The correct answer is A; statement (1) alone is sufficient.
7. If x > 0, what is the value of x5?
1. (1) √x = 32
2. (2) x2 = 220
Algebra Exponents
3. (1) Given that 
= 32, it follows that x = 322 and x5 = (322)5; SUFFICIENT.
4. (2) Given that x2 = 220, since x is positive, it follows that x = 
= 210 and x5 = (210)5; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
8. What is the value of the integer N?
1. (1) 101 < N < 103
2. (2) 202 < 2N < 206
Arithmetic Inequalities
3. (1) Given that N is an integer and 101 < N < 103, it follows that N = 102; SUFFICIENT.
4. (2) Given that N is an integer and 202 < 2N < 206, it follows that 101 < N < 103 and N = 102; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
9. Is zw positive?
1. (1) z + w3 = 20
2. (2) z is positive.
Arithmetic Properties of numbers
3. (1) Given that z + w3 = 20, if z = 1 and w = 
, then z + w3 = 20 and zw is positive. However, if z = 20 and w = 0, then z + w3 = 20 and zw is not positive; NOT sufficient.
4. (2) Given that z is positive, if z = 1 and w = , then zw is positive. However, if z = 20 and w = 0, then zw is not positive; NOT sufficient.
Taking (1) and (2) together is of no more help than either (1) or (2) taken separately because the same examples were used in both (1) and (2).
The correct answer is E; both statements together are still not sufficient.
10. In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equal area, what are the coordinates of point R?
1. (1) The coordinates of point P are (0,12).
2. (2) OP = OQ and QS = RS.
Geometry Coordinate geometry; triangles
Since the area of ΔOPQ is equal to the area of ΔQRS, it follows that 
(OQ)(OP) = (QS) (SR), or (OQ)(OP) = (QS)(SR). Also, if both OS and SR are known, then the coordinates of point R will be known.
3. (1) Given that the y-coordinate of P is 12, it is not possible to determine the coordinates of point R. For example, if OQ = QS = SR = 12, then the equation (OQ)(OP) = (QS)(SR) becomes (12)(12) = (12)(12), which is true, and the x-coordinate of R is OQ + QS = 24 and the y-coordinate of R is SR = 12. However, if OQ = 12, QS = 24, and SR = 6, then the equation (OQ)(OP) = (QS)(SR) becomes (12)(12) = (24)(6), which is true, and the x-coordinate of R is OQ + QS = 36 and the y-coordinate of R is SR = 6; NOT sufficient.
4. (2) Given that OP = OQ and QS = RS, it is not possible to determine the coordinates of point R, since everything given would still be true if all the lengths were doubled, but doing this would change the coordinates of point R; NOT sufficient.
Taking (1) and (2) together, it follows that OP = OQ = 12. Therefore, (OQ)(OP) = (QS)(SR) becomes (12)(12) = (QS)(SR), or 144 = (QS)(SR). Using QS = RS in the last equation gives 144 = (QS)2, or 12 = QS. Thus, OQ = QS = SR = 12 and point R has coordinates (24,12).
The correct answer is C; both statements together are sufficient.
11. If y is greater than 110 percent of x, is y greater than 75?
1. (1) 
2. (2) 
Arithmetic; Algebra Percents; Inequalities
3. (1) It is given that 
and 
. Therefore, 
, and so y is greater than 75; SUFFICIENT.
4. (2) Although it is given that 
, more information is needed to determine if y is greater than 75. For example, if 
and 
, then y is greater than 110 percent of x, 
, and y is greater than 75. However, if 
and 
, then y is greater than 110 percent of x, , and y is not greater than 75; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
12. What is the average (arithmetic mean) of x and y?
1. (1) The average of x and 2y is 10.
2. (2) The average of 2x and 7y is 32.
Algebra Statistics
The average of x and y is 
, which can be determined if and only if the value of x + y can be determined.
3. (1) It is given that the average of x and 2y is 10. Therefore, 
, or 
. Because the value of 
is desired, rewrite the last equation as 
, or 
. This shows that the value of can vary. For example, if 
and 
, then 
and 
. However, if 
and 
, then 
and 
; NOT sufficient.
4. (2) It is given that the average of 2x and 7y is 32. Therefore, 
, or 
. Because the value of is desired, rewrite the last equation as 
, or 
. This shows that the value of can vary. For example, if 
and 
, then and 
. However, if 
and 
, then and 
; NOT sufficient. Given (1) and (2), it follows that 
and 
. These two equations can be solved simultaneously to obtain the individual values of x and y, which can then be used to determine the average of x and y. From 
it follows that 
. Substituting 
for x in 
gives 
, or 
, or 
, or 
. Thus, using , the value of x is 
. Alternatively, it can be seen that unique values for x and y are determined from (1) and (2) by the fact that the equations and represent two nonparallel lines in the standard (x, y) coordinate plane, which have a unique point in common.
The correct answer is C; both statements together are sufficient.
13. What is the value of 
?
1. (1) 
2. (2) 
Arithmetic Operations with rational numbers
Since 
, the value of 
can be determined exactly when either the value of 
can be determined or the value of 
can be determined.
3. (1) It is given that 
. Therefore, 
; SUFFICIENT.
4. (2) It is given that 
. Therefore, 
; SUFFICIENT.
The correct answer is D; each statement alone is sufficient
14. If 
and w represent the length and width, respectively, of the rectangle above, what is the perimeter?
1. (1) 
2. (2) 
Geometry Perimeter
The perimeter of the rectangle is 
, which can be determined exactly when the value of 
can be determined.
3. (1) It is given that 
. Therefore, 
, or 
. Therefore, different values of can be obtained by choosing different values of 
. For example, if 
and 
, then 
. However, if 
and 
, then 
; NOT sufficient.
4. (2) It is given that ; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
15. For all x, the expression x∗ is defined to be ax + a, where a is a constant. What is the value of 2∗?
1. (1) 3∗ = 2
2. (2) 5∗ = 3
Algebra Linear equations
Determine the value of 2∗ = (a)(2) + a = 3a, or equivalently, determine the value of a.
3. (1) Given that 3∗ = 2, it follows that (a)(3) + a = 2, or 4a = 2, or a = ; SUFFICIENT.
4. (2) Given that 5∗ = 3, it follows that (a) (5) + a = 3, or 6a = 3, or a = ; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
16. Is k + m < 0?
1. (1) k < 0
2. (2) km > 0
Arithmetic Properties of numbers
3. (1) Given that k is negative, it is not possible to determine whether k + m is negative. For example, if k = −2 and m = 1, then k + m is negative. However, if k = −2 and m = 3, then k + m is not negative; NOT sufficient.
4. (2) Given that km is positive, it is not possible to determine whether k + m is negative. For example, if k = −2 and m = −1, then km is positive and k + m is negative. However, if k = 2 and m = 1, then km is positive and k + m is not negative; NOT sufficient.
Taking (1) and (2) together, k is negative and km is positive, it follows that m is negative. Therefore, both k and m are negative, and hence k + m is negative.
The correct answer is C; both statements together are sufficient.
17. A retailer purchased a television set for x percent less than its list price, and then sold it for y percent less than its list price. What was the list price of the television set?
1. (1) 
2. (2) 
Arithmetic Percents
3. (1) This provides information only about the value of x. The list price cannot be determined using x because no dollar value for the purchase price is given; NOT sufficient.
4. (2) This provides information about the relationship between x and y but does not provide dollar values for either of these variables; NOT sufficient.
The list price cannot be determined without a dollar value for either the retailer”s purchase price or the retailer”s selling price. Even though the values for x and y are given or can be determined, taking (1) and (2) together provides no dollar value for either.
The correct answer is E; both statements together are still not sufficient.
18. If Ann saves x dollars each week and Beth saves y dollars each week, what is the total amount that they save per week?
1. (1) Beth saves $5 more per week than Ann saves per week.
2. (2) It takes Ann 6 weeks to save the same amount that Beth saves in 5 weeks.
Algebra Simultaneous equations
Determine the value of .
3. (1) It is given that 
. Therefore, 
, which can vary in value. For example, if 
and 
, then and 
. However, if 
and 
, then and 
; NOT sufficient.
4. (2) It is given that 
, or 
. Therefore, 
, which can vary in value. For example, if and 
, then 
and 
. However, if and 
, then and 
; NOT sufficient.
Given (1) and (2), it follows that and . These two equations can be solved simultaneously to obtain the individual values of x and y, which can then be used to determine . Equating the two expressions for y gives 
, or 
, or 
. Therefore, 
and 
.
The correct answer is C; both statements together are sufficient.
19. What is the total number of executives at Company P?
1. (1) The number of male executives is 
the number of female executives.
2. (2) There are 4 more female executives than male executives.
Algebra Simultaneous equations
Let M be the number of male executives at Company P and let F be the number of female executives at Company P. Determine the value of M + F.
3. (1) Given that M = 
F, it is not possible to determine the value of M + F. For example, if M = 3 and F = 5, then M = F and M + F = 8. However, if M = 6 and F = 10, then M = F and M + F = 16; NOT sufficient.
4. (2) Given that F = M + 4, it is not possible to determine the value of M + F. For example, if M = 3 and F = 7, then F = M + 4 and M + F = 10. However, if M = 4 and F = 8, then F = M + 4 and M + F = 12; NOT sufficient.
Taking (1) and (2) together, then F = M + 4 and M = F, so F = F + 4. Now solve for F to get 
F = 4 and F = 10. Therefore, using F = 10 and F = M + 4, it follows that M = 6, and hence M + F = 6 + 10 = 16.
The correct answer is C; both statements together are sufficient.
20. What is the ratio of c to d?
1. (1) The ratio of 3c to 3d is 3 to 4.
2. (2) The ratio of c + 3 to d + 3 is 4 to 5.
Arithmetic Ratio and proportion
Determine the value of 
.
3. (1) Given that 
, it follows that 
; SUFFICIENT.
4. (2) Given that 
, then it is not possible to determine the value of 
. For example, if c = 1 and d = 2, then 
and = 
. However, if c = 5 and d = 7, then 
and 
; NOT sufficient.
The correct answer is A; statement (1) alone is sufficient.
21. A certain dealership has a number of cars to be sold by its salespeople. How many cars are to be sold?
1. (1) If each of the salespeople sells 4 of the cars, 23 cars will remain unsold.
2. (2) If each of the salespeople sells 6 of the cars, 5 cars will remain unsold.
Algebra Simultaneous equations
Let T be the total number of cars to be sold and S be the number of salespeople. Determine the value of T.
3. (1) Given that 
, it follows that the positive integer value of T can vary, since the positive integer value of S cannot be determined; NOT sufficient.
4. (2) Given that 
, it follows that the positive integer value of T can vary, since the positive integer value of S cannot be determined; NOT sufficient.
(1) and (2) together give a system of two equations in two unknowns. Equating the two expressions for T gives 
, or 
, or 
. From this the value of T can be determined by 
.
The correct answer is C; both statements together are sufficient.
22. Committee member W wants to schedule a one-hour meeting on Thursday for himself and three other committee members, X, Y, and Z. Is there a one-hour period on Thursday that is open for all four members?
1. (1) On Thursday W and X have an open period from 9:00 a.m. to 12:00 noon.
2. (2) On Thursday Y has an open period from 10:00 a.m. to 1:00 p.m. and Z has an open period from 8:00 a.m. to 11:00 a.m.
Arithmetic Sets
3. (1) There is no information about Y and Z, only information about W and X; NOT sufficient.
4. (2) Similarly, there is no information about W and X, only information about Y and Z; NOT sufficient.
1. Together, (1) and (2) detail information about all four committee members, and it can be determined that on Thursday all four members have an open one-hour period from 10:00 a.m. to 11:00 a.m.
The correct answer is C; both statements together are sufficient.
23. Some computers at a certain company are Brand X and the rest are Brand Y. If the ratio of the number of Brand Y computers to the number of Brand X computers at the company is 5 to 6, how many of the computers are Brand Y?
1. (1) There are 80 more Brand X computers than Brand Y computers at the company.
2. (2) There is a total of 880 computers at the company.
