GMAT Classic Mock Test - 4 - GMAT MCQ

1. We are given two conversion factors and are asked to start with 1 mile and convert it to inches.
2. The first conversion relates miles to feet: 1 mile = 5,280 ft. Thus, the 1 mile we began with is equivalent to 5,280 feet.
3. In order to convert from feet to inches, we must set up a conversion ratio. Since 12 in = 1 ft, (12 in)/(1 ft) = 1. Likewise, (1 ft)/(12 in) = 1 since 1ft = 12in.
4. Taking the 5,280 ft that we have in 1 mile, multiply it by the correct factor of 1, making sure the units cancel correctly:
   5,280ft * (12in/1ft) = 5,280(12) in
   The equation below aligns the units wrong. Do not make this mistake:
   5,280ft * (1ft/12in) = 5,280*1 ft2/in
5. The first calculation yields the units we are looking for. Thus, the answer is C, 12*5,280.
6. Another way to think of this problem is to realize that if there are 12 inches in 1 foot and 5280 foot in 1 mile, then:
   12 inch = 1 foot
   5280 feet = 1 mile
   Via substitution:
   5280 [12 inches] = 1 mile
   5280*12 = 1 mile

1. We have two inequalities and we need to find an upper and a lower bound for the value of n. Start with the first inequality to find one of these bounds:
   5n + 2 > 12
   5n > 10
   n > 2
2. Subtract 2 from both sides to yield: 5n > 10.
3. Divide by 5 on both sides to yield: n > 2. Thus, 2 is one of the limits. This rules out choices A and C.
4. Take the second inequality to find the upper bound:
   7n - 5 < 44
   7n < 49
   n < 7
5. Add 5 to both sides of the inequality to yield: 7n < 49.
6. Divide both sides by 7 to yield the upper bound: n < 7.
7. Combining the answers from both inequalities yields that n is between 2 and 7. Thus, choice D is correct.

There are two main methods of solving this question.
Algebra
Multiply both sides by n³ to yield n5 (16k - 8) = n⁵:
n⁵ (16k - 8) (n ^{- 3}) = n²
n⁵(16k-8)=n⁵
16k-8)=1
Cancel the n⁵ by dividing both sides by n⁵ to yield 16k-8=1. Dividing by a variable is only possible since the value of that variable is not zero. Dividing by zero would otherwise result in a lost solution to the equation.
16k - 8 = 1
16k = 9
k = 9/16
Plugging-In
We are interested in the value of k but we do not know the value of n.
Since the answer choices we are given are not expressed in terms of n, the value of k must be the same regardless of the value of n.
Since this is true, choose a non-zero value for n such as 1. 1(any integer)=1. So plugging in n=1, the equation simplifies down to 16k - 8 = 1
Add 8 to both sides of the equation to yield: 16k = 9
Now divide by 16 to get the value of k = 9/16
Thus, answer B is correct.

1. The median of a set of numbers is the "middle" number when the set of numbers is ordered sequentially. To find the middle, first order the numbers:
   8, 12, 32, 40, 48, 62
2. Since there is an even number of integers in the set, there are two middle numbers: 32 and 40.
3. To find the median, average the two middle numbers:
   (32+40)/2 = 36
   Thus, answer C is correct.

The correct response is (B). It’s important to first understand what this question is asking. “On the way to Beachside” means the way there. The question is asking the average speed for a portion of the total trip. To find it, we’ll need to know the distance for that part of the trip and the time spent on that part of the trip.

If the average speed of the entire journey was 2/3 miles per hour, then every 3 hours 2 miles were travelled. Since the total distance was 2 miles, the total time must have been 3 hours. If the way back took half as much time as the way there, then for every 3 hours, 2 hours was spent on the way there, and 1 hour was spent on the way back.

Average Speed = Distance/Time = 1 mile / 2 hours. The average speed for the way to Beachside was ½ mph.

