Ready? Let’s move along.
7. If p + q = 2r + 3, what is the value of p?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Whenever the question stem presents you with an equation with multiple variables and asks you to solve for one of them, you should immediately rewrite the equation to find a more useful question. Specifically, you should isolate the variable you’re asked to solve for and then focus on the expression equal to that variable.
Here’s what that would look like for this problem:
p + q = 2r + 3
Subtract q from each side of the equation to isolate p.
p = 2r – q + 3
Because this is Data Sufficiency, ask, “What do I need to know to know the value of 2r – q + 3?” The answer to that question is simply “2r – q.”
Statement (1) includes p (which we don’t want) but not q (which we do want), so it is not sufficient. Eliminate A and D.
Turn to Statement (2) and try to isolate 2r – q.
r = (q/2) + 2
Multiply each side by 2 to clear the fraction.
2r = q + 4
Subtract q from each side to isolate 2r – q.
2r – q = 4
Statement (2) is sufficient. The correct answer is B.
8. What is the value of x?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Because both (1) and (2) involve y as well as x, each may allow you to solve for x in terms of y, but neither by itself will yield a unique constant value for x. Eliminate A, B, and D.
Are the two statements sufficient together? Probably not, since (1) is not linear equation, and a system of a single linear equation with a single nonlinear equation doesn’t ordinarily yield a solution. Just in case this an exception, though, let’s try to solve this system for x.
(If we’re very clever we might notice that solving for y is just as good as solving for x, since it leads to a value for x. Solving for y might also be easier here. Let’s suppose that we missed that shortcut, though, and just solve for x.)
Solving the system for x begins with solving for y in terms of x in the second equation.
x + 2y = 8
Subtract x from each side.
2y = x + 8
Divide each side by 2.
y = (x/2) + 4
Substitute the expression (x/2) + 4 for y in the first equation.
x – (4/((x/2) + 4)) = 2
Rewrite the denominator (x/2) + 4 as (x/2) + (8/2) or simply as (x + 8)/2.
x – (4/((x + 8)/2) = 2
Simplify the compound fraction.
x – (8/(x + 8)) = 2
Multiply each term by x + 8 to clear the fraction.
x(x + 8) – 8 = 2(x + 8)
Distribute to clear the parentheses.
x^2 + 8x – 8 = 2x + 16
Transpose to arrange in the usual quadratic form
x^2 + 6x + 8 = 0
At this point you might recognize that this is a quadratic but not a perfect quadratic square, and so must have two solutions. If you don’t recognize that, go ahead and solve it.
(x + 2)(x + 4) = 0
x = {-2, -4}
Since the two statements together don’t yield a unique constant value, they are not sufficient. The correct answer is E.
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Be sure to check out the explanations for questions #1-2, and questions #3-4 before moving on to today’s post. Let’s get started.
5. What is the value of m+n?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
There’s no obviously useful way to rephrase the question, so we’ll probably need to solve for m and n and then add the values together. Keep an eye open, though, for an opportunity to solve for the sum m + n directly.
Statement (1) doesn’t allow any good method of breaking up the single term mn, so will not yield a value for m, n, or m + n. Eliminate A and D.
Was that too fast? Suppose that it wasn’t obvious to you that the statement “doesn’t allow any good method of breaking up the single term mn.” What could you do? You could either attempt to isolate the expression for which you’re solving or test values. Let’s try the latter.
To test values for Statement (1) make a simple table. Use this table to ask, “Does every pair of values legal for Statement (1) yield the same answer to our question, that is the same value for m + n?” My table begins with a pair of values m and n whose product is -8, and adds a third column for the sum of those values. If the sum is the same for every row, then the statement is sufficient to answer the question, or we’ve just overlooked a counterexample.
Since different values consistent with Statement (1) yield different answers to our question, (1) is not sufficient. Eliminate A and D.
We could use a similar method with Statement (2), but it’s probably easier in this case to isolate the expression for which we’re solving.
-2m = n
Add m to each side to get m+n on the right-hand side of the equation.
-m = m + n
Since the expression m + n is not equal to a unique constant, unless we’ve missed some more fruitful way to manipulate this equation, Statement (2) doesn’t answer the question. Eliminate B.
Now consider the two statements together. Since Statement (1) is not linear, even together the two statements probably won’t yield a unique constant solution. There are a couple of exceptional cases though, so let’s solve this system of equations.
Since (2) gives us n in terms of m, it makes solving by substitution pretty manageable. Substitute -2m for n in (1).
m(-2m) = -8
Clear the parentheses.
