Without further ado, today’s sample problem comes from our friends at Manhattan GMAT:

Problem:

x6 – y6 =

(A) (x3 + y3)(x2 + y2)(x – y)

(B) (x3 – y3)(x3 – y3)

(C) (x2 + y2)(x2 + y2)(x + y)(x – y)

(D) (x4 + y4)(x + y)(x – y)

(E) (x3 + y3)(x2 + xy + y2)(x – y)

Solution:

An efficient way to attack this problem is to rephrase the expression in the stem. Since a sixth power is a square, we can look at the difference of sixth powers as the difference of squares, and factor:

x6 – y6 = (x3)2 – (y3)2 = (x3 + y3)(x3 – y3)

This new expression is not among the answer choices, but as a search strategy, we should focus on these factors.

Choice (A) contains (x3 + y3), so we should determine whether the rest of (A) works out to (x3 – y3). The terms (x2 + y2)(x – y) look promising, since they give us +x3 and –y3, but we get cross-terms that don’t cancel: (x2 + y2)(x – y) = x3 – x2y + xy2 – y3.

Choice (B) is close but not right. (x3 – y3)(x3 – y3) = (x3 – y3)2, which also gives us cross-terms that don’t cancel: (x3 – y3)2 = x6 – 2x3y3 + y6. (Moreover, the sign is wrong on the y6 term.)

Let’s skip to choice (E), since we see (x3 + y3). The remaining terms multiply out as follows: (x2 + xy + y2)(x – y) = x3 + x2y + xy2 – x2y – xy2 – y3 = x3 – y3. This is what we were looking for. We can verify that the other answer choices do not multiply out to x6 – y6 exactly.

The correct answer is (E).

Of course, you can also plug simple numbers. Don’t pick 0 for either x or y, because too many terms will cancel. Also, don’t pick the same number for x and y, because then the result is 0 (and every answer choice gives you 0 as well). But if you pick 2 and 1, you can keep the quantities relatively small and still eliminate wrong answers.

26 – 16 = 64 – 1 = 63. Our target is 63.

(A) (8 + 1)(4 + 2)(2 – 1) = 48 (B) (8 – 1)(8 – 1) = 49 (C) (4 + 2)(4 + 2)(2 + 1)(2 – 1) = 108 (D) (16 + 1)(2 + 1)(2 . . . → Continue Reading]]> In an effort to meet the needs of our growing audience, we’re pleased to announce that this site will now regularly feature GMAT practice problems, test-taking advice and more.  Content will be provided by a range of test preparation companies, many of which were profiled in the Clear Admit Guide to GMAT Test Preparation Companies (a free publication available for immediate download).  As such, stay tuned to this site for expanded coverage of the GMAT exam, helpful test-taking strategies and more!

Without further ado, today’s sample problem comes from our friends at Manhattan GMAT:

Problem:

x6 – y6 =

(A) (x3 + y3)(x2 + y2)(x – y)

(B) (x3 – y3)(x3 – y3)

(C) (x2 + y2)(x2 + y2)(x + y)(x – y)

(D) (x4 + y4)(x + y)(x – y)

(E) (x3 + y3)(x2 + xy + y2)(x – y)

Solution:

An efficient way to attack this problem is to rephrase the expression in the stem. Since a sixth power is a square, we can look at the difference of sixth powers as the difference of squares, and factor:

x6 – y6 = (x3)2 – (y3)2 = (x3 + y3)(x3 – y3)

This new expression is not among the answer choices, but as a search strategy, we should focus on these factors.

Choice (A) contains (x3 + y3), so we should determine whether the rest of (A) works out to (x3 – y3). The terms (x2 + y2)(x – y) look promising, since they give us +x3 and –y3, but we get cross-terms that don’t cancel: (x2 + y2)(x – y) = x3 – x2y + xy2 – y3.

Choice (B) is close but not right. (x3 – y3)(x3 – y3) = (x3 – y3)2, which also gives us cross-terms that don’t cancel: (x3 – y3)2 = x6 – 2x3y3 + y6. (Moreover, the sign is wrong on the y6 term.)