1. Begin with the inequality (x/y)>1. We can rewrite this by multiplying both sides by y to yield x > y. Note that since x and y are stated to be positive, the direction of the inequality does not change. (If we did not know that x and y were positive, we could not multiply y or x since we would not know whether we need to flip the inequality sign.)
2. Now go through the possible answers, simplifying them so that the answer is more apparent. Remember that taking a number to a negative power is the same as dividing by the number with a positive power:
   A: x/y
   B: x³/y⁴
   C: x³y
   D: x(y³)
   E: y/x
3. Now consider each choice, starting with the simplest two: A and E.
   A is simply a repeat of the original inequality (x/y) > 1, so this value is clearly not less than 1.
4. Considering choice E: if you divide y by x and x>y, this results in any proper fraction: for example 1/2, 2/5, or 9/11. Any proper fraction is less than 1, resulting in E being the correct choice.In other words, choice E guarantees that we will be dealing with a fraction whose numerator is smaller than its denominator, meaning a smaller number will be divided by a larger number. This always results in a value less than one.
5. Just to be sure, you can evaluate the other answers, disproving them by showing a single counterexample in each case. The values x=4 and y=2 happen to work for all of these choices:
   B: 4³/24=64/16=4, which is not less than 1.
   C: 4³*2=64*2=128, which is not less than 1.
   D: 4(2³)=4*8=32, which is not less than 1.
   We already eliminated A, so this confirms that E must be the correct answer.

1. We know the after a 20% gratuity the bill came to $960. Set up an equation to figure out the total bill before gratuity (x):(1 + gratuity percent)(pre-gratuity bill) = total bill 1.20x = $960 {1.2 is the same as a 20% increase}
2. Divide both sides by 1.20 to yield a total bill before gratuity of $800.
3. Since there were 10 people dining (5 bankers + 5 clients), in order to find the average cost per person before gratuity, we divide the total bill before gratuity by the total number of people:
   $800/10, which equals $80. The correct answer is D.

1. There are 10 teams, so any given team will play all of the other 9 teams, resulting in 9 games for each of the 10 teams.
2. Multiply the 10 teams by the 9 games each team will play to yield 90 games. However, do not be tricked into picking answer B at this point: you are not yet done.
3. 90 games includes duplicates (i.e., double-counting). In the counting of 90 games, we included the games team 1 played against teams 2 to 10. However, when we counted the number of games played by team 2, and likewise by teams 3 to 10, we also counted the games that these teams played against team 1.
4. In other words, you cannot count as a unique game both Team 1 vs. Team 2 and Team 2 vs. Team 1. Based upon the question ("how many games are played in total"), we must find the number of unique games.
5. Based upon this double-counting, 90 games is twice the number of games actually played. If we include only half the games we counted (including games such as team 1 vs team 2 while eliminating non-unique games such as team 2 vs team 1), we end up with a total count of 45 games (=90/2). The correct answer is D.

1. Assign letters to each man's bank account:
   Peter = p.
   John = j.
   Fred = f.
2. Now translate the sentences from the question into equations:
   j = 5p "John's bank account has 5 times what Peter's bank account has"
   j = (1/3)f "John's bank account has ... 1/3 what Fred's bank account has"
3. Combine the two equations to yield:
   5p = j = (1/3)f
   5p = (1/3)f.
4. Divide both sides by 1/3 (the same as multiplying by 3) to yield:
   15p = f
   Translated back into words, this means Fred has 15p dollars in his account.
5. We are interested in the difference between the amount Fred has (15p) and the amount Peter has (p). Subtract p from 15p to yield 14p.
   Fred - Peter =
   15p - p = 14p
   Thus, B is correct.

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.

Begin by finding the area of the circle:
Area_circle = πr²
Area_circle = π(AB)² = π(BC)²
In dealing with triangle ABC, BC = AB since both are radii. At this point, some students make a mistake and assume that AB is the height of the triangle and BC is the base of the triangle (or vice versa). However, we cannot assume that BC is the base and AB is the height since we have not yet shown that ABC is a right triangle. You could only make BC the base and AB the height if triangle ABC were a right triangle (in which case AB would be a perpendicular segment drawn from a vertex, A, to the side opposite that vertex, B).
By definition, the height of a triangle is the length of a segment drawn from a vertex perpendicular to the side opposite that vertex. A line that is perpendicular to the side opposite a vertex will, by definition, form a 90 degree angle. Consequently, for line AB to be the height of triangle ABC, angle ABC must be a right angle (i.e., 90 degrees).
Since the question states that "the measurement of angle ABC is not a multiple of 30," angle ABC cannot be 30, 60, 90, 120, etc. Consequently, angle ABC is not a right angle and line AB is not the height of triangle ABC.
Without the height, you cannot determine the area of the triangle. Without the area of the triangle, you do not have enough information to solve the problem. The correct answer is It Cannot Be Determined.
Note: If the question omitted the words "the measurement of angle ABC is not a multiple of 30" and instead said that the length of line AC is 2^1/2 times larger than the radius, you would be dealing with a 45-45-90 right triangle with sides r, r, and r*2^1/2. In this instance with a right triangle, the area of the triangle would be (1/2)bh = (1/2)(r)(r) = .5r² and the ratio of the area of the circle centered at point B to the area of triangle ABC would be 2π.