Algebra Simultaneous equations
Let x and y be the numbers of Brand X computers and Brand Y computers, respectively, at the company. Then 
, or after cross multiplying, 
. Determine the value of y.
3. (1) Given that 
, it follows that 
. Substituting 6y for 5x on the left side of the last equation gives 
, or 
. Alternatively, it can be seen that unique values for x and y are determined by the fact that 
and 
represent the equations of two nonparallel lines in the standard (x, y) coordinate plane, which have a unique point in common; SUFFICIENT.
4. (2) Given that 
, it follows that 
. Substituting 6y for 5x on the left side of the last equation gives 
, or 
, or 
. Alternatively, it can be seen that unique values for x and y are determined by the fact that 
and 
represent the equations of two nonparallel lines in the standard (x, y) coordinate plane, which have a unique point in common; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
24. In the figure shown, lines k and m are parallel to each other. Is x = z?
1. (1) x = w
2. (2) y = 180 − w
Geometry Angles
Since lines k and m are parallel, it follows from properties of parallel lines that in the diagram above x is the degree measure of 
ABC in quadrilateral ABCD. Therefore, because y = 180 − x, the four interior angles of quadrilateral ABCD have degree measures (180 − x), x, w, and (180 − z).
3. (1) Given that x = w, then because the sum of the degree measures of the angles of the quadrilateral ABCD is 360, it follows that (180 − x) + x + x + (180 − z) = 360, or x − z = 0, or x = z; SUFFICIENT.
4. (2) Given that y = 180 − w, then because y = 180 − x, it follows that 180 − w = 180 − x, or x = w. However, it is shown in (1) that x = w is sufficient; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
25. When the wind speed is 9 miles per hour, the wind-chill factor w is given by
w = −17.366 + 1.19t,
where t is the temperature in degrees Fahrenheit. If at noon yesterday the wind speed was 9 miles per hour, was the wind-chill factor greater than 0?
1. (1) The temperature at noon yesterday was greater than 10 degrees Fahrenheit.
2. (2) The temperature at noon yesterday was less than 20 degrees Fahrenheit.
Algebra Applied problems
Determine whether −17.366 + 1.19t is greater than 0.
3. (1) Given that t > 10, it follows that −17.366 + 1.19t > −17.366 + 1.19(10), or −17.366 + 1.19t > −5.466. However, it is not possible to determine whether −17.366 + 1.19t is greater than 0. For example, if t = 19, then −17.366 + 1.19t = 5.244 is greater than 0. However, if t = 11, then −17.366 + 1.19t = −4.276, which is not greater than 0; NOT sufficient.
4. (2) Given that t < 20, the same examples used in (1) show that it is not possible to determine whether −17.366 + 1.19t is greater than 0; NOT sufficient.
Taking (1) and (2) together is of no more help than either (1) or (2) taken separately because the same examples were used in both (1) and (2).
The correct answer is E; both statements together are still not sufficient.
26. What is the volume of the cube above?
1. (1) The surface area of the cube is 600 square inches.
2. (2) The length of diagonal AB is 10
inches.
Geometry Volume
This problem can be solved by determining the side length, s, of the cube.
3. (1) This indicates that 6s2 = 600, from which it follows that s2 = 100 and s = 10; SUFFICIENT.
4. (2) To determine diagonal AB, first determine diagonal AN by applying the Pythagorean theorem to ΔAMN: AN = 
. Now determine AB by applying the Pythagorean theorem to ΔANB: AB = 
= s
. It is given thatAB = 10, and so s = 10; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
27. Of the 230 single-family homes built in City X last year, how many were occupied at the end of the year?
1. (1) Of all single-family homes in City X, 90 percent were occupied at the end of last year.
2. (2) A total of 7,200 single-family homes in City X were occupied at the end of last year.
Arithmetic Percents
3. (1) The percentage of the occupied single-family homes that were built last year is not given, and so the number occupied cannot be found; NOT sufficient.
4. (2) Again, there is no information about the occupancy of the single-family homes that were built last year; NOT sufficient.
Together (1) and (2) yield only the total number of the single-family homes that were occupied. Neither statement offers the needed information as to how many of the single-family homes built last year were occupied at the end of last year.
The correct answer is E; both statements together are still not sufficient.
28. If J, S, and V are points on the number line, what is the distance between S and V?
1. (1) The distance between J and S is 20.
2. (2) The distance between J and V is 25.
Arithmetic Properties of numbers
3. (1) Since no restriction is placed on the location of V, the distance between S and V could be any positive real number; NOT sufficient.
4. (2) Since no restriction is placed on the location of S, the distance between S and V could be any positive real number; NOT sufficient.
Given (1) and (2) together, it follows that 
and 
. However, V could be on the left side of S or V could be on the right side of S. For example, suppose J is located at 0 and S is located at 20. If V were on the left side of S, then V would be located at −25, and thus SV would be 
, as shown below.
However, if V were on the right side of S, then V would be located at 25, and thus SV would be 
, as shown below.
The correct answer is E; both statements together are still not sufficient.
29. If x is a positive integer, what is the value of x?
1. (1) 
2. (2) 
.
Algebra Operations with radicals
3. (1) It is given that x is a positive integer. Then,
|
x2 = |
given |
|
x4 = x |
square both sides |
|
x4 − x = 0 |
subtract x from both sides |
|
x(x − 1) (x2 + x + 1) = 0 |
factor left side |
4. Thus, the positive integer value of x being sought will be a solution of this equation. One solution of this equation is x = 0, which is not a positive integer. Another solution is x = 1, which is a positive integer. Also, x2 + x + 1 is a positive integer for all positive integer values of x, and so x2 + x + 1 = 0 has no positive integer solutions. Thus, the only possible positive integer value of x is 1; SUFFICIENT.
5. (2) It is given that n ≠ 0. Then,
|
|
given |
|
n = nx |
multiply both sides by x |
|
1 = x |
divide both sides by n, where n ≠ 0 |
= nThus, x = 1; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
30. During a certain bicycle ride, was Sherry”s average speed faster than 24 kilometers per hour? (1 kilometer = 1,000 meters)
1. (1) Sherry”s average speed during the bicycle ride was faster than 7 meters per second.
2. (2) Sherry”s average speed during the bicycle ride was slower than 8 meters per second.
Arithmetic Applied problems
This problem can be solved by converting 24 kilometers per hour into meters per second. First, 24 kilometers is equivalent to 24,000 meters and 1 hour is equivalent to 3,600 seconds. Then, traveling 24 kilometers in 1 hour is equivalent to traveling 24,000 meters in 3,600 seconds, or 
meters per second.
3. (1) This indicates that Sherry”s average speed was faster than 7 meters per second, which is faster than 
meters per second and, therefore, faster than 24 kilometers per hour; SUFFICIENT.
4. (2) This indicates that Sherry”s average speed was slower than 8 meters per second. Her average speed could have been 7 meters per second (since 7 < 8), in which case her average speed was faster than meters per second and, therefore, faster than 24 kilometers per hour. Or her average speed could have been 5 meters per second (since 5 < 8), in which case her average speed was not faster than meters per second and, therefore, not faster than 24 kilometers per hour; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
31. If x and y are integers, what is the value of x?
1. (1) xy = 1
2. (2) x ≠ −1
Arithmetic Properties of integers
Given that x and y are integers, determine the value of x.
3. (1) If x = y = −1, then xy = 1, and if x = y = 1, then xy = 1; NOT sufficient.
4. (2) Given that x ≠ −1, the value of x could be any other integer; NOT sufficient.
Taking (1) and (2) together, since the two possibilities for the value of x are x = −1 or x = 1 by (1), and x ≠ −1 by (2), then x = 1.
The correct answer is C; both statements together are sufficient
32. If p, s, and t are positive, is |ps − pt| > p(s − t)?
1. (1) p < s
2. (2) s < t
Algebra Absolute value
Since p is positive, it follows that |p(s − t)| = |p||s − t| = p|s − t|. Therefore, the task is to determine if |s − t| > s − t. Since |s − t| = s − t if and only if s − t ≥ 0, it follows that |s − t| > s − t if and only if s − t < 0.
3. (1) This indicates that p < s but does not provide information about the relationship between s and t. For example, if p = 5, s = 10, and t = 15, then p < s and s < t, but if p = 5, s = 10, and t = 3, then p < s and s > t; NOT sufficient.
4. (2) This indicates that s < t, or equivalently, s − t < 0; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
33. What were the gross revenues from ticket sales for a certain film during the second week in which it was shown?
1. (1) Gross revenues during the second week were $1.5 million less than during the first week.
2. (2) Gross revenues during the third week were $2.0 million less than during the first week.
Arithmetic Arithmetic operations
3. (1) Since the amount of gross revenues during the first week is not given, the gross revenues during the second week cannot be determined; NOT sufficient.
4. (2) No information is provided, directly or indirectly, about gross revenues during the second week; NOT sufficient.
With (1) and (2) taken together, additional information, such as the amount of gross revenues during either the first or the third week, is still needed.
The correct answer is E; both statements together are still not sufficient.
34. The total cost of an office dinner was shared equally by k of the n employees who attended the dinner. What was the total cost of the dinner?
1. (1) Each of the k employees who shared the cost of the dinner paid $19.
2. (2) If the total cost of the dinner had been shared equally by 
of the n employees who attended the dinner, each of the employees would have paid $18.
Algebra Simultaneous equations
3. (1) Given that each of the k employees paid $19, it follows that the total cost of the dinner, in dollars, is 19k. However, since k cannot be determined, the value of 19k cannot be determined; NOT sufficient.
4. (2) Given that each of 
employees would have paid $18, it follows that the total cost of the dinner, in dollars, is 
. However, since k cannot be determined, the value of cannot be determined; NOT sufficient.
Given (1) and (2) together, it follows that 
, or 
, or 
. Therefore, the total cost of the dinner is 
.
The correct answer is C; both statements together are sufficient.
35. For a recent play performance, the ticket prices were $25 per adult and $15 per child. A total of 500 tickets were sold for the performance. How many of the tickets sold were for adults?
1. (1) Revenue from ticket sales for this performance totaled $10,500.
2. (2) The average (arithmetic mean) price per ticket sold was $21.
Algebra Simultaneous equations
Let A and C be the numbers of adult and child tickets sold, respectively. Given that 
, or 
, determine the value of A.
3. (1) Given that 
, or 
, it follows by substituting 
for C that 
, which can be solved to obtain a unique value for A. Alternatively, it can be seen that unique values for A and C are determined by the fact that 
and 
represent the equations of two nonparallel lines in the standard (x, y) coordinate plane, which have a unique point in common; SUFFICIENT.
4. (2) It is given that 
, or 
, which is the same information given in (1). Therefore, A can be determined, as shown in (1) above; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
36. What is the value of x?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
3. (1) Transposing terms gives the equivalent equation 
, or 
; SUFFICIENT.
4. (2) Multiplying both sides by 2x gives the equivalent equation 
, or 
; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
37. If x and y are positive integers, what is the remainder when 
is divided by 3?
1. (1) 
2. (2) 
Arithmetic Properties of numbers
3. (1) Given that 
, then 
. More than one remainder is possible when 
is divided by 3. For example, by long division, or by using the fact that 
, the remainder is 2 when 
and the remainder is 0 when 
; NOT sufficient.
4. (2) Given that , then 
. Since the sum of the digits of 
, which is divisible by 3, it follows that 
is divisible by 3, and hence has remainder 0 when divided by 3. This can also be seen by writing as 
, which is divisible by 3; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
38. What was the amount of money donated to a certain charity?
1. (1) Of the amount donated, 40 percent came from corporate donations.
2. (2) Of the amount donated, $1.5 million came from noncorporate donations.
Arithmetic Percents
The statements suggest considering the amount of money donated to be the total of the corporate donations and the noncorporate donations.
3. (1) From this, only the portion that represented corporate donations is known, with no means of determining the total amount donated; NOT sufficient.
4. (2) From this, only the dollar amount that represented noncorporate donations is known, with no means of determining the portion of the total donations that it represents; NOT sufficient.
