CHAPTER 19
Advanced Math Practice
In this chapter, you’ll find two 20-question practice sets consisting entirely of questions most test takers find challenging. The first practice set contains Problem Solving questions; the second contains Data Sufficiency questions. Following each practice set, you’ll find the answer key, as well as complete explanations for every question.
Use these question sets to hone your critical thinking skills if you are aiming for a very high GMAT score. If you are not yet comfortable with the practice questions in the preceding chapters of this book, then continue to review the arithmetic, algebra, geometry, and other topics presented there until you are able to find the correct answers to most of those practice questions. Then you’ll be ready to tackle the tougher questions here.
As always, be sure to review the explanation to every problem you do in this chapter. Noting a different way to work through a problem, even if you got that problem right, is a powerful way to enhance your critical thinking skills—and it’s ultimately those skills that you will be relying on in order to achieve a very high score on Test Day.
Ready to take your Quantitative score to the next level? Then turn the page and begin work!
ADVANCED PROBLEM SOLVING PRACTICE SET
The numbers m, n, and T are all positive, and m > n. A weekend farm stand sells only peaches. On Saturday, the farm stand has T peaches to sell, at a profit of m cents each. Any peaches remaining for sale on Sunday will be marked down and sold at a profit of (m − n) cents each. If all peaches available for sale on Saturday morning are sold by Sunday evening, how many peaches, in terms of T, m, and n, does the stand need to sell on Saturday in order to make the same profit on each day?
Set X consists of at least 2 members and is a set of consecutive odd integers with an average (arithmetic mean) of 37.
Set Y consists of at least 10 members and is also a set of consecutive odd integers with an average (arithmetic mean) of 37.
Set Z consists of all of the members of both set X and set Y.
Which of the following statements must be true?
The standard deviation of set Z is not equal to the standard deviation of set X.
The standard deviation of set Z is equal to the standard deviation of set Y.
The average (arithmetic mean) of set Z is 37.
I only
II only
III only
I and III
II and III
The number 10,010 has how many positive integer factors?
31
32
33
34
35
A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10. Which of the following could NOT be the sum of their numbers?
1,254
1,428
3,972
4,316
8,010
For all values of x, y, and z, x ◊ y ◊ z = x2(y − 1)(z + 2).
If a < 0, which of the following shows R, S, and T arranged in order from least to greatest?
R: 1 ◊ a ◊ 3
S: 3 ◊ a ◊ 1
T: a ◊ 3 ◊ 1
R, S, T
T, S, R
R, T, S
T, R, S
S, R, T
If a and b are integers, and 2a + b = 17, then 8a + b cannot equal which of the following?
−1
33
35
65
71
Two workers have different pay scales. Worker A receives $50 for any day worked plus $15 per hour. Worker B receives $27 per hour. Both workers may work for a fraction of an hour and be paid in proportion to their respective hourly rates. If worker A arrives at 9:21 a.m. and receives the $50 upon arrival and Worker B arrives at 10:09 a.m., assuming both work continuously, at what time would their earnings be identical?
11:09 a.m.
12:01 p.m.
2:31 p.m.
3:19 p.m.
3:36 p.m.
In the figure above, ABDE is a square, and arc ATD is part of a circle with center B. The measure of angle BCD is 60 degrees, and line segment CD has a length of 4. What is the area of the shaded region?
16π − 32
12π − 24
12π − 48
48π − 24
48π − 48
When the cube of a non-zero number y is subtracted from 35, the result is equal to the result of dividing 216 by the cube of that number y. What is the sum of all the possible values of y?
5
6
10
12
One letter is selected at random from the 5 letters V, W, X, Y, and Z, and event A is the event that the letter V is selected. A fair six-sided die with sides numbered 1, 2, 3, 4, 5, and 6 is to be rolled, and event B is the event that a 5 or a 6 shows. A fair coin is to be tossed, and event C is the event that a head shows. What is the probability that event A occurs and at least one of the events B and C occurs?
A car traveled from Town A to Town B. The car traveled the first
of the distance from Town A to Town B at an average speed of x miles per hour, where x > 0. The car traveled the remaining distance at an average speed of y miles per hour, where y > 0. The car traveled the entire distance from Town A to Town B at an average speed of z miles per hour. Which of the following equations gives y in terms of x and z?An ornithologist has studied a particular population of starlings and discovered that their population has increased by 400% every ten years starting in 1890. If the initial population in 1890 was 256 birds, how large was the population of starlings in 1970?
102,400
10,000,000
16,777,216
20,000,000
100,000,000
If (x2 + 8)yz < 0, wz > 0, and xyz < 0, the which of the following must be true?
x < 0
wy < 0
yz < 0
II only
III only
I and III only
II and III only
I, II, and III
If
is an integer, then h could be divisible by each of the following EXCEPT:8
12
15
18
31
If a, b, and c are integers such that 0 < a < b < c, and a is even, b is prime, and c is odd, which of the following is a possible value for abc?
5
12
16
34
54
A fair die with sides numbered 1, 2, 3, 4, 5, and 6 is to be rolled 4 times. What is the probability that on at least one roll the number showing will be less than 3?
There are 816 students in enrolled at a certain high school. Each of these students is taking at least one of the subjects economics, geography, and biology. The sum of the number of students taking exactly one of these subjects and the number of students taking all 3 of these subjects is 5 times the number of students taking exactly 2 of these subjects. The ratio of the number of students taking only the two subjects economics and geography to the number of students taking only the two subjects economics and biology to the number of students taking only the two subjects geography and biology is 3:6:8. How many of the students enrolled at this high school are taking only the two subjects geography and biology?
35
42
64
136
240
Working alone at a constant rate, machine P produces a widgets in 3 hours. Working alone at a constant rate, machine Q produces b widgets in 4 hours. If machines P and Q work together for c hours, then in terms of a, b, and c, how many widgets will machines P and Q produce?
4ac + 3bc
In the figure above, the perimeter of triangle I is 16 feet greater than the perimeter of triangle II. What is the length of PQ, in feet?
27
51
68
75
85
If x > 0, y > 0, and
, what is the value of 
?
ADVANCED DATA SUFFICIENCY PRACTICE SET
If x and y are positive even integers, is (40x)x divisible by y?
(1)
(2) y is a multiple of 160.
If
, and 
, is 
?(1) a2 − 2a − 3 = 0
(2) b2 − 4b + 4 = 1
In the figure above, what is the area of semicircle DPA?
(1) The area of quadrilateral ABCD is 140.
(2) The length of the line segment whose endpoints are B and D is 25.
Is x divisible by 39?
(1) x divided by 65 results in a remainder of 7.
(2) x divided by 36 results in a remainder of 15.
If a and b are positive integers with different units digits, and b is the square of an integer, is a also the square of an integer?
(1) The units digit of a + b is 8.
(2) b = 121
What is the value of 8x + y?
(1) 3x − 2y + z = 10
(2) 2(x + 3y) − (y + 2z) = −25
If b and c are two-digit positive integers and b − c = 22d, is d an integer?
(1) The tens digit and the units digit of b are identical.
(2)
, and x is an integer.
The integers x and y are positive, x > y + 8, and y > 8. What is the remainder when x2 − y2 is divided by 8?
(1) The remainder when x + y is divided by 8 is 7.
(1) The remainder when x − y is divided by 8 is 5.
If x > y > 0, does 3x + 1 + 3(2y) = 12v?
(1)
(2) 2(3x + 2) + 9(2y + 1) = 72v
In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?
(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is 
.
(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.
In the sequence T, the first term is the non-zero number a, and each term after the first term is equal to the non-zero number r multiplied by the previous term. What is the value of the fourth term of the sequence?
(1) The sum of the first 2 terms of the sequence is 16.
(2) The 18th term of the sequence is 81 times the 14th term of the sequence.
A person is to be selected at random from the group T of people. What is the probability that the person selected is a member of club E?
(1) The probability that a person selected at random from group T is not a member of club D and is not a member of club E is 
.
(2) The probability that a person selected at random from group T is a member of club D and not a member of club E is 
.
The population of Town X on January 1, 2010, was 56 percent greater than the population of the same town on January 1, 2005. The population of Town X on January 1, 2015, was 75 percent greater than the population of the same town on January 1, 2010. What was the population of Town X on January 1, 2005?
(1) The population of Town X on January 1, 2015, was 21,840.
(2) The increase in the population of Town X from January 1, 2010, to January 1, 2015, was 4,880 greater than the increase in the population of Town X from January 1, 2005, to January 1, 2010.
What is the value of x − 3z?
(1) x + 4y = 3
(2) x2 + 4xy − 3xz − 12yz = 24
In the figure above, the center of the circle is O, and the measure of angle CAO is 60 degrees. What is the perimeter of triangle OAC?
(1) The length of arc CDA is 16π greater than the length of arc ABC.
(2) The area of triangle OAC is 
.
If x and y are integers and 3x > 8y, is y > −18?
(1) −9 < x < 20
(2)
Over the course of 5 days, Monday through Friday, Danny collects a total of 76 baseball cards. Each day, he collects a different number of cards. If Danny collects the largest number of cards on Friday and the second largest number of cards on Thursday, did Danny collect more than 8 cards on Thursday?
(1) On Friday, Danny collected 49 cards.
(2) On one of the first 3 days, Danny collected 6 cards.
If y = x2, is the equation (25y)(4120) = 1620x true?
(1) (4 − x)(12 − x) = 0
(2) (x − 4)(x − 8)(x − 24) = 0
The ratio of the number of students in an auditorium who are seniors to the number of students in the auditorium who are not seniors is 7:5. How many students are there in the auditorium?
(1) The ratio of the number of students who are seniors who are taking history to the number of students who are not seniors who are taking history is 21:5.
(2) Of the students in the auditorium who are seniors, 
are taking history; of the students in the auditorium who are not seniors, 
are taking history; and the number of seniors in the auditorium who are taking history is 208 greater than the number of students in the auditorium who are not seniors and taking history.
If y > 0, is
< 3?(1) x(x + y) − 4y(x + y) < 0
(2) 5(y + 8) < 20y − 3x + 40
Answer Key
ADVANCED PROBLEM SOLVING PRACTICE SET
E
C
B
D
E
B
D
B
B
D
E
E
D
D
E
A
C
B
C
E
Answer Key
ADVANCED DATA SUFFICIENCY PRACTICE SET
A
E
B
A
A
C
C
C
D
B
E
C
D
C
D
B
A
A
B
E
Answers and Explanations
ADVANCED PROBLEM SOLVING PRACTICE SET
1. (E)
The numbers m, n, and T are all positive, and m > n. A weekend farm stand sells only peaches. On Saturday, the farm stand has T peaches to sell, at a profit of m cents each. Any peaches remaining for sale on Sunday will be marked down and sold at a profit of (m − n) cents each. If all peaches available for sale on Saturday morning are sold by Sunday evening, how many peaches, in terms of T, m, and n, does the stand need to sell on Saturday in order to make the same profit on each day?