1. Probability problems require the knowledge of all possible combinations (mathematicians call this the probability space), the total number of combinations that satisfy the condition you are looking for and the probability that the combination will occur. The die is fair all possibilities are equally likely. Let A, B, C represent all possible values for the 3 die tosses. A = {1,2,3,4,5,6} , B = {1,2,3,4,5,6}, C = {1,2,3,4,5,6} and a,b,c represent possible outcomes of A,B,C respectively. (mathematicians say a,b,c are elements of the set A,B,C). We are to solve for the sum of 3 die tosses to equal 16. In other words, we are looking for all possible combinations such that a + b + c = 16.
2. Each die toss has six possible outcomes. Tossing the die once gives six possible values. Tossing the die twice gives a total of ( 6 possibilities for the first toss) times ( 6 possibilities for the second toss) = 36 possible outcomes . Similarly, for three die tosses one has (6)(6)(6) = 216 possible outcomes.
3. The first die toss is from 1 to 6. Since the maximum sum of the final 2 die tosses is 6 + 6 = 12, the first dice toss must be at least 4 for the sum of all three tosses to be 16.
4. If the first die toss is 4 then the last two must add up to 12. This will occur for the case {4,6,6} which is a total of one possibility.
5. If the first die toss is 5 then the last two must add up to 11. This will occur for the cases {5,5,6}, {5,6,5} which is a total of two possibilities.
6. If the first die toss is 6 then the last two must add up to 10. This will occur for the case {6,4,6}, {6,5,5}, {6,6,4} which is a total of three possibilities.
7. Out of the 216 total possible combinations, 1+2+3=6 add up to 6.
8. The answer is given by (number of outcomes that add to 16 ) divided by (the total number of possible outcomes) which is 6 / 216 or 1 / 36

1. Since the order in which the group is selected does not matter, we are dealing with a combinations problem (and not a permutations problem).
2. The formula for combinations is:
   N!/((N-K)!K!)
   Where N = the total number of elements from which we will select, 8 people in this case.
   Where K = the total number of elements to select, 4 people in this case.
3. The total number of combinations is therefore:
   8!/((8-4)!4!) = 70
4. However, two of these combinations are not valid since they have all members of one gender.
5. The correct answer is 70-2 = 68

There are two common ways of solving this problem. One involves algebra and the other involves statistical formulas.
Method 1: Use Algebra
Assign variables to the groups of interest:
Let b = the percent of students who took both classes (what we are interested in).
Let f = the percent of students who only took a finance class.
Let m = the percent of students who only took a marketing class.
We know that 40% of the students did not take either class, so 60% (=100% - 40%) must have taken either a finance class, a marketing class, or both.
This 60% is made up of those three distinct groups: those who took a finance class only, those who took a marketing class only, and those who took both:
m+f+b=60%.
We know that 25% of the students took a finance class, which is made up of those who only took this class and those who took both classes:
f+b=25%.
Likewise, 50% of the students took a marketing class, made up of those who only took marketing and those who took both:
m+b=50%.
We are interested in finding the value of b (percent who took both classes). So solve these last two equations for f and m by subtracting b from both sides of each equation:
f=25%-b.
m=50%-b.
Now plug these values of f and m into the first equation:
m+f+b=60%
50%-b + 25%-b + b = 60%.
Combine like terms to simplify:
75% - b = 60%.
Add b to both sides:
75%= 60% + b.
Subtract 60% from both sides:
15%= b.
Thus the correct answer is D.
Method 2: Use Statistical Formulas
In general, the probability of event M or F occurring is P(M∪F) = P(M) + P(F) - P(M∩F) where P(M∩F) is the probability of M and F simultaneously occurring.
In this problem:
P(M) = the probability of a student taking marketing
P(F) = the probability of a student taking finance
P(M∪F) = the probability of a student taking marketing or finance
P(M∩F) = the probability of a student taking marketing and finance; this is the variable we are trying to solve for
Fill in what we know:
P(M) = 50%
P(F) = 25%
An important insight into this problem is to realize that (the probability of a student taking marketing or finance) + (the probability of a student taking neither marketing nor finance) = 1 since these two events are complementary and complementary events must sum to one.
The question tells us that "40% took neither a finance nor a marketing class last semester." As a result, we know that 40% + P(M∪F) = 100%
Consequently: (M∪F) = 60%
Filling all that we know into the fundamental equation:
P(M∪F) = P(M) + P(F) - P(M∩F)
60% = 50% + 25% - P(M∩F)
-15% = - P(M∩F)
P(M∩F) = 15%
Thus the correct answer is D.

