GMAT Classic Mock Test - 5 - GMAT MCQ

1. Plug in n = 2 and z = 2
2. Evaluate the expression:
   n # z = n² - n^z - 2(z-2)ⁿ
   2 # 2 = 2² – 2² – 2(2-2)²
   = 4 – 4 – 2(0)²
   = 4 - 4 - 0 = 0 – 0 = 0

1. Begin with the two important circle formulas.
   Area = πr²
   Circumference = 2πr, where r is the radius.
2. Since circles x and y have the same area, the length of their respective radii must be the same.
   Area_x = πr_x² = Area_y = πr_y²
   π(r_x)² = π(r_y)²
   r_x² = r_y² divide by π
   r_x = r_y
3. With the information given in the question, you can write the following equation:
   Circumference_x = 2πr_x = 16π
   2r_x = 16 divide by π r_x = 8
4. Remembering that r_x = r_y, you know that r_y = 8. Consequently, "half of the radius of circle y" is (1/2)(8) = 4.
   Answer C is correct.

There are two ways to solve this problem.

1. Solve the equation algebraically.
   1. Start with the original fraction and simplify:
      
   2. Since the question simply asks what value of x will make the equation an integer, the equation can be simplified to ⁵⁰/_x. This is true because, if ⁵⁰/_x is an integer, adding 1 to that value will not change whether it is an integer. Likewise, if ⁵⁰/_x is not an integer, adding 1 to that value will not make it an integer.
   3. Simplified version of original question: is ⁵⁰/_x an integer?
   4. Use the divisibility rules to quickly evaluate this question:
      A: x = 0 → Not an integer. You cannot divide by 0
      B: x = 4 → Not an integer. You cannot divide 50 by 2 twice, so 50/4 is not an integer.
      C: x = 5 → An integer; 50/5 = 10
      At this point, you could stop. However, just to be safe, it does not hurt to keep checking.
      D: x = 6 → Not an integer. You cannot divide 50 by both 2 and 3, so 50/6 is not an integer. (Remember that for a number to be divisible by 3, its digits must sum to a number divisible by 3. The sum of the digits of 50 is 5, which is not divisible by 3).
      E: x = 7 → Not an integer. No quick rule for divisibility by 7. However, since 7*7 is 49, there is no way 50 will be divisible by 7.
2. Solve the equation using Backsolving.
   1. Simply plug in the answer choices and see what answer works.
      
   2. Since only C provides an integer, it is the correct answer
   3. Although this method provides the correct answer, it is considerably longer and should be used only as a last resort.

1. The original cost of the computer (from the store's perspective) was $1,000.
2. The retail price was 20% higher than the original cost, so multiply by 1.20; recall that 1.2 is equivalent to (100% + 20%):
   Retail Price: $1000(1.20) = $1,200.
3. The employee discount price was 20% lower than the retail price, so multiply by 0.80; recall that 0.8 is equivalent to (100% - 20%):
   Employee Discount Price: $1200(0.80) = $960.
4. Find the difference between the employee discount price and the retail price:
   $1200 - $960 = $240.
   The correct answer is C.

1. In order to find how many integers are in set M, it may help to start listing the contents of both sets to try to identify a pattern:
   K = {2, 4, 6, 8, ...}
   M = {1.41, 2.00, 2.45, 2.83, ...}
2. We see that K is the set of even integers from 2 to 40 and M is the set of the square roots of each number in K. Conversely, given the set M, K is the set of squares of numbers in M.
3. To see if a certain integer is in M, we can check to see if its square is in K. In other words, we are looking for positive integers x, such that x² is an even perfect square less than 40.
4. Listing the perfect squares in K from 1 to 40 yields:
   1, 4, 9, 16, 25, and 36.
5. Eliminating any odd numbers that would not be included in K leaves us with 4, 16, and 36.
6. This means that the integers 2, 4, and 6 are included in M since 2²=4, 4²=16, and 6²=36. No negative values are in M because square roots are always positive. This is an exhaustive list of the integers in M; thus the answer is 3, which is choice D.

1. Start by setting up two equations to solve for the number of men and women on the team.
2. The total number of people is 16, so:
   Men + Women = 16
3. There are 4 more women than there are men on the team. This means that if you take the number of men and add 4, you get the number of women:
   Women = Men + 4
4. Plug in the number of women in terms of men from the second equation into the first to yield:
   Men + Women = 16
   Men + (Men + 4) = 16
5. Combine like terms to yield:
   Men + (Men + 4) = 16
   2(Men) + 4 = 16
6. Subtract 4 from each side to yield:
   2(Men) = 12.
7. Divide by 2 to yield that the number of men on the team is 6.
8. Of the 16 players, if 6 are men, 10 must be women. Alternatively, if 6 are men and there are 4 more women than there are men on the team, 6+4, or 10, must be women.
9. Since there are 6 men and 10 women, the ratio of men to women is 6/10. The answer is D.

1. Starting with 12,000 ballets, 1/3 were thrown out, leaving 2/3 still valid:
   12,000 x (2/3) = 8,000 valid ballets.
2. Of those valid ballets, since 1/4 went to A, the other 3/4 must have gone to B since this is a head-to-head run-off election.
   8,000 x (3/4) = 6,000 valid ballets for Candidate B.
   The correct answer is C.