Step 1: Analyze the Question
There are a total of T peaches, all of which are to be sold during the course of the weekend. On Saturday, the profit for each peach is m cents. On Sunday, the profit for each peach is (m − n) cents.
Step 2: State the Task
How many peaches must be sold on Saturday to make Saturday’s profit equal Sunday’s profit?
Step 3: Approach Strategically
Start by setting up expressions for Saturday’s profit and Sunday’s profit and then set those expressions equal. There’s no variable given for the number of peaches sold on either day, so call the number of peaches to be sold on Saturday x. The total profit for Saturday will be (m cents per peach) times (x peaches), which is mx cents. On Sunday, there will be (T − x) peaches left, each of which will bring a profit of (m − n) cents. The total profit for Sunday will be [(m − n) cents per peach] times [(T − x) peaches], which is (m − n)(T − x) cents. Saturday’s profit must equal Sunday’s profit: mx = (m − n)(T − x). The question asks for the number of peaches to be sold on Saturday, so solve for x:
None of the answer choices are written 
. However, 
. Choice (E) is correct.
Step 4: Confirm Your Answer
Confirm that your answer makes sense. Rewrite the denominator to test the size of the fraction: 
. It should be fairly easy to see that this is less than 
. So 
, which equals 
, must be less than 
, or 
. It follows that the number of peaches sold on Saturday, which is 
, is less than half of the total number of peaches sold. That makes sense because the profit per peach is lower on Sunday than on Saturday—so fewer than half the peaches would have to be sold on Saturday in order for both days’ profits to be equal.
2. (C)
Set X consists of at least 2 members and is a set of consecutive odd integers with an average (arithmetic mean) of 37.
Set Y consists of at least 10 members and is also a set of consecutive odd integers with an average (arithmetic mean) of 37.
Set Z consists of all of the members of both set X and set Y.
Which of the following statements must be true?
The standard deviation of set Z is not equal to the standard deviation of set X.
The standard deviation of set Z is equal to the standard deviation of set Y.
The average (arithmetic mean) of set Z is 37.
I only
II only
III only
I and III
II and III
Step 1: Analyze the Question
Set X has at least two members and consists of consecutive odd integers that have a mean of 37. For any set of evenly spaced numbers, the mean equals the median, so the median of set X is 37. The question says that set X has at least 2 members. However, for a set of consecutive odd integers to have an average that is an odd integer, there must be an odd number of members. So set X must contain at least 3 members. Thus, set X could be, for example, {35, 37, 39} or {33, 35, 37, 39, 41}.
Set Y has the same mean and median as set X. However, set Y has a greater minimum number of members. The question says that set Y has at least 10 members. But again, for a set of consecutive odd integers to have an average that is an odd integer, there must be an odd number of members, so set Y must contain at least 11 members. Note that although set Y has a higher minimum number of terms, neither set has a maximum. This means that it’s possible for set X to have more members than set Y or even for the sets to be equal to one another.
Set Z contains all the members of both set X and set Y. When sets X and Y have different numbers of members, set Z equals either set X or set Y, whichever has the greater number of members. For example, if set X = {35, 37, 39} and set Y = {27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47}, then set Z would be equal to set Y, as the entirety of set X is contained within set Y. When sets X and Y have the same number of members (and therefore are equal), set Z equals both sets X and Y.
Step 2: State the Task
The task is to evaluate the three statements and determine which are always true about sets X, Y, and Z.
Step 3: Approach Strategically
Of the Roman numeral statements, III is the most frequently mentioned in the answer choices and, conveniently, also the easiest to consider. Set Z will always equal set X and/or Y, and both set X and set Y have a mean of 37. Therefore, set Z will also always have a mean of 37, and statement III is part of the correct answer. Eliminate (A) and (B).
The other two options involve the standard deviation of set Z. Recall that standard deviation refers to how spread out a set of numbers is. Like many GMAT questions involving standard deviation, this one does not require calculation of the actual value. Instead, recognize that sets of equally spaced integers will have different standard deviations depending on how many members are in the set. When two sets have equally spaced numbers and the spacing in both sets is the same, the set with the greater number of numbers has the greater standard deviation. While the standard deviation of set Z will always be equal to either (or both) the standard deviation of set X or Y, it is not necessarily different than the standard deviation of set X, nor is it necessarily equal to the standard deviation of set Y. This eliminates (D) and (E), making (C) the correct answer.
Step 4: Confirm Your Answer
Quickly confirm by recognizing that when Set X has more members than Set Y, both Statements I and II are not true. Statement III is the only one that is true under all circumstances.
3. (B)
The number 10,010 has how many positive integer factors?
31
32
33
34
35
Step 1: Analyze the Question
There’s no information to catalogue in the question stem; there is just the number 10,010. However, the answer choices provide a clue: you need to find a way to count more than 30 factors.
Step 2: State the Task
The question asks for the total number of positive factors, not just prime factors.
Step 3: Approach Strategically
Making a chart of all factors of a number works well for two-digit numbers, but it would require a great deal of arithmetic in this case. Instead, start by finding the prime factorization of 10,010, and then use those prime factors to find the total number of factors. The number 10,010 ends in zero, so it’s divisible by 10, or 2 × 5. So far, you have 2 × 5 × 1,001. Now use divisibility rules to investigate the factor 1,001. This is not an even number, so it isn’t divisible by 2. Its digits do not sum to 3, so it isn’t divisible by 3. It doesn’t end in 5 or 0, so 5 is also not a factor. What about 7? To check whether a given number is divisible by 7, multiply the units digit by 2, subtract the result from the remaining digits, and check if the result is divisible by 7. If yes, then the original number is also divisible by 7. The units digit of 1,001 is 1. Multiply by 2 and subtract from 100: 100 − 2 = 98. Because 98 is divisible by 7 (98 = 7 × 14), 1,001 is also divisible by 7. Do the long division to find that 1,001 ÷ 7 = 143. Now again apply divisibility rules to 143. It’s not divisible by 2, 3, 5, or 7. What about 11? A number is divisible by 11 if the difference between the sum of its even-placed digits and the sum of its odd-placed digits is divisible by 11 without leaving a remainder. In the case of 143, (1 + 3) − 4 = 0. Zero divided by 11 is 0 without any remainder, so 143 is divisible by 11. Do the division to find that 143 ÷ 11 = 13.
The full prime factorization of 10,010, then, is as follows: 2 × 5 × 7 × 11 × 13. Once you have its prime factorization, the fastest way to determine how many factors a number has is to add one to each exponent in the prime factorization and multiply the resulting values. You can write the prime factorization of 10,010 as 21 × 51 × 71 × 111 × 131. Now add one to each exponent and multiply: (1+1)(1+1)(1+1)(1+1)(1+1) = (2)(2)(2)(2)(2) = 32. The correct answer is (B).
Step 4: Confirm Your Answer
Do a quick check to make sure you included all the prime factors you discovered and check the exponents.
4. (D)
A department of motor vehicles asks visitors to draw numbered tickets from a dispenser so that they can be served in order by number. Six friends have graduated from truck-driving school and go to the department to get commercial driving licenses. They draw tickets and find that their numbers are a set of evenly spaced integers with a range of 10. Which of the following could NOT be the sum of their numbers?
1,254
1,428
3,972
4,316
8,010
Step 1: Analyze the Question
Six people draw 6 evenly spaced integers with a range of 10. You could pick two numbers with a range of 10, such as 1 and 11 or 2 and 12, and experiment to see how six evenly spaced integers could fit in the range. Clearly, the integers must be 2 apart.
Alternatively, the formula for the number of members in a set of evenly spaced integers is this:
In this case:
Anytime the GMAT uses a small set of evenly spaced integers, remember that a single variable is sufficient to represent the series. Thus, x can represent the first integer, and the six integers can be written x, x + 2, x + 4, x + 6, x + 8, x + 10. The sum of these would be 6x + 30, or 6(x + 5). While you do not know the value of this expression (since you do not know the value of x), you do know it is a multiple of 6.
Step 2: State the Task
You need to find the answer choice that cannot equal 6(x + 5). The correct answer is the one that is not a multiple of 6.
Step 3: Approach Strategically
Use the rule for divisibility by 6, which combines the rules for divisibility by 2 and 3. To be divisible by 2, the number must be even, and to be divisible by 3, the sum of the digits of the number must be divisible by 3.
All the answer choices are even, so use the rule for divisibility by 3.
(A): 1 + 2 + 5 + 4 = 12. Eliminate.
(B): 1 + 4 + 2 + 8 = 15. Eliminate.
(C): 3 + 9 + 7 + 2 = 21. Eliminate.
(D): 4 + 3 + 1 + 6 = 14, which is not divisible by 3. This is the correct answer.
For the record …
(E): 8 + 0 + 1 + 0 = 9. Eliminate.
Step 4: Confirm Your Answer
If you divide 4,316 by 6, you get 719 with a remainder of 2. Therefore, 4,316 is not divisible by 6, and choice (D) is correct.
5. (E)
For all values of x, y, and z, x ◊ y ◊ z = x2(y − 1)(z + 2).
If a < 0, which of the following show R, S, and T arranged in order from least to greatest?
R: 1 ◊ a ◊ 3
S: 3 ◊ a ◊ 1
T: a 3 ◊ 1
R, S, T
T, S, R
R, T, S
T, R, S
S, R, T
Step 1: Analyze the Question
You are given a symbolic algebra equation that describes the relationship of variables x, y, and z. Using this symbolism as a function, substitute 1, 3, and a for x, y, and z in the various orders shown. Keep in mind that a is negative.
Step 2: State the Task
The question requires you to use the symbolic equation given and evaluate R, S, and T, placing them in order from smallest to largest.
Step 3: Approach Strategically
There are no obvious answer choices to eliminate, so calculate the values of R, S, and T and rank them in order from smallest to largest. Again, x ◊ y ◊ z = x2(y − 1)(z + 2).
You know that T must be a positive value because the a is squared and multiplied by a positive number. Keeping in mind that a is negative, it should be clear that the expressions that result for R and S subtract a positive value from a negative value, making them both negative numbers. Therefore, T must be the greatest of the choices. Eliminate answer choices (B), (C), and (D).
Next compare R and S. Both involve multiplying the negative value a by a constant and then subtracting a positive integer. However, both terms of S have a greater absolute value than those of R, making S less than R and (E) the correct answer.
Step 4: Confirm Your Answer
You could Pick Numbers for a to confirm your answer. For example, if a = −1, then R = 5a − 5 = 5(−1) − 5 = −10, S = 27a − 27 = 27(−1) − 27 = −54, and T = 6(−12) = 6. For any permissible value of a, S is always less than R and T is always positive, again making (E) the correct answer.