1. The old ratio of x to y was 4:6. If this ratio is cut in half, then the new ratio of x to y is 2:6.
2. The old ratio of x to z was 4:10. If this ratio is tripled, then the new ratio of x to z is 12:10.
3. In order to combine these two ratios into a new ratio of x:y:z, we must rewrite them so that the element in common, x, has the same coefficient. With the same x-coefficient, we can compare the ratios of x:y and x:z. Using a multiplier of 6 on the first ratio (x:y = 2:6) yields x:y = 12:36.
4. Since the new ratio of x:z is 12:10, we can combine the new x:y ratio that we multiplied by 6/6 with the new x:z ratio in order to arrive at an x:y:z ratio of 12:36:10. In other words:
   x:z = 12:10
   x:y = 12:36
   With 12 as a common term:
   x:y:z = 12:36:10
5. In order to find the weight of substance x in the total changed compound, set up an equation of the combination:
   x + y + z = 58 lbs
6. Now substitute an unknown multiplier, m, for each quantity to ensure that the ratios are enforced in the equation (x=12m, y=36m, z=10m):
   12m + 36m + 10m = 58 lbs
   Note: The unknown multiplier is the ratio by which the 12:36:10 ratio holds true. In other words, if m = 1, the substances will be in the ratio of 12:36:10. If m = 2, the substances will be in the ratio of 12(2):36(2):10(2) = 24:72:20
7. Combining like terms simplifies the equation to 58m=58.
8. Dividing through by 58 shows a multiplier of m=1.
9. Using this multiplier in the original equation we set up, we can see that the weight of x=12(1)=12. Thus, 12lbs or D is correct.

Statement 1:

 

A regular hexagon can be divided into six equilateral triangles, each with a central angle measure of 60 degrees. Remember that we can inscribe a regular hexagon in a circle such that each vertex is a point on the circumference of the circle. Central Angle Theorem states that the central angle in a circle is always twice an inscribed angle when the point of the inscribed angle (in this case point C) is on the major arc. For this figure, this means angle FCE = 30. Since angle CFE = 60, we have a 30-60-90 triangle. If BE = 14, then FC = 14 as well. We’d now be able to solve for the height (CE) and base (FE) using the 30-60-90 side ratios. Since this is a data sufficiency question, we can stop here and conclude that statement 1 is sufficient.

However, if you actually had to calculate the area of triangle FCE, here is how you might proceed: The sides of a 30-60-90 are in the ratio 1 : √3 : 2 , so the sides of triangle FCE will have the following lengths: FE = 7, CE= 7√3 , FC=14.

Statement 2:
If the hexagon is regular, we can calculate the area of triangle FCE using a process similar to the one illustrated above. However, statement 2 does not tell us that the hexagon is regular. If the hexagon is not regular, we can come up with different areas for triangle FCE, depending on what we assume. Statement 2 alone is not sufficient.

The correct response is (D). This question is provides a great opportunity to try out numbers.
Statement 1: Let’s say m = 9 and n = 1, then m – n = 9 – 1 = 8. There are 7 integers between 1 and 9. If we choose a different set of integers: let’s say m = -2 and n = -10. There are still 7 integers between these two numbers.
Statement 2: If there are 7 integers between m -1 and n -1, then there will still be 7 integers between m and n. This is sufficient for the same reason that statement 1 was sufficient.