Divide each side by -2.
Take the square root of each side of the equation.
That’s two possible solutions, so even together (1) and (2) are not sufficient to solve for m. It might occur to you that we’re after the sum m + n not just after m. If you want to be extra-careful, substitute the two possible values for m into either of the given equations and see whether they yield the same value for m + n. They do not. The correct answer is E.
6. If xy = 12, what is the value of x – y?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
In this case, the question includes an “if” statement, xy = 12.
Once again there is no obviously useful way to rephrase the question. We could, say, use the “if” equation to solve for y in terms of x, and substitute the resulting expression for y in the question. That’s not vey promising, though, so let’s turn to (1).
Even if we restrict ourselves to positive integers, the “if” statement and (1) together allow a number of solutions, each of which yields a different value for x – y. Specifically, x and y could be 12 and 1, 6 and 2, or 4 and 3, and could therefore be 11, 4, or 1. Eliminate A and D.
Statement (2) and the “if” statement together might seem to allow just one solution, x=4, y=3, and x-y=1. In fact, though, (2) and the “if” statement also allow x=3, y=4, =-1. So (2) is also not sufficient. Eliminate B.
Together (1) and (2) (along with the “if” statement) allow just one solution, x=4, y=3, and x-y=1. The correct answer is C.
Remember to check back tomorrow for the explanations to questions #7 and #8. In the meantime, you can check your answers here.
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Welcome back! I missed you!
This week in GMAT Tuesdays, Kevin breaks down a common critical reasoning question on the test—complete the argument. He covers how to identify the question, the three different tasks to expect with these question types, strategy for approaching these questions, and the common wrong answers to look out for.
And here’s a closer look at the board:
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3. If 2x = 2y – 3z, what is the value of z?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Whenever the question stem presents you with an equation with multiple variables and asks you to solve for one of them, you should immediately rewrite the equation to find a more useful question. Specifically, you should isolate the variable you’re asked to solve for and then focus on the expression equal to that variable.
Here’s what that would look like for this problem:
2x = 2y – 3z
Add 3z and subtract 2x from each side of the equation to isolate the z term.
3z = 2y – 2x
Factor the two from the right-hand side of the equation.
3z = 2(y – x)
Divide each side of the equation by 3.z
Because this is Data Sufficiency, ask, “What do I need to know to know the value of ?” The answer to that question is simply “y -x.”
Turn to Statement (1) and try to isolate y – x.
y = x + 2
Subtract y from each side of the equation.
y – x = 2
Statement (1) is sufficient. Eliminate B, C, and E.
Turn to Statement (2) and try to isolate y – x.
x = y – 2
Subtract x from and add 2 to each side of the equation.
y – x = 2
Statement (2) is sufficient. Eliminate A. The correct answer is D.
4. On each lab that René completed he received either 100 points or 85 points. On how many labs did he score 100 points?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
This problem yields a system of linear equations.
Let the number of labs on which René earned a score of 100 points=x.
The total score for these labs=100x.
Let the number of labs on which René earned a score of 80 points=y.
The total score for these labs=80y.
The total score for all labs=100x+80y.
The question asks us to solve for x.
Statement (1) can be rewritten as 100x+80y=940.
Statement (2) can be rewritten as x+y=12.
When a DS story problem yields a system of distinct linear equations but implicitly requires that solutions be integers, the smart thing to do is to test values. Generally the numbers involved won’t be very large, so the arithmetic won’t be too daunting.
Statement (1): First, stipulate an integer value for y, then calculate 85y, then subtract that product from 1140. If the difference left isn’t a multiple of 100, don’t consider it further:
Since only one integer value for y yields an integer value for x, and since both x and y must be integers, Statement (1) is sufficient. Eliminate B, C, and E.
Statement (2): Any pair of integers that sum to 12 is a possible pair of values for x and y, so Statement (2) is not sufficient. Eliminate D. The correct answer is A.
Remember to check back on Wednesday for the explanations to questions #5 and #6. In the meantime, you can check your answers here.
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1. What is the value of x?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
(hint)
Since (1) is a linear equation with two variables, it will allow you to solve for x only in terms of y. So that statement is insufficient by itself, eliminate answers A and D. This leaves B, C, and E.
Since (2) is also a linear equation with two variables, it will allow you to solve for x only in terms of y. So that statement is insufficient by itself, eliminate B. This leaves C and E.