Let’s skip to choice (E), since we see (x3 + y3). The remaining terms multiply out as follows:
(x2 + xy + y2)(x – y) = x3 + x2y + xy2 – x2y – xy2 – y3 = x3 – y3. This is what we were looking for. We can verify that the other answer choices do not multiply out to x6 – y6 exactly.

The correct answer is (E).

Of course, you can also plug simple numbers. Don’t pick 0 for either x or y, because too many terms will cancel. Also, don’t pick the same number for x and y, because then the result is 0 (and every answer choice gives you 0 as well). But if you pick 2 and 1, you can keep the quantities relatively small and still eliminate wrong answers.

26 – 16 = 64 – 1 = 63. Our target is 63.

(A) (8 + 1)(4 + 2)(2 – 1) = 48
(B) (8 – 1)(8 – 1) = 49
(C) (4 + 2)(4 + 2)(2 + 1)(2 – 1) = 108
(D) (16 + 1)(2 + 1)(2 – 1) = 51
(E) (8 + 1)(4 + 2 + 1)(2 – 1) = 63

Again, the correct answer is (E).

Incidentally, a “zenzicube” is a very old word for the sixth power of a number. An even funnier word exists for the eighth power of a number: zenzizenzizenzic. The “zenzi” part means “square,” so a “zenzicube” is the square of a cube, or (x3)2 = x6. Likewise, a “zenzizenzizenzic” is the square of a square of a square, or ((x2)2)2 = x8. Modern exponential notation has its advantages.

http://en.wikipedia.org/wiki/Zenzizenzizenzic

Question If p, x, and y are positive integers, y is odd, and p = x2 + y2, is x divisible by 4?

(1) When p is divided by 8, the remainder is 5.

(2) x – y = 3

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Solution Statement (1): SUFFICIENT. We know that p is odd. We know from the problem stem that y is odd, which means that y2 is odd. Therefore, x2 must be even (because O = E + O), so x must be even.

However from all of this, we can infer something else—specifically, that x is NOT a multiple of 4. Here’s why:

y2 = (y2 — 1) + 1, which using the quadratic property, yields: y = (y+1)(y—1) +1. Because y is odd, then (y+1)(y—1) is even times even, which is a multiple of 4. However we ALSO know that either y+1 or y—1 is a multiple of 4, because they are consecutive multiples of 2. Therefore, y2 is 1 greater than a multiple of 8. (You can confirm this by thinking about all squared odd numbers: 12 = 1, 32 = 9, 52 = 25, 72 = 49, etc.)

Since y2 divided by 8 yields a remainder of 1, from statement (1) we need x2 divided by 8 to yield a remainder of 4. This means that x must be even, but NOT be a multiple of 4, because if it were a multiple of 4, then x2 would be a multiple of 16, and therefore a multiple of 8.

Statement (2): INSUFFICIENT. We know that x is even. However, we can come up with two different examples which lead to different answers. For example, if x = 8 and y = 5, then x is a multiple of 4, but if x = 6 and y = 3, then x is not a multiple of 4.

The answer is A: Statement (1) is sufficient to answer the question, but statement (2) is insufficient.

Question If p, x, and y are positive integers, y is odd, and p = x2 + y2, is x divisible by 4?

(1) When p is divided by 8, the remainder is 5.

(2) x – y = 3

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Question If n is a positive integer and x does not equal zero, is x^n > x^(n+1)?

1) x < 1

2) n is even.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Solution The answer to the question depends on the values of both x and n. Specifically, we care about the value of x^n, since this will determine how we can rephrase the question.

If n is even, then x^n > 0, no matter what the value of x is (remember that x is nonzero). Likewise, if x > 0, then x^n > 0, no matter what the value of n is.

The reason that we care about the value of x^n is that we can simplify the question by dividing by x^n:

After we divide both sides of the inequality by x^n, the question “Is x^n > x^(n+1)?” becomes “Is 1 > x?” ONLY IF x^n > 0, which is true if x > 0 OR if n is even. (Recall that x is nonzero; thus, we are allowed to divide by x^n.) On the other hand, if x^n < 0, then the question rephrases to “Is 1 < x?”