The correct response is (B). We can plug in these values for “w” and “y” in the original equation:
(-1/x)(x)(1/z)(z) = -1. The x’s and z’s cancel out so that we get: (-1)(1) = -1. The answer will always be “Yes” no matter what the actual values are for w, x, y, and z. This is sufficient.
If you chose (A), remember this is a Y/N question. The stem does not tell us any information about what numbers w, x, y, and z are. This is a great question to pick numbers! As long as we can choose two sets of numbers: one that gives us an outcomes of “Yes” and the other that gives an outcome of “No,” we know the statement is insufficient.
Let’s choose w = 1, x = 1, y = 1, and z = -1. These numbers satisfy the condition in Statement (1) and allow us the answer the question “Yes.” However, if we chose w = 2, x = 2, y = 2, and z = -2, we would answer the question “No.” Therefore, this Statement must be insufficient.
If you chose (C), you may not have realized that we could substitute the value in Statement (2) to simplify wxyz = -1. By doing this, we realize that for ALL integers, b and d will cancel out leaving us with (-1)(1) = -1, which is always true.
If you chose (D), you correctly saw that substitution allows Statement (2) to be sufficient, but Statement (1) is not sufficient. Depending on what integers we select for the variables we can make their product equal or not equal to -1.
If you chose (E), you missed that Statement (2) is sufficient once we substitute it into the original equation. Sometimes statements will be unexpectedly sufficient in this way on the GMAT, even though we don’t know the actual values for the variables!

The correct response is (D). Each statement alone is sufficient.
Statement 1: If (-j, k) lie above the x-axis and to the right of the y-axis (that is, in the first quadrant), then we can write the following two inequalities: -j>0, and k>0 OR j<0 and k>0. Statement 1 is sufficient to answer the question.
Statement (2): If (j, -k) is in the third quadrant (as the information in statement 2 implies), then j = negative, and –k = negative.
We can write the following inequalities: j<0, and –k<0, OR j<0, and k>0. Statement 2 alone is sufficient to answer the question.

Write out the formula for the mean and arrange it in several different ways so that you can spot algebraic substitutions:
Mean = (w + x + y + z + 10)/5
5*Mean = w + x + y + z + 10
Evaluate Statement (1) alone.
Translate each piece of information into algebra:
"the average (arithmetic mean) of w and y is 7.5"
(w + y)/2 = 7.5
w + y = 15

"the average (arithmetic mean) of x and z is 2.5"
(x + z)/2 = 2.5
x + z = 5
Combine the two equations by adding them together:
(x + z) + (w + y) = (5) + (15)
x + z + w + y = 15 + 5
w + x + y + z = 20
Substitute into the equation from the top:
Equation from top: 5*Mean = w + x + y + z + 10
5*Mean = 20 + 10 = 30
Mean = 6
Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
Simplify the algebra:
-[-z - y -x - w] = 20
z + y + x + w = 20
This can be substituted into the mean formula:
z + y + x + w = 20
w + x + y + z = 20 {rearrange left side to make substitution easier to see}

Equation from top: 5*Mean = w + x + y + z + 10
5*Mean = (w + x + y + z) + 10
5*Mean = 20 + 10 = 30 {substitute information from Statement (2)}
Mean = 6
Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

1. Simplify the question:
Since multiplying a number by 13 does not change its sign, the question can be simplified to: "is N a positive number?"
2. Evaluate Statement (1) alone.
Write out algebraically:
-21N = negative
21N = positive {divided by -1}
N = positive
Since N is a positive number, 13N will always be a positive number.
Statement (1) alone is SUFFICIENT.
3. Evaluate Statement (2) alone.
Any time you are dealing with a number raised to an even exponent, you must remember that the even exponent hides the sign of the base (e.g., x² = 16; x = 4 AND -4).
Solve the inequality:
N2 < 1
-1 < N < 1 {take the square root, remembering that there is a positive and negative root}
Since N can be both positive (e.g., .5) or negative (e.g., -.5), Statement (2) is not sufficient.
Statement (2) alone is NOT SUFFICIENT.
4. Since Statement (1) alone is SUFFICIENT but Statement (2) alone is NOT SUFFICIENT, answer A is correct.