6. (B)
If a and b are integers, and 2a + b = 17, then 8a + b cannot equal which of the following?
−1
33
35
65
71
Step 1: Analyze the Question
Because the question asks about what 8a + b cannot equal, try to find some rule about what the expression must equal and go from there. The correct answer will violate some property represented in the given information.
Step 2: State the Task
To find the value in the answer choices that would not work with the given information, use the equation 2a + b = 17 in conjunction with the expression 8a + b.
Step 3: Approach Strategically
Since you have only one equation, you can’t solve for either variable, but you can still substitute and get one variable in the expression you’re evaluating. Substituting for either variable would work, but it is easier to substitute for b, since all you have to do is put 2a on the other side of the equation. If b = −2a + 17, then the expression 8a + b becomes 8a + (−2a + 17), which simplifies to 6a + 17. Because you know that a is an integer, 6a must be a multiple of 6. Then because you’re adding 17 to a multiple of 6, the correct answer must be just that—the result of adding 17 to a multiple of 6. (If you notice that this is 1 less than a multiple of 6, evaluating the answer choices is even faster.)
(A): −1 − 17 = −18, a multiple of 6. Eliminate.
(B): 33 − 17 = 16, which is not a multiple of 6. (B) is the correct answer.
For the record …
(C): 35 − 17 = 18, a multiple of 6. Eliminate.
(D): 65 − 17 = 48, a multiple of 6. Eliminate.
(E): 71 − 17 = 54, a multiple of 6. Eliminate.
Step 4: Confirm Your Answer
Using substitution, you know that 8a + b = 6a + 17. If 6a + 17 = 33, then 6a = 16 and a = 
, which is not an integer. Choice (B) is correct.
7. (D)
Two workers have different pay scales. Worker A receives $50 for any day worked plus $15 per hour. Worker B receives $27 per hour. Both workers may work for a fraction of an hour and be paid in proportion to their respective hourly rates. If worker A arrives at 9:21 a.m. and receives the $50 upon arrival and Worker B arrives at 10:09 a.m., assuming both work continuously, at what time would their earnings be identical?
11:09 a.m.
12:01 p.m.
2:31 p.m.
3:19 p.m.
3:36 p.m.
Step 1: Analyze the Question Stem
The stem gives you a lot of information. There are two workers, each with a different pay scale as well as time arrived. It would be worth writing down the basics on your noteboard, like this:
A: 9:21 $50 + 15/hr
B: 10:09 $27/hr
Step 2: State the Task
Even though this is about wages and money, it’s a combined rates problem, just like a question that asks when two trains pass each other. The correct answer will be the time that the different rates produce the same sum of money.
Step 3: Approach Strategically
While it’s possible to Backsolve, it would be necessary to plug each time into two different formulas, making this approach both time-consuming and potentially error-prone. Furthermore, combining rates is usually very fast if done correctly. However, if combining the rates as described below seems too tricky and you feel confident with the arithmetic involved (and understand what result would indicate a time too late or too early), Backsolving may be a good option.
Anytime the GMAT gives you different rates and asks you to find when or where they produce the same result, you can use four steps to get the answer—often in a minute or less.
The first step is to calculate any distance/work/earnings that one entity achieves when the other is not active. In this case, A works for 48 minutes before B begins, so you want to know how much money A has already made. In combined work problems, calculations are easiest if you state time as a fraction, and 48 minutes is 
of an hour. Thus, $15 × = $12 earned. Also, A gets $50 right off the bat, so A is $12 + $50 = $62 up when B begins working.
The second step is to combine rates. In this case, think about B catching up to A. Worker A makes $15 an hour and Worker B makes $27 an hour, so for each hour they both work, B gains on A by $12.
The third step is to divide the distance/work/earnings by the combined rate to get the time, so divide $62 by $12 per hour, and get 5
or 5
hours. A sixth of an hour is 10 minutes, so they are both working for 5 hours 10 minutes.
The last step is to add the time worked to the starting time. In this case, add 5 hours 10 minutes to the first time when both started working, 10:09 a.m., and get 3:19 p.m. Choice (D) is the correct answer.
Step 4: Confirm Your Answer
When taking the four steps described above leads to an answer that exactly matches one of the choices, you can feel confident that you have solved correctly. However, to confirm, you could find that Worker A worked for 5 hours 58 minutes and so, at $15 an hour, made $89.50 + $50, or $139.50. Worker B worked for 5 hours 10 minutes at $27 an hour, which also yields $139.50. Again, choice (D) is correct.
8. (B)
In the figure above, ABDE is a square, and arc ATD is part of a circle with center B. The measure of angle BCD is 60 degrees, and line segment CD has a length of 4. What is the area of the shaded region?
16π − 32
12π − 24
12π − 48
48π − 24
48π − 48
Step 1: Analyze the Question
In a shaded region question, it is important to recognize what shapes are interacting to produce the shaded region. In this case, the shaded region is the result of removing triangle ABD, whose area is half the area of square ABDE, from the sector of a circle. Additionally, you are given two important measurements for triangle BCD: angle BCD measures 60 degrees and side CD has a length of 4.
Step 2: State the Task
Subtract half the area of the square from the area of the circle sector to calculate the area of the shaded region.
Step 3: Approach Strategically
The area of a sector of a circle has the same proportional relationship to the area of the full circle as its central angle does to 360 degrees. This sector has a central angle which is interior angle ABD of the square. Since B is a vertex of a square, the measure of angle ABD is 90 degrees. Because 90 degrees is one-fourth of 360 degrees, the sector defined by arc ATD has an area one-fourth of that of the full circle. To calculate the area of both the half square and quarter circle, determine the length of BD, which is both a radius of the circle and a side of the square. Triangle BCD is a 30-60-90 triangle, and its sides therefore have the proportion 
. Side CD is opposite the 30-degree angle and is therefore represented by the x in this proportional relationship. Side BD thus has a length of 
. The area of the square is 
, and the area of triangle ABD is 24. The area of the circle is 
, and the quarter circle has an area of 
. Therefore, the area of the shaded region is 12π − 24, and the correct answer is (B).
Step 4: Confirm Your Answer
Check that you correctly applied the 30-60-90 triangle side length ratio and performed the calculations accurately.
9. (B)
When the cube of a non-zero number y is subtracted from 35, the result is equal to the result of dividing 216 by the cube of that number y. What is the sum of all the possible values of y?
5
6
10
12
Step 1: Analyze the Question
Start by translating the English into math. The cube of the number y is y3. The result of subtracting the cube of the number y from 35 is 35 − y3. When the 216 is divided by y3, the result is 
. So 
.
Step 2: State the Task
The task is to solve for all possible values of y and add them.
Step 3: Approach Strategically
Solve the equation you’ve written for y.
Now simplify the equation y6 − 35y3 + 216 = 0. It helps to recognize that y6 = y3 × 2 = (y3)2. Replacing y6 with (y3)2 in the equation y6 − 35y3 + 216 = 0 produces (y3)2 − 35y3 + 216 = 0. Now notice that you have a quadratic equation if you consider y3 to be the unknown. The equation (y3)2 − 35y3 + 216 = 0 factors to (y3 − 8)(y3 − 27) = 0. So y3 = 8 or y3 = 27. Now 8 = 23 and 27 = 33. So if y3 = 8, then y = 2, while if y3 = 27, then y = 3. The possible values of y are 2 and 3. The sum of all the possible values of y is 2 + 3 = 5.
Choice (B) is correct.
Step 4: Confirm Your Answer
You can check that you have the correct values of y by plugging them into the equation 
. If y = 2, you get 35 − 8 = 
, or 27 = 27. If y = 3, you get 35 − 27 = 
, or 8 = 8.
10. (D)
One letter is selected at random from the 5 letters V, W, X, Y, and Z, and event A is the event that the letter V is selected. A fair six-sided die with sides numbered 1, 2, 3, 4, 5, and 6 is to be rolled, and event B is the event that a 5 or a 6 shows. A fair coin is to be tossed, and event C is the event that a head shows. What is the probability that event A occurs and at least one of the events B and C occurs?
Step 1: Analyze the Question
This is a probability question with three separate and independent events.
Step 2: State the Task
You need to calculate the probability that event A will occur and that either or both events B and C will occur.
Step 3: Approach Strategically
The probability of an event is 
. For event A, one letter is selected at random from the five letters V, W, X, Y, and Z, so the probability that event A occurs is 
. For event B, a 5 or 6 must occur from the six possible outcomes 1, 2, 3, 4, 5, and 6, so the probability that event B occurs is 
= 
. For event C, a head comes up out of two possible outcomes, heads and tails, so the probability that event C occurs is 
.
This can be written as P(A) = 
, P(B) = , and P(C) = .
The probability that event A occurs and at least one of the events B and C occurs is P(A) × P(B or C).
Find P(B or C): Applying the generalized formula for two events, P(B or C) = P(B) + P(C) − P(B and C).
The events B and C are independent, so P(B and C) = P(B)P(C).
Replacing P(B and C) with P(B)P(C), you have P(B or C) = P(B) + P(C) − P(B)P(C). Then substitute in P(B) = and P(C) = :
Now find the probability that event A occurs and at least one of the events B and C occurs:
Choice (D) is correct.
Step 4: Confirm Your Answer
You can check that you have calculated P(B or C) correctly by calculating the probability that neither B nor C occurs and subtracting from 1. P(~B) = 
and P(~C) = , so 
.
11. (E)
A car traveled from Town A to Town B. The car traveled the first 
of the distance from Town A to Town B at an average speed of x miles per hour, where x > 0. The car traveled the remaining distance at an average speed of y miles per hour, where y > 0. The car traveled the entire distance from Town A to Town B at an average speed of z miles per hour. Which of the following equations gives y in terms of x and z?
Step 1: Analyze the Question
The car moves at x mph for the first 
of the trip and y mph for the final 
. Its average speed over the entire distance traveled is z mph.
Step 2: State the Task
You want y in terms of x and z.
Step 3: Approach Strategically
Remember that Distance = Speed × Time, and that Total Distance = Average Speed × Total Time. Call the distance from Town A to Town B d miles. Then the first of the distance from Town A to Town B is 
miles. Since the car traveled the first 
miles at an average speed of x miles per hour, the time that it took the car to travel the first miles was 
hours. The remainder of the distance from Town A to Town B, in miles, was 
. Since the car traveled miles at an average speed of y miles per hour, the time that it took the car to travel the remaining miles was 
hours. The total time for the trip was 
hours. Since the car traveled a distance of d miles in 
hours, the average speed of the car was 
, or 
miles per hour. You know that the average speed of the car was z miles per hour. So 
.