1. In working the problem, it is extremely important to remember that, by the rules of exponents, x^Y = 1/(x^-Y)
   
2. Alternatively, you could do the problem like this:
3. Plug in 1 for n to yield (.1*1)⁷⁹⁸/(.1*1)⁸⁰⁰, which is (.1)⁷⁹⁸/(.1)⁸⁰⁰ since any number multiplied by 1 is simply that number. In other words, .1*1 = .1
4. Cancel 798 0.1's from the numerator and denominator. It helps to think of the expression as x⁷⁹⁸/x⁸⁰⁰, where x=0.1. This yields 1/x², or 1/(.1)².
5. 0.1 = 1/10, so (.1)²=1²/10², or 1/100.
6. The number we are interested in is the reciprocal of (.1)², that is 1/(.1)², so the reciprocal of 1/100 is 100.
   The answer is B.

1. Since all we know about the relationship between the original and renovated greens is the doubling of the radius (i.e., distance from center to edge), we must find the radius of the new green based off the area of the original green.
2. The formula for the area of a circle is A = πr². The area A = 100π, so we must solve for r in the equation 100π = πr².
3. Divide both sides by pi to yield 100 = r².
4. Thus, the radius of the original is the square root of 100, which could be -10 or 10, but since we are dealing with distances, it must be positive. So the radius of the original green is 10 feet. A = 100π = πr²
   100 = r²
   r = 10
5. Since the original radius is 10 ft, the renovated green has a radius double that: 20 ft.
6. Now to calculate the total area of the renovated green:
   A = πr²
   A = π20²
   A = 400π
   Thus, the correct answer is B.
   Note: The question asked for the "total area" not the additional area.

1. To determine f(-1), we must first know (-1)¹⁹⁷ and (-1)²⁴⁸. You should never multiply out lots of numbers.
2. Instead, look for a pattern:
   (-1)¹=-1
   (-1)²=+1
   (-1)³=-1
   (-1)⁴=+1
   ...
   (-1)^{(odd number)}=-1
   (-1)^{(even number)}=+1
3. This means (-1)¹⁹⁷=-1 since 197 is odd and (-1)²⁴⁸=+1 since 248 is even.
4. This pattern simplifies the equation to:
   -5(-1) - 8(1) + 1, which equals 5 - 8 + 1, or -2.
   Thus, C is the correct answer.

1. The ratio of a batting average is a fraction. As you decrease the numerator or increase the denominator, the fraction becomes smaller. Likewise, as you increase the numerator or decrease the denominator, the fraction becomes larger.
2. In the case of a batting average, the numerator is "hits" (H) while the denominator is "at bats" (B). Thus, the ratio we are looking at is:
   H/B, where 2 < H < 3 and 4 < B < 6.
3. To find the lowest value that the batting average could be, we want to assume the lowest numerator (hits of 2) and the highest denominator (at bats of 6): 2/6 = 0.333.
4. Likewise, to find the highest value that the batting average could be, we want to assume the highest numerator (hits of 3) and the lowest denominator (at bats of 4): 3/4 = 0.75.
5. Combining these answers yields the correct answer C: between 0.33 and 0.75.

1. This problem requires working backward from solutions to a quadratic equation to the equation itself.
   Instead of factoring a quadratic equation to solve it (the process many questions require), this question requires you to go the other way—from solutions to factors.
   f(x) = x² + bx + c = 0
2. The key to unlocking this problem is recognizing that if f(a) = 0 and f(x) is a quadratic equation in the form x² + bx + c = 0, (x – a) is a factor of the equation.
   (x + d)(x + e) = x² + dex + de = 0
3. If this past step does not make sense immediately, take a look at a few examples and allow them to convince you of this truth.
   x² + 3x - 4 = 0
   (x - 1)(x + 4)
   Solutions: f(1) = 0 or f(-4) = 0
   x² -9x + 20 = 0
   (x – 4)(x – 5)
   Solutions: f(4) = 0 or f(5) = 0
4. Following this same principle, you know that since f(1) = 0 and f(-4) = 0, (x – 1) and (x + 4) are factors.
   (x – 1)(x + 4) = 0
5. Multiply these factors together to form a quadratic equation.
   (x – 1)(x + 4) = 0
   x² – x + 4x – 4 = 0
   x² + 3x – 4 = 0
6. For any quadratic equation in the form ax² + bx + c = 0, the y-intercept (i.e., the place where the equation crosses the y-axis) is c.
   Even if you did not know this, you could still find the y-axis. The line will cross the y-axis where x = 0.
   f(0) = (0)² + 3(0) – 4 = -4
7. Y-Intercept is -4