Since the sum of the measure of the interior angles of a triangle equals 180 degrees, you can write the following equation:
The measure of angles A + B + C = 180
Evaluate Statement (1) alone.
Translate Statement (1) into algebra:
A + C = 120
Use the foundational triangle equation (i.e., all angles add up to 180):
A + B + C = 180
(A + C) + B = 180
Substitute A + C = 120 into the equation.
120 + B = 180
B = 60
It is impossible to determine the value of angle C. Angle A could be 60 degrees and angle C could be 60 degrees. However, angle A could be 20 degrees and angle C could be 100 degrees.
Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Translate Statement (2) into algebra:
A + B = 80
Use the foundational triangle equation (i.e., all angles add up to 180):
A + B + C = 180
(A + B) + C = 180
Substitute A + B = 80 into the equation.
80 + C = 180
C = 100
Statement (2) is SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

Draw a diagram of the problem with the information from the question:

AC = 15
You are dealing with a right-triangle since a right angle will be formed by going straight north and then turning straight east.
Evaluate Statement (1) alone.
Translate the information from Statement (1) into algebra:
(AB)/(BC) = 4/3
3(AB) = 4(BC)
Set up a Pythagorean theorem equation:
(AB)² + (BC)² = (AC)²
(AB)² + (BC)² = (15)²
You now have two equations and two unknowns:
Equation (1): (AB)² + (BC)² = (15)²
Equation (2): 3(AB) = 4(BC)
With two unique equations and two unknowns, a solution must exist. With this solution, you can subtract 15 from the detour distance and arrive at an answer.
Statement (1) is SUFFICIENT.
Note: You should not do these calculations on the test since they are not necessary for determining sufficiency. However, to demonstrate that there is a solution, we show how you would arrive at a numerical answer:
(AB)² + (BC)² = (15)²
(AB)² + (.75AB)² = (15)²
{rearrange Equation 2, solving for BC and substitute in BC=.75AB}
(AB)²(1 + .752) = 225 {factor out (AB)²}
(AB)² = 144
AB = 12
Substitute Back into Equation 2:
3(12) = 4(BC)
BC = 9
With these two distances, you can calculate the distance traveled on the detour.
Direct Route: 15
Detour: 9 + 12 = 21
Extra Distance: 21 - 15 = 6
Evaluate Statement (2) alone.
Set up a Pythagorean theorem equation:
(AB)² + (BC)² = (AC)²
(AB)² + (BC)2 = (15)²
You are told that AB = 12. Substitute this information in and solve for BC.
(12)² + (BC)² = (15)²
(BC)² = 81
BC = 9
Statement (2) is SUFFICIENT.
Note: You should not do these calculations on the test since t

There are two possible cases (or conditions) under which the product of X and Y could be positive:
Case (1): Positive(Positive) = Positive
Case (2): Negative(Negative) = Positive
Evaluate Statement (1) alone.
Since Y is raised to an odd exponent, the sign of the base (i.e., the sign of Y) is the same as the sign of the entire expression (i.e., the sign of Y5).
There is no way of distinguishing whether we are in Case (1) or (2) and the answer to the resulting question of whether X + Y is negative can be different depending on the chosen values of X and Y. Consider two examples, one from each case.
Case (1):
X = 100
Y = 2
XY = (100)(2) = 200 = Positive
X > Y5
X + Y = 100 + 2 = 102 = Positive
Case (2):
X = -10
Y = -2
XY = (-10)(-2) = 20 = Positive
X > Y5
X + Y = (-10) + (-2) = -12 = Negative
Since there is no way to determine whether X + Y is positive, Statement (1) is not sufficient.
Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Y6 must be a positive number since, even if Y were negative, raising it to an even exponent would make the entire quantity positive.
Substituting this into the information given in Statement (2):
X > Y6
X > (positive number)
X must be positive since any number that is larger than a positive number is itself positive.
Since X is positive, in order for XY to be positive, Y must also be positive (i.e., we are dealing with Case (1) from above). Consequently, a positive number (i.e., X) plus a positive number (i.e., Y) must itself be positive.
X + Y = ?
Positive + Positive = Positive
We can definitively answer "no" to the original question.
Statement (2) is SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT but Statement (2) alone is SUFFICIENT, answer B is correct.

