Quant Problems

The population of locusts in a certain swarm doubles every two hours. If 4 hours ago the swarm just doubled to 1,000 locusts, in approximately how many hours will the swarm population exceed 250,000 locusts? 6 8 10 12 14

A population problem on the GMAT is best solved with a population chart that illustrates the swarm population at each unit of time. An example of a population chart is shown below: Time Population 4 hours ago 1,000 2 hours ago 2,000 NOW 4,000 in 2 hours 8,000 in 4 hours 16,000 in 6 hours 32,000 in 8 hours 64,000 in 10 hours 128,000 in 12 hours 256,000 As can be seen from the chart, in 12 hours the swarm population will be equal to 256,000 locusts. Thus, we can infer that the number of locusts will exceed 250,000 in slightly less than 12 hours. Since we are asked for an approximate value, 12 hours provides a sufficiently close approximation and is therefore the correct answer. The correct answer is D.

What is the distance between x and y on the number line? (1) |x| - |y| = 5 (2) |x| + |y| = 11

One tip on this problem is that whenever you are asked about the distance between two points on a number line, it is mathematically equivalent to the absolute value of the difference between those two points. Thus in the problem, the question can be rephrased as: What is |x - y|? Depending upon the clues given in the statements, one of the two interpretations may be easier to use in solving the problem. (1) INSUFFICIENT: This tells us that the difference between the absolute value of x and the absolute value of y is 5. Let's pick some numbers to prove that this is insufficient. Say, for example, x = 6 and y = 1. Then |x| - |y| = 5 and the distance between x and y is 6 - 1 = 5. However, let's pick x = 6 and y = -1. Then |x| - |y| = 5 and the distance between x and y is 6 - (-1) = 7. Since we picked two sets of numbers that fit the criteria and got different answers, the statement is insufficient. (2) INSUFFICIENT: This tells us that the sum of the absolute value of x and the absolute value of y is 5. Once again, let's pick some numbers to prove that this is insufficient. Say, for example, x = 6 and y = 5. Then |x| + |y| = 11 and the distance between x and y is 6 - 5 = 1. However, let's pick x = 8 and y = 3. Then |x| + |y| = 11 and the distance between x and y is 8 - 3 = 5. Since we picked two sets of numbers that fit the criteria and got different answers, the statement is insufficient. (1) and (2) INSUFFICIENT: We know from Statement 1 that |x| - |y| = 5. We know from Statement 2 that |x| + |y| = 11. We can use elimination to produce: |x| - |y| = 5 +(|x| + |y| = 11) 2|x| = 16 |x| = 8 Therefore, x = 8 or -8. Substituting, we get: 8 + |y| = 11 |y| = 3 Therefore, y = 3 or -3. This table shows that we still can't get a consistent value for |x - y| |x| |y| |x - y| 8 3 5 8 -3 11 -8 3 11 -8 -3 5 The correct answer is E

The manufacturer's suggested retail price (MSRP) of a certain item is $60. Store A sells the item for 20 percent more than the MSRP. The regular price of the item at Store B is 30 percent more than the MSRP, but the item is currently on sale for 10 percent less than the regular price. If sales tax is 5 percent of the purchase price at both stores, how much more will someone pay in sales tax to purchase the item at Store A? $0 $0.09 $0.18 $0.30 $1.80

First, determine the price of the item at each store. The MSRP is $60. Store A adds 20%. Take 10%, $6, and multiply by 2 to get 20%, or $12. Store A's price is $60 + $12 = $72. Store B adds 30% and then discounts by 10%. First, find the original price: 10% multiplied by 3 is 6 × 3 = 18 (or 30%). The non-sale price is $60 + $18 = $78. Then subtract 10% of the new number: $78 - $7.80 = $70.20. The sales price is 5% at both stores. The question asks how much more someone will pay in sales tax at Store A. Don't calculate the full sales tax for the item at each store! The sales tax is identical at the two stores for the first $70.20 spent. Instead, calculate only the sales tax on the difference between the two prices. The difference is $1.80. 10% of that number is $0.18. 5% is half of that figure, or $0.09. The correct answer is (B).

Line l is defined by the equation y - 5x = 4 and line w is defined by the equation 10y + 2x + 20 = 0. If line k does not intersect line l, what is the degree measure of the angle formed by line k and line w? (Assume that all lines lie in one coordinate plane.) 0 30 60 90 It cannot be determined from the information given.

First, let's rewrite both equations in the standard form of the equation of a line: Equation of line l: y = 5x + 4 Equation of line w: y = -(1/5)x - 2 Note that the slope of line w, -1/5, is the negative reciprocal of the slope of line l. Therefore, we can conclude that line w is perpendicular to line l. Next, since line k does not intersect line l, lines k and l must be parallel. Since line w is perpendicular to line l, it must also be perpendicular to line k. Therefore, lines k and w must form a right angle, and its degree measure is equal to 90 degrees. The correct answer is D.