Letting x represent the total dollar amount donated, it follows from (1) that the amount donated from corporate sources can be represented as 0.40x. Combining the information from (1) and (2) yields the equation 
, which can be solved to obtain exactly one solution for x.
The correct answer is C; both statements together are sufficient.
39. What is the value of the positive integer n?
1. (1) 
2. (2) 
Arithmetic Arithmetic operations
3. (1) If n is a positive integer and 
, then n can be either 1 or 2, since 
and 
; NOT sufficient.
4. (2) Since the only positive integer equal to its square is 1, each positive integer that is not equal to 1 satisfies (2); NOT sufficient.
Using (1) and (2) together, it follows from (1) that 
or 
, and it follows from (2) that 
, and hence the value of n must be 2.
The correct answer is C; both statements together are sufficient.
40. If the set S consists of five consecutive positive integers, what is the sum of these five integers?
1. (1) The integer 11 is in S, but 10 is not in S.
2. (2) The sum of the even integers in S is 26.
Arithmetic Sequences
3. (1) This indicates that the least integer in S is 11 since S consists of consecutive integers and 11 is in S, but 10 is not in S. Thus, the integers in S are 11, 12, 13, 14, and 15, and their sum can be determined; SUFFICIENT.
4. (2) This indicates that the sum of the even integers in S is 26. In a set of 5 consecutive integers, either two of the integers or three of the integers are even. If there are three even integers, then the first integer in S must be even. Also, since 
, the three even integers must be around 8. The three even integers could be 6, 8, and 10, but are not because their sum is less than 26; or they could be 8, 10, and 12, but are not because their sum is greater than 26. Therefore, S cannot contain three even integers and must contain only two even integers. Those integers must be 12 and 14 since 12 + 14 = 26. It follows that the integers in S are 11, 12, 13, 14, and 15, and their sum can be determined; SUFFICIENT.
Alternately, if n, n + 1, n + 2, n + 3, and n + 4 represent the five consecutive integers and three of them are even, then n + (n + 2) + (n + 4) = 26, or 3n = 20, or 
, which is not an integer.
On the other hand, if two of the integers are even, then (n +1) + (n +3) = 26, or 2n = 22, or n = 11. It follows that the integers are 11, 12, 13, 14, and 15, and their sum can be determined; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
41. A total of 20 amounts are entered on a spreadsheet that has 5 rows and 4 columns; each of the 20 positions in the spreadsheet contains one amount. The average (arithmetic mean) of the amounts in row i is Ri (1 ≤ i ≤ 5). The average of the amounts in column j isCj (1 ≤ j ≤ 4). What is the average of all 20 amounts on the spreadsheet?
1. (1) R1 + R2 + R3 + R4 + R5 = 550
2. (2) C1 + C2 + C3 + C4 = 440
Arithmetic Statistics
It is given that Ri represents the average of the amounts in row i. Since there are four amounts in each row, 4Ri represents the total of the amounts in row i. Likewise, it is given that Cj represents the average of the amounts in column j. Since there are five amounts in each column, 5Cj represents the total of the amounts in column j.
3. (1) It is given that R1 + R2 + R3 + R4 + R5 = 550, and so 4(R1 + R2 + R3 + R4 + R5) = 4R1 + 4R2 + 4R3 + 4R4 + 4R5 = 4(550) = 2,200. Therefore, 2,200 is the sum of all 20 amounts (4 amounts in each of 5 rows), and the average of all 20 amounts is 
= 110; SUFFICIENT.
4. (2) It is given that C1 + C2 + C3 + C4 = 440, and so 5(C1 + C2 + C3 + C4) = 5C1 + 5C2 + 5C3 + 5C4 = 5(440) = 2,200. Therefore, 2,200 is the sum of all 20 amounts (5 amounts in each of 4 columns), and the average of all 20 amounts is = 110; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
42. Was the range of the amounts of money that Company Y budgeted for its projects last year equal to the range of the amounts of money that it budgeted for its projects this year?
1. (1) Both last year and this year, Company Y budgeted money for 12 projects and the least amount of money that it budgeted for a project was $400.
2. (2) Both last year and this year, the average (arithmetic mean) amount of money that Company Y budgeted per project was $2,000.
Arithmetic Statistics
Let G1 and L1 represent the greatest and least amounts, respectively, of money that Company Y budgeted for its projects last year, and let G2 and L2 represent the greatest and least amounts, respectively, of money that Company Y budgeted for its projects this year. Determine if the range of the amounts of money Company Y budgeted for its projects last year is equal to the range of amounts budgeted for its projects this year; that is, determine if G1 − L1 = G2 − L2.
3. (1) This indicates that L1 = L2 = $400, but does not give any information about G1 or G2; NOT sufficient.
4. (2) This indicates that the average amount Company Y budgeted for its projects both last year and this year was $2,000 per project, but does not give any information about the least and greatest amounts that it budgeted for its projects either year; NOT sufficient.
Taking (1) and (2) together, it is known that L1 = L2 = $400 and that the average amount Company Y budgeted for its projects both last year and this year was $2,000 per project, but there is no information about G1 or G2. For example, if, for each year, Company Y budgeted $400 for each of 2 projects and $2,320 for each of the 10 others, then (1) and (2) are true and the range for each year was $2,320 − $400 = $1,920. However, if, last year, Company Y budgeted $400 for each of 2 projects and $2,320 for each of the 10 others, and, this year, budgeted $400 for each of 11 projects and $19,600 for 1 project, then (1) and (2) are true, but the range for last year was $1,920 and the range for this year was $19,600 − $400 = $19,200.
The correct answer is E; both statements together are still not sufficient.
43. If a, b, c, and d are numbers on the number line shown and if the tick marks are equally spaced, what is the value of a + c?
1. (1) a + b = −8
2. (2) a + d = 0
Algebra Sequences
It is given that the distance between a and b is the same as the distance between b and c, which is the same as the distance between c and d. Letting q represent this distance, then b = a + q, c = a + 2q, and d = a + 3q. The value of a + c can be determined if the value of a + (a + 2q) = 2a + 2q can be determined.
3. (1) It is given that a + b = −8. Then, a + (a + q) = 2a + q = −8. From this, the value of 2a + 2q cannot be determined. For example, the values of a and q could be −5 and 2, respectively, or they could be −6 and 4, respectively; NOT sufficient.
4. (2) It is given that a + d = 0. Then, a + (a + 3q) = 2a + 3q = 0. From this, the value of 2a + 2q cannot be determined. For example, the values of a and q could be −3 and 2, respectively, or they could be −6 and 4, respectively; NOT sufficient.
Taking (1) and (2) together, adding the equations, 2a + q = −8 and 2a + 3q = 0 gives 4a + 4q = −8 and so 2a + 2q = 
= −4.
The correct answer is C; both statements together are sufficient.
44. In the triangle above, does 
?
1. (1) 
2. (2) 
Geometry Triangles
The Pythagorean theorem states that 
for any right triangle with legs of lengths a and b and hypotenuse of length c. A right triangle is a triangle whose largest angle has measure 
. The converse of the Pythagorean theorem also holds: If 
, then the triangle is a right triangle.
3. (1) The sum of the degree measures of the three interior angles of a triangle is 
. It is given that 
. Thus, the remaining interior angle (not labeled) has degree measure 180 − 90 = 90. Therefore, the triangle is a right triangle, and hence it follows from the Pythagorean theorem that ; SUFFICIENT.
4. (2) Given that 
, the triangle could be a right triangle (for example, 
) or fail to be a right triangle (for example, 
), and hence can be true (this follows from the Pythagorean theorem) or can be false (this follows from the converse of the Pythagorean theorem); NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
45. If 
, is 
?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
3. (1) Dividing each side of the equation 
by rs gives 
, or 
, or 
; SUFFICIENT.
4. (2) If 
, then 
, but if 
, then 
; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
46. If x, y, and z are three integers, are they consecutive integers?
1. (1) 
2. (2) 
Arithmetic Properties of numbers
3. (1) Given 
, it is possible to choose y so that x, y, and z are consecutive integers (for example, 
, 
, and 
) and it is possible to choose y so that x, y, and z are not consecutive integers (for example, , 
, and ); NOT sufficient.
4. (2) Given that 
, the three integers can be consecutive (for example, , , and ) and the three integers can fail to be consecutive (for example, , 
, and 
); NOT sufficient.
Using (1) and (2) together, it follows that y is the unique integer between x and z and hence the three integers are consecutive.
The correct answer is C; both statements together are sufficient.
47. A collection of 36 cards consists of 4 sets of 9 cards each. The 9 cards in each set are numbered 1 through 9. If one card has been removed from the collection, what is the number on that card?
1. (1) The units digit of the sum of the numbers on the remaining 35 cards is 6.
2. (2) The sum of the numbers on the remaining 35 cards is 176.
Arithmetic Properties of numbers
The sum 
can be evaluated quickly by several methods. One method is to group the terms as 
, and therefore the sum is 
. Thus, the sum of the numbers on all 36 cards is 
.
3. (1) It is given that the units digit of the sum of the numbers on the remaining 35 cards is 6. Since the sum of the numbers on all 36 cards is 180, the sum of the numbers on the remaining 35 cards must be 179, 178, 177, . . . , 171, and of these values, only 176 has a units digit of 6. Therefore, the number on the card removed must be 
; SUFFICIENT.
4. (2) It is given that the sum of the numbers on the remaining 35 cards is 176. Since the sum of the numbers on all 36 cards is 180, it follows that the number on the card removed must be 
; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
48. In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of 
?
1. (1) The circle has radius 2.
2. (2) The point 
lies on the circle.
Geometry Simple coordinate geometry
Let R be the radius of the circle. A right triangle with legs of lengths 
and 
can be formed so that the line segment with endpoints (r, s) and (0,0) is the hypotenuse. Since the length of the hypotenuse is R, the Pythagorean theorem for this right triangle gives 
. Therefore, to determine the value of 
, it is sufficient to determine the value of R.
3. (1) It is given that 
; SUFFICIENT.
4. (2) It is given that 
lies on the circle. A right triangle with legs each of length 
can be formed so that the line segment with endpoints 
and (0,0) is the hypotenuse. Since the length of the hypotenuse is the radius of the circle, which isR, where 
, the Pythagorean theorem for this right triangle gives 
. Therefore, 
; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
49. If r, s, and t are nonzero integers, is r 5s3t 4 negative?
1. (1) rt is negative.
2. (2) s is negative.
Arithmetic Properties of numbers
Since 
and (rt)4 is positive, r5s3t4 will be negative if and only if rs3 is negative, or if and only if r and s have opposite signs.
3. (1) It is given that rt is negative, but nothing can be determined about the sign of s. If the sign of s is the opposite of the sign of r, then 
will be negative. However, if the sign of s is the same as the sign of r, then will be positive; NOT sufficient.
4. (2) It is given that s is negative, but nothing can be determined about the sign of r. If r is positive, then 
will be negative. However, if r is negative, then will be positive; NOT sufficient.
Given (1) and (2), it is still not possible to determine whether r and s have opposite signs. For example, (1) and (2) hold if r is positive, s is negative, and t is negative, and in this case r and s have opposite signs. However, (1) and (2) hold if r is negative, s is negative, and t is positive, and in this case r and s have the same sign.
The correct answer is E; both statements together are still not sufficient.
50. If x and y are integers, what is the value of y?
1. (1) 
2. (2) 
Arithmetic Arithmetic operations
3. (1) Many different pairs of integers have the product 27, for example, (−3)(−9) and (1)(27). There is no way to determine which pair of integers is intended, and there is also no way to determine which member of a pair is x and which member of a pair is y; NOT sufficient.
4. (2) Given that 
, more than one integer value for y is possible. For example, y could be 1 (with the value of x being 1) or y could be 2 (with the value of x being 4); NOT sufficient.
Using both (1) and (2), y2 can be substituted for the value of x in (1) to give 
, which has exactly one solution, 
.
The correct answer is C; both statements together are sufficient.
51. How many newspapers were sold at a certain newsstand today?
1. (1) A total of 100 newspapers were sold at the newsstand yesterday, 10 fewer than twice the number sold today.