Statement 1: INSUFFICIENT. We know that x < 1, but x could be positive or negative. Moreover, we do not know whether n is even or odd. As a result, we do not know the sign of x^n, and thus we do not know the answer to either the rephrased question or to the original question.

Alternatively, you can choose positive and negative values of x and an odd n, in order to test the question. If n = 1 and x is positive (but less than 1), then x^n > x^(n+1). But if n = 1 and x is negative, then x^n > x^(n+1).

Statement 2: INSUFFICIENT. We know that n is even, so we know that x^n > 0, and therefore we can rephrase the question as “Is 1 > x?” However, we do not know the answer to that question.

Statements 1 & 2 TOGETHER: SUFFICIENT. Using Statement (2), we can rephrase the question as “Is 1 > x?”, to which Statement (1) gives us a definitive answer.

The answer is C: BOTH statements TOGETHER are sufficient to answer the question, but neither statement alone is sufficient.

Question If n is a positive integer and x does not equal zero, is x^n > x^(n+1)?

1) x < 1

2) n is even.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Question A certain military vehicle can run on pure Fuel X, pure Fuel Y, or any mixture of X and Y. Fuel X costs $3 per gallon; the vehicle can go 20 miles on a gallon of Fuel X. In contrast, Fuel Y costs $5 per gallon, but the vehicle can go 40 miles on a gallon of Fuel Y. What is the cost per gallon of the fuel mixture currently in the vehicle’s tank?

1) Using fuel currently in its tank, the vehicle burned 8 gallons to cover 200 miles.

2) The vehicle can cover 7 and 1/7 miles for every dollar of fuel currently in its tank.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Solution This problem has to do with weighted averages. To find the cost per gallon of the fuel mixture currently in the vehicle’s tank, we need to know the ratio (by gallons) of Fuel X to Fuel Y in that mixture.

Statement 1: SUFFICIENT. The statement tells us that the vehicle covered 200 miles on 8 gallons of the fuel mixture; that is, the fuel mixture delivers 25 miles per gallon. As a result, we can find the ratio of Fuel X (20 mpg) to Fuel Y (40 mpg) in the mixture (this ratio turns out to be 3:1, but we do not need to find the exact ratio; we simply need to know that there will be one unique ratio). Finally, we can use this ratio to find the weighted average cost per gallon of the fuel mixture (this weighted average turns out to be $3.50 per gallon, but again, we do not need the exact number).

Statement 2: SUFFICIENT. The statement tells us that $1 of fuel “buys” 7 and 1/7 miles. Multiplying through by 7, we can rephrase the ratio as $7 for every 50 miles. Let’s now express the weighted average cost per gallon as $3w + $5(1-w) = 5 – 2w, where w represents the percent of Fuel X in the mixture (and 1-w represents the percent of Fuel Y in the mixture). The number of gallons bought by the $7 is then $7 divided by the average cost per gallon, or 7/(5 – 2w). Likewise, we can express the weighted average fuel efficiency (miles per gallon) as 20w + 40(1-w) = 40 – 20w. The number of gallons burned to cover 50 miles is then 50 divided by the average miles per gallon, or 50/(40 – 20w). We can now set these numbers of . . . → Continue Reading]]> Check below to see what was paid at the pump in yesterday’s GMAT question!

Question
A certain military vehicle can run on pure Fuel X, pure Fuel Y, or any mixture of X and Y. Fuel X costs $3 per gallon; the vehicle can go 20 miles on a gallon of Fuel X. In contrast, Fuel Y costs $5 per gallon, but the vehicle can go 40 miles on a gallon of Fuel Y. What is the cost per gallon of the fuel mixture currently in the vehicle’s tank?

1) Using fuel currently in its tank, the vehicle burned 8 gallons to cover 200 miles.

2) The vehicle can cover 7 and 1/7 miles for every dollar of fuel currently in its tank.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Solution
This problem has to do with weighted averages. To find the cost per gallon of the fuel mixture currently in the vehicle’s tank, we need to know the ratio (by gallons) of Fuel X to Fuel Y in that mixture.