You might be tempted to choose C, because (1) and (2) together give you two linear equations with the same two variables. As we’ve seen though, we must first make sure that the two equations are distinct. To see whether they are in fact the same equation in different guises, rewrite one of them—say Statement (1)—in the form that other already has.
So let’s solve for y using (1).
5x + 3y = 15
Subtract 5x from each side.
3y=15-5x
Divide each side by 3.
y = 5 – (5/3)x
So (1) and (2) are in fact the same equation. The correct answer is E.
Rather than checking in this way to see whether the two equations are equivalent, you could instead solve the system of equation by substitution, especially since (2) has already isolated y.
Take the value of y given in Statement and substitute that value for y in Statement (1).
Distribute the 3 to clear the parentheses.
5x – 15 + 5x = 15
Subtract 15 to each side to isolate the x terms on the left-hand side of the equation.
5x – 5x = 15 – 15
Simplify.
0=0
That doesn’t give us any information, so the correct answer is E.
2. If y = 2(x+1) what is the value of x + y?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
There’s no obviously useful way to rephrase the question, so we’ll probably need to solve for x and y and then add the values together. Keep an eye open, though, for an opportunity to solve for the sum x + y directly.
You may want to simplify the given equation—the “if” equation—before turning to the statements. Distribute the 2 on the right-hand of the equation to clear the fraction: y = 2x – 2. Now turn to the statements.
You might notice right away that (1) has a variable in a denominator as well as a one outside any denominator. This means that it is not a linear equation and so that information that at first appears sufficient will not be.
Whether you notice that or not, you’ll want to start with the simpler statement (2). Well, that a pleasant surprise! Statement (2) is a linear equation, as is the “if” equation. So unless they’re equivalent equations, (2) should be sufficient. Since both the “if” equation and (2) already isolate y, solving by substitution means setting equal the right hand sides of each equation.
x + 2 = 2x – 2
Subtract x from and add 2 to each side.
x = 4
You could solve for y and add that to 4 to solve x + y, but there’s no need to. If you can see that substituting 4 for x in either of the original equations will yield some constant value for y, then you can conclude that (2) allows you to solve for x + y, and so that (2) is sufficient. Eliminate A, C, and E.
Now let’s turn to Statement (1). A system of a single linear equation (the “if” equation) and a single nonlinear equation (Statement (1)) will not usually yield a unique constant value for each variable, but if we have the time to attempt a solution we should, unless we can rule out the exceptional cases some other way.
Since the “if” equation gives us y in terms of x, we can substitute 2x -2 for y in Statement (1).
Simplify the fraction by dividing the numerator and denominator by 2.
Multiply all terms by to clear the fraction.
x (x – 1) + 3 = 3 (x – 1)
Distribute to clear the parentheses.
Transpose to arrange in the usual quadratic form.
Because this is a quadratic but not a quadratic square, it will have two solutions. Eliminate D. The correct answer is B.
Remember to check back on Monday for the explanations to questions #3 and #4. In the meantime, you can check your answers here. Have a good weekend!
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Today, we present the answers to those questions, so that you can check your work. Over the next week or so, we will publish posts with in-depth explanations of how to arrive at the correct answer to each tricky data sufficiency practice question. Look for the links here:
1. What is the value of x?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: E
2. If y = 2(x + 1) what is the value of x + y?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: B
3. If 2x = 2y – 3z, what is the value of z?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: D
4. On each lab that René completed he received either 100 points or 85 points. On how many labs did he score 100 points?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: A
5. What is the value of m+n?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: E
6. If xy = 12, what is the value of x – y?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: C
7. If p + q = 2r + 3, what is the value of p?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: B
8. What is the value of x?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Correct Answer: E
Come back tomorrow for the first two explanations.
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1) Two teachers, Ms. Ames and Mr. Betancourt, each had N cookies. Ms. Ames was able to give the same number of cookies to each one of her 24 students, with none left over. Mr. Betancourt also able to give the same number of cookies to each one of his 18 students, with none left over. What is the value of N?
Statement #1: N < 100
Statement #2: N > 50
2) In a certain company, 25% of the women and 17% of the men participate in the voluntary equity program. Let M be the number of male employees. If there are 600 employees total, what is the value of M?
Statement #1: M > 100
Statement #2: more than 130 employees participate in the voluntary equity program
Statement #1: AB = 12
Statement #2: AD = CD
Statement #1: k + m = 9
Statement #2: k = 2m
5) Let abc denote the digits of a three-digit number. Is this number divisible by 7?