2. (2) The number of newspapers sold at the newsstand yesterday was 45 more than the number sold today.
Algebra First- and second-degree equations
Let t be the number of newspapers sold today.
3. (1) The given information can be expressed as 
, which can be solved for a unique value of t; SUFFICIENT.
4. (2) It is given that 
newspapers were sold at the newsstand yesterday. Since the number sold yesterday is unknown, t cannot be determined; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
52. John took a test that had 60 questions numbered from 1 to 60. How many of the questions did he answer correctly?
1. (1) The number of questions he answered correctly in the first half of the test was 7 more than the number he answered correctly in the second half of the test.
2. (2) He answered 
of the odd-numbered questions correctly and 
of the even-numbered questions correctly.
Arithmetic Fractions
3. (1) Let f represent the number of questions answered correctly in the first half of the test and let s represent the number of questions answered correctly in the second half of the test. Then the given information can be expressed as 
, which has several solutions in which f and s are integers between 1 and 60, leading to different values of f + s. For example, f could be 10 and s could be 3, which gives 
, or f could be 11 and s could be 4, which gives 
; NOT sufficient.
4. (2) Since there are 30 odd-numbered questions and 30 even-numbered questions in a 60-question test, from the information given it follows that the number of questions answered correctly was equal to 
; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
53. If x is a positive integer, is 
an integer?
1. (1) 
is an integer.
2. (2) 
is not an integer.
Algebra Radicals
3. (1) It is given that 
, or 
, for some positive integer n. Since 4x is the square of an integer, it follows that in the prime factorization of 4x, each distinct prime factor is repeated an even number of times. Therefore, the same must be true for the prime factorization of x, since the prime factorization of x only differs from the prime factorization of 4x by two factors of 2, and hence by an even number of factors of 2; SUFFICIENT.
4. (2) Given that 
is not an integer, it is possible for 
to be an integer (for example, 
) and it is possible for to not be an integer (for example, 
); NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
54. Is the value of n closer to 50 than to 75?
1. (1) 
2. (2) 
Algebra Inequalities
Begin by considering the value of n when it is at the exact same distance from both 50 and 75. The value of n is equidistant between 50 and 75 when n is the midpoint between 75 and 50, that is, when 
. Alternatively stated, n is equidistant between 50 and 75 when the distance that n is below 75 is equal to the distance that n is above 50, i.e., when 
, as indicated on the number line below.
3. (1) Since here 
, it follows that the value of n is closer to 50 than to 75; SUFFICIENT.
4. (2) Although n is greater than 60, for all values of n between 60 and 62.5, n is closer to 50, and for all values of n greater than 62.5, n is closer to 75. Without further information, the value of n relative to 50 and 75 cannot be determined; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
55. Last year, if Elena spent a total of $720 on newspapers, magazines, and books, what amount did she spend on newspapers?
1. (1) Last year, the amount that Elena spent on magazines was 80 percent of the amount that she spent on books.
2. (2) Last year, the amount that Elena spent on newspapers was 60 percent of the total amount that she spent on magazines and books.
Arithmetic Percents
Let n, m, and b be the amounts, in dollars, that Elena spent last year on newspapers, magazines, and books, respectively. Given that 
, determine the value of n.
3. (1) Given that m is 80% of b, or 
, it follows from that 
, or 
. Since more than one positive value of b is possible, the value of n cannot be determined; NOT sufficient.
4. (2) Given that n is 60% of the sum of m and b, or 
, or 
, it follows from that 
, which can be solved to obtain a unique value of n; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
56. If p, q, x, y, and z are different positive integers, which of the five integers is the median?
1. (1) 
2. (2) 
Arithmetic Statistics
Since there are five different integers, there are two integers greater and two integers less than the median, which is the middle number.
3. (1) No information is given about the order of y and z with respect to the other three numbers; NOT sufficient.
4. (2) This statement does not relate y and z to the other three integers; NOT sufficient.
Because (1) and (2) taken together do not relate p, x, and q to y and z, it is impossible to tell which is the median. For example, if 
, , 
, 
, and 
, then the median is 8, but if , , , 
, and 
, then the median is 3.
The correct answer is E; both statements together are still not sufficient.
57. If 
, what is the value of wz?
1. (1) w and z are positive integers.
2. (2) w and z are consecutive odd integers.
Arithmetic Arithmetic operations
3. (1) The fact that w and z are both positive integers does not allow the values of w and z to be determined because, for example, if 
and 
, then 
, and if 
and 
, then 
; NOT sufficient.
4. (2) Since w and z are consecutive odd integers whose sum is 28, it is reasonable to consider the possibilities for the sum of consecutive odd integers: 
, 
, 
, 
, 
, 
, 
, 
. From this list it follows that only one pair of consecutive odd integers has 28 for its sum, and hence there is exactly one possible value for wz.
This problem can also be solved algebraically by letting the consecutive odd integers w and z be represented by 
and 
, where n can be any integer. Since 
, it follows that
|
|
|
|
|
simplify |
|
|
subtract 4 from both sides |
|
|
divide both sides by 4 |
Thus, 
, 
, and hence exactly one value can be determined for wz; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
58. What is the value of 
?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
3. (1) If 
then, when b is subtracted from both sides, the resultant equation is 
; SUFFICIENT.
4. (2) Since 
, either or 
. There is no further information available to determine a single numerical value of 
; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
59. Machine X runs at a constant rate and produces a lot consisting of 100 cans in 2 hours. How much less time would it take to produce the lot of cans if both Machines X and Y were run simultaneously?
1. (1) Both Machines X and Y produce the same number of cans per hour.
2. (2) It takes Machine X twice as long to produce the lot of cans as it takes Machines X and Y running simultaneously to produce the lot.
Arithmetic Rate problems
The problem states that the job is to produce 100 cans and that Machine X can do the job in 2 hours. Thus, to determine how much less time it would take for both of them running simultaneously to do the job, it is sufficient to know the rate for Machine Y or the time that Machines X and Y together take to complete the job.
3. (1) This states that the rate for Y is the same as the rate for X, which is given; SUFFICIENT.
4. (2) Since double the time corresponds to half the rate, the rate for X is 
the combined rate for X and Y running simultaneously, it can be determined that X and Y together would take the time, or 1 hour, to do the job; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
60. Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?
1. (1) 
2. (2) p is odd.
Arithmetic Properties of numbers
3. (1) This statement implies that p can be only among the integers 32, 33, 34, 35, and 36. Because each of these integers can be expressed as the product of two integers, each of which is greater than 1 (e.g., 
, etc.), the question can be answered even though the specific value of p is not known; SUFFICIENT.
4. (2) If , then p cannot be expressed as the product of two integers, each of which is greater than 1. However, if 
, then p can be expressed as the product of two integers, each of which is greater than 1; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
61. Is 
?
1. (1) 
2. (2) 
Algebra Inequalities
3. (1) This gives no information about x and its relationship to y; NOT sufficient.
4. (2) This gives no information about y and its relationship to x; NOT sufficient.
From (1) and (2) together, it can be determined only that z is less than both x and y. It is still not possible to determine the relationship of x and y, and x might be greater than, equal to, or less than y.
The correct answer is E; both statements together are still not sufficient.
62. If 
, is 
?
1. (1) 
2. (2) 
Algebra Fractions
Since 
and 
, it is to be determined whether 
.
3. (1) Given that a = 1, the equation to be investigated, 
, is 
. This equation can be true for some nonzero values of b and c (for example, 
) and false for other nonzero values of b and c (for example, 
and 
); NOT sufficient.
4. (2) Given that 
, the equation to be investigated, , is 
. This equation is true for all nonzero values of a and b; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
63. A certain list consists of 400 different numbers. Is the average (arithmetic mean) of the numbers in the list greater than the median of the numbers in the list?
1. (1) Of the numbers in the list, 280 are less than the average.
2. (2) Of the numbers in the list, 30 percent are greater than or equal to the average.
Arithmetic Statistics
In a list of 400 numbers, the median will be halfway between the 200th and the 201st numbers in the list when the numbers are ordered from least to greatest.
3. (1) This indicates that 280 of the 400 numbers in the list are less than the average of the 400 numbers. This means that both the 200th and the 201st numbers, as well as the median, are less than the average and, therefore, that the average is greater than the median; SUFFICIENT.
4. (2) This indicates that (0.3)(400) = 120 of the numbers are greater than or equal to the average. This means that the other 400 − 120 = 280 numbers are less than the average, which is the same as the information in (1); SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
64. What is the area of rectangular region R?
1. (1) Each diagonal of R has length 5.
2. (2) The perimeter of R is 14.
Geometry Rectangles
Let L and W be the length and width of the rectangle, respectively. Determine the value of LW.
3. (1) It is given that a diagonal”s length is 5. Thus, by the Pythagorean theorem, it follows that 
. The value of LW cannot be determined, however, because 
and 
satisfy 
with 
, and 
and 
satisfy with 
; NOT sufficient.
4. (2) It is given that 
, or 
, or 
. Therefore, 
, which can vary in value. For example, if 
and 
, then 
and 
. However, if 
and 
, then and 
; NOT sufficient.
Given (1) and (2) together, it follows from (2) that 
, or 
. Using (1), 25 can be substituted for 
to obtain 
, or 
, or . Alternatively, 
can be substituted forL in to obtain the quadratic equation 
, or 
, or 
, or 
. The left side of the last equation can be factored to give 
. Therefore, 
, which gives 
and 
, or 
, which gives 
and 
. Since in either case, a unique value for LW can be determined.
The correct answer is C; both statements together are sufficient.
65. If Q is an integer between 10 and 100, what is the value of Q?
1. (1) One of Q”s digits is 3 more than the other, and the sum of its digits is 9.
2. (2) 
Algebra Properties of numbers
3. (1) While it is quite possible to guess that the two integers satisfying these stipulations are 36 and 63, these two integers can also be determined algebraically. Letting x and y be the digits of Q, the given information can be expressed as 
and 
. These equations can be solved simultaneously to obtain the digits 3 and 6, leading to the integers 36 and 63. However, it is unknown which of these two integers is the value of Q; NOT sufficient.
4. (2) There is more than one integer between 10 and 49; NOT sufficient.
When the information from (1) and (2) is combined, the value of Q can be uniquely determined, because, of the two possible values for Q, only 36 is between 10 and 49.
The correct answer is C; both statements together are sufficient.
66. If p and q are positive integers and 
, what is the value of p?
1. (1) 
is an integer.
2. (2) 
is an integer.
Arithmetic Arithmetic operations
There are four pairs of positive integers whose product is 24: 1 and 24, 2 and 12, 3 and 8, and 4 and 6.
3. (1) The possible values of q are therefore 6, 12, and 24, and for each of these there is a different value of p (4, 2, and 1); NOT sufficient.
4. (2) The possible values of p are therefore 2, 4, 6, 8, 12, and 24; NOT sufficient.
From (1) and (2) together, the possible values of q can only be narrowed down to 6 or 12, with corresponding values of p being either 4 or 2.
The correct answer is E; both statements together are still not sufficient.
67. What is the value of 
?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
3. (1) If 
, then 
. When this expression for x is substituted in 
, the result is 
, which can vary in value. For example, if (and hence, ), then 
. However, if (and hence, ), then 
; NOT sufficient.
4. (2) Since 
, 
, or 
. Thus, the value of is 1; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
68. How many integers n are there such that 
?
1. (1) 
2. (2) r and s are not integers.
Arithmetic Properties of numbers
3. (1) The difference between s and r is 5. If r and s are integers (e.g., 7 and 12), the number of integers between them (i.e., n could be 8, 9, 10, or 11) is 4. If r and s are not integers (e.g., 6.5 and 11.5), then the number of integers between them (i.e., n could be 7, 8, 9, 10, or 11) is 5. No information is given that allows a determination of whether s and r are integers; NOT sufficient.
4. (2) No information is given about the difference between r and s. If 
and 
, then r and s have no integers between them. However, if and s = 3.5, then r and s have 3 integers between them; NOT sufficient.