1. This is a typical permutation-probability problem. To make this problem easily understandable we will break into two parts:
   i. First, we will find out all the seven lettered words from the letters of word CLASSIC
   ii. Next, we will find out how many of these words will have the two C's together.
2. The total number of words formed using the seven letters from the word CLASSIC is found by using the multiplication principle. There are seven places for each of the seven letters. The first place has 7 choices, the second place has (7-1) =6 choices, the third places has 5 choices and the seventh place has 1 choice. Hence, the total number of words formed is:
   = 7 x 6 x 5 x 4 x ... x 1 = 7!
   Notice that there are two C's and two S' in the word, which can be treated as repeated elements. To adjust for the repeated elements we will divide 7! by the product of 2! x 2!
   So, the total number of words formed is:
   7!/(2! x 2!)
3. We need to find how many of these words will have the two C's together. To do this, let us treat the two C's as a single entity. So, now we have six spaces to fill. Continuing the same way as in the step above, we can fill the first place in 6 ways, the second place in 5 ways and the sixth place in 1 way. Hence there are 6! ways of forming the words. Once again, we will need to adjust for the two S' which can be done by dividing 6! by 2!.
   Total number of 7 lettered words such that the two C's are always together = 6!/2!
4. The fraction of seven lettered words such that the two C's are always together is:
   = (number of words with two C's together/total number of words) = (6!/2!)/(7!/[2! x 2!]) = (2!/7) = 2/7
5. Hence the correct answer choice is A.

1. At first, this question looks daunting. Do we have to determine what all of these numbers are, raised to their respective powers, and then multiply them by one another? Recognize that if we just figure out what the units (ones) digit is for all of these numbers, we then just have to multiply the units digits by one another, which is much more manageable. This is because, to obtain the units digit of a product of two numbers, we only need to multiply the units digit of these two numbers by each other. Then the units digit of that product will be the units digit of the whole product. For example, the units digit of 38 x 9317 = 6, because 8 x 7 = 56, and the units digit of 56 is 6.
2. Thus, let us first determine the units digit of the numbers in this expression.
3. (2)³ = 2 x 2 x2 = 8
   So the units digit of (2)³ is 8
4. (3)³ = 3 x 3 x 3
   = 9 x 3
   = 27
   So the units digit of (3)³ = 7
5. (4)³ = 4  x 4 x  4
   = 16  x  4
   = 64
   The units digit of (4)³ = 4
6. For (5)⁷, recognize that 5 to any power greater than 0 must have a 5 in the units digit. For example, 5 x 5 = 25; 25 x 5 = 125, etc.
   Therefore the units digit of (5)⁷ = 5
7. At this point, we can stop, if we recognize that 5 (i.e., the units digit of (5)⁷)  x  4 (i.e., the units digit of (4)³) = 20. Remember that after determining all of the units digits, we were going to multiply them by one another to determine the units digit of the whole product. The fact that 5 x 4 = 20 is significant because, taking 0 as the units digits of this product, we then will multiply 0 by all of the other units digits we determine. This product, however, no matter what the other units digits are, must be 0, because any number multiplied by 0 is 0. Therefore, our units digit must be 0, answer choice (A).

1. In this combinations problem, order is not important (i.e., Mariah having an interview with John Smith, Mary Jones and David Arlington is the same as an interview with Mary Jones, David Arlington and John Smith). Because order is not important, the solution involves the combination formula and not the permutation formula.
2. The formula for a combination is:
   _nC_r= n!/((n - r)!r!)
   where n is the total number of selections available and r is the number of items to be selected.
3. For this problem the total number of selections, n, is 10 and the total number of items to be selected, r, is 3. So the combination formula is written and calculated as:
   ₁₀C₃ = 10! /( (10 – 3)! x 3! )
   = (10 x 9 x 8) /( 3 x 2) = 720 / 6 = 120
4. Since there are 120 combinations, it will take her 120 days to interview all possible combinations of job candidates in groups of three. So the correct answer is B.
5. A common mistake is to use the permutation formula which would yield six times as many days or 720 days. Other common mistakes are to reverse the order for n and r in the combination formula.

1. Evaluate Statement (1) alone.
   1. Statement (1) says that the sum of any two factors is even. The sum of two integers is even under two circumstances:
      odd + odd = even
      even + even = even
   2. Since the sum of any two factors is even, all the factors must have the same parity. If x had both even and odd factors, then it would be possible for two factors to add together and be odd (remember that an odd number + an even number = an odd number and, in Statement 1, the sum of any two positive factors must be even).
   3. Since the problem says "the sum of any two positive factors of x is even" and the number 1 is a factor of any number, x must only contain odd factors. If x contained one even factor, it would be possible to add that even factor with the number one and the result would be an odd number. Since the number 2 is a factor of every even number, x cannot be even. Otherwise, it would be possible to add the factors 1 and 2 together and their sum would not be even.
   4. Statement (1), when inspected carefully, says that x is an odd number that only contains odd factors. Since there are many possibilities (x = 1, 3, 5, 7, 9, 11, 15, ...), Statement (1) is NOT SUFFICIENT.
2. Evaluate Statement (2) alone.
   1. Statement (2) says that x is a prime number less than 4. Remember that x must also be a positive integer (as per the original question). Although this narrows the possibilities for x, because there are still two possibilities (x = 2 or x = 3; both these values are prime, less than 4, and positive integers), Statement (2) is NOT SUFFICIENT. Please remember that the number one is not prime.
3. Evaluate Statements (1) and (2) together.
   1. Statements (1) and (2), when taken together, definitively show that x = 3. Statements (1) and (2), when taken together, are SUFFICIENT. Answer choice C is correct.