A number divided by 5 will be an odd integer if and only if that number contains only odd factors, one of which is 5. In other words, there are two conditions under which x divided by 5 will be an odd integer:
(1) x is a multiple of 5
(2) x contains only odd factors
Evaluate Statement (1) alone.
If x contains only odd factors, there is no guarantee that one of those factors is 5. Consequently, there is no guarantee that x will be divisible by 5.
For example:
9 = 3*3 → not an odd integer when divided by 5 since 5 is not a factor
21 = 3*7 → not an odd integer when divided by 5 since 5 is not a factor
15 = 3*5 → an odd integer when divided by 5 since 5 is a factor
105 = 3*7*5 → an odd integer when divided by 5 since 5 is a factor
Statement (1) alone is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Simply because x is a multiple of 5 does not guarantee that x only contains odd factors. Consequently, there is no guarantee that x is divisible by 5.
5 = 5*1 → an odd integer when divided by 5 because 5 is a factor and there are only odd factors
15 = 5*3 → an odd integer when divided by 5 because 5 is a factor and there are only odd factors
25 = 5*5 → an odd integer when divided by 5 because 5 is a factor and there are only odd factors
20 = 5*4 → not an odd integer when divided by 5 because there is at least one even factor
30 = 6*5 → not an odd integer when divided by 5 because there is at least one even factor
Statement (2) alone is NOT SUFFICIENT.
Evaluate Statements (1) and (2) together.
With only odd factors and x as a multiple of 5 (i.e., with 5 as a factor), you know that x divided by 5 must be an odd number since the two conditions laid out earlier are fulfilled.
Consider the following examples:
5 = 5*1 → an odd integer when divided by 5
15 = 5*3 → an odd integer when divided by 5
25 = 5*5 → an odd integer when divided by 5
35 = 5*7 → an odd integer when divided by 5
Statements (1) and (2), when taken together, are SUFFICIENT.
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.

Simplify the equation:
Is (2y+z)(3x)(5y)(7z) < (90y)(14z)?
Simplified: is (2y+z)(3x)(5y)(7z) < ((2*5*3*3)y)((7*2)z)?
Simplified: is (2y+z)(3x)(5y)(7z) < (2y)(5y)(32y)(7z)(2z)?
Simplified: is (2y+z)(3x)(5y)(7z) < (2y+z)(5y)(32y)(7z)?
Cancel out 2y+z, 5y, and 7z
Simplified: is 3x < 32y?
Evaluate Statement (1) alone.
Statement (1) says that x = 1. So, plug that information in and work from there.
Simplified Question: is 31 < 32y where y is a positive integer?
Further Simplified: is 1 < 2y where y is a positive integer?
At this point, some students can see that Statement (1) is SUFFICIENT. However, a more thorough analysis is provided just to be clear.
Since x and y are given as positive integers, the smallest possible value for y is 1. In this case 1 < 2(1). Since the inequality held true when y=1, it will hold true for any legal value of y since y will only get larger and x will not change.
Thus, 3x will always be less than 32y.
Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
Statement (2) says y = 1 and x and z are positive integers. So, plug that information and work from there.
Is 3x < 32(1)?
Or: is 3x < 32?
Or: is x < 2?
Since the only restriction on x is that it is a positive integer, x could be 1 (in which case the inequality would be true and the answer to the question would be "Yes") or, x could be 2 (in which case the inequality would not be true and the answer to the question would be "No").
Since different answers to the question "is x < 2?" are possible, there is no definitive answer to the question. Statement (2) is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

Simplify the original question by factoring:
x² + 2x > x² + x
2x > x
x > 0
Simplified question: is x > 0?
Evaluate Statement (1) alone.
When dealing with a number that is raised to an even exponent, it is important to remember that the sign of the base number can be either positive or negative (i.e., if x² = 16, x = -4 and 4). Moreover, it is important to remember that raising a fraction to a larger exponent makes the resulting number smaller:
(1/2)² > (1/2)³ > (1/2)⁴
There are three possible cases:
Case (1): x < 0
If x were negative, x^odd would be negative while x^even would be positive. This would make x^odd {=negative} < x^even {=positive}, which is an explicit contradiction of Statement (1). As a result, we know x cannot be negative. Statement (1) is SUFFICIENT. At this point, you should not keep evaluating since you know that Statement (1) provides enough information to answer the question "is x > 0?"

Case (2): 0 < x < 1
In this case, based upon what was shown above, for x^{odd integer} > x^even integer to hold true, odd integer must be less than even integer.

Case (3): x > 1
This case is the opposite of Case (2). In other words, for x^{odd integer} > x^even integer, the odd integer must be greater than the even integer.
Since Statement (1) eliminates the possibility of x being a negative number, we can definitively answer the question: is x > 0?
Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
Factor x² + x - 12 = 0
(x - 3)(x + 4) = 0
x = 3, -4
Since x can be either positive or negative, Statement (2) is not sufficient.
Statement (2) alone is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT but Statement (2) alone is NOT SUFFICIENT, answer A is correct.

Evaluate Statement (1) alone.
Substitute the value of x from Statement (1) into the equation and manipulate it algebraically.