Now solve this equation for y in terms of x and z. The variable d is in the way, but it looks like this variable can be canceled. Factoring d out of the denominator in the left side of the equation 
yields 
. Canceling d from the numerator and denominator of the left side results in 
. Multiplying both sides by the denominator 
, you get 
. Multiplying out the right side produces 
. Multiply both sides by the least common denominator on the right side, which is 8xy, to get rid of the denominators: 
, so 8xy = 3yz + 5xz. Since you are trying to solve for y in terms of x and z, get all the expressions involving y on one side of the equation, and all expressions not involving y on the other side of the equation. Subtracting 3yz from both sides yields 8xy − 3yz = 5xz. Factoring y out of the left side, y(8x − 3z) = 5xz. Dividing both sides by 8x − 3z, 
.
Choice (E) is correct.
Step 4: Confirm Your Answer
Check to make sure you applied the Average Speed formula correctly, and that you solved for the correct variable.
12. (E)
An ornithologist has studied a particular population of starlings and discovered that their population has increased by 400% every ten years starting in 1890. If the initial population in 1890 was 256 birds, how large was the population of starlings in 1970?
102,400
10,000,000
16,777,216
20,000,000
100,000,000
Step 1: Analyze the Question
You know that the initial population of starlings in 1890 is 256, and you know that the population increases by 400% every 10 years.
Step 2: State the Task
Find the size of the population of starlings in 1970.
Step 3: Approach Strategically
Although this question does not deal with money, it is useful to notice that the given facts fit the compound interest formula perfectly. The formula, as applied to money, is (Total of principal and interest) = Principal × (1 + r)t, where r is the interest rate per time period and t is the number of time periods. In this question, the total of “principal and interest” is the final population of starlings in 1970, the “principal” is the initial population (256), the “interest rate” is the population growth rate expressed as a decimal (4.00), and t is the number of 10-year periods from 1890 to 1970 (8). Thus, you have the following:
1970 population = 256 × (1 + 4)8 = 256 × 58
At this point, you could do the arithmetic, but without a calculator, this approach would be time-consuming and potentially prone to errors. Remember that when seemingly complicated arithmetic arises on the GMAT, there is often a more strategic way to get to the answer. If you notice that 256 is a power of 2, you can use the exponent rules to make your task much simpler:
256 × 58 = 28 × 58 = (2 × 5)8 = 108 = 100,000,000
The correct answer is (E).
Step 4: Confirm Your Answer
Confirm that you have applied the compound interest formula correctly.
13. (D)
If (x2 + 8)yz < 0, wz > 0, and xyz < 0, the which of the following must be true?
x < 0
wy < 0
yz < 0
II only
III only
I and III only
II and III only
I, II, and III
Step 1: Analyze the Question
This question concerns positive and negative number properties. Since wz > 0 and xyz < 0, each of the variables w, x, y, and z is non-zero. In the first inequality, which is (x2 + 8)yz < 0, x2 must be greater than zero because from the third inequality xyz < 0, you know that x is not zero. Since x2 is positive and 8 is positive, x2 + 8 is positive. Since (x2 + 8)yz < 0, and since x2 + 8 is positive, it must be that yz is negative. Thus, y and z have opposite signs. Since wz > 0, both w and z must have the same sign. Finally, in order for xyz to be negative, either one or all of the three variables x, y, and z must be negative.
Step 2: State the Task
Determine which of three inequalities must be true.
Step 3: Approach Strategically
Usually, the most efficient way to solve this type of question is to start with the Roman numeral that appears most often in the answer choices; in this case, that is III, yz < 0. You already determined that y and z have opposite signs, so this statement must be true. You can eliminate choice (A).
Look at statement II next. You determined that w and z have the same sign. Therefore, if y times z is negative, then y times w must also be negative, and statement II must be true. You can now eliminate choices (B) and (C).
Finally, if xyz is negative and yz is also negative, then x must be positive. Eliminate choice (E). Answer choice (D) is correct.
Step 4: Confirm Your Answer
Check your reasoning to confirm that y has the opposite sign as w, that y has the opposite sign as z, and that x is positive.
14. (D)
If 
is an integer, then h could be divisible by each of the following EXCEPT:
8
12
15
18
31
Step 1: Analyze the Question
Calculating the actual value of the numerator would be too time-consuming, but it’s presented in the form of the difference of two perfect squares (612 and 12). This is one of the classic quadratics: 612 − 12 = (61 + 1)(61 − 1), or (62)(60).
Step 2: State the Task
In order for 
to be an integer, 612 − 12 must be a multiple of h. So the question essentially asks which factor of h would not divide evenly into the numerator. Four of the choices will divide evenly into the numerator; the correct answer will not.
Step 3: Approach Strategically
It would take far too much time to find the product of 60 and 62 and then divide that by each of the answer choices. Instead, remember that for h to divide evenly into the product of 60 and 62, h must share factors with 60 and/or 62. Moreover, the answer choices represent factors of h. Looking at the choices, 8 has factors of 2 and 4. Because 2 is a factor of 62 and 4 is a factor of 60, you can cross out choice (A). Choices (B), (C), and (E) are a little more straightforward, since 12 itself is a factor of 60, 15 is also a factor of 60, and 31 is a factor of 62. The only choice that is not a factor of either number is 18. The correct answer is (D).
Step 4: Confirm Your Answer
You can quickly check that 18 is not a factor of the product of 60 and 62 by breaking down 18, 60, and 62 into their prime factors: 
. Notice that the 2 in the denominator will cancel out one of the 2s in the numerator, but only one of the 3s in the denominator will cancel out. So the denominator will have a 3 remaining that cannot be canceled out. Therefore, the product of 60 and 62 is not divisible by 18, and choice (D) is correct.
15. (E)
If a, b, and c are integers such that 0 < a < b < c, and a is even, b is prime, and c is odd, which of the following is a possible value for abc?
5
12
16
34
54
Step 1: Analyze the Question
You are told that a, b, and c are integers such that 0 < a < b < c, and you know that a is even, b is prime, and c is odd.
Step 2: State the Task
You need to determine which of the answer choices is a possible value for abc. Note that this doesn’t mean that the correct choice is the only possible value for abc.
Step 3: Approach Strategically
Given the structure of the problem, you will have to eliminate any answer choice that cannot be a possible value for abc rather than solve for a single value of abc. If you think of abc as a × bc, then, regardless of the values of b and c, abc must be even since a is an even number. Eliminate choice (A).
To deal with the remaining answer choices, since the question contains variables and the answer choices are numbers, Picking Numbers is a good approach. The least value possible for a is 2, the first even number greater than 0. The next largest prime number, 3, can be substituted for b. (Note that b is a prime number that has to be greater than a positive even number, meaning it can’t be 2, the only even prime number.) The product of a and b would be 6, and this could be multiplied by 2 to make abc = 12. However, 2 is not a permissible value for c, so eliminate (B). 2 × 3 = 6, 2 × 5 = 10, and 2 × 7 = 14 are not factors of 16, and moving up to the next even number, 4, makes the smallest possible value for ab 20, so eliminate (C). Similarly, if a is 2, the only possible values of ab that are less than 34 are 6, 10, 14, 22, and 26. None of these are factors of 34, so you’ll have to consider a = 4. Since ab = 20 and ab = 28 are not factors of 34, eliminate choice (D). Since 6 × 9 = 54 and 9 is a valid value for c, 54 is indeed one possible value for abc, and choice (E) is correct.
Step 4: Confirm Your Answer
Check that you used permissible values when checking the answer choices.
16. (A)
A fair die with sides numbered 1, 2, 3, 4, 5, and 6 is to be rolled 4 times. What is the probability that on at least one roll the number showing will be less than 3?
Step 1: Analyze the Question
Note that “less than 3” means a 1 or 2.
Step 2: State the Task
You need the probability that at least one roll is a 1 or 2. That means: find the probability that one, two, three, or four rolls result in a 1 or 2.
Step 3: Approach Strategically
Instead of calculating all those probabilities separately and adding them together, it will be easier to find the probability that no rolls result in a 1 or 2, and then subtract that probability from 1. Start by finding the probability that each of the 4 rolls results in at least a 3. The probability formula is 
. When rolling the die once, there are 4 desired outcomes, which are 3, 4, 5, and 6, and there are 6 possible outcomes, which are 1, 2, 3, 4, 5, and 6. The probability that when the die is rolled once, a number greater than or equal to 3 results is 
. Since the results of the 4 rolls of the dice are independent of each other, the probability that all 4 rolls result in a number greater than or equal to 3 is 
. It follows that the probability that at least one roll results in a number less than 3 is 
.
Choice (A) is correct.
Step 4: Confirm Your Answer
Check that you’ve calculated the probability of rolling a 3, 4, 5, or 6 correctly; check that you’ve accounted for four rolls; and check your subtraction.
17. (C)
There are 816 students in enrolled at a certain high school. Each of these students is taking at least one of the subjects economics, geography, and biology. The sum of the number of students taking exactly one of these subjects and the number of students taking all 3 of these subjects is 5 times the number of students taking exactly 2 of these subjects. The ratio of the number of students taking only the two subjects economics and geography to the number of students taking only the two subjects economics and biology to the number of students taking only the two subjects geography and biology is 3:6:8. How many of the students enrolled at this high school are taking only the two subjects geography and biology?
35
42
64
136
240
Step 1: Analyze the Question
This is a tough overlapping sets question. You are given the total number of students (816), sufficient information to set up two equations about students taking one, two, and three of the subjects, and ratios of the various two-subject combinations.
Step 2: State the Task
You need to find the number of students taking one particular two-subject combination: geography and biology.
Step 3: Approach Strategically
Start by setting up a couple of equations concerning students taking one, two, and three subjects. Say that the number of students taking exactly one of the subjects is x, say that the number of students taking exactly 2 of the subjects is y, and say that the number of students taking all 3 of the subjects is z. Since there are 816 students in the high school, x + y + z = 816. Since the sum of the number of students taking exactly one of the subjects and the number of students taking all 3 of the subjects is 5 times the number of students taking exactly 2 of the subjects, x + z = 5y. Now rewrite the equation x + y + z = 816 as (x + z) + y = 816. Substitute 5y for x + z to find that 5y + y = 816, 6y = 816, and y = 
= 36. Thus, the number of students taking exactly 2 of the subjects is 136.
Now make use of the ratios provided. The ratio of the number of students taking only economics and geography to the number of students taking only economics and biology to the number of students taking only geography and biology is 3:6:8, so the number of students taking only the two subjects geography and biology is 
.
Choice (C) is correct.
Step 4: Confirm Your Answer
One way to confirm your answer is to calculate how many students are taking the other two-subject combinations, and then add the three values to see if they in fact sum to 136: 
(136) = (3)(8) = 24 and 
(136) = (6)(8) = 48. It is in fact the case that 64 + 24 + 48 = 136.