1. Simplify the question: since raising a number to an odd power does not change the sign, x is a positive integer.
2. The question, is w - z > 5(7^x-1 - 5^x)?, can be simplified to: is w - z > 5*7^x-1 - 5^x+1?
3. Evaluate Statement (1) alone.
   1. Statement (1) allows the following substitution:
      Is 7^x - (a number less than 25) > 5(7^x-1) - 5^x+1?
      Equivalently: Is 7^x - (a number less than 5²) > 5(7^x-1) - 5^x+1?
   2. If this question can be answered definitively for all legal values of x (i.e., positive integers), Statement (1) is sufficient. Although this statement is difficult to evaluate algebraically, a little logic makes Statement (1) plainly sufficient. It is helpful to step back and see the logic about to be employed.
      a - b will always be greater than c - d if these numbers are positive and a > c and b < d. In this situation, a smaller number (b is smaller than d) is being subtracted from a larger number (a is greater than c). Consequently, if the left side of the equation starts from a larger number and subtracts a smaller number than the right side of the equation, it is quite clear that the difference on the left side will be larger than the difference on the right side of the inequality.
      For example: 10 - 2 > 5 - 4
      You are starting with a larger number on the left (i.e., 10 > 5) and subtracting a smaller number on the left (2 < 4). Consequently, it only makes sense that the number on the left is going to be larger.
   3. This same logic holds true in the inequality derived in Statement (1). Since x is a positive integer (it is essential to know this), 7^x will be bigger than 5(7^x-1). You know this is true because there will be x sevens on the left side of the inequality and (x-1) sevens on the right side of the inequality. The extra 7 on the left will out-weight the extra 5 on the right, making the left side start with a larger number.
   4. Continuing with this logic, (a number less than 5²) will be less than 5^x+1 since x is a positive integer and the smallest possible value for x (i.e., 1) makes 5^x+1 = 5¹⁺¹ = 5² = 25. Since 5^x+1 will always be at least 25, it will always be greater than (a number less than 25). Statement (1) is SUFFICIENT.
      Note: If z were a negative number, which it could be, the inequality would still hold true. It would make the left side of the inequality even larger as we would effectively be adding a number to 7^x.
4. Evaluate Statement (2) alone.
   1. Statement (2) says that x = 4. The inequality can now be re-written:
      is w - z > 5(7^4-1 - 5⁴⁺¹)?
      In other words, is w - z > 1,715 - 3,125?
      Or, to simplify it as much as possible:
      is w - z > -1,410?
      If w = 7⁴ = 2,401 and z = 1, the answer is YES. However, if w = -100,000 (nothing in Statement (2) precludes this possibility—do not import information over from Statement (1)) and z = 1, the answer is NO. Since Statement (2) does not provide enough information to definitively answer the original question, it is NOT SUFFICIENT.
5. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

1. In order to solve this question efficiently, it is necessary to begin with number properties. For a product of any number of terms to be odd, all the terms must be odd. If there is but one even term, the product will be even. To see this, consider the following examples:
   All Terms Odd → Odd Product
   3 x 7 x 9 x 5 = 945
   7 x 9 x 3 = 189
   1 x 3 x 5 = 15
   But: One or More Even Terms → Even Product
   3 x 7 x 9 x 2 = 378
   7 x 9 x 4 = 252
   1 x 3 x 5 x 6 = 90
2. In order for (x³ + 19837)(x² + 5)(x – 3) to be an odd number, all the terms must be odd.
3. To determine under what conditions each term will be odd, it is important to remember the following relationships:
   odd + odd = even
   odd - odd = even
   even + even = even
   even - even = even
   even + odd = odd
   even - odd = odd
   odd + even = odd
   odd - even = odd
4. The only way for each term of (x³ + 19837)(x² + 5)(x – 3) to be odd is if an even and an odd number are added or subtracted together within the parenthesis of each term. In other words:
   even + odd = odd: For (x³ + 19837) to be odd, since 19837 is odd, x³ will need to be even. This will happen only when x is even.
   even + odd = odd: For (x² + 5) to be odd, since 5 is odd, x² will need to be even. This will happen only when x is even.
   even - odd = odd: For (x – 3) to be odd, since 3 is odd, x will need to be even.
5. When combining the results from the analysis of the three terms above, the only way for (x³ + 19837)(x² + 5)(x – 3) to be odd is if each term is odd. This will only happen if x is even. Consequently, the original question can be simplified to: is x even? Another version of the simplified question is: what is the parity of x?
6. Evaluate Statement (1) alone.
   1. In order for the sum of any prime factor of x and x to be even, it must follow one of two patterns:
      Pattern (1): even + even = even
      Pattern (2): odd + odd = even
   2. There are two possible cases:
      Case (1): x is even. In this case, Pattern (1) must hold. Since x is even in Case (1), any and every prime factor of x must be even (otherwise we could choose an odd prime factor of x and the sum of x and the odd prime factor would be odd). Let's consider some examples:
      Let x = 12: However, x cannot equal 12 since one prime factor of 12 is 3 and 12 + 3 = odd number.
      Let x = 26: However, x cannot equal 26 since one prime factor of 26 is 13 and 26 + 13 = odd number.
      Let x = 14: However, x cannot equal 14 since one prime factor of 14 is 7 and 14 + 7 = odd number.
      Let x = 16: x can equal 16 since every prime factor of 16 is even and as a result we know that and 16 + any prime factor = even number.
      It is clear that Statement (1) allows x to be even (e.g., 16 is a possible value of x).
      Case (2): x is odd. In this case, Pattern (2) must hold. Since x is odd in Case (2), any and every prime factor of x must be odd (otherwise we could choose an even prime factor of x and the sum of x and the even prime factor would be odd). Since all the prime factors of x are odd, x must be odd in Case (2). Let's consider some examples:
      Let x = 11: Every prime factor of 11 is odd, so: 11 + prime factor of 11 = even number.
      Let x = 15: Every prime factor of 15 is odd, so: 15 + prime factor of 15 = even number.
   3. Since Statement (1) allows x to be either even (e.g., 16) or odd (e.g., 15), we cannot determine the parity of x.
   4. Statement (1) is NOT SUFFICIENT.
7. Evaluate Statement (2) alone.
   1. 3x = Even Number
      (odd)(x)=(even)
      x must be even because, as shown above, if x were odd, 3x would be odd.
   2. Statement (2) is SUFFICIENT since it definitively tells the parity of x.
8. Since Statement (1) alone is NOT SUFFICIENT but Statement (2) alone is SUFFICIENT, answer B is correct.