Using the information from both (1) and (2), it can be determined that, because r and s are not integers, there are 5 integers between them.
The correct answer is C; both statements together are sufficient.
69. If the total price of n equally priced shares of a certain stock was $12,000, what was the price per share of the stock?
1. (1) If the price per share of the stock had been $1 more, the total price of the n shares would have been $300 more.
2. (2) If the price per share of the stock had been $2 less, the total price of the n shares would have been 5 percent less.
Arithmetic Arithmetic operations; Percents
Since the price per share of the stock can be expressed as 
, determining the value of n is sufficient to answer this question.
3. (1) A per-share increase of $1 and a total increase of $300 for n shares of stock mean together that 
. It follows that 
; SUFFICIENT.
4. (2) If the price of each of the n shares had been reduced by $2, the total reduction in price would have been 5 percent less or 0.05($12,000). The equation 
expresses this relationship. The value of n can be determined to be 300 from this equation; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
70. If n is positive, is 
?
1. (1) 
2. (2) 
Algebra Radicals
Determine if 
, or equivalently, if 
.
3. (1) Given that 
, or equivalently, 
, it follows from
1. that 
is equivalent to 
, or 
. Since 
allows for values of n that are greater than 10,000 and 
allows for values of n that are not greater than 10,000, it cannot be determined if 
; NOT sufficient.
4. (2) Given that 
, or equivalently, 
, it follows from
1. that 
is equivalent to 
, or 
. Since 
, it can be determined that ; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
71. Is 
?
1. (1) 
and 
.
2. (2) 
Algebra Inequalities
3. (1) While it is known that 
and 
, xy could be 
, which is greater than 5, or xy could be 
, which is not greater than 5; NOT sufficient.
4. (2) Given that 
, xy could be 6 (when and 
), which is greater than 5, and xy could be 4 (when and 
), which is not greater than 5; NOT sufficient.
Both (1) and (2) together are not sufficient since the two examples given in (2) are consistent with both statements.
The correct answer is E; both statements together are still not sufficient.
72. In Year X, 8.7 percent of the men in the labor force were unemployed in June compared with 8.4 percent in May. If the number of men in the labor force was the same for both months, how many men were unemployed in June of that year?
1. (1) In May of Year X, the number of unemployed men in the labor force was 3.36 million.
2. (2) In Year X, 120,000 more men in the labor force were unemployed in June than in May.
Arithmetic Percents
Since 8.7 percent of the men in the labor force were unemployed in June, the number of unemployed men could be calculated if the total number of men in the labor force was known. Let t represent the total number of men in the labor force.
3. (1) This implies that for May 
, from which the value of t can be determined; SUFFICIENT.
4. (2) This implies that 
or 
. This equation can be solved for t; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
73. If 
, what is the value of 
?
1. (1) 
2. (2) 
Arithmetic; Algebra Arithmetic operations; Simplifying expressions
3. (1) Since 
, it follows that 
; SUFFICIENT.
4. (2) Since 
(and, therefore, 
) and the values of p or q are unknown, the value of the expression 
cannot be determined; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
74. On Monday morning a certain machine ran continuously at a uniform rate to fill a production order. At what time did it completely fill the order that morning?
1. (1) The machine began filling the order at 9:30 a.m.
2. (2) The machine had filled 
of the order by 10:30 a.m. and 
of the order by 11:10 a.m.
Arithmetic Arithmetic operations
3. (1) This merely states what time the machine began filling the order; NOT sufficient.
4. (2) In the 40 minutes between 10:30 a.m. and 11:10 a.m., 
of the order was filled. Therefore, the entire order was completely filled in 
minutes, or 2 hours. Since half the order took 1 hour and was filled by 10:30 a.m., the second half of the order, and thus the entire order, was filled by 11:30 a.m.; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
75. If 
, is 
?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
The equation 
can be manipulated to obtain the following equivalent equations:
|
|
multiply both sides by m |
|
|
remove parentheses |
|
|
subtract xm2 |
3. (1) When cross multiplied, 
becomes 
, or 
when both sides are then multiplied by k. Thus, the equation 
is equivalent to the equation 
, and hence equivalent to the equation 
, which can be true or false, depending on the values of x, n, m, and y; NOT sufficient.
4. (2) When cross multiplied, 
becomes , or when both sides are then multiplied by n. Thus, the equation is equivalent to the equation , and hence equivalent to the equation , which can be true or false, depending on the values of x, k, m, and z; NOT sufficient.
Combining the information in both (1) and (2), it follows from (1) that is equivalent to , which is true by (2).
The correct answer is C; both statements together are sufficient.
76. What is the radius of the circle above with center O?
1. (1) The ratio of OP to PQ is 1 to 2.
2. (2) P is the midpoint of chord AB.
Geometry Circles
3. (1) It can be concluded only that the radius is 3 times the length of OP, which is unknown; NOT sufficient.
4. (2) It can be concluded only that 
, and the chord is irrelevant to the radius; NOT sufficient.
Together, (1) and (2) do not give the length of any line segment shown in the circle. In fact, if the circle and all the line segments were uniformly expanded by a factor of, say, 5, the resulting circle and line segments would still satisfy both (1) and (2). Therefore, the radius of the circle cannot be determined from (1) and (2) together.
The correct answer is E; both statements together are still not sufficient.
77. What is the number of 360-degree rotations that a bicycle wheel made while rolling 100 meters in a straight line without slipping?
1. (1) The diameter of the bicycle wheel, including the tire, was 0.5 meter.
2. (2) The wheel made twenty 360-degree rotations per minute.
Geometry Circles
For each 360-degree rotation, the wheel has traveled a distance equal to its circumference. Given either the circumference of the wheel or the means to calculate its circumference, it is thus possible to determine the number of times the circumference of the wheel was laid out along the straight-line path of 100 meters.
3. (1) The circumference of the bicycle wheel can be determined from the given diameter using the equation 
, where 
the diameter; SUFFICIENT.
4. (2) The speed of the rotations is irrelevant, and no dimensions of the wheel are given; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
78. If 
, what is the value of t?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
Because 
, the value of t can be determined exactly when the value of xn can be determined.
3. (1) Given that 
, more than one value of xn is possible. For example, xn could be 0 (if 
and 
) and xn could be 4 (if 
and 
); NOT sufficient.
4. (2) Given that 
, or 
, more than one value of xn is possible, since 
, which will vary in value when n varies in value; NOT sufficient.
The value of x determined from equation (2) can be substituted in equation (1) to obtain 
, or 
. Therefore, 
.
The correct answer is C; both statements together are sufficient.
79. In the equation 
, x is a variable and b is a constant. What is the value of b?
1. (1) 
is a factor of 
.
2. (2) 4 is a root of the equation .
Algebra First- and second-degree equations
3. (1) Method 1: If 
is a factor, then 
for some constant c. Equating the constant terms (or substituting 
), it follows that 
, or 
. Therefore, the quadratic polynomial is 
, which is equal to 
, and hence 
.
1. Method 2: If 
is a factor of 
, then 3 is a root of 
. Therefore, 
, which can be solved to get 
.
2. Method 3: The value of b can be found by long division:
3. These calculations show that the remainder is 
. Since the remainder must be 0, it follows that 
, or ; SUFFICIENT.
4. (2) If 4 is a root of the equation, then 4 can be substituted for x in the equation 
, yielding 
. This last equation can be solved to obtain a unique value for b; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
80. A Town T has 20,000 residents, 60 percent of whom are female. What percent of the residents were born in Town T?
1. (1) The number of female residents who were born in Town T is twice the number of male residents who were not born in Town T.
2. (2) The number of female residents who were not born in Town T is twice the number of female residents who were born in Town T.
Arithmetic Percents
Since 60 percent of the residents are female, there are 
female residents. The remaining residents are male, so there are 
male residents. Let N be the number of residents who were born in Town T. The percent of the residents who were born in Town T is 
, which can be determined exactly when N can be determined.
This information is displayed in the following table:
|
Table 1 |
|||
|
Male |
Female |
Total |
|
|
Born in Town T |
N |
||
|
Not Born in Town T |
|||
|
Total |
8,000 |
12,000 |
20,000 |
3. (1) Let x represent the number of male residents who were not born in Town T. Then the number of female residents who were born in Town T is 2x. Adding this information to Table 1 gives
|
Table 2 |
|||
|
Male |
Female |
Total |
|
|
Born in Town T |
2x |
N |
|
|
Not Born in Town T |
x |
||
|
Total |
8,000 |
12,000 |
20,000 |
1. Other cells in the table can then be filled in as shown below.
|
Table 3 |
|||
|
Male |
Female |
Total |
|
|
Born in Town T |
|
2x |
N |
|
Not Born in Town T |
x |
|
|
|
Total |
8,000 |
12,000 |
20,000 |
1. Then, it can be seen from Table 3 that 
and also that 
. However, without a value for x, the value of N cannot be determined; NOT sufficient.
4. (2) Let y represent the number of female residents who were born in Town T. Then the number of female residents who were not born in Town T is 2y. Adding this information to Table 1 gives
|
Table 4 |
|||
|
Male |
Female |
Total |
|
|
Born in Town T |
y |
N |
|
|
Not Born in Town T |
2y |
||
|
Total |
8,000 |
12,000 |
20,000 |
1. From Table 4 it can be seen that 
, so 
and 
. With this value, the table can be expanded to
|
Table 5 |
|||
|
Male |
Female |
Total |
|
|
Born in Town T |
4,000 |
N |
|
|
Not Born in Town T |
8,000 |
||
|
Total |
8,000 |
12,000 |
20,000 |
2. However, there is not enough information to determine the value of N; NOT sufficient.
Given (1) and (2) together, the information from Table 3, which uses the information given in (1), can be combined with Table 5, which uses the information given in (2), to obtain 
. Therefore, 
and thus 
.
The correct answer is C; both statements together are sufficient.
81. In 
, what is the length of YZ?
1. (1) The length of XY is 3.
2. (2) The length of XZ is 5.
Geometry Triangles
Given the length of one side of a triangle, it is known that the sum of the lengths of the other two sides is greater than that given length. The length of either of the other two sides, however, can be any positive number.
3. (1) Only the length of one side, XY, is given, and that is not enough to determine the length of YZ; NOT sufficient.
4. (2) Again, only the length of one side, XZ, is given and that is not enough to determine the length of YZ; NOT sufficient.
Even by using the triangle inequality stated above, only a range of values for YZ can be determined from (1) and (2). If the length of side YZ is represented by k, then it is known both that 
and that 
, or 
. Combining these inequalities to determine the length of k yields only that 
.
The correct answer is E; both statements together are still not sufficient.
82. If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?
1. (1) The range of the n integers is 14.
2. (2) The greatest of the n integers is 17.
Arithmetic Statistics
Let k be the least of the n consecutive odd integers. Then the n consecutive odd integers are k, 
, 
, . . . , 
, where 
is the greatest of the n consecutive odd integers and 
is the range of the nconsecutive odd integers. Determine the value of k.
3. (1) Given that the range of the odd integers is 14, it follows that 
, or 
, or 
. It is also given that the average of the 8 consecutive odd integers is 10, and so 
, from which a unique value for k can be determined; SUFFICIENT.
4. (2) Given that the greatest of the odd integers is 17, it follows that the n consecutive odd integers can be expressed as 17, 
, 
, . . . , 
. Since the average of the n consecutive odd integers is 10, then
,
1. or
(i)
The n consecutive odd integers can also be expressed as k, 
, , . . . , 
. Since the average of the n consecutive odd integers is 10, then
,
or
(ii)
Adding equations (i) and (ii) gives
Alternatively, because the numbers are consecutive odd integers, they form a data set that is symmetric about its average, and so the average of the numbers is the average of the least and greatest numbers. Therefore, 
, from which a unique value for k can be determined; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
83. What was the ratio of the number of cars to the number of trucks produced by Company X last year?
1. (1) Last year, if the number of cars produced by Company X had been 8 percent greater, the number of cars produced would have been 150 percent of the number of trucks produced by Company X.