18. (B)
Working alone at a constant rate, machine P produces a widgets in 3 hours. Working alone at a constant rate, machine Q produces b widgets in 4 hours. If machines P and Q work together for c hours, then in terms of a, b, and c, how many widgets will machines P and Q produce?
4ac + 3bc
Step 1: Analyze the Question
The question gives the production rates for three machines. The answer choices are given in algebraic terms.
Step 2: State the Task
Find the number of widgets produced by two of the three machines.
Step 3: Approach Strategically
Since machine P produces a widgets in 3 hours, it produces widgets at the rate of 
widgets per hour. By the same logic, machine P produces widgets at the rate of 
widgets per hour. So working together, Machine P and Machine Q produce widgets at the rate of 
widgets per hour. Recall that Work = Rate × Time. Thus, in c hours, working together, Machine P and Machine Q produce 
widgets.
None of the answer choices is written as 
. So rewrite this expression:
Choice (B) is correct.
Step 4: Confirm Your Answer
Double-check your calculations to make sure your algebra is correct.
19. (C)
In the figure above, the perimeter of triangle I is 16 feet greater than the perimeter of triangle II. What is the length of PQ, in feet?
27
51
68
75
85
Step 1: Analyze the Question
The question shows two right triangles arranged so that the hypotenuse of one triangle is also one of the legs of the other triangle. Dimensions of some of the sides are provided in terms of the variable y. The question also provides information about the difference between the perimeters of the two triangles. The answer choices are numbers, not expressed in terms of y.
Step 2: State the Task
You must find the length of the unshared hypotenuse.
Step 3: Approach Strategically
Describe the lengths of the hypotenuses of right triangles I and II in terms of y and feet. Looking at right triangle II, you see that the leg of length 9y feet can be expressed as 3 × (3y feet) and the leg of length 12y feet is 4 × (3y feet). So right triangle II is a 3-4-5 right triangle with each member of the 3 to 4 to 5 ratio multiplied by 3y feet. That means the length of the hypotenuse of right triangle II is 5 × (3y feet) = 15y feet.
Now find the length of PQ. Triangle I is a right triangle with one leg of length 8y feet and the other leg (the hypotenuse of triangle II) of 15y feet. You may have the ratio 8:15:17 memorized, and if so, you know that PQ is 17y feet. Alternatively, you can use the Pythagorean theorem: a2 + b2 = c2, or 82 + 152 = PQ2.
You now know the lengths of all the sides of triangles I and II.
The perimeter of triangle I is 8y + 15y + 17y = 40y.
The perimeter of triangle II is 9y + 12y + 15y = 36y.
The perimeter of triangle I is 16 feet greater than the perimeter of triangle II. So 40y = 36y + 16, 4y = 16, and y = 4. The length of PQ, in feet, is 17y = 17(4) = 68. Answer choice (C) is correct.
Step 4: Confirm Your Answer
Since PQ is the hypotenuse of the triangle that has a leg that is the hypotenuse of the other triangle, PQ must be the longest side of either triangle. The greatest length of the other sides is 15y, so 17y is a logical value.
20. (E)
If x > 0, y > 0, and 
, what is the value of 
?
Step 1: Analyze the Question
The problem presents an equation in two variables, x and y, with quadratic expressions in both the numerator and denominator. The answer choices are in the form of fractions, with no variables.
Step 2: State the Task
Your task is to find the numeric value of an expression stated in terms of x and y.
Step 3: Approach Strategically
Begin by simplifying the equation 
.
7x2 + 72xy + 4y2 = 4(4x2 + 12xy + 5y2)
7x2 + 72xy + 4y2 = 16x2 + 48xy + 20y2
72xy + 4y2 = 9x2 + 48xy + 20y2
24xy + 4y2 = 9x2 + 20y2
24xy = 9x2 + 16y2
9x2 − 24xy + 16y2 = 0
(3x − 4y)(3x − 4y) = 0
Thus, 3x − 4y = 0.
You are seeking the value of 
. Rewrite this as 
.
Solve the equation 3x − 4y = 0 for the value of 
:
Thus, 
. Answer choice (E) is correct.
Step 4: Confirm Your Answer
Unfortunately, there is no easy way to plug the solution back into the problem. Check the math used to derive your answer.
Answers and Explanations
ADVANCED DATA SUFFICIENCY PRACTICE SET
1. (A)
If x and y are positive even integers, is (40x)x divisible by y?
(1) 
(2) y is a multiple of 160.
Step 1: Analyze the Question Stem
This is a Yes/No question. Essentially, the question asks whether 
is an integer. Note that x and y are both even. Also note that because x is even and 40 is an integer, the expression (40x)x must be even. Finally, note that x is a factor of the base as well as the exponent.
To determine that y divides evenly into (40x)x, you need to know that y has no prime factors that are not found in 40 or x and that it has no prime factor raised to an exponent that is greater than the number of times that prime number is a factor of (40x)x. If a statement allows you to establish both these conditions, you can answer the question with a yes, and the statement is sufficient. If you can show that either of these conditions does not exist, then the statement allows you to answer the question with a no and is likewise sufficient.
Step 2: Evaluate the Statements Using 12TEN
Statement (1) tells you that y = 128x. Taking this statement into account, the question now becomes whether 
is an integer. The number 128 has only 2 as a prime factor, breaking down to 27. So focus on the number of times 2 is a factor of the expression (40x)x. Remember that x is positive and even, so x ≥ 2. Since x is the exponent in the expression (40x)x, a greater value of x will make the number of times 2 is a factor of (40x)x greater. It follows that if x = 2 makes 128x a factor of (40x)x, the answer to the question is definitively yes, and you have sufficiency. (Note that if x > 2, all other factors of x besides 2 will cancel in the denominator because there will be a greater number of them in the numerator due to the exponent. For instance, if x = 6 = (2)(3), then there will be one factor of 3 in the denominator and 6 factors of 3 in the numerator.)
Substituting 2 for x yields:
So x = 2 makes 128x a factor of (40x)x, and Statement (1) is sufficient. Eliminate (B), (C), and (E).
Statement (2) tells you that y is a multiple of 160 and thus has 2 and 5 as its distinct prime factors. While that may seem to align with the 40 in the exponent, y being a multiple of 160 means that it could be infinitely large and thus potentially too big to be a factor, given that we don’t know the value of x. Statement (2) is insufficient.
Choice (A) is the correct answer.
2. (E)
If 
, and 
, is 
?
(1) a2 − 2a − 3 = 0
(2) b2 − 4b + 4 = 1
Step 1: Analyze the Question Stem
The question stem gives two equations and asks whether a third algebraic fraction is greater than a constant, making this a Yes/No question. You can simplify the given information by combining the two equations. Note that multiplying them together will make the y cancel out: 
results in 
. The question asks about 
, so if you can find the values of a and b, you will have sufficiency. You will also have sufficiency if you can find the value of 
. Since 
is the reciprocal of , you will also have sufficiency if you can find the value of . Finally, you will have sufficiency if you can find that, in some other way, there is only one answer to the question regarding 
and .
Step 2: Evaluate the Statements Using 12TEN
Since each statement uses only one of the variables needed, (A), (B), and (D) can be crossed off immediately. The task then becomes determining whether the two statements taken together are sufficient or not.
Neither of the statements is simpler than the other, so begin by factoring either of them. In Statement (1), factoring the left side of the equation a2 − 2a − 3 = 0 yields (a + 1)(a − 3) = 0, making a = −1 or 3. In Statement (2), subtract 1 from both sides, resulting in b2 − 4b + 3 = 0. Factoring the left side of the equation b2 − 4b + 3 = 0 produces (b − 1)(b − 3) = 0, making b = 1 or b = 3.
As noted in Step 1, in order to answer the question is 
, you must know the values of a and b, or you must know the value 
, or you must know the value 
, or you must be able to compare 
and in some other way. The four possible combinations of a and b are these:
a = −1 and b = 1
a = −1 and b = 3
a = 3 and b = 1
a = 3 and b = 3
Test the first pair, as it is the easiest. If a = −1 and b = 1, then 
, which is not greater than . In this case, the answer to the question is no.
To try to avoid having to test all four combinations, notice that with both a and b positive, 
will be negative, and the answer will still be no. So stay with a = −1. You can expect a greater value for 
when b = 3 than when b = 1, so test b = 3. If a = −1 and b = 3, 
. In this case, 
and the answer to the question is yes.
One possibility gives you an answer of yes to the question and a different possibility gives you an answer of no to the question, making the statements taken together insufficient to answer the question definitively. The correct answer is (E).
3. (B)
In the figure above, what is the area of semicircle DPA?
(1) The area of quadrilateral ABCD is 140.
(2) The length of the line segment whose endpoints are B and D is 25.
Step 1: Analyze the Question Stem
This is a Value question. The question stem contains a diagram with multiple figures, and you need to be able to find a single possible area of the semicircle DPA to have sufficiency. In order to do that, you would need to know the length of diameter AD of the semicircle. AD is also a side of quadrilateral ABCD. Because a semicircle has half the area of a circle with the same diameter, knowing the length of AD would allow you to calculate the area of semicircle DPA.
Step 2: Evaluate the Statements Using 12TEN
Statement (1) tells you that the area of the quadrilateral ABCD is 140. Using the formula for the area of a trapezoid, area = 
, it would appear that you could plug in the information you know from the diagram and Statement (1) (the area, the height, and b1) to solve for b2. However, it’s important to remember that figures in Data Sufficiency questions are not necessarily drawn to scale. You are not given enough information to determine whether sides AD and BC are parallel; therefore, ABCD is not necessarily a trapezoid. Statement (1) is thus insufficient, and you can eliminate choices (A) and (D).
Statement (2) tells you that the length of the line segment whose endpoints are B and D is 25. This line segment, if added to the figure, would be the hypotenuse of right triangle ABD. Since you now know two sides of a right triangle, it would be possible to calculate the length of the third side, AD, by using the Pythagorean theorem. As mentioned above, knowing the length of AD would allow you to calculate the area of the semicircle, and Statement (2) is sufficient. Therefore, the correct answer is (B).
4. (A)
Is x divisible by 39?
(1) x divided by 65 results in a remainder of 7.
(2) x divided by 36 results in a remainder of 15.
Step 1: Analyze the Question
This is a Yes/No question. Unlike other remainder questions in which you can pick numbers, the remainders in the statements would be very tedious and time-consuming to test multiple times individually, let alone to combine. Thus, there must be a simpler way to determine divisibility. Whenever the numbers given are unwieldy, the easiest and most efficient way to establish divisibility is to find the prime factors. In this case, 39 breaks down to 3 × 13, so for x to be divisible by 39, it must be divisible by both 3 and 13.