1. For some students, the theoretical nature of this question makes it intimidating. For these individuals, we recommend picking numbers as a means of determining sufficiency.
2. Evaluate Statement (1) alone.
   1. Draw a table to quickly pick numbers in order to determine whether Statement (1) is sufficient. It is quickest to choose numbers for D+1 and C+1 that work (i.e., produce a remainder of 5) and then infer the values of D and C.
      Let R(X/Y) = the remainder of 
   2. Different legitimate values of D+1 and C+1 yield different remainders for D/C. Consequently, the information in Statement (1) is not sufficient to determine the remainder when D is divided by C.
   3. Algebraically, we know that D+1 divided by C+1 will not have the same remainder as D divided by C since fractions do not stay equivalent when you add to them (i.e., x divided by y does not equal x+1 divided by y+1).
   4. Statement (1) alone is NOT SUFFICIENT.
3. Evaluate Statement (2) alone.
   1. Before evaluating Statement (2), it is essential to simplify by factoring the numerator:
      ND + NC = N(D+C)
      Cancel out the N in both the numerator and denominator. Statement (2) can be simplified to: If D+C is divided by C, the remainder is 5.
   2. We can further simplify by noticing that D+C divided by C is equal to D divided by C plus C divided by C.
   3. There are two parts to this equation: (1) D divided by C (2) the number 1
      The sum of parts (1) and (2) will always have a remainder of 5 (this is what Statement 2 says). This remainder cannot come from the second part (i.e., C divided by C equals +1 and there is no remainder).
      Consequently, the remainder of 5 must come from D divided by C. So, we know that D divided by C will always produce a remainder of 5, which provides sufficient information to answer the original question.
   4. Statement (2) alone is SUFFICIENT.
4. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

1. Evaluate Statement (1) alone.
   1. Based upon the formula for the average, you know that:
      (5 + x² + 2 + 10x + 3)/5 = -3
      x² + 10x + 5 + 2 + 3 = -15
      x² + 10x + 5 + 2 + 3 + 15 = 0
      x² + 10x + 25 = 0
      (x + 5)² = 0
      x = -5
   2. Statement (1) alone is SUFFICIENT.
2. Evaluate Statement (2) alone.
   1. Order the numbers in ascending order without x:
      -43, -32, -30, -15, 10, 7, 109, 208
   2. Consider the possible placements for x and whether these would make the median equal to -5:
      Case (1): x, -43, -32, -30, -15, 10, 7, 109, 208
      Median: -15
      Not a possible case since the median is not -5.
      Case (2): -43, x, -32, -30, -15, 10, 7, 109, 208
      Median: -15
      Not a possible case since the median is not -5.
      Case (3): -43, -32, x, -30, -15, 10, 7, 109, 208
      Median: -15
      Not a possible case since the median is not -5.
      Case (4): -43, -32, -30, x, -15, 10, 7, 109, 208
      Median: -15
      Not a possible case since the median is not -5.
      Case (5): -43, -32, -30, -15, x, 10, 7, 109, 208
      Median: x
      A possible case since the median is x, which can legally be -5.
      In this case, x must be -5 in order for the median of the set to be -5, which must be according to Statement (2).
      Case (6): -43, -32, -30, -15, 10, x, 7, 109, 208
      Median: 10
      Not a possible case since the median is not -5.
      Case (7): -43, -32, -30, -15, 10, 7, x, 109, 208
      Median: 10
      Not a possible case since the median is not -5.
      Case (8): -43, -32, -30, -15, 10, 7, 109, x, 208
      Median: 10
      Not a possible case since the median is not -5.
      Case (9): -43, -32, -30, -15, 10, 7, 109, 208, x
      Median: 10
      Not a possible case since the median is not -5.
   3. Since Statement (2) tells us that the median must be -5, we know that x must be a value such that the median is -5. This can only happen in Case 5. Specifically, it can only happen when x = -5. Since the median must equal -5 and this can only happen when x = -5, we know that x = -5.
   4. Statement (2) alone is SUFFICIENT.
3. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