Statement 1: SUFFICIENT. The statement tells us that the vehicle covered 200 miles on 8 gallons of the fuel mixture; that is, the fuel mixture delivers 25 miles per gallon. As a result, we can find the ratio of Fuel X (20 mpg) to Fuel Y (40 mpg) in the mixture (this ratio turns out to be 3:1, but we do not need to find the exact ratio; we simply need to know that there will be one unique ratio). Finally, we can use this ratio to find the weighted average cost per gallon of the fuel mixture (this weighted average turns out to be $3.50 per gallon, but again, we do not need the exact number).

Statement 2: SUFFICIENT. The statement tells us that $1 of fuel “buys” 7 and 1/7 miles. Multiplying through by 7, we can rephrase the ratio as $7 for every 50 miles. Let’s now express the weighted average cost per gallon as $3w + $5(1-w) = 5 – 2w, where w represents the percent of Fuel X in the mixture (and 1-w represents the percent of Fuel Y in the mixture). The number of gallons bought by the $7 is then $7 divided by the average cost per gallon, or 7/(5 – 2w). Likewise, we can express the weighted average fuel efficiency (miles per gallon) as 20w + 40(1-w) = 40 – 20w. The number of gallons burned to cover 50 miles is then 50 divided by the average miles per gallon, or 50/(40 – 20w). We can now set these numbers of gallons equal to each other:

7/(5 – 2w) = 50/(40 – 20w)
280 – 140w = 250 – 100w
30 = 40w
3/4 = w

Knowing that the mixture is 75% Fuel X and 25% Fuel Y, we can now in theory calculate the average cost per gallon of the mixture. (In fact, we should have stopped before calculating w – simply knowing that we could calculate that percentage is sufficient.)

The answer is D: EACH statement ALONE is sufficient to answer the question.

A certain military vehicle can run on pure Fuel X, pure Fuel Y, or any mixture of X and Y. Fuel X costs $3 per gallon; the vehicle can go 20 miles on a gallon of Fuel X. In contrast, Fuel Y costs $5 per gallon, but the vehicle can go 40 miles on a gallon of Fuel Y. What is the cost per gallon of the fuel mixture currently in the vehicle’s tank?

1) Using fuel currently in its tank, the vehicle burned 8 gallons to cover 200 miles.

2) The vehicle can cover 7 and 1/7 miles for every dollar of fuel currently in its tank.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Question A medical test for a certain liver enzyme can be given in the morning, in the afternoon, or in the evening; moreover, the result of the test can be low, average, or high. At least three-quarters of low and medium readings are not given in the evening. Sixty percent of exams are given in the morning or in the afternoon, and 20% of exams result in a high reading. What percent of exams given in the evening result in low or medium readings?

(A) 20% (B) 30% (C) 40% (D) 50% (E) 60%

Answer The key to this problem is to realize that you can collapse certain categories together. The distinction between low and medium readings does not matter, because we are never given data about just low or just medium readings. Likewise, morning and afternoon tests can be combined, because the given information never distinguishes those times of day. Thus, we only have two categories for each dimension: time of day is either “morning+afternoon” or evening, and result is either “low+medium” or high. We can now set up a 2×2 table, plus a total row and column (and labels):

We choose 100 for the total of all tests, since we are only dealing with percents. Filling in the table with the given information and completing the total row and column, we get the following:

Since the number of high morning+afternoon exams cannot be less than zero, it must actually be zero. This means that the table fills in this way:

Thus, the percentage of evening exams that do NOT result in a high reading is 20/40 x 100%, or 50%.

The correct answer is (D).

Question A medical test for a certain liver enzyme can be given in the morning, in the afternoon, or in the evening; moreover, the result of the test can be low, average, or high. At least three-quarters of low and medium readings are not given in the evening. Sixty percent of exams are given in the morning or in the afternoon, and 20% of exams result in a high reading. What percent of exams given in the evening result in low or medium readings?