Step 2: Evaluate the Statements Using 12TEN
Statement (1) gives the number 65. Consider that 65 is 5 × 13; one of the prime factors of 39 is present. For x to be divisible by 13, the remainder itself must be a multiple of 13. The remainder 7 is not a multiple of 13, so it is impossible for x to be divisible by 13 or 39. This makes the answer to the question always no, and Statement (1) is sufficient. Eliminate choices (B), (C), and (E).
In evaluating Statement (2), you can use the same principle. The number 36 is divisible by 3, one of the necessary primes, so the remainder must also be divisible by 3. The remainder 15 is divisible by 3, meaning that x is a multiple of 3. Since you don’t know anything else about x, it’s entirely possible that x is also divisible by 13, while there are certainly values that would make x not divisible by 13. Statement (2) is insufficient.
The correct answer is (A).
5. (A)
If a and b are positive integers with different units digits, and b is the square of an integer, is a also the square of an integer?
(1) The units digit of a + b is 8.
(2) b = 121
Step 1: Analyze the Question Stem
This is a Yes/No question. The given information provides a clue about the two numbers having different units digits and tells you that b is a perfect square. On advanced number properties questions, it’s worth checking if two different types of clues work together. Units digits of certain groups of numbers often fall into consistent patterns, and squares certainly fit into that category. The units digits of the squares of 1–10 are {1, 4, 9, 6, 5, 6, 9, 4, 1, 0}, and this pattern will continue, since the units digits of 11–20, 21–30, and so on will be the same as those of 1–10. So a perfect square can only have one of six units digits: 0, 1, 4, 5, 6, or 9. For a to be a perfect square, its units digit must be one of those numbers.
Step 2: Evaluate the Statements Using 12TEN
Evaluating Statement (1), you see that the sum of the integers a and b has a units digit of 8. Work your way through the list of possible units digits of squares of integers. Again, that list is 0, 1, 4, 5, 6, and 9. If the units digit of b is 0, then the units digit of a is 8, and 8 is not on the list. If the units digit of b is 1, the units digit of a is 7, and 7 is also not on the list. If the units digit of b is 4, the units digit of a is 4, but this is not permitted because the units digits of a and b must be different. If the units digit of b is 5, the units digit of a is 3, and 3 is not on the list. If the units digit of b is 6, the units digit of a is 2, and 2 is not on the list. If the units digit of b is 9, the units digit of a is 9, but again, the units digits of a and b must be different. It follows that the units digit of a cannot be the units digit of the square of an integer, so a cannot be the square of an integer. The answer to the question is definitively no, and Statement (1) is sufficient. Eliminate choices (B), (C), and (E).
Statement (2) precludes a from having 1 as its units digit, but it does nothing else, since the only relevant limitation is that the two numbers have different units digits. Statement (2) is insufficient.
The correct answer is (A).
6. (C)
What is the value of 8x + y?
(1) 3x − 2y + z = 10
(2) 2(x + 3y) − (y + 2z) = −25
Step 1: Analyze the Question Stem
This is a Value question: a statement that allows the calculation of a single value for 8x + y is sufficient. Simplify the equation in Statement (2):
2(x + 3y) − (y + 2z) = −25
2x + 6y − y − 2z = −25
2x + 5y − 2z = −25
Step 2: Evaluate the Statements Using 12TEN
Each of the two statements is an equation with the same three variables. As neither of these equations can be solved for 8x + y, neither statement on its own is sufficient, and you can eliminate choices (A), (B), and (D).
Taken together, the two statements represent a system of equations. To solve for each variable in a system of linear equations with 3 variables, 3 three distinct equations are required. Since the statements provide only two equations, choice (E) may be tempting. However, even when there are fewer equations than variables, sometimes the value of a particular variable, or the value of an expression, can be found. When a question concerns solving a system of equations for an expression, consider whether multiplying one or both equations by a constant and combining them will allow you to solve. Multiplying the equation in Statement (1) by 2 results in 6x − 4y + 2z = 20. Adding the corresponding sides of this equation and the simplified equation in Statement (2), 2x + 5y − 2z = −25, gives 8x + y = −5. Since both statements together allow the calculation of 8x + y, the correct answer is (C).
7. (C)
If b and c are two-digit positive integers and b − c = 22d, is d an integer?
(1) The tens digit and the units digit of b are identical.
(2) 
, and x is an integer.
Step 1: Analyze the Question Stem
This is a Yes/No question. The question stem tells us that the variables b and c are integers between 10 and 99 inclusive and asks whether b − c is a multiple of 22.
Step 2: Evaluate the Statements Using 12TEN
Statement (1) merely tells you that the two digits of b are the same, so the value of b must be 11, 22, 33, etc. Since you are given no further information about c, Statement (1) by itself is insufficient, and answer choices (A) and (D) can be eliminated.
Statement (2) provides the information that b + c is a multiple of 22, since b + c = 22x and x is an integer. However, the question asks about b − c, not b + c, so this statement, too, is insufficient. If you were unsure, you could plug in some number pairs to verify that b − c could be a multiple of 22 but need not be. Set b = 12 and c = 10; b + c = 22, which is divisible by 22, but b − c = 2, which clearly is not divisible by 22. Now try b = 11 and c = 33; b + c = 44 is divisible by 22, while b − c = −22 is divisible by 22 as well. Eliminate answer choice (B).
Now look at both statements together along with the information in the question stem. Statement (1) says that the tens and units digits of b are the same, so b is a multiple of 11. From statement (2), you know that b + c is a multiple of 22. Every multiple of 22 is a multiple of 11, so b + c is a multiple of 11. Since b + c is a multiple of 11 and b is a multiple of 11, (b + c) − b = c, which is a multiple of 11. Since b and c are both multiples of 11, b − c is also a multiple of 11. Since b + c is a multiple of 22, b + c is even. This means that both b and c are even or both b and c are odd. In either case, b − c is even, or, in other words, b − c is a multiple of 2. Since b − c is a multiple of 11, b − c is a multiple of 2, and the integers 2 and 11 have no common factor greater than 1, b − c is a multiple of 11 × 2 = 22. The statements taken together are sufficient to answer the question definitively yes. Choice (C) is correct.
8. (C)
The integers x and y are positive, x > y + 8, and y > 8. What is the remainder when x2 − y2 is divided by 8?
(1) The remainder when x + y is divided by 8 is 7.
(2) The remainder when x − y is divided by 8 is 5.
Step 1: Analyze the Question Stem
This is a Value question. You are given two positive integers. Since y is an integer and y > 8, the minimum value of y is 9. Since x > y + 8, and the minimum value of y is 9, x must be greater than 9 + 8 = 17. Thus, x > 17. Since x is an integer, the minimum value of x is 18. The question asks for the remainder when the difference of the squares of the two numbers is divided by 8. You can factor the expression x2 − y2 to its equivalent (x + y)(x − y).
Step 2: Evaluate the Statements Using 12TEN
Statement (1): Since the remainder when x + y is divided by 8 is 7, you can represent x + y as 8m + 7, where m is an integer. Substituting this into the factored expression, (8m + 7)(x − y) still doesn’t tell you enough about x − y to get a definite answer, so Statement (1) is insufficient. Eliminate (A) and (D).
Statement (2): This time, you can substitute 8n + 5 for x − y, where n is an integer, but now you don’t know enough about x + y, so Statement (2) is also insufficient and you can eliminate (B).
Combining the results above, can you determine with certainty the remainder when (8m + 7)(8n + 5) is divided by 8?
Using FOIL, you can write the expression (8m + 7)(8n + 5) as 64mn + 40m + 56n + 35. Since the first three terms contain the coefficients 64, 40, and 56, respectively, and m and n are integers, each of these terms is evenly divisible by 8. Thus, the remainder resulting from dividing the expression x2 − y2 by 8 is the remainder when 35 is divided by 8, which is 3. So the statements together are sufficient, and answer choice (C) is correct.
9. (D)
If x > y > 0, does 3x + 1 + 3(2y) = 12v?
(1) 
(2) 2(3x + 2) + 9(2y + 1) = 72v
Step 1: Analyze the Question Stem
This is a Yes/No question. A statement is sufficient if it allows you to answer the question with either a definite yes or a definite no.
Begin by simplifying the equation 3x + 1 + 3(2y) = 12v. Using the law of exponents that says that babc = ba + c, you can rewrite 3x + 1 as (3x)(31) = 3(3x). So replace 3x + 1 with 3(3x): 3(3x) + 3(2y) = 12v. Now factor a 3 out of the left side to yield 3(3x + 2y) = 12v. Dividing both sides of the equation by 3 results in 3x + 2y = 4v. So the question stem is asking, “If x > y > 0, does 3x + 2y = 4v?”
Step 2: Evaluate the Statements Using 12TEN
Statement (1): Rewrite the equation 
to evaluate it. There is a law of exponents that says that (ba)c = bac, so 32x = (3x)2 and 22y = (2y)2. It follows that 32x − 22y = (3x)2 − (2y)2. This is one of the classic quadratics: (3x)2 − (2y)2 = (3x − 2y)(3x + 2y). So Statement (1) can be written as 
. Cancel a factor of 3x − 2y from the numerator and denominator of the left side of the equation to find that 3x + 2y = 4v. Since 3x + 2y = 4v is equivalent to 3x + 1 + 3(2y) = 12v, Statement (1) allows you to answer the question with a definite yes and is sufficient. Eliminate choices (B), (C), and (E).
Statement (2): Because babc = ba + c, 3x + 2 = (3x)(32) = (3x)(9) = 9(3x). Also, 2y + 1 = (2y)(21) = (2y)(2) = 2(2y). Thus, 2(3x + 2) + 9(2y + 1) = 72v can be written as 2[9(3x)] + 9[2(2y)] = 72v, or 18(3x) + 18(2y) = 72v. Factoring 18 out of the left side of the equation gives 18(3x + 2y) = 72v. Divide both sides of the equation by 18: 3x + 2y = 4v. Since this is the same equation shown to be sufficient for Statement (1), Statement (2) allows you to answer the question with a definite yes and is sufficient.
Choice (D) is correct.
10. (B)
In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?
(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is 
.
(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.
Step 1: Analyze the Question Stem
This is a Value question. The figure shows a right triangle whose hypotenuse is one side of a square, and a statement that is sufficient would allow you to find the length of side AB.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): The area of right triangle ABE is 
. Since triangle ABE is a right triangle, the Pythagorean theorem says that (BE)2 = (AB)2 + 122 = (AB)2 + 144. Thus, 
. The area of square BCDE is 
. Since the ratio of the area of triangle ABE to the area of square BCDE is 
. Solve this equation for AB:
Factor y2 − 25y + 144. With some testing, you find that y2 − 25y + 144 = (y − 9)(y − 16). So (y − 9)(y − 16) = 0. Thus, it is possible that the length of AB is 9 or 16. Because more than one answer to the question is possible, Statement (1) is insufficient. Eliminate choices (A) and (D).