1. In evaluating this problem, it is important to keep in mind the list of possible prime numbers:
   7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53
2. Evaluate Statement (1) alone.
   1. The prime numbers between 15 and 34, not-inclusive, include:
      17, 19, 23, 29, 31
   2. Since there is no definitive information about the value of X, we do not know how many prime numbers exist between 7 and X.
      If X = 17, there would be 2 prime numbers between 7 and X (i.e., 11 and 13).
      If X = 18, there would be 3 prime numbers between 7 and X (i.e., 11, 13, and 17).
      If X = 21, there would be 4 prime numbers between 7 and X (i.e., 11, 13, 17, and 19).
      There is not enough information to definitively answer the question.
   3. Statement (1) alone is NOT SUFFICIENT.
3. Evaluate Statement (2) alone.
   1. List the multiples of 11 and their sums (stopping when the sum is no longer less than 7).
      x = 11; sum of digits is 1 + 1 = 2
      x = 22; sum of digits is 2 + 2 = 4
      x = 33; sum of digits is 3 + 3 = 6
      x = 44; sum of digits is 4 + 4 = 8, which is too high so x cannot be greater than 33.
   2. Since X can be 11, 22, or 33, there are different possible answers to the question of how many prime numbers are there between the integers 7 and X:
      If X = 11, there would be 0 prime numbers between 7 and X.
      If X = 22, there would be 4 prime numbers between 7 and X (i.e., 11, 13, 17, and 19).
      There is not enough information to definitively answer the question.
   3. Statement (2) alone is NOT SUFFICIENT.
4. Evaluate Statements (1) and (2) together.
   1. Putting Statements (1) and (2) together, X must meet the following conditions:
      (1) 15 < X < 34
      (2) X = 11, 22, 33
      This means that possible values for X include:
      X = 22 or 33
   2. The two possible values for X give different answers to the original question:
      If X = 22, there would be 4 prime numbers between 7 and X (i.e., 11, 13, 17, and 19).
      If X = 33, there would be 7 prime numbers between 7 and X (i.e., 11, 13, 17, 19, 23, 29, and 31).
   3. Statements (1) and (2), even when taken together, are NOT SUFFICIENT.
5. Since Statement (1) alone is NOT SUFFICIENT, Statement (2) alone is NOT SUFFICIENT, and Statements (1) and (2), even when taken together, are NOT SUFFICIENT, answer E is correct.

1. If "the value of every item in Country X plummeted by 50% from 1990 to 1995," the value in 1995 would be 100%-50% = 50% of the value in 1990.
   Translate the information in the question into an algebraic equation.
   P₉₅ = (.5)P₉₀; P₉₀ = ?
2. Evaluate Statement (1) alone.
   1. Statement (1) says that P₉₃ = $30.
   2. It may be tempting to assume that the value of the book changed the same amount each year. If this were true, Statement (1) would be sufficient since $30 would be the result of the value falling an equal percent for a known number of years. But, you cannot make this assumption. All you know is that the value in 1995 was half the value in 1990 and the value in 1993 was $30. It is possible that the value could have risen substantially from 1990 to 1993, only to fall dramatically enough during 1993, 1994, and 1995 that the value decreased by 50% from 1990 to 1995.
   3. Consider the following two examples, which are both possible under the constraints of Statement (1) yet give different values for P₉₀.
      (1) P₉₀ = $15 and the value doubled to P₉₃ = $30, only to fall to P₉₅ = $7.5
      (2) P₉₀ = $25 and the value grew 20% to P₉₃ = $30, only to fall to P₉₅ = $7.5
      Since both examples satisfy the conditions (i.e., P₉₃ = $30 and P₉₅ = (.5)P₉₀) yet produce different values for P₉₀, Statement (1) is not definitive.
   4. Statement (1) alone is NOT SUFFICIENT.
3. Evaluate Statement (2) alone.
   1. Translate Statement (2) into algebra.
      (.5)P₉₅ = P₂₀₀₀
      (.5)P₉₅ = $25
      Therefore: P₉₅ = $50
   2. The original question states the following relationship:
      P₉₅ = (.5)P₉₀
      Since we know that P₉₅ = $50, by substitution, we also know that:
      $50 = (.5)P₉₀
      Therefore: P₉₀ = $100
      Statement (2) is SUFFICIENT.
4. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

1. Evaluate Statement (1) alone.
   1. Substitute Z into the equation:
      10x + 10y + 16x² + 25y² = 10 + Z
      10x + 10y + 16x² + 25y² = 10 + (4x)² + (5y)²
      10x + 10y + 16x² + 25y² = 10 + 16x² + 25y²
      10x + 10y = 10
      10(x+y) = 10
      x + y = 1
   2. Statement (1) is SUFFICIENT.
2. Evaluate Statement (2) alone.
   1. Substitute the information you know (i.e, x = 1) into the equation:
      10x + 10y + 16x² + 25y² = 10 + Z
      10(1) + 10y + 16(1)² + 25y² = 10 + Z
      10y + 25y² + 10 + 16 = 10 + Z
      25y² + 10y + 16 - Z = 0
   2. At this point, we reach a wall. Since we do not know what Z equals, we cannot solve for Y. Without a value for Y, we cannot find x + y.
   3. Statement (2) is NOT SUFFICIENT.
3. Since Statement (1) alone is SUFFICIENT but Statement (2) alone is NOT SUFFICIENT, answer A is correct.