Statement #1: The two-digit number bc is divisible by 7
Statement #2: a + b + c = S, and S is divisible by 7
6) In the diagram above, AC = AB, and angle DAB = angle DBC. What is the measure of angle BCD?
Statement #1: angle BDC = 2*(angle DAB)
Statement #2: AD = BD
7) If N is a positive integer, does N have exactly three factors?
Statement #1: the integer N^2 has exactly five factors
Statement #2: only one factor of N is a prime number
8) Right now, a barrel of water is exactly 40% full of water. What is the volume of the barrel?
Statement #1: If the water in the barrel right now were increased by 25%, then the barrel would be exactly half full.
Statement #2: If a volume of water equal to half of what is in the barrel right now were added to the barrel, the empty space left in the barrel then would equal 75% of the volume of the water now.
9) In a set of twenty numbers, 19 of the 20 numbers are between 40 and 50. Is the median greater than the mean?
Statement #1: the standard deviation is greater than 15
Statement #2: the 20^{th} number is greater than 100
10) In the diagram above, ABCD is a square with side M, and EFGH is a square with side K. What is the value of (M + K)?
Statement #1: (M – K) = 9
Statement #2: the area of triangle AEH is 36
11) When 900 is divided by positive integer d, the remainder is r. For some integer N > 5000, when N is divided by positive integer D, the remainder is R. Is R > d?
Statement #1: r = 1
Statement #2: D = 23
12) If F is the prime factorization of N!, how many factors in F have an exponent of 1?
Statement #1: 30 ≤ N ≤ 40
Statement #2: 25 ≤ N ≤ 35
On the GMAT, part of the challenge is deciphering what is being asked. This is particularly the case on the Data Sufficiency questions, because insofar as the prompt question is obscure, it’s hard to determine what information would be sufficient to answer this question. Part of our task on GMAT DS questions is to re-interpret the prompt question given, often discerning a much simply way to ask the same thing. Once we have a simple version, it’s that much easy to find the answer.
Unfortunately, there’s no simple rule for how to simplify such DS prompt. You have to know the fundamental mathematical definitions well, and you have to be agile in interpreting the particular combination of restraints given in a problem. Be mindful of this when you study solutions to problems: if the solutions begin by making a simplification of the prompt, probably everything after that point is relatively easy to follow, but pay attention to what the author of the solution had to notice about the problem in order to make that simplification. As I have discussed in other blogs, the primary question should not just be “what to do? what steps do I take to get to a solution?” but more “how to see? how does one reframe the problem and what it is asking?” In many cases, once we can frame the problem correctly, what to do becomes much more straightforward.
Here are some more blogs on the topic of GMAT DS.
2) GMAT Sample Data Sufficiency Practice Questions
3) GMAT Data Sufficiency: More Practice Questions
If you have any insights about rephrasing the question that you would like to share, please let us know in the comments section!
1) This question is really about common multiples and the LCM. If Ms. Ames can give each of her 24 students k cookies, so that they all get the same and none are left over, then 24k = N. Similarly, in Mr. Betencourt’s class, 18s = N.
What are the common multiples of 18 and 24?
18 = 2*9 = 2*3*3 = 6*3
24 = 3*8 = 2*2*2*3 = 6*4
From the prime factorizations, we see that GCF = 6, so the LCM is
LCM = 6*3*4 = 72
and all other common multiples are multiples of 72: {72, 144, 216, 288, 360, …}
Statement #1: if N < 100, the only possibility is N = 72. This statement, alone and by itself, is sufficient.
Statement #2: if N > 50, then N could be 72, or 144, or 216, or etc. Many possibilities. This statement, alone and by itself, is not sufficient.
Answer = (A)
2) In this problem, it’s very important to realize: when the GMAT say “17% of the men,” it doesn’t mean 16.93% or 17.02%. It means exactly, precisely, 17%. This 17% of M must be a whole number, a positive integer. The only way this can happen is if M equals either 100 or a multiple of 100. That’s a huge insight. Both M and F (the number of female employees) must be multiples of 100 that add up to 600. In other words, there aren’t that many possibilities for the gender breakdown in this company.
Statement #1: M > 100. Well, M could be 200 or 300 or 400 or etc. Many possibilities. This statement, alone and by itself, is not sufficient.
Now, forget about statement #1!
Statement #2: Now, we have to think about the number in the equity program.
If M = 0, and F = 600, then there are 150 in the equity program. A possibility.
If M = 100 & F = 500, then 17 males & 125 females, which results in 142 in the equity program. A possibility.
If M = 200 & F = 400, then 34 males & 100 females, which results in 134 in the equity program. A possibility.