(A) 20% (B) 30% (C) 40% (D) 50% (E) 60%

Question Line k lies in a coordinate plane. Is the slope of line k positive?

(1) Line k and the graph of the function f(x) = x^2 – bx, where b is positive, intersect on the x-axis.

(2) Line k and the graph of the function g(x) = –x^2 – c, where c is positive, intersect on the y-axis.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer The given information simply sets the stage: a coordinate plane.

Statement (1) states that the graph of the function f(x) = x^2 – bx, where b is positive, and line k intersect on the x-axis. Since the function is a quadratic, we know that the graph of the function is a parabola. Let’s find the places that this function intersects the x-axis. This is equivalent to finding the values of x for which the function equals zero. f(x) = x^2 – bx = x(x – b) = 0. Thus, x = 0 or x = b, a positive number. Therefore, line k touches the x-axis at one (or possibly both) of the points (0,0) and (b, 0). However, we do not know another point guaranteed to be on the line. Thus, we do not know the slope of line k. INSUFFICIENT.

Statement (2) states that the graph of the function g(x) and line k intersect on the y-axis. Again, we know that the graph of the function is a parabola, since the function itself is quadratic. To find where the parabola intersects the y-axis, we find the value of the function at x = 0. g(x) = –x^2 – c g(0) = –0^2 – c = –c Since c is positive, –c is negative. Thus, the line k touches the y-axis at the point (0, –c). However, we do not know another point guaranteed to be on the line. Thus, we do not know the slope of line k. INSUFFICIENT.

Statements (1) and (2) together: Line k goes through (0, –c) AND either (0,0) or (b, 0). If line k goes through (0, –c) and (b, 0), where both b and c are positive, then we know that the slope of line k is positive (both the rise and the run are positive). However, line k could go through (0, –c) and (0, 0), which would mean that line k is vertical (it actually would coincide with the y-axis). A vertical line has an undefined slope. Since the slope could either be positive or undefined, the two statements together are INSUFFICIENT.

The correct answer is E: . . . → Continue Reading]]> Check out the answer to yesterday’s Challenge Problem below!

Question
Line k lies in a coordinate plane. Is the slope of line k positive?

(1) Line k and the graph of the function f(x) = x^2 – bx, where b is positive, intersect on the x-axis.

(2) Line k and the graph of the function g(x) = –x^2 – c, where c is positive, intersect on the y-axis.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer
The given information simply sets the stage: a coordinate plane.

Statement (1) states that the graph of the function f(x) = x^2 – bx, where b is positive, and line k intersect on the x-axis. Since the function is a quadratic, we know that the graph of the function is a parabola. Let’s find the places that this function intersects the x-axis. This is equivalent to finding the values of x for which the function equals zero.
f(x) = x^2 – bx = x(x – b) = 0. Thus, x = 0 or x = b, a positive number. Therefore, line k touches the x-axis at one (or possibly both) of the points (0,0) and (b, 0). However, we do not know another point guaranteed to be on the line. Thus, we do not know the slope of line k. INSUFFICIENT.

Statement (2) states that the graph of the function g(x) and line k intersect on the y-axis. Again, we know that the graph of the function is a parabola, since the function itself is quadratic. To find where the parabola intersects the y-axis, we find the value of the function at x = 0.
g(x) = –x^2 – c
g(0) = –0^2 – c = –c
Since c is positive, –c is negative. Thus, the line k touches the y-axis at the point (0, –c). However, we do not know another point guaranteed to be on the line. Thus, we do not know the slope of line k. INSUFFICIENT.

Statements (1) and (2) together: Line k goes through (0, –c) AND either (0,0) or (b, 0). If line k goes through (0, –c) and (b, 0), where both b and c are positive, then we know that the slope of line k is positive (both the rise and the run are positive). However, line k could go through (0, –c) and (0, 0), which would mean that line k is vertical (it actually would coincide with the y-axis). A vertical line has an undefined slope. Since the slope could either be positive or undefined, the two statements together are INSUFFICIENT.

The correct answer is E: Statements (1) and (2) TOGETHER are INSUFFICIENT to answer the question.