1. Evaluate Statement (1) alone.
   1. The equation in Statement (1) can be factored.
      x² + 4x + 4 = 0
      (x + 2)(x + 2) = 0
      Consequently, x = -2.
   2. With one specific value of x, the inequality can be definitively evaluated:
      Is -2|(-2)|³ < (|-2|)^-2?
   3. Since this will give a definitive answer, the data are sufficient. (Note: Although the answer to the question here is yes, it does not need to be yes in order for sufficiency to exist. In other words, if the answer to our question were always no, that would be sufficient.) Statement (1) is SUFFICIENT.
2. Evaluate Statement (2) alone.
   1. With the information in Statement (2), plug in the sign of x:
      is (negative)(|negative|)³ < (|negative|)^negative?
      Simplified:
      Is (negative)(positive)³ < (positive)^(negative)?
      Since a positive number raised to an odd exponent is always positive and (negative)(positive) = negative, we can simplify further:
      Is (negative) < (positive)^(negative)?
      Since a positive number raised to a negative number is simply a smaller positive number, we can simplify further:
      Is (negative) < (positive)?
   2. Statement (2) enables the question to be definitively answered. Statement (2) is SUFFICIENT.
3. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

1. A number is divisible by any of its prime factors or any combination of its prime factors. For a number to be divisible by four, it must have two 2s in its prime factorization since 2 x 2 = 4 and, if 4 is a factor of X, X will be divisible by 4.
2. Evaluate Statement (1) alone.
   1. Since X has two 2s in its prime factorization, 4 must be a factor of X and, consequently, X must be divisible by 4. Statement (1) is SUFFICIENT.
   2. If this seems too abstract, consider the following examples which show that whenever X has at least two 2s in its prime factorization (which it must as per Statement (1)), X is divisible by 4:
      X = 4: has two 2s in its prime factorization and, as a result, is divisible by 4
      X = 6: has only one 2 in its prime factorization and, as a result, is not divisible by 4
      X = 8: has at least two 2s in its prime factorization and, as a result, is divisible by 4
      X = 10: has only one 2 in its prime factorization and, as a result, is not divisible by 4
   3. Since X cannot be 6, 10, etc. as these values do not have at least two 2s as prime factors (as is required by Statement (1)), X can only be 4, 8, etc. and will always be divisible by 4.
   4. Statement (1) alone is SUFFICIENT.
3. Evaluate Statement (2) alone.
   1. If X is divisible by 2, you know that X must have at least one 2 in its prime factorization. However, you do not know that X has two 2s in its prime factorization and, as a result, you cannot be sure that X is divisible by 4.
   2. If this seems too abstract, consider the following examples, all of which are divisible by 2 in keeping with the requirements of Statement (2):
      X = 4: has two 2s in its prime factorization and, as a result, is divisible by 4
      X = 6: has only one 2 in its prime factorization and, as a result, is not divisible by 4
      X = 8: has at least two 2s in its prime factorization and, as a result, is divisible by 4
      X = 10: has only one 2 in its prime factorization and, as a result, is not divisible by 4
   3. Statement (2) alone is NOT SUFFICIENT.
4. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

1. The value of a fraction is less than one if its numerator is smaller than its denominator. For example, 4/6 is less than one because 4 < 6. So, the question at hand can be simplified to: is z < n?
2. Evaluate Statement (1) alone.
   1. Statement (1) can be re-arranged:
      z - n > 0
      z > n
   2. Since z > n, you can definitively answer no to the question: "is z < n?"
   3. Statement (1) is SUFFICIENT.
3. Evaluate Statement (2) alone.
   1. Based upon the question, since z-15 is a positive number, the following inequality must hold:
      z - 15 > 0
      z > 15
   2. Statement (2) says:
      n < 15
   3. Since z > 15 and n < 15, you know that z > n
   4. You can definitively answer no to the question: "is z < n?"
   5. Statement (2) is SUFFICIENT.
4. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

1. Evaluate Statement (1) alone.
   1. Simplify the inequality:
      4x² – 8x > (2x)² – 7x
      4x² – 8x > 4x² – 7x
      -8x > -7x
      -8x + 8x > -7x + 8x
      0 > x
      x < 0
   2. Since X is less than zero, X is a negative number. This means that negative X is a positive number since multiplying a negative number by a negative number (i.e., -1) results in a positive number.
   3. Statement (1) alone is SUFFICIENT.
2. Evaluate Statement (2) alone.
   1. Simplify the inequality:
      x + 2 > 0
      x > - 2
   2. Since we cannot be sure whether X is negative (e.g., -1) or positive (e.g., 2), we cannot be sure whether negative X is positive or negative.
   3. Statement (2) alone is NOT SUFFICIENT.
3. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