Statement (2): The perimeter of square ABCD is 80, so the length of one side of the square is 
= 20. Because the ratio of AE to AB is 3 to 4, right triangle ABE is a 3-4-5 right triangle. Knowing the ratios of the sides and the length of the hypotenuse, you could calculate the length of side AB. However, since this is a Data Sufficiency question, there is no need to perform the calculations. Statement (2) is sufficient. (For the record, right triangle ABE has side lengths of 12, 16, and 20. AB is the longer leg and equals 16.)
Choice (B) is correct.
11. (E)
In the sequence T, the first term is the non-zero number a, and each term after the first term is equal to the non-zero number r multiplied by the previous term. What is the value of the fourth term of the sequence?
(1) The sum of the first 2 terms of the sequence is 16.
(2) The 18th term of the sequence is 81 times the 14th term of the sequence.
Step 1: Analyze the Question Stem
This is a Value question involving a geometric sequence. Each term after the first term is r times the previous term. So if the nth term is an, where n is an integer and n ≥ 1, then a1 = a, a2 = ar, a3 = ar2, a4 = ar3, a5 = ar4, and so on. In general, if n is an integer and n ≥ 1, then the nth term is the product of one factor of a and n − 1 factors of r, or an = arn − 1.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): The sum of the first two terms is 16. Thus, a + ar = 16. This is one equation with two variables, and there are many possible values for a and r. Statement (1) is insufficient. Eliminate choices (A) and (D).
Statement (2): This statement says that a18 = 81a14. You know the following:
a15 = a14r
a16 = a15r
a17 = a16r
a18 = a17r
So a18 = a17r = (a16r)r = a16r2 = (a15r)r2 = a15r3 = (a14r)r3 = a14r4.
Since a18 = 81a14, a14r4 = 81a14. According to the question stem, a ≠ 0 and r ≠ 0, so a14 does not equal 0. Divide by a14: r4 = 81. Now 81 = 34 or (−3)4, so r = 3 or r = −3. This statement provides two values for r and no information about a, so the fourth term cannot be calculated. For example, if a = 4 and r = 3, then a4 = 4(34 − 1) = 4(33) = 4(27) = 108. But if a = 2 and r = −3, then a4 = 2(−3)4 − 1 = 2(−3)3 = 2(−27) = −54. Since different answers to the question are possible, statement (2) is insufficient. Eliminate choice (B).
The statements taken together: from statement (1), which says that the sum of the first two terms is 16, you know that a + ar = 16. From statement (2), which says that a18 = 81a14, you know that r = 3 or r = −3.
Consider the case where r = 3. Substituting 3 for r into the equation a + ar = 16, a + a(3) = 16, a + 3a = 16, 4a = 16, and a = 4. In this case, the fourth term is ar4 − 1 = 4(34 − 1) = 4(33) = 4(27) = 108. Now consider the case where r = −3. Substituting −3 for r into the equation a + ar = 16, a + a(−3) = 16, a − 3a = 16, −2a = 16, and a = −8. With these values of r and a, the fourth term is (−8)(−3)4 − 1 = (−8)(−3)3 = (−8)(−27) = 216. Because there is more than one possible answer to the question, the statements taken together are insufficient.
Choice (E) is correct.
12. (C)
A person is to be selected at random from the group T of people. What is the probability that the person selected is a member of club E?
(1) The probability that a person selected at random from group T is not a member of club D and is not a member of club E is 
.
(2) The probability that a person selected at random from group T is a member of club D and not a member of club E is 
.
Step 1: Analyze the Question Stem
This is a Value question involving the probability of a single event. A person is to be selected at random from group T, and the question asks for the probability that the person selected is a member of club E. The probability that the person selected is a member of club D can be written as P(D) and the probability that the person is a member of club E as P(E).
Step 2: Evaluate the Statements Using 12TEN
Statement (1): The probability that the person selected is not a member of either club is 
. The probability that an event does not occur is equal to 1 minus the probability that the event does occur, so the probability that the person chosen is a member of at least one of the two clubs—that is, P(D or E)—is 
. In general, P(D or E) = P(D) + P(E) − P(D and E). (This formula is worth memorizing.) In this case, you can substitute 
for P(D or E), so P(D) + P(E) − P(D and E) = 
. However, without any other information about P(D) or P(D and E), you cannot find P(E). Statement (1) is insufficient. Eliminate choices (A) and (D).
Statement (2): Because the probability that the chosen person is a member of club D and not club E is 
, P(D and not E) = 
. Since P(D) = P(D and E) + P(D and not E), P(D and not E) = P(D) − P(D and E). Thus, P(D) − P(D and E) = 
. There is no way to find P(E) from this information. Statement (2) is insufficient, and choice (B) can be eliminated.
The statements taken together: from statement (1), P(D) + P(E) − P(D and E) = 
. From statement (2), P(D) − P(D and E) = 
. You can rearrange the equation from statement (1):
P(D) − P(D and E) + P(E) = 
. Using P(D) − P(D and E) = 
, substitute 
for P(D) − P(D and E) in P(D) − P(D and E) + P(E) = 
. Then 
+ P(E) = 
, and you can solve for the probability that the person selected is a member of club E. There is no need to do the calculation, but for the record, it would be 
. The statements taken together are sufficient.
Choice (C) is correct.
13. (D)
The population of Town X on January 1, 2010, was 56 percent greater than the population of the same town on January 1, 2005. The population of Town X on January 1, 2015, was 75 percent greater than the population of the same town on January 1, 2010. What was the population of Town X on January 1, 2005?
(1) The population of Town X on January 1, 2015, was 21,840.
(2) The increase in the population of Town X from January 1, 2010, to January 1, 2015, was 4,880 greater than the increase in the population of Town X from January 1, 2005, to January 1, 2010.
Step 1: Analyze the Question Stem
This is a Value question involving multiple percent increases. Say the population of Town X on January 1, 2005, was N. Because the population of Town X on January 1, 2010, was 56% greater than the population on January 1, 2005, the population in 2010 was N + (56% of N) = N + 0.56N = 1.56N. Then because the population on January 1, 2015, was 75% greater than the population on January 1, 2010, the population in 2015 can be written as follows:
Note that because you know the percent increase from each of the three named years to the next, a statement that provides the population for any of those three years will be sufficient.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): Because the population on January 1, 2015, was 2.73N, you can write the equation 2.73N = 21,840. Then 
. This equation produces just one value for N, so statement (1) is sufficient. Eliminate choices (B), (C), and (E).
Statement (2): The increase in the population from 2005 to 2010 can be written as 1.56N − N = 0.56N. The increase in the population from 2010 to 2015 is 2.73N −1.56N = 1.17N. Because the increase from 2010 to 2015 was 4,880 greater than the increase from 2005 to 2010, you can write the equation 1.17N = 0.56N + 4,880. There is one equation with one variable, so you could solve for N and statement (2) is sufficient.
Choice (D) is correct.
14. (C)
What is the value of x − 3z?
(1) x + 4y = 3
(2) x2 + 4xy − 3xz − 12yz = 24
Step 1: Analyze the Question Stem
This is a Value question. Knowing the values of x and z separately would be sufficient to answer the question, as would knowing the value of the entire expression x − 3z.
Step 2: Evaluate the Statements Using 12TEN
Statement (1) is clearly insufficient, as the equation x + 4y = 3 does not include the variable z. Eliminate choices (A) and (D).
Statement (2) provides one equation with 3 variables, so there will be different possible values for x − 3z. For example, if you let x = 0 and y = 8, then the equation x2 + 4xy − 3xz − 12yz = 24 leads to 
. Plugging x = 0 and 
into x − 3z, you have 
. In this case, the answer to the question is 
. But if you let x = 3 and y = 0, the equation x2 + 4xy − 3xz − 12yz = 24 leads to 
. Plugging x = 3 and 
into x − 3z gives you the following: 
. In this case, the answer to the question is 8. Since different answers to the question are possible, statement (2) is insufficient. Eliminate choice (B).
To combine the statements, start by simplifying the left side of the equation in Statement (2):
x2 + 4xy − 3xz − 12yz = x(x + 4y) − 3z(x + 4y) = (x + 4y)(x − 3z)
Thus, statement (2) is equivalent to (x + 4y)(x − 3z) = 24. Statement (1) says that x + 4y = 3. Substitute 3 for x + 4y in the equation (x + 4y)(x − 3z) = 24 to produce 3(x − 3z) = 24. Dividing both sides by 3 yields x − 3z = 8. The statements together lead to a single possible value of 8 for x − 3z. The statements taken together are sufficient.
Choice (C) is correct.
15. (D)
In the figure above, the center of the circle is O, and the measure of angle CAO is 60 degrees. What is the perimeter of triangle OAC?
(1) The length of arc CDA is 16π greater than the length of arc ABC.
(2) The area of triangle OAC is 
.
Step 1: Analyze the Question Stem
This question asks for the perimeter of the triangle, so it is a Value question. Sides OA and OC of triangle OAC are both radii of the circle, so OA = OC, making the triangle isosceles, so that ∠OCA must also equal 60°. Because the three angles of a triangle must sum to 180°, the central angle is also 60°—in other words, triangle OAC is equilateral. So each of its three sides is equal to the circle’s radius. If you can find the radius, you can find the perimeter of the triangle, and you have sufficiency.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): In a circle with radius r, the length L of an arc intercepted by a central angle whose measure is n degrees is given by the formula 
. Here, the central angle intercepting arc CDA has a measure of 360° − 60° = 300°, and the central angle intercepting arc ABC has a measure of 60°. So the length of arc CDA is 
, and the length of arc ABC is 
. Since Statement (1) says that the length of arc CDA is 16π greater than the length of arc ABC, you know that 
. That’s a single equation in one variable, so you can solve for r. Statement (1) is sufficient. (For the record, dividing both sides of this equation by π and multiplying both sides by 3 yields 5r = r + 48, so r = 12, and the perimeter is 3(12) = 36.) Eliminate choices (B), (C), and (E).
Statement (2) says that the area of triangle OAC is 
. Triangle OAC is equilateral. Call the length of each side of triangle OAC s.
Drop an altitude from point C to side OA:
Altitude CD divides triangle OAC into 2 identical 30°-60°-90° degree right triangles.