2. (2) Last year Company X produced 565,000 cars and 406,800 trucks.
Arithmetic Ratio; Percents
Let c equal the number of cars and t the number of trucks produced by Company X last year. The ratio of cars to trucks produced last year can be expressed as 
.
3. (1) An 8 percent increase in the number of cars produced can be expressed as 108 percent of c, or 1.08c. Similarly, 150 percent of the number of trucks produced can be expressed as 1.5t. The relationship between the two can be expressed in the equation 
. From this:
|
|
divide both sides by t |
|
|
divide both sides by 1.08 |
4. Thus the ratio of cars to trucks produced last year can be determined; SUFFICIENT.
5. (2) The values of c and t are given; so the ratio can be determined; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
84. If x, y, and z are positive numbers, is 
?
1. (1) 
2. (2) 
Algebra Inequalities
3. (1) Dividing both sides of the inequality by z yields 
. However, there is no information relating z to either x or y; NOT sufficient.
4. (2) Dividing both sides of the inequality by y yields only that 
, with no further information relating y to either x or z; NOT sufficient.
From (1) and (2) it can be determined that x is greater than both y and z. Since it still cannot be determined which of y or z is the least, the correct ordering of the three numbers also cannot be determined.
The correct answer is E; both statements together are still not sufficient.
85. K is a set of numbers such that
1. (i) if x is in K, then −x is in K, and
2. (ii) if each of x and y is in K, then xy is in K.
Is 12 in K?
3. (1) 2 is in K.
4. (2) 3 is in K.
Arithmetic Properties of numbers
5. (1) Given that 2 is in K, it follows that K could be the set of all real numbers, which contains 12. However, if K is the set {. . . , −16, −8, −4, −2, 2, 4, 8, 16, . . .}, then K contains 2 and K satisfies both (i) and (ii), but K does not contain 12. To see that K satisfies (ii), note that K can be written as {. . . , −24, −23, −22, −21, 21, 22, 23, 24, . . .}, and thus a verification of (ii) can reduce to verifying that the sum of two positive integer exponents is a positive integer exponent; NOT sufficient.
6. (2) Given that 3 is in K, it follows that K could be the set of all real numbers, which contains 12. However, if K is the set {. . . , −81, −27, −9, −3, 3, 9, 27, 81, . . .}, then K contains 3 and K satisfies both (i) and (ii), but K does not contain 12. To see that K satisfies (ii), note that K can be written as {. . . , −34, −33, −32, −31, 31, 32, 33, 34, . . .}, and thus a verification of (ii) can reduce to verifying that the sum of two positive integer exponents is a positive integer exponent; NOT sufficient.
Given (1) and (2), it follows that both 2 and 3 are in K. Thus, by (ii), 
is in K. Therefore, by (ii), 
is in K.
The correct answer is C; both statements together are sufficient.
86. If 
, what is the value of 
?
1. (1) 
2. (2) 
Algebra Simplifying algebraic expressions
Since 
and it is given that 
, it follows that 
.
Therefore, the value of 
can be determined if and only if the value of xy can be determined.
3. (1) Since the value of xy is given, the value of can be determined; SUFFICIENT.
4. (2) Given only that 
, it is not possible to determine the value of xy. Therefore, the value of cannot be determined; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
87. If x, y, and z are numbers, is 
?
1. (1) The average (arithmetic mean) of x, y, and z is 6.
2. (2) 
Arithmetic Statistics
3. (1) From this, it is known that 
4. or, when both sides are multiplied by 3, 
.
1. Since nothing is known about the value of 
, no conclusion can be drawn about the value of z; NOT sufficient.
5. (2) This implies that 
but gives no further information about the values of x, y, and z; NOT sufficient.
Taking (1) and (2) together is sufficient since 0 can be substituted for 
in the equation to yield 
.
The correct answer is C; both statements together are sufficient.
88. After winning 50 percent of the first 20 games it played, Team A won all of the remaining games it played. What was the total number of games that Team A won?
1. (1) Team A played 25 games altogether.
2. (2) Team A won 60 percent of all the games it played.
Arithmetic Percents
Let r be the number of the remaining games played, all of which the team won. Since the team won 
of the first 20 games and the r remaining games, the total number of games the team won is 
. Also, the total number of games the team played is 
. Determine the value of r.
3. (1) Given that the total number of games played is 25, it follows that 
, or 
; SUFFICIENT.
4. (2) It is given that the total number of games won is 
, which can be expanded as 
. Since it is also known that the number of games won is , it follows that 
. Solving this equation gives 
, or 
, or ; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
89. Is x between 0 and 1?
1. (1) x2 is less than x.
2. (2) x3 is positive.
Arithmetic Arithmetic operations
3. (1) Since x 2 is always positive, it follows that here x must also be positive, that is, greater than 0. Furthermore, if x is greater than 1, then x 2 is greater than x. If 
or 1, then 
. Therefore, x must be between 0 and 1; SUFFICIENT.
4. (2) If x 3 is positive, then x is positive, but x can be any positive number; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
90. Is p2 an odd integer?
1. (1) p is an odd integer.
2. (2) 
is an odd integer.
Arithmetic Properties of numbers
The product of two or more odd integers is always odd.
3. (1) Since p is an odd integer, 
is an odd integer; SUFFICIENT.
4. (2) If 
is an odd integer, then 
is an odd integer. Therefore, is also an odd integer; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
91. If m and n are nonzero integers, is mn an integer?
1. (1) nm is positive.
2. (2) nm is an integer.
Arithmetic Properties of numbers
It is useful to note that if 
and 
, then 
, and therefore mn will not be an integer. For example, if 
and 
, then 
.
3. (1) Although it is given that nm is positive, mn can be an integer or mn can fail to be an integer. For example, if 
and 
, then 
is positive and 
is an integer. However, if and , then 
is positive and 
is not an integer; NOT sufficient.
4. (2) Although it is given that nm is an integer, mn can be an integer or mn can fail to be an integer. For example, if and , then 
is an integer and 
is an integer. However, if and , then 
is an integer and 
is not an integer; NOT sufficient.
Taking (1) and (2) together, it is still not possible to determine if mn is an integer, since the same examples are used in both (1) and (2) above.
The correct answer is E; both statements together are still not sufficient.
92. What is the value of xy?
1. (1) 
2. (2) 
Algebra First- and second-degree equations; Simultaneous equations
3. (1) Given 
, or 
, it follows that 
, which does not have a unique value. For example, if , then 
, but if 
, then 
; NOT sufficient.
4. (2) Given 
, or 
, it follows that 
, which does not have a unique value. For example, if , then 
, but if , then 
; NOT sufficient.
Using (1) and (2) together, the two equations can be solved simultaneously for x and y. One way to do this is by adding the two equations, and , to get 
, or 
. Then substitute 
into either of the equations to obtain an equation that can be solved to get 
. Thus, xy can be determined to have the value 
. Alternatively, the two equations correspond to a pair of nonparallel lines in the (x, y) coordinate plane, which have a unique point in common.
The correct answer is C; both statements together are sufficient.
93. Is x2 greater than x?
1. (1) x2 is greater than 1.
2. (2) x is greater than −1.
Arithmetic; Algebra Exponents; Inequalities
3. (1) Given 
, it follows that either 
or 
. If , then multiplying both sides of the inequality by the positive number x gives 
. On the other hand, if 
, then x is negative and x2 is positive (because ), which also gives 
; SUFFICIENT.
4. (2) Given 
, x2 can be greater than x (for example, 
) and x2 can fail to be greater than x (for example, ); NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
94. Michael arranged all his books in a bookcase with 10 books on each shelf and no books left over. After Michael acquired 10 additional books, he arranged all his books in a new bookcase with 12 books on each shelf and no books left over. How many books did Michael have before he acquired the 10 additional books?
1. (1) Before Michael acquired the 10 additional books, he had fewer than 96 books.
2. (2) Before Michael acquired the 10 additional books, he had more than 24 books.
Arithmetic Properties of numbers
If x is the number of books Michael had before he acquired the 10 additional books, then x is a multiple of 10. After Michael acquired the 10 additional books, he had 
books and is a multiple of 12.
3. (1) If 
, where x is a multiple of 10, then 
, 20, 30, 40, 50, 60, 70, 80, or 90 and 
, 30, 40, 50, 60, 70, 80, 90, or 100. Since is a multiple of 12, then 
and 
; SUFFICIENT.
4. (2) If 
, where x is a multiple of 10, then x must be one of the numbers 30, 40, 50, 60, 70, 80, 90, 100, 110, . . . , and must be one of the numbers 40, 50, 60, 70, 80, 90, 100, 110, 120, . . . . Since there is more than one multiple of 12 among these numbers (for example, 60 and 120), the value of , and therefore the value of x, cannot be determined; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
95. If 
, does 
?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
By expanding the product 
, the question is equivalent to whether 
, or 
, when 
.
3. (1) If 
, then , and hence by the remarks above, 
; SUFFICIENT.
4. (2) If 
, then 
can be true 
and can be false 
; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
96. Last year in a group of 30 businesses, 21 reported a net profit and 15 had investments in foreign markets. How many of the businesses did not report a net profit nor invest in foreign markets last year?
1. (1) Last year 12 of the 30 businesses reported a net profit and had investments in foreign markets.
2. (2) Last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both.
Arithmetic Concepts of sets
Consider the Venn diagram below in which x represents the number of businesses that reported a net profit and had investments in foreign markets. Since 21 businesses reported a net profit, 
businesses reported a net profit only. Since 15 businesses had investments in foreign markets, 
businesses had investments in foreign markets only. Finally, since there is a total of 30 businesses, the number of businesses that did not report a net profit and did not invest in foreign markets is 
.
Determine the value of 
, or equivalently, the value of x.
3. (1) It is given that 
; SUFFICIENT.
4. (2) It is given that 
. Therefore, 
, or 
.
1. Alternatively, the information given is exactly the number of businesses that are not among those to be counted in answering the question posed in the problem, and therefore the number of businesses that are to be counted is 
; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
97. If k and n are integers, is n divisible by 7?
1. (1) 
2. (2) 
is divisible by 7.
Arithmetic Properties of numbers
3. (1) This is equivalent to the equation 
. By picking various integers to be the value of k, it can be shown that for some values of k (e.g., 
), 
is divisible by 7, and for some other values of k (e.g., 
), is not divisible by 7; NOT sufficient.
4. (2) While 
is divisible by 7, this imposes no constraints on the integer n, and therefore n could be divisible by 7 (e.g., 
) and n could be not divisible by 7 (e.g., 
); NOT sufficient.
Applying both (1) and (2), it is possible to answer the question. From (1), it follows that 
can be substituted for 2k. Carrying this out in (2), it follows that 
, or 
, is divisible by 7. This means that 
for some integer q. It follows that
, and so n is divisible by 7.
The correct answer is C; both statements together are sufficient.
98. Is the perimeter of square S greater than the perimeter of equilateral triangle T?
1. (1) The ratio of the length of a side of S to the length of a side of T is 4:5.
2. (2) The sum of the lengths of a side of S and a side of T is 18.
Geometry Perimeter
Letting s and t be the side lengths of square S and triangle T, respectively, the task is to determine if 
, which is equivalent (divide both sides by 4t) to determining if 
.
3. (1) It is given that 
. Since 
, it follows that ; SUFFICIENT.
4. (2) Many possible pairs of numbers have the sum of 18. For some of these (s, t) pairs it is the case that 
(for example, 
), and for others of these pairs it is not the case that (for example, 
and 
); NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
99. If 
, is 
?
1. (1) 
2. (2) 
Algebra Inequalities
3. (1) The inequality 
gives 
. Adding this last inequality to the given inequality, 
, gives 
, or 
, which suggests that (1) is not sufficient. Indeed, z could be 2 (
and 
satisfy both and ), which is greater than 1, and z could be 
(
and 
satisfy both and ), which is not greater than 1; NOT sufficient.
4. (2) It follows from the inequality that 
. It is given that 
, or 
, or 
. Therefore, 
and 
, from which it follows that 
; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
100. Can the positive integer n be written as the sum of two different positive prime numbers?
1. (1) n is greater than 3.