If M = 300 & F = 300, then 51 males & 750 females, which results in 126 in the equity program. This is not possible, and as we add more men and subtract women, this number will drop even lower, so there are no more possible cases.
Right now, we have many possibilities. This statement, alone and by itself, is not sufficient.
Combined: As you probably anticipated, with both statements, things are looking good. If M > 100, then the only case that allows for more than 130 in the equity program is the case with M = 200 & F = 400. We now know that M = 200, so we have definitively answered the prompt question. Combined, the statements are sufficient.
Answer = (C)
3) In order to find the area of the circle, we would need the radius, AD.
Statement #1: AB = 12. Notice that AD is the base of triangle ABD. AB is the height, now known, and we know the area. If we know the height and the area, we can find the base by A = 0.5*bh. Therefore, we can find AD, which would allow us to find the area of the circle. This statement, alone and by itself, is sufficient.
Statement #2: We know that AD = AC, because all radii of a circle are equal. If CD also equals these two, then ACD is an equilateral triangle. That means that the angle ADC = 60°, which means that triangle ABD is a 30-60-90 triangle. If AD = x is the base, then this times the square-root of 3 is the height, and we could create an equation because we know the area:
This is an equation we could solve for x, which would allow us to find the area of the circle. This statement, alone and by itself, is sufficient.
Answer = (D)
4) We can re-arrange the numerator to simplify:
We really need to find the value of that smaller fraction, with just k in the numerator.
Statement #1: k + m = 9
This doesn’t help us. We cannot plug this in as is. If we solve for one variable, say k, and plug in, we will get an expression that doesn’t cancel. This does not lead to an answer. This statement, alone and by itself, is not sufficient.
Statement #2: k = 2m
We can plug this into the final form of the expression.
That final value we could write as a mixed numeral or change to an improper fraction, but it doesn’t matter: we have a numerical value for the expression. This statement, alone and by itself, is sufficient.
Answer = (B)
5) Statement #1, by itself, doesn’t help us. The number 49 is divisible by 7, but think about 149, which equals (100 + 49): the divisor 7 would divide evenly into 49, but it wouldn’t divide evenly in to 100, so 149 can’t possibly be divisible by 7.
Another way to think about it: 140 is a multiple of 7, so 147 must be, and 154 must be as well. The number 149 falls between the multiples of 7. (In fact, you would not need to know this for the GMAT, but 149 is a prime number.)
Of course, if the hundred’s digit is 7, then all the number of this form would be divisible by 7: 735, 742, 749, 756, etc. So, we can get either a yes or a no answer with this statement.
This statement, alone and by itself, is not sufficient.
Statement #2: a + b + c = S, and S is divisible by 7
This is reminiscent of the trick for divisibility by 3. That trick works for 3, but it doesn’t work for 7!!
Yes, we could find a number, such as 777, which is divisible by 7 and whose digits add up to 21, also divisible by 7. That number would produce a “yes” answer with this prompt.
But we also could find a number such as 223. The digits add up to 7, so this could be the number according to statement #2. Think about the multiples of 7 in that vicinity. Clearly, 21 is a multiple of 7, so 210 must also be. From there, 210, 217, 224, 231, etc. The number 223 is between the multiples of 7, so it not a multiple of 7. (In fact, you would not need to know this for the GMAT, but 223 is also a prime number.) This produces a “no” answer. Two different prompt answer possible.
This statement, alone and by itself, is not sufficient.
Combined statements:
For our “yes” representative, we can pick 777: the last two digits form 77, a number divisible by 7; the sum of the digits is 21, divisible by 7; and the number 777 is most certainly divisible by 7.
For our “no” representative, we can pick 149: the last two digits form 49, a number divisible by 7; the sum of the digits is 149, divisible by 7; and, as discussed above, the number 149 is not divisible by 7.
Even with both statements, two prompt answers are possible. Even together, the statements are not sufficient.
Answer = (E)
(This is totally beyond what you would need to know for the GMAT, but the numbers 149 and 421 and 491 and 563 are the three digit numbers that satisfy both statements but are prime!!)
6) From the prompt, we know that triangle ABC is isosceles, with AB = AC and angle ABC = angle DCB. Because angle DAB = angle DBC, and they share the angle at C, we know triangle BCD is similar to triangle ABC; therefore, triangle BCD must also be isosceles, with BC = BD and angle BDC = angle BCD. For simplicity, let’s say that
x = angle DAB = angle DBC
y = angle ABC = angle DCB = angle BDC
We know that (x + 2y) = 180°, and the prompt is asking for the value of y.