The 2 identical 30°-60°-90° degree right triangles are triangles ACE and OCE. The side lengths in a 30°-60°-90° degree right triangle are in a ratio of 1 to 
to 2. Since the length of hypotenuse AC of right triangle ACE is s, the length of leg AE, which is opposite the 30-degree angle must be 
, and the length of leg CE, which is opposite the 60-degree angle, must be 
. You can now describe the area of triangle OAC in terms of s. The area of any triangle is 
. Call the base is OA and the height CE. Then OA = s and 
. The area of triangle OAC is 
. Since the area of triangle OAC is given in statement (2) to be 
, you can write the equation 
. That’s again a single equation in one variable. It does have a squared term, but as negative lengths are disallowed, you can stop here and declare Statement (2) sufficient. (For the record: 
so s2 = 144 and s = 12. The perimeter is again 3(12) = 36.)
Choice (D) is correct.
16. (B)
If x and y are integers and 3x > 8y, is y > −18?
(1) −9 < x < 20
(2) 
Step 1: Analyze the Question Stem
This is a Yes/No question with two integer variables. Simplify the inequality given in the stem, 3x > 8y, to get 
.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): The possible values of x are limited by the range of this inequality, so examine the endpoints of the range. If x > −9, the least value that is permissible for x is −8. Since y is less than 
, it follows that y < −3. Some values of y could be > −18, but there is no lower limit for y, so it could also be < −18. There is no need to evaluate the upper boundary of x. Statement (1) is insufficient, so eliminate choices (A) and (D).
Statement (2): Simplify the equation by multiplying both sides by 5 to get 5y = x − 42. Add 42 to both sides and x = 5y + 42. Substitute this value for x into the inequality 3x > 8y:
3(5y + 42) > 8y
15y + 126 > 8y
7y + 126 > 0
7y > −126
y > −18
Statement (2) is sufficient to answer definitively yes.
Choice (B) is correct.
17. (A)
Over the course of 5 days, Monday through Friday, Danny collects a total of 76 baseball cards. Each day, he collects a different number of cards. If Danny collects the largest number of cards on Friday and the second largest number of cards on Thursday, did Danny collect more than 8 cards on Thursday?
(1) On Friday, Danny collected 49 cards.
(2) On one of the first 3 days, Danny collected 6 cards.
Step 1: Analyze the Question Stem
This is a Yes/No question that is packed with information in the stem. Danny collects a different number of cards each of the 5 days. He collects the largest number of cards on Friday and the second largest number on Thursday. The question asks if Danny collected more than 8 cards on Thursday.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): The largest number of cards collected on Thursday that would result in a “no” answer to the question is 8. Since the number of cards collected Monday, Tuesday, and Wednesday is different each day and must be less than 8, the maximum numbers of cards he could collect on those days are 5, 6, and 7. Totaling up all the days, 5 + 6 + 7 + 8 + 49 = 75. Collecting 8 cards on Thursday does not enable Danny to collect 76 cards in total given the constraints of the question, so the information in Statement (1) means that Danny must have collected more than 8 cards on Thursday. Statement (1) is sufficient to answer “always yes,” so eliminate choices (B), (C), and (E).
Statement (2): If Danny collects 6 cards on one of the first 3 days, the fewest cards he could have collected on those days would be 6 + 2 + 1 = 9, leaving − 76 − 9 = 67 cards to be collected on Thursday and Friday. Danny could have collected 7 cards on Thursday and 60 on Friday, in which case the answer to the question is “no.” Alternatively, he could have collected as many as 33 cards on Thursday and 34 on Friday and the answer would be “yes.” Therefore, statement (2) is insufficient.
Choice (A) is correct.
18. (A)
If y = x2, is the equation (25y)(4120) = 1620x true?
(1) (4 − x)(12 − x) = 0
(2) (x − 4)(x − 8)(x − 24) = 0
Step 1: Analyze the Question Stem
This question asks whether an equation is true, making it a Yes/No question. Start by simplifying the equation in the question stem. Notice that you can write all three powers with a base of 2, since 4 = 22 and 16 = 24. Substituting these values produces (25y)((22)120) = (24)20x, which simplifies to (25y)(22 × 120) = 24 × 20x, or (25y)(2240) = 280x. This, in turn, can be rewritten as 25y + 240 = 280x. When equal powers have the same base, the exponents must be equal. So 5y + 240 = 80x. Remember that y = x2, so you can substitute x2 for y to yield 5x2 + 240 = 80x. Subtracting 80x from both sides, 5x2 − 80x + 240 = 0. Dividing both sides by 5, x2 − 16x + 48 = 0. This factors to (x − 4)(x − 12) = 0. So x − 4 = 0 or x − 12 = 0, and x = 4 or x = 12.
So if x = 4 or x = 12, then the equation (25y)(4120) = (1620x) is true, and the answer to the question is definitively yes. If x is neither of these values, the answer is definitively no. Either of these situations would be sufficient.
Step 2: Evaluate the Statements Using 12TEN
Statement (1) says that (4 − x)(12 − x) = 0. When the product of a group of numbers is 0, at least one of the numbers must be 0. So if (4 − x)(12 − x) = 0, then 4 − x = 0 or 12 − x = 0. If 4 − x = 0, then 4 = x, or x = 4. If 12 − x = 0, then 12 = x, or x = 12. You’ve already determined that the equation (25y)(4120) = (1620x) is true when x = 4 or x = 12. Statement (1) is sufficient, as the answer to the question is always “yes.” Eliminate choices (B), (C), and (E).
Statement (2) says that (x − 4)(x − 8)(x − 24) = 0. Again, when the product of a group of numbers is 0, at least one of the numbers must be 0. So in this case, x − 4 = 0, x − 8 = 0, or x − 24 = 0. If x − 4 = 0, then x = 4. If x − 8 = 0, then x = 8. If x − 24 = 0, then x = 24. The possible values of x are 4, 8, and 24. You know that the equation (25y)(4120) = (1620x) is true when x = 4 or x = 12. If x = 4, then the answer to the question is “yes.” If x = 8, or if x = 24, the answer to the question is “no.” Since more than one answer to the question is possible, statement (2) is insufficient.
Choice (A) is correct.
19. (B)
The ratio of the number of students in an auditorium who are seniors to the number of students in the auditorium who are not seniors is 7:5. How many students are there in the auditorium?
(1) The ratio of the number of students who are seniors who are taking history to the number of students who are not seniors who are taking history is 21:5.
(2) Of the students in the auditorium who are seniors, 
are taking history; of the students in the auditorium who are not seniors, 
are taking history; and the number of seniors in the auditorium who are taking history is 208 greater than the number of students in the auditorium who are not seniors and taking history.
Step 1: Analyze the Question Stem
This is a Value question. Since the question asks for the total number of students, start by rewriting the part to part ratio in the question stem as a part to whole ratio. The ratio of seniors to non-seniors is 7:5, so the ratio of the students who are seniors to the total number of students is 
, and the ratio of the students in the auditorium who are not seniors to the total number of students is 
. Remember that there is a common multiplier in the numerator and denominator of every ratio, so you could write the ratio of seniors to total students as 
and the ratio of non-seniors to total students as 
. If you can find the common multiplier x, you can find the total number of students, and you have sufficiency.
Step 2: Evaluate the Statements Using 12TEN
Statement (1): You are given the ratio of the number of students who are seniors who are taking history to the number of students who are not seniors who are taking history. There is no actual number of students given, and no way to calculate the common multiplier in the original ratio and thus calculate the total number of students. Statement (1) is insufficient. Eliminate choices (A) and (D).
Statement (2): From the ratio given in the question stem, you know that there are 7x seniors, 5x non-seniors, and 12x total students present in the auditorium. From Statement (2), you know that of the 7x seniors, 
are taking history. In other words, 
seniors are taking history. Similarly, 
of the 5x non-seniors are taking history, so 
non-seniors are taking history. Furthermore, the number of seniors taking history is 208 greater than the number of non-seniors taking history, so you can write an equation to solve for 
. This is a single equation in one variable, so you can stop here and declare Statement (2) to be sufficient. (For the record: 21x = 5x + 208(5), 16x = 208(5), x = 13(5) = 65. There are 12x = (12)(65) = 780 total students.)
Choice (B) is correct.
20. (E)
If y > 0, is 
< 3?
(1) x(x + y) − 4y(x + y) < 0
(2) 5(y + 8) < 20y − 3x + 40
Step 1: Analyze the Question Stem
Since y is positive, you can multiply both sides of the inequality 
< 3 by the positive number y to obtain the equivalent inequality x < 3y. So the question is equivalent to “If y > 0, is x < 3y?”
Step 2: Evaluate the Statements Using 12TEN
Statement (1): Factor x + y out of the left side of the inequality to yield x(x + y) − 4y(x + y) = (x + y)(x − 4y). So statement (1) is equivalent to (x + y)(x − 4y) < 0. When the product of two quantities is negative, one of the quantities must be negative and the other quantity must be positive. So either (i) x + y < 0 and x − 4y > 0, or (ii) x + y > 0 and x − 4y < 0. (Keep in mind that y is positive.) In case (i), if x + y < 0, then x < −y. If x − 4y > 0, then x > 4y. So in case (i), x < −y and x > 4y. Since y is positive, x < −y means that x is less than −y, where −y is a negative number. Since y is positive, x > 4y means that x is greater than 4y, where 4y is a positive number. So (i) requires that x be less than the negative number −y and also requires that x be greater than the positive number 4y. This is impossible. So case (i), which is x + y < 0 and x − 4y > 0, cannot happen. On to case (ii), which is x + y > 0 and x − 4y < 0. (Again, keep in mind that y > 0.) If x + y > 0, then x > −y. If x − 4y < 0, then x < 4y. So this time, x > −y and x < 4y. Since y is positive, x > −y means that x is greater than the negative number −y. Since y is positive, x < 4y means that x is less than the positive number 4y. So case (ii), x + y > 0 and x − 4y < 0, is possible. In case (ii), you can conclude that −y < x < 4y. Because case (i) is impossible, Statement (1) is equivalent to −y < x < 4y. Since y is positive, 3y < 4y. You know that x < 4y. However, you do not know whether or not x < 3y.
More than one answer to the question is possible, so Statement (1) is insufficient. Eliminate choices (A) and (D).
Statement (2): Simplify the inequality 5(y + 8) < 20y − 3x + 40.
5(y + 8) < 20y − 3x + 40
5y + 40 < 20y − 3x + 40
5y < 20y − 3x
3x < 15y
x < 5y
Statement (2) is equivalent to x < 5y. Since y is positive, 3y < 5y. You know that x < 5y. However, you do not know whether or not x < 3y. Statement (2) is insufficient. Eliminate choice (B).
The statements taken together: Statement (1) is equivalent to −y < x < 4y and Statement (2) is equivalent to x < 5y. Because the question requires that y > 0, 4y < 5y and the range in Statement (1) is a subset of the range in Statement (2). Therefore, the statements taken together require that −y < x < 4y, which you have already determined to be insufficient.
Choice (E) is correct.