1. Evaluate Statement (1) alone.
   1. Statement (1) simply says that B > 10. It provides no information about the value of A, making a comparison between B and A impossible.
   2. If B = 12 and A = 5, then the answer to the question "is B > A?" would be yes. However, if B = 15 and A = 20, then the answer to the question "is B > A?" would be no.
   3. Since different legitimate values of A and B produce different answers to the question, Statement (1) is NOT SUFFICIENT.
   4. Note: Some students are thrown off by setting A = 20 or A = 5. You can do this in evaluating whether Statement (1) alone is sufficient since there is nothing in Statement (1) that prevents this. However, A cannot be 20 in evaluating statement 2 because Statement (2) clearly says that A must be less than 10. But, for now we are evaluating Statement (1).
2. Evaluate Statement (2) alone.
   1. Statement (2) simply says that A < 10. It provides no information about the value of B, making a comparison between B and A impossible.
   2. If B = 12 and A = 5, then the answer to the question "is B > A?" would be yes. However, if B = 1 and A = 9, then the answer to the question "is B > A?" would be no.
   3. Since different legitimate values of A and B produce different answers to the question, Statement (2) is NOT SUFFICIENT.
   4. Note: Some students are thrown off by setting B = 1 or B = 12. You can do this in evaluating whether Statement (2) alone is sufficient since there is nothing in Statement (2) that prevents this.
3. Evaluate Statements (1)and (2) together.
   1. When taking Statements (1) and (2) together, you know:
      B > 10 and A < 10
   2. So, you know that B > A. Statements (1) and (2), when taken together, are SUFFICIENT.
4. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT yet Statements (1) and (2), when taken together, are SUFFICIENT, answer C is correct.

1. Factor the original equation:
   xn – ny – nz = n(x - y - z)
2. If we know the value of both n and x - y - z, we can determine the value of xn – ny – nz.
3. Evaluate Statement (1) alone.
   1. Since x – y – z = 10, based upon the above factoring:
      xn – ny – nz = n(10)
      However, we do not know the value of n so we cannot solve for the value of xn – ny – nz.
   2. Statement (1) is NOT SUFFICIENT.
4. Evaluate Statement (2) alone.
   1. Since n = 5, based upon the above factoring:
      xn – ny – nz = 5(x - y - z)
      However, we do not know the value of x - y - z so we cannot solve for the value of xn – ny – nz.
   2. Statement (2) is NOT SUFFICIENT.
5. Evaluate Statements (1) and (2) together.
   1. Since n = 5 and x - y - z = 10, based upon the above factoring:
      xn – ny – nz = n(x - y - z)=5(10)=50
   2. Statements (1) and (2), when taken together, are SUFFICIENT.
6. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, but Statements (1) and (2), when taken together, are SUFFICIENT, answer C is correct.

1. Do not be distracted by "X and Y are both positive integers whose combined factors include 3 and 7." The factors given do not allow you to conclude that X or Y is either odd or even. To conclude that X and Y are even, X and Y need to have at least one even factor. To conclude that X and Y are odd, X and Y must only have odd factors.
2. For X + Y + 1 to be odd, the sum X + Y must be even since adding one to an even integer makes it odd. Said algebraically:
   X + Y + 1 = odd
   X + Y = even
3. The sum of two integers will be even if and only if the parity of the two numbers is the same. In other words, odd + odd = even and even + even = even. However, the sum of two numbers of different parity is odd (i.e., odd + even = odd). Consequently, in order for X + Y = even, both X and Y must be of the same parity. There are two possibilities:
   X_odd + Y_odd = even
   X_even + Y_even = even
4. Evaluate Statement (1) alone.
   1. A number is divisible by 2 if and only if it is even. Consider the following examples:
      4 is even and divisible by 2
      5 is not even and not divisible by 2
      6 is even and divisible by 2
      7 is not even and not divisible by 2
      8 is even and divisible by 2
      9 is not even and not divisible by 2
      10 is even and divisible by 2
      11 is not even and not divisible by 2
   2. Since Statement (1) tells us that both X and Y are divisible by 2, both X and Y are even. Since X and Y have the same parity, the sum X + Y is even and the sum X + Y + 1 is odd; Statement (1) is SUFFICIENT.
   3. Statement (1) alone is SUFFICIENT.
5. Evaluate Statement (2) alone.
   1. If you take a number and add 2, you do not change the parity of that number. Consider the following examples:
      4 {i.e., even} + 2 = 6 {i.e., even}
      5 {i.e., odd} + 2 = 7 {i.e., odd}
      6 {i.e., even} + 2 = 8 {i.e., even}
      7 {i.e., odd} + 2 = 9 {i.e., odd}
      8 {i.e., even} + 2 = 10 {i.e., even}
      9 {i.e., odd} + 2 = 11 {i.e., odd}
   2. Statement (2) indicates that the parity of X and Y are the same since adding 2 to X will not change the parity of X.
      X + 2 = Y
      Parity_X + 2 = Parity_Y
      Parity_X = Parity_Y
      Statement (2) is SUFFICIENT.
   3. Statement (2) alone is SUFFICIENT.
6. Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.