Statement #1: angle BDC = 2*(angle DAB)
In other words, y = 2x. Then
x + 2y = x + 4x = 5x = 180°
This means x = 36° and y = 72°. This statement leads directly to the numerical value sought in the prompt. This statement, alone and by itself, is sufficient.
Statement #2: AD = BD
This tell us that triangle ABD is also isosceles. This means that angle DAB = angle ABD. Think about angle ABD. That angle is the “leftover” between two angles we have already discussed:
angle ABD = (angle ABC) – (angle CBD) = y – x
Well, angle DAB = x, so if these two are equal, this means:
y – x = x
y = 2x
This turns out to be the exact same information that was given in statement #1, which we already know is full sufficient.
Answer = (D)
(BTW, more than you need to know for the GMAT, but these are Golden Triangles, because the ratio AB/BC equals the Golden Ratio! )
7) This is a question about prime numbers in disguise. Of course, any prime number has exactly two factors: 1 and itself. If we multiple two different primes, say 2 and 5, we get a number with four factors: the factors of 10 are {1, 2, 5, 10}. The only way to get a number with exactly three factors is if the number is the perfect square of a prime number. For example, 9 is the square of 3, and the factors of 9 are {1, 3, 9}; 25 is the square of 5, and the factors of 25 are {1, 5, 25}. If P is a prime number, then the factors of P squared are (a) 1, (b) P, and (c) P squared. Three factors. That’s what the question is asking. Is N the perfect square of a prime number.
Statement #1: very interesting.
If N is a prime number itself, then its square only has three factors. Squaring a prime doesn’t produce enough factors. This doesn’t meet the condition of this statement.
If N is the product of two primes, then its square has 7 factors. For example, 2*5 = 10, and 10 square is 100 which has seven factors: {1, 2, 4, 5, 10, 20, 25, 100}. Squaring a product of primes produces too many factors. This also doesn’t meet the condition of this statement.
The only way the square of N could have five factors is if N is the square of a prime number. Suppose N = 2 square, which is 4. Then N squared would be 16, which has five factors: {1, 2, 4, 8, 16}. In general, if P is a prime number, and N equals P squared, then N squared would equal P to the 4^{th} power, which has five factors:
Therefore, N must be the square of a prime number, so we can give a clear “yes” answer to the prompt question. This statement, alone and by itself, is sufficient.
Statement #2:
This is tricky. If we know that N is the square of a prime number, then this statement would be true, but that’s backwards logic. We want to know: if this statement is true, does it allow us to conclude that N is the square of a prime number? If only factor of N is a prime number, then N could be:
(a) a prime number: the only prime factor of 7 is 7.
(b) any power of a prime number: the only prime factor of 7 to the 20^{th} is 7
So N could be the square of a prime number, or just the prime number itself, or the cube of the prime number, or the fourth power, or etc. Any number of factors would be possible, so we have no way to answer the question. This statement, alone and by itself, is not sufficient.
Answer = (A)
8) In this problem, both statements are tautological. That is to say, each one, in a less-than-obvious way, rephrases the information in the prompt. In a way, we simply have the same information, the information already given in the prompt, recycled in three different ways. No new information is added by either statement.
From the prompt, we know that the barrel is 40% full of water right now.
Well, suppose a barrel is 40%, and we increase the water by 25%, or 1/4. That would mean adding a quarter of 40%, or 10% of the barrel to what is already in the barrel. This would increase the barrel to 50%, which is exactly what statement #1 tells us.
Go back to the 40% full barrel: the empty space is 60%. Half of that is 30%. Suppose we add 30% more: then the water would be 70% of the barrel, and the new empty space would be 30%. The new empty space, 30%, is 3/4, or 75%, of the old amount of water, 40%. This is exactly what statement #2 tells us.
Of course, the prompt by itself is never sufficient, and if neither statement adds any new information, then even altogether, we can’t deduce anything.
A totally different way to say this is: the prompt is asking for an actual quantity, the real volume of the barrel, but the prompt & statements give us nothing but ratio information (including percents & fractions). We need a real measurement to get a real measurement, and we never get one in this problem.
Nothing is sufficient.
Answer = (E)
9) For a symmetrical distribution, the mean = median, and if the median is close to symmetry, the mean and the median are close in value. When the distribution of numbers is radical asymmetrical, with one outlier or several outliers on only one side of the distribution, then the mean is pulled in the direction of the outliers. The median, resistant to outliers, stays in the middle of the majority of numbers, but the mean is sensitive to outliers, gets pulled in their direction. High outliers pull the mean up, and low outliers pull the mean down.