2. (2) n is odd.
Arithmetic Properties of numbers
The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, etc., that is, those integers 
whose only positive factors are 1 and p.
3. (1) If 
, then n can be written as the sum of two different primes 
. If 
, however, then n cannot be written as the sum of two different primes. (Note that while 
, neither of these sums satisfies both requirements of the question.) This value of n does not allow an answer to be determined; NOT sufficient.
4. (2) While some odd integers can be written as the sum of two different primes (e.g., 
), others cannot (e.g., 11). This value of n does not allow an answer to be determined; NOT sufficient.
Since the sum of two odd integers is always even, for an odd integer greater than 3 to be the sum of two prime numbers, one of those prime numbers must be an even number. The only even prime number is 2. Thus, the only odd integers that can be expressed as the sum of two different prime numbers are those for which 
is an odd prime number. Using the example of 11 (an odd integer greater than 3), 
, which is not a prime number. Statements (1) and (2) together do not define n well enough to determine the answer.
The correct answer is E; both statements together are still not sufficient.
101. In the figure above, segments RS and TU represent two positions of the same ladder leaning against the side SV of a wall. The length of TV is how much greater than the length of RV?
1. (1) The length of TU is 10 meters.
2. (2) The length of RV is 5 meters.
Geometry Triangles
The Pythagorean theorem 
can be applied here. Since the triangle TUV is a 
triangle, the lengths of the sides are in the ratio 
; so the length of any one side determines the length of the other two sides. Similarly, the triangle RSV is a 
triangle with the lengths of the sides in the ratio 
; so the length of any one side determines the length of the other two sides. Also, the length of the hypotenuse is the same in both triangles, because it is the length of the ladder. Hence, the length of any one side of either triangle determines the lengths of all sides of both triangles.
3. (1) Since the length of one side of triangle TUV is given, the length of any side of either triangle can be found. Therefore, the difference between TV and RV can also be found; SUFFICIENT.
4. (2) Since the length of one side of triangle RSV is given, the length of any side of either triangle can be found. Therefore, the difference between TV and RV can also be found; SUFFICIENT.
The correct answer is D; both statements alone are sufficient.
|
Cancellation Fees |
|
|
Days Prior to Departure |
Percent of Package Price |
|
46 or more |
10% |
|
45−31 |
35% |
|
30−16 |
50% |
|
15−5 |
65% |
|
4 or fewer |
100% |
102. The table above shows the cancellation fee schedule that a travel agency uses to determine the fee charged to a tourist who cancels a trip prior to departure. If a tourist canceled a trip with a package price of $1,700 and a departure date of September 4, on what day was the trip canceled?
1. (1) The cancellation fee was $595.
2. (2) If the trip had been canceled one day later, the cancellation fee would have been $255 more.
Arithmetic Percents
3. (1) The cancellation fee given is 
of the package price, which is the percent charged for cancellation 45−31 days prior to the departure date of September 4. However, there is no further information to determine exactly when within this interval the trip was cancelled; NOT sufficient.
4. (2) This implies that the increase in the cancellation fee for canceling one day later would have been 
of the package price. The cancellation could thus have occurred either 31 days or 16 days prior to the departure date of September 4 because the cancellation fee would have increased by that percentage either 30 days before departure or 15 days before departure. However, there is no further information to establish whether the interval before departure was 31 days or 16 days; NOT sufficient.
Taking (1) and (2) together establishes that the trip was canceled 31 days prior to September 4.
The correct answer is C; both statements together are sufficient.
103. If P and Q are each circular regions, what is the radius of the larger of these regions?
1. (1) The area of P plus the area of Q is equal to 
.
2. (2) The larger circular region has a radius that is 3 times the radius of the smaller circular region.
Geometry Circles
The area of a circle with a radius of r is equal to 
. For this problem, let r represent the radius of the smaller circular region, and let R represent the radius of the larger circular region.
3. (1) This can be expressed as 
. Dividing both sides of the equation by π gives 
, but this is not nough information to determine R; NOT sufficient.
4. (2) This can be expressed as 
, which by itself is not enough to determine R; NOT sufficient.
Using (1) and (2), the value of R, or the radius of the larger circular region, can be determined. In (2), , and thus 
. Therefore, 
can be substituted for r in the equation 
from (1). The result is the equation 
that can be solved for a unique value of R2, and thus for a unique positive value of R. Remember that it is only necessary to establish the sufficiency of the data; there is no need to actually find the value of R.
The correct answer is C; both statements together are sufficient.
104. For all z, 
denotes the least integer greater than or equal to z. Is 
?
1. (1) 
2. (2) 
Algebra Operations with real numbers
Determining if 
is equivalent to determining if 
. This can be inferred by examining a few representative examples, such as 
, 
, 
, 
, 
, and 
.
3. (1) Given 
, it follows that 
, since 
represents all numbers x that satisfy 
along with all numbers x that satisfy 
; SUFFICIENT.
4. (2) Given 
, it follows from the same reasoning used just before (1) above that this equality is equivalent to 
, which in turn is equivalent to 
. Since from among these values of x it is possible for 
to be true (for example, 
) and it is possible for to be false (for example, 
), it cannot be determined if 
; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
105. The circular base of an above-ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool”s base is point A?
1. (1) The base has area 250 square feet.
2. (2) The center of the base is 20 feet from point C.
Geometry Circles
Let Q be the center of the pool”s base and r be the distance from Q to A, as shown in the figure below.
Since A is a point on the circular base, QA is a radius (r) of the base.
3. (1) Since the formula for the area of a circle is area = πr2, this information can be stated as 
or 
; SUFFICIENT.
4. (2) Since 
is tangent to the base, 
is a right triangle. It is given that 
, but there is not enough information to use the Pythagorean theorem to determine the length of 
; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
106. If 
, what is the value of 
?
1. (1) 
2. (2) 
Algebra First- and second-degree equations
By substituting −6 as the value of xy, the question can be simplified to “What is the value of 
?”
3. (1) Adding y to both sides of 
gives 
. When 
is substituted for x in the equation 
, the equation yields 
, or 
. Factoring the left side of this equation gives 
. Thus, y may have a value of 
. Since a unique value of y is not determined, neither the value of x nor the value of xy can be determined; NOT sufficient.
4. (2) Since 
and 
, it follows that 
. When −6 is substituted for xy, this equation yields 
, and hence 
. Since and , it follows that 
, or . Therefore, the value of 
, and hence the value of 
can be determined; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
107. If the average (arithmetic mean) of 4 numbers is 50, how many of the numbers are greater than 50?
1. (1) None of the four numbers is equal to 50.
2. (2) Two of the numbers are equal to 25.
Arithmetic Statistics
Let w, x, y, and z be the four numbers. The average of these 4 numbers can be represented by the following equation:
.
3. (1) The only information about the 4 numbers is that none of the numbers is equal to 50. The 4 numbers could be 25, 25, 26, and 124, which have an average of 50, and only 1 of the numbers would be greater than 50. The 4 numbers could also be 25, 25, 75, and 75, which have an average of 50, and 2 of the numbers would be greater than 50; NOT sufficient.
4. (2) Each of the examples in (1) has exactly 2 numbers equal to 25; NOT sufficient.
Taking (1) and (2) together, the examples in (1) also illustrate the insufficiency of (2). Thus, there is more than one possibility for how many numbers are greater than 50.
The correct answer is E; both statements together are still not sufficient.
108. [y] denotes the greatest integer less than or equal to y. Is 
?
1. (1) 
2. (2) 
Algebra Operations with real numbers
3. (1) It is given 
. If y is an integer, then 
, and thus 
, which is less than 1. If y is not an integer, then y lies between two consecutive integers, the smaller of which is equal to [y]. Since each of these two consecutive integers is at a distance of less than 1 from y, it follows that [y] is at a distance of less than 1 from y, or 
. Thus, regardless of whether y is an integer or y is not an integer, it can be determined that 
; SUFFICIENT.
4. (2) It is given that 
, which is equivalent to 
. This can be inferred by examining a few representative examples, such as 
, 
, 
, 
, and 
. From , it follows that 
; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
109. If m is a positive integer, then m3 has how many digits?
1. (1) m has 3 digits.
2. (2) m2 has 5 digits.
Arithmetic Properties of numbers
3. (1) Given that m has 3 digits, then m could be 100 and 
would have 7 digits, or m could be 300 and 
would have 8 digits; NOT sufficient.
4. (2) Given that m2 has 5 digits, then m could be 100 (because 
has 5 digits) or m could be 300 (because 
has 5 digits). In the former case, has 7 digits and in the latter case, has 8 digits; NOT sufficient.
Given (1) and (2), it is still possible for m to be 100 or for m to be 300, and thus m3 could have 7 digits or m3 could have 8 digits.
The correct answer is E; both statements together are still not sufficient.
110. For each landscaping job that takes more than 4 hours, a certain contractor charges a total of r dollars for the first 4 hours plus 0.2r dollars for each additional hour or fraction of an hour, where r > 100. Did a particular landscaping job take more than 10 hours?
1. (1) The contractor charged a total of $288 for the job.
2. (2) The contractor charged a total of 2.4r dollars for the job.
Algebra Applied problems
If y represents the total number of hours the particular landscaping job took, determine if y > 10.
3. (1) This indicates that the total charge for the job was $288, which means that r + 0.2r(y − 4) = 288. From this it cannot be determined if y > 10. For example, if r = 120 and y = 11, then 120 + 0.2(120)(7) = 288, and the job took more than 10 hours. However, if r = 160 and y = 8, then 160 + 0.2(160)(4) = 288, and the job took less than 10 hours; NOT sufficient.
4. (2) This indicates that r + 0.2r(y − 4) = 2.4r, from which it follows that
|
r + 0.2ry − 0.8r = 2.4r |
use distributive property |
|
0.2ry = 2.2r |
subtract (r − 0.8r) from both sides |
|
y = 11 |
divide both sides by 0.2r |
Therefore, the job took more than 10 hours; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
111. The sequence s1, s2, s3, . . . , sn, . . . is such that sn 
for all integers 
. If k is a positive integer, is the sum of the first k terms of the sequence greater than 
?
1. (1) 
2. (2) 
Arithmetic Sequences
The sum of the first k terms can be written as
.
Therefore, the sum of the first k terms is greater than 
if and only if 
, or 
, or 
. Multiplying both sides of the last inequality by 
gives the equivalent condition 
, or 
.
3. (1) Given that 
, then it follows that 
; SUFFICIENT.
4. (2) Given that 
, it is possible to have (for example, 
) and it is possible to not have (for example, 
); NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
112. If x and y are nonzero integers, is 
?
1. (1) 
2. (2) 
Arithmetic; Algebra Arithmetic operations; Inequalities
It is helpful to note that 
.
3. (1) Given 
, then 
and 
. Compare xy to yx by comparing y2y to yy2 or, when the base y is greater than 1, by comparing the exponents 2y and y2. If 
, then 
is less than 
, and hence xy would be less thanyx. However, if 
, then 
is not less than 
, and hence xy would not be less than yx; NOT sufficient.
4. (2) It is known that 
, but no information about x is given. For example, let . If , then 
is less than 
, but if , then 
is not less than 
; NOT sufficient.
If both (1) and (2) are taken together, then from (1) 2y is compared to y2 and from (2) it is known that . Since 
when , it follows that 
.
The correct answer is C; both statements together are sufficient.
113. In the sequence S of numbers, each term after the first two terms is the sum of the two immediately preceding terms. What is the 5th term of S?
1. (1) The 6th term of S minus the 4th term equals 5.
2. (2) The 6th term of S plus the 7th term equals 21.
Arithmetic Sequences
If the first two terms of sequence S are a and b, then the remaining terms of sequence S can be expressed in terms of a and b as follows.
|
n |
nth term of sequence S |
|
1 |
a |
|
2 |
b |
|
3 |
|
|
4 |
|
|
5 |
|
|
6 |
|
|
7 |
|
For example, the 6th term of sequence S is 
because 
. Determine the value of the 5th term of sequence S, that is, the value of 
.