Since the question says that y is a positive integer, you know that the cube root of y³, which equals y, will also be a positive integer. Statement (1) is SUFFICIENT.
Evaluate Statement (1) alone (Alternative Method).
For the cube root of a number to be an integer, that number must be an integer cubed. Consequently, the simplified version of this question is: "is x + y² equal to an integer cubed?"
Statement (1) can be re-arranged as follows:
x = y³ - y²
y³ = x + y²
Since y is an integer, the cube root of y³, which equals y, will also be an integer.
Since y³ = x + y², the cube root of x + y² will also be an integer. Therefore, the following will always be an integer:

Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
Statement (2) provides minimal information. The question can be written as: "is the following cube root an integer?"

If y = 4, x + y² = 2 + 42 = 18 and the cube root of 18 is not an integer. However, if y = 5, x + y² = 2 + 52 = 27 and the cube root of 27 is an integer. Statement (2) is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer A is correct.

In order for a number, n, to be divisible by 9, its digits must add to nine. Likewise, the remainder of the sum of the digits of n divided by 9 is the remainder when n is divided by 9. In other words:

To see this, consider a few examples:
Let N = 901
901/9 = 100 + (R = 1)
(9+0+1)/9 = 10/9 = 1 + (R = 1)

Let N = 85
85/9 = 9 + (R = 4)
(8+5)/9 = 1 + (R = 4)

Let N = 66
66/9 = 7 + (R = 3)
(6+6)/9 = 1 + (R = 3)

Let N = 8956
8956/9 = 995 + (R = 1)
(8+9+5+6)/9 = 28/9 = 3 + (R = 1)
Evaluate Statement (1) alone.
Based upon what was shown above, since the sum of the digits of N is always 13, we know that remainder of N/9 will always be the remainder of 13/9, which is R=4.
In case this is hard to believe, consider the following examples:
4540/9 = 504 + (R = 4)
(4+5+4+0)/9 = 13/9 = 1 + (R = 4)

1390/9 = 154 + (R = 4)
(1+3+9+0)/9 = 13/9 = 1 + (R = 4)

7231/9 = 803 + (R = 4)
(7+2+3+1)/9 = 13/9 = 1 + (R = 4)

1192/9 = 132 + (R = 4)
(1+1+9+2)/9 = 13/9 = 1 + (R = 4)
Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
If adding 5 to a number makes it divisible by 9, there are 9-5=4 left over from the last clean division. In other words, N/9 will have a remainder of 4.
To help see this, consider the following examples:
Let N = 4
N+5=9 is divisible by 9 and N/9 -> R = 4

Let N = 13
N+5=18 is divisible by 9 and N/9 -> R = 4

Let N = 724
N+5=729 is divisible by 9 and N/9 -> R = 4

Let N = 418
N+5=423 is divisible by 9 and N/9 -> R = 4
Since N + 5 is divisible by 9, we know that the remainder of N/9 will always be 4. Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

Before even evaluating the statements, simplify the question. In a more complicated data sufficiency problem, it is likely that some rearranging of the terms will be necessary in order to see the correct answer.
Use the formula for a difference of squares (a² - b²) = (a + b)(a - b). However, let x² equal a, meaning a² = x⁴.
x⁴ - y⁴ = (x² + y²)(x² - y²)
Recognize that the expression contains another difference of squares and can be simplified even further.
(x² + y²)(x² – y²) = (x² + y²)(x – y)(x + y)
The question can now be simplified to: "If x and y are distinct positive integers, what is the value of (x² + y²)(x – y)(x + y)?" If you can find the value of (x² + y²)(x - y)(x + y) or x⁴ - y⁴, you have sufficient data.
Evaluate Statement (1) alone.
Statement (1) says (y² + x²)(y + x)(x - y) = 240. The information in Statement (1) matches exactly the simplified question. Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
Statement (2) says xy = yx and x > y. In other words, the product of multiplying x together y times equals the product of multiplying y together x times.
The differences in the bases must compensate for the fact that y is being multiplied more times than x (since x > y and y is being multiplied x times while x is being multiplied y times).
4 and 2 are the only numbers that work because only 4 and 2 satisfy the equation n2 = 2n, which is the condition that would be necessary for the equation to hold true.
Observe that this is true: 42 = 24 = 16.
Remember that x > y, so x = 4 and y = 2. Consequently, you know the value of x⁴ - y⁴ from Statement (2). So, Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.