Statement #1: the standard deviation is greater than 15
This tells us that there’s large variation, suggesting that the 20^{th} number is far away from the other 19, but far away in which direction? Much higher or much lower than the rest of the numbers? We don’t know. A high outlier would pull the mean up, and a low outlier would pull the mean down. Here, we know we have an outlier, but we don’t know its directions, so we don’t know in which direction the mean is affected. We cannot answer the question. This statement, alone and by itself, is not sufficient.
Statement #2: the 20^{th} number is greater than 100
Now, we know that the outlier is a high outlier, much bigger than the other numbers in the set. A high outlier pulls the mean up, away from the median, so the mean is higher than the median. We can give a definitive “yes” answer to the prompt question. This statement, alone and by itself, is sufficient.
Answer = (B)
10) In the diagram, we literally have two squares, M squared and K squared. The four right triangle are what we get if we subtract one square from the other: they are literally the difference of two squares, so we can use the Difference of Two Squares formula:
We could find (M + K) if we knew (M – K) and the difference of the squares.
Statement #1: this gives us (M – K), but we don’t know the difference of the two squares. This statement, alone and by itself, is not sufficient.
Statement #2: this gives us the area of one triangle, and if we multiply by 4, we have the difference of the two squares. But, now we don’t know (M – K), so we can solve. This statement, alone and by itself, is not sufficient.
Combined statements. With the two statements, we know both (M – K) and the difference of the squares, so we can solve for (M + K). Together, the statements are sufficient.
Answer = (C)
11) This is a tricky one about remainders.
Statement #1: If r = 1, then we divide 900 by d, and the remainder is 1. This means that d is a factor of 899. That’s interesting, but at the moment, we know zilch about R, which could be anything. This statement, alone and by itself, is not sufficient.
Statement #2: If D = 23, then when we divide by 23, the remainder has to be smaller than the divisor. We know R < 23. But, now, the only thing we know about d is that it’s not a factor of 900: d could be 7 or 97. We have no idea of its size, so we can’t compare it to R. This statement, alone and by itself, is not sufficient.
Combined:
From the second statement, we know R < 23. From the first, we know d must be a factor of 899. What are the factors of 899? For this we will use an advanced factoring technique. Notice that 899 = 900 – 1. This means, we can express 899 as the Difference of Two Squares, because 900 is 30 squared. We can use that algebraic pattern to factors numbers.
So, it turns out that 899 is the product of two prime numbers, 29 and 31. This means that 899 has four factors: {1, 29, 31, and 899}. Those are the candidate values for d. Obviously, d cannot equal 1, because when we divide any integer by 1, we never get a remainder of any sort: 1 goes evenly into every integer. That means, d could be 29 or 31 or 899. Well, if R < 23, this means that R must be less than d. We can give a definitive “yes” answer to the prompt question. Combined, the statements are sufficient.
Answer = (C)
12) This is a tricky one.
Let’s think about, say, 40! This number, 40!, is the product of all the integers from one to 40. Let’s think about its prime factorization. It would have at least one factor of 2 for every even number from 2 to 40, and a second factor for every multiple of 4, and a third factor for every multiple of 8, etc.; a lot of factors of two. Think about the factors, say, of 7: there are five multiples of 7 from 7 to 35, so in the prime factorization of 40!, the factor 7 would have an exponent of 5. Which factors would have exponents of 1? Well, the prime numbers that are less than N, but have no other multiples less than N. For example, in 40!, the factor 37 would have an exponent of 1 since it appears once and no other multiple of it is less than 40.
Statement #1: 32 ≤ N ≤ 40
As we move through different N’s in this region, we cross the prime number 37, which will have an exponent of 1 if it appears. Some N’s include this prime number and some don’t, so the number of factors with an exponent of 1 is different for different values of N. This statement, alone and by itself, is not sufficient.
Statement #2: 27 ≤ N ≤ 35
As we move through different N’s in this region, we cross two prime numbers, 29 and 31, each of which will have an exponent of 1 if it appears. Some N’s include neither, some include 29 and not 31, and some include both, so the number of factors with an exponent of 1 is different for different values of N. This statement, alone and by itself, is not sufficient.