3. (1) Given that the 6th term of S minus the 4th term of S is 5, it follows that 
. Combining like terms, this equation can be rewritten as 
, and thus the 5th term of sequence S is 5; SUFFICIENT.
4. (2) Given that the 6th term of S plus the 7th term of S is 21, it follows that 
. Combining like terms, this equation can be rewritten as 
. Letting e represent the 5th term of sequence S, this last equation is equivalent to 
, or 
, which gives a direct correspondence between the 5th term of sequence S and the 2nd term of sequence S. Therefore, the 5th term of sequence S can be determined if and only if the 2nd term of sequenceS can be determined. Since the 2nd term of sequence S cannot be determined, the 5th term of sequence S cannot be determined. For example, if 
and 
, then 
and the 5th term of sequence S is 
. However, if 
and 
, then 
and the 5th term of sequence S is 
; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
114. If d is a positive integer, is 
an integer?
1. (1) d is the square of an integer.
2. (2) 
is the square of an integer.
Arithmetic Properties of numbers
The square of an integer must also be an integer.
3. (1) This can be expressed as d = x 2, where x is a nonzero integer. Then, 
which in turn equals x or −x, depending on whether x is a positive integer or a negative integer, respectively. In either case, 
is also an integer; SUFFICIENT.
4. (2) This can be expressed as 
, where x is a nonzero integer. The square of an integer (x 2) must always be an integer; therefore, must also be an integer; SUFFICIENT.
The correct answer is D; each statement alone is sufficient.
115. Is the positive integer n a multiple of 24?
1. (1) n is a multiple of 4.
2. (2) n is a multiple of 6.
Arithmetic Properties of numbers
3. (1) This says only that n is a multiple of 4 (i.e., n could be 8 or 24), some of which would be multiples of 24 and some would not; NOT sufficient.
4. (2) This says only that n is a multiple of 6 (i.e., n could be 12 or 48), some of which would be multiples of 24 and some would not; NOT sufficient.
Both statements together imply only that n is a multiple of the least common multiple of 4 and 6. The smallest integer that is divisible by both 4 and 6 is 12. Some of the multiples of 12 (e.g., n could be 48 or 36) are also multiples of 24, but some are not.
The correct answer is E; both statements together are still not sufficient.
116. If 75 percent of the guests at a certain banquet ordered dessert, what percent of the guests ordered coffee?
1. (1) 60 percent of the guests who ordered dessert also ordered coffee.
2. (2) 90 percent of the guests who ordered coffee also ordered dessert.
Arithmetic Concepts of sets; Percents
Consider the Venn diagram below that displays the various percentages of 4 groups of the guests. Thus, x percent of the guests ordered both dessert and coffee and y percent of the guests ordered coffee only. Since 75 percent of the guests ordered dessert, 
of the guests ordered dessert only. Also, because the 4 percentages represented in the Venn diagram have a total sum of 100 percent, the percentage of guests who did not order either dessert or coffee is 
. Determine the percentage of guests who ordered coffee, or equivalently, the value of .
3. (1) Given that x is equal to 60 percent of 75, or 45, the value of cannot be determined; NOT sufficient.
4. (2) Given that 90 percent of is equal to x, it follows that 
, or 
. Therefore, 
, or 
. From this the value of cannot be determined. For example, if 
and 
, then all 4 percentages in the Venn diagram are between 0 and 100, , and . However, if 
and , then all 4 percentages in the Venn diagram are between 0 and 100, 
, and 
; NOT sufficient.
Given both (1) and (2), it follows that 
and . Therefore, 
, or 
, and hence 
.
The correct answer is C; both statements together are sufficient.
117. A tank containing water started to leak. Did the tank contain more than 30 gallons of water when it started to leak? (Note: 
)
1. (1) The water leaked from the tank at a constant rate of 6.4 ounces per minute.
2. (2) The tank became empty less than 12 hours after it started to leak.
Arithmetic Rate problems
3. (1) Given that the water leaked from the tank at a constant rate of 6.4 ounces per minute, it is not possible to determine if the tank leaked more than 30 gallons of water. In fact, any nonzero amount of water leaking from the tank is consistent with a leakage rate of 6.4 ounces per minute, since nothing can be determined about the amount of time the water was leaking from the tank; NOT sufficient.
4. (2) Given that the tank became empty in less than 12 hours, it is not possible to determine if the tank leaked more than 30 gallons of water because the rate at which water leaked from the tank is unknown. For example, the tank could have originally contained 1 gallon of water that emptied in exactly 10 hours or the tank could have originally contained 31 gallons of water that emptied in exactly 10 hours; NOT sufficient.
Given (1) and (2) together, the tank emptied at a constant rate of 
for less than 12 hours.
If t is the total number of hours the water leaked from the tank, then the total amount of water emptied from the tank, in gallons, is 3t, which is therefore less than 
. From this it is not possible to determine if the tank originally contained more than 30 gallons of water. For example, if the tank leaked water for a total of 11 hours, then the tank originally contained (3)(11) gallons of water, which is more than 30 gallons of water. However, if the tank leaked water for a total of 2 hours, then the tank originally contained (3)(2) gallons of water, which is not more than 30 gallons of water.
The correct answer is E; both statements together are still not sufficient.
118. If x is an integer, is y an integer?
1. (1) The average (arithmetic mean) of x, y, and 
is x.
2. (2) The average (arithmetic mean) of x and y is not an integer.
Arithmetic Statistics; Properties of numbers
3. (1) From this, it is known that
|
|
|
|
|
multiply both sides by 3 |
|
|
combine like terms; subtract x from both sides |
|
|
divide both sides by 2 |
, or:
4. This simplifies to 
. Since x is an integer, this equation shows that x and y are consecutive integers; SUFFICIENT.
5. (2) According to this, y might be an integer (e.g., and 
, with an average of 5.5), or y might not be an integer (e.g., and 
, with an average of 5.6); NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
119. If 2 different representatives are to be selected at random from a group of 10 employees and if p is the probability that both representatives selected will be women, is 
?
1. (1) More than of the 10 employees are women.
2. (2) The probability that both representatives selected will be men is less than 
.
Arithmetic Probability
Let m and w be the numbers of men and women in the group, respectively. Then 
and the probability that both representatives selected will be a woman is
. Therefore, determining if 
is equivalent to determining if 
. Multiplying both sides by (10)(9)(2) gives the equivalent condition 
, or 
. By considering the values of (2)(1), (3)(2), . . . , (10)(9), it follows that 
if and only if w is equal to 8, 9, or 10.
3. (1) Given that 
, it is possible that w is equal to 8, 9, or 10 (for example, 
) and it is possible that w is not equal to 8, 9, or 10 (for example, 
); NOT sufficient.
4. (2) Given the probability that both selections will be men is less than 
, it follows that 
. Multiplying both sides by (9)(10) gives 
. Thus, by numerical evaluation, the only possibilities for m are 0, 1, 2, and 3. Therefore, the only possibilities for w are 10, 9, 8, or 7. However, it is still possible that w is equal to 8, 9, or 10 (for example, ) and it is still possible that w is not equal to 8, 9, or 10 (for example, ); NOT sufficient.
Given (1) and (2), it is not possible to determine if w is equal to 8, 9, or 10. For example, if , then both (1) and (2) are true and w is equal to 8, 9, or 10. However, if , then both (1) and (2) are true and w is not equal to 8, 9, or 10.
The correct answer is E; both statements together are still not sufficient.
120. In the xy-plane, lines k and ℓ intersect at the point (1,1). Is the y-intercept of k greater than the y-intercept of ℓ?
1. (1) The slope of k is less than the slope of ℓ.
2. (2) The slope of ℓ is positive.
Algebra Coordinate geometry
Let m1 and m2 represent the slopes of lines k and ℓ, respectively. Then, using the point-slope form for the equation of a line, an equation of line k can be determined: y − 1 = m1(x − 1), or y = m1x + (1 − m1). Similarly, an equation for line ℓ is y = m2x + (1 − m2). Determine if (1 − m1) > (1 − m2), or equivalently if m1 < m2.
3. (1) This indicates that m1 < m2; SUFFICIENT.
4. (2) This indicates that m2 > 0. If m1 = −1, for example, then m1 < m2, but if m2 = 4 and m1 = 5, then m1 > m2; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
121. Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. If a book is to be selected at random from the books on the shelf, is the probability less than 
that the book selected will be a paperback written in Spanish?
1. (1) Of the books on the shelf, 30 are paperbacks.
2. (2) Of the books on the shelf, 15 are written in Spanish.
Arithmetic Probability
3. (1) This indicates that 30 of the 45 books are paperbacks. Of the 30 paperbacks, 25 could be written in Spanish. In this case, the probability of randomly selecting a paperback book written in Spanish is 
. On the other hand, it is possible that only 5 of the paperback books are written in Spanish. In this case, the probability of randomly selecting a paperback book written in Spanish is 
; NOT sufficient.
4. (2) This indicates that 15 of the books are written in Spanish. Then, at most 15 of the 45 books on the shelf are paperbacks written in Spanish, and the probability of randomly selecting a paperback book written in Spanish is at most 
; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.
122. If S is a set of four numbers w, x, y, and z, is the range of the numbers in S greater than 2?
1. (1) 
2. (2) z is the least number in S.
Arithmetic Statistics
The range of the numbers w, x, y, and z is equal to the greatest of those numbers minus the least of those numbers.
3. (1) This reveals that the difference between two of the numbers in the set is greater than 2, which means that the range of the four numbers must also be greater than 2; SUFFICIENT.
4. (2) The information that z is the least number gives no information regarding the other numbers or their range; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
123. Stations X and Y are connected by two separate, straight, parallel rail lines that are 250 miles long. Train P and train Q simultaneously left Station X and Station Y, respectively, and each train traveled to the other”s point of departure. The two trains passed each other after traveling for 2 hours. When the two trains passed, which train was nearer to its destination?
1. (1) At the time when the two trains passed, train P had averaged a speed of 70 miles per hour.
2. (2) Train Q averaged a speed of 55 miles per hour for the entire trip.
Arithmetic Applied problem; rates
3. (1) This indicates that Train P had traveled 2(70) = 140 miles when it passed Train Q. It follows that Train P was 250 − 140 = 110 miles from its destination and Train Q was 140 miles from its destination, which means that Train P was nearer to its destination when the trains passed each other; SUFFICIENT.
4. (2) This indicates that Train Q averaged a speed of 55 miles per hour for the entire trip, but no information is given about the speed of Train P. If Train Q traveled for 2 hours at an average speed of 55 miles per hour and Train P traveled for 2 hours at an average speed of 70 miles per hour, then Train P was nearer to its destination when the trains passed. However, if Train Q traveled for 2 hours at an average speed of 65 miles per hour and Train P traveled for 2 hours at an average speed of 60 miles per hour, then Train Q was nearer to its destination when the trains passed. Note that if Train Q traveled at 
miles per hour for the remainder of the trip, then its average speed for the whole trip was 55 miles per hour; NOT sufficient.
The correct answer is A; statement 1 alone is sufficient.
124. In the xy-plane shown, the shaded region consists of all points that lie above the graph of y = x2 − 4x and below the x-axis. Does the point (a, b) (not shown) lie in the shaded region if b < 0?
1. (1) 0 < a < 4
2. (2) a2 − 4a < b
Algebra Coordinate geometry
In order for (a, b) to lie in the shaded region, it must lie above the graph of y = x2 − 4x and below the x-axis. Since b < 0, the point (a, b) lies below the x-axis. In order for (a, b) to lie above the graph of y = x2 − 4x, it must be true that b > a2 − 4a.
3. (1) This indicates that 0 < a < 4. If a = 2, then a2 − 4a = 22 − 4(2) = −4, so if b = −1, then b > a2 − 4a and (a, b) is in the shaded region. But if b = −5, then b < a2 − 4a and (a, b) is not in the shaded region; NOT sufficient.
4. (2) This indicates that b > a2 − 4a, and thus, (a, b) is in the shaded region; SUFFICIENT.
The correct answer is B; statement 2 alone is sufficient.