Combined: 32 ≤ N ≤ 35
Now, there are no prime values in the range specified. But, here’s a tricky thing. If N = 32 or 33, then either 32! or 33! contains exactly one factor of the prime numbers {17, 23, 29, 31}: four prime factors with an exponent of one. BUT, if N = 34 or 35, there are now two factors of 17 (one from 17 and one from 34), either 34! or 35! contains exactly one factor of the prime numbers {23, 29, 31}: three prime factors with an exponent of one. Even in this narrow range, different choices lead to different answers for the prompt question. Even together, the statements are not sufficient.
Answer = (E)
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Hello!! GMAT + Tuesday = GMAT Tuesdays!
Active reading is a tricky concept and not as obvious as many people make it seem. This week, Kevin unearths another useful strategy to help you activate your reading. To prepare for the video, first read the article If an Algorithm Wrote This, How Would You Even Know?
Check out a super close-up pic of this week’s board:
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Today’s post is a short test to see whether you can avoid the traps I’ve recently written about. So that your eyes aren’t drawn to the answer key, we’ll publish it as a separate post. The best way to use the answer key is to read your answers to someone else, and have them tell you what you got wrong. Then retry those questions you got wrong, without first learning what the right answer is. In case you want a hint, after each question below we’ve included a link to the relevant recent post. Don’t use the hint on your first attempt at a question, and don’t use it at all if you don’t need it.
Over my next four blog posts I’ll also publish explanations, just a couple of questions at a time.
1. What is the value of x?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
(hint)
2. If y = 2(x + 1) what is the value of x + y?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
3. If 2x = 2y – 3z, what is the value of z?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
4. On each lab that René completed he received either 100 points or 85 points. On how many labs did he score 100 points?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
5. What is the value of m+n?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
6. If xy = 12, what is the value of x – y?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
7. If p + q = 2r + 3, what is the value of p?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
8. What is the value of x?
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Ready for the answers? Here’s the key.
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The battle for the highest GMAT score in the world, that is.
If you took the GMAT test in 2014, then you weren’t just competing against those in your test center–you were also competing with every other test taker in the world (how’s that for some high stakes? Cue suspenseful music!). Your score was combined with the GMAT score of every other test taker in your nation and then compared to the average GMAT score of every other nation in the world.
Which country do you think came out on top? India, Iran, the U.S., Spain? Maybe none of the above? Read on to find out!
I was also very curious about this question. My initial guess was–surprise, surprise–the U.S.! But before I got too carried away, our designer Mark used the GMAC’s Citizenship Report for 2014 to put together an interactive map that solved the mystery.
Scroll over the map below to see if your guess was also wrong or right! You’ll find information on the average GMAT score for each country, the average age of all test takers in that country, and the total number of people in that country who took the GMAT in 2014.
And the award for highest GMAT score in the world goes to…New Zealand! The same land that gave us Lord of the Rings and The Hobbit takes the crown when it comes to GMAT scores. With a whopping average score of 608, New Zealand was also one of the only 3 other nations that broke the 600 score range (Australia and Singapore are in that group as well, with average scores of 605 and 603 respectively).
Bringing in the rear with the lowest GMAT scores in the world, we find Sierra Leone with 317, the Republic of Congo with 314, and Saudi Arabia with 307.
My pick for highest score, the United States, came in at 47th place on the list with an average score of 537. So my prediction was so very wrong–but that’s alright, America. There’s always 2015!
So what should we do with all this information? Well for starters, is anyone in favor of moving to New Zealand with me? We’d improve our GMAT scores and enjoy living among lots of sheep.
But on a more serious note, it’s hard to say exactly how much we can conclude about a country’s academic level based on this average test score data. For one, some of the disparity in scores can be explained by the differing cultural views of standardized tests which we’d find in each country. In the U.S., for example, many approach the GMAT as a sort of decision maker–they take the exam in an effort to figure out of business school is really right for them. But in another country, the approach to the GMAT would be very different. Especially since many top ranked b-schools are in the U.S., test takers from other countries would likely be the cream of the crop in their countries. They would have already committed to a b-school track and would be taking the test to compete for a spot at a top ranked U.S. school.
Another explanation for this test score disparity is that each country also differs in its ability to provide a climate conducive to higher education and effective test preparation. So one could argue that a better and fairer method of comparing GMAT scores around the world would involve bracketing test scores from countries that have similar resources and environments and then determining the highest and lowest scoring nations in those brackets. That way, we’d be able to make more accurate statements about the academic level of people in a given nation.
But until then, congrats, New Zealand!
If you’ve got any thoughts on this GMAT test score distribution, or if have predictions about how things might change in 2015, please leave us a comment below!
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