A cylindrical tank has a base with a circumference of meters and an isosceles right triangle painted on the interior side of the base. A grain of sand is dropped into the tank, and has an equal probability of landing on any particular point on the base. If the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4, which of the following is the length of at least one side of the triangle? 3 12 root 2 root 3 root 6

First, visualize what the problem describes. You can ignore the cylinder; the important portion is the circular base with the triangle inside. Note that an "isosceles right triangle" is a 45-45-90 triangle: The circumference of the circle is . Use this information to find the area of the circle. First, find the radius. The circumference equals . Don't solve for r! Remember that the goal is to find the area of the circle, and the area equals . Because the probability of the grain of sand landing outside the triangle is 3/4 , the triangle must comprise 1/4 of the area of the circle. The area of the triangle, then, is 3. Call the two shorter legs of the 45-45-90 triangle s. The base is s and the height is also s. The area of a triangle = 1/2 × base × height, so the area of this triangle can be expressed as: . This triangle has an area of 3, so: The correct answer is E.

10 students took a chemistry exam that was graded on a scale of 0 to 100. Five of the students were in Dr. Adams' class and the other five students were in Dr. Brown's class. Is the median score for Dr. Adams' students greater than the median score for Dr. Brown's students? (1) The range of scores for students in Dr. Adams' class was 40 to 80, while the range of scores for students in Dr. Brown's class was 50 to 90. (2) If the students are paired in study teams such that each student from Dr. Adams' class has a partner from Dr. Brown's class, there is a way to pair the 10 students such that the higher scorer in each pair is one of Dr. Brown's students.

The median of a set of numbers is the middle number when the numbers are arranged in increasing order. For a set of 5 scores, the median is the 3rd score. We will call the set of scores A = {A1, A2, A3, A4, A5} and B = {B1, B2, B3, B4, B5} for Dr. Adams' and Dr. Brown's students, respectively, where the scores are arranged in increasing order within each set. Rephrasing the question using this notation yields "Is A3 > B3?" (1) INSUFFICIENT: This statement tells us only the highest and lowest score for each set of students, but the only thing we know about the scores in between is that they are somewhere in that range. Since the median is one of the scores in between, this uncertainty means that the statement is insufficient. To illustrate, A3 could be greater than B3, making the answer to the question "yes": A = {40, 50, 60, 70, 80} B = {50, 55, 55, 80, 90} However, A3 could be less than or equal to B3, making the answer to the question "no": A = {40, 50, 60, 70, 80} B = {50, 60, 70, 80, 90} (2) SUFFICIENT: This statement tells us that for every student pair, the B student scored higher than the A student, or Bn > An. This statement can be considered qualitatively. Every student in set B scored higher than at least one student in set A. The students in set B not only scored higher individually, but also as a group, so one can reason that the median score for set B is higher than the median score for set A. Therefore, B3 > A3, and the answer to the question is "no." But let's prove conclusively that the answer cannot be "yes." Constrain A3 to be greater than B3, then try to pair the students according to the restriction that Bn > An. For example, pick any number x between 0 and 100, and let's say that A3 > x, or high (H), and that B3 < x, or low (L). Since the scores are increasing order, the 1st and 2nd scores must be less than or equal to the 3rd, while the 4th and 5th scores must be greater than or equal to the 3rd. Thus we know whether all the other scores are high or low. A = {A1, A2, H, A4, A5} = {L, L, H, H, H} B = {B1, B2, L, B4, B5} = {L, L, L, H, H} In order to meet the restriction that Bn > An, each of the 3 high scorers (H) in set A must be paired with a high(er) scorer, but there are only 2 high scorers (H) in set B—not enough to go around! Conversely, the 3 low scorers (L) in set B cannot be paired with a high scorer (H) from set A, leaving only 2 potential study partners for them from set A—not enough to go around! There is no way for A3 to be greater than B3 and still meet the restriction that Bn > An , so A3 < B3. Thus, the answer can never be "yes," it is always "no," and this statement is sufficient. The correct answer is B.

If n is a positive integer greater than 6, what is the remainder when n is divided by 6? (1) n2 - 1 is not divisible by 3. (2) n2 - 1 is even.

The problem can be solved algebraically or using smart numbers (testing cases). Both methods are shown below. (1) NOT SUFFICIENT: Algebra The expression n2 - 1 = (n - 1)(n + 1). Together with n, the three expressions represent three consecutive integers: n - 1, n, n + 1. Statement 2 indicates that the product (n - 1)(n + 1) is not divisible by the prime number 3; therefore, neither n - 1 nor n + 1 is a multiple of 3. Since every third integer is a multiple of 3, it follows that n must be a multiple of 3. This fact alone is not sufficient, though, since different multiples of 3 can give different remainders upon division by 6. For example, 9/6 gives a remainder of 3, and 12/6 gives a remainder of 0. Testing Cases Find values of n > 6 that satisfy statement 1. The value n = 7 does not satisfy the statement, since 72 - 1 = 48 is divisible by 3. The value n = 8 does not satisfy the statement, since 82 - 1 = 63 is divisible by 3. The value n = 9 satisfies the statement, since 92 - 1 = 80 is not divisible by 3. The remainder of 9/6 is 3. The value n = 10 does not satisfy the statement, since 102 - 1 = 99 is divisible by 3. The value n = 11 does not satisfy the statement, since 112 - 1 = 120 is divisible by 3. The value n = 12 satisfies the statement, since 122 - 1 = 143 is not divisible by 3. The remainder of 12/6 is 0. At least two different remainders are possible, so the statement is insufficient. (2) NOT SUFFICIENT: Algebra If n2 - 1 is even, then n2 must be odd, so n itself is odd. This fact alone is not sufficient, though, since different odd numbers can give different remainders upon division by 6. For example, 7/6 gives a remainder of 1, and 9/6 gives a remainder of 3. Testing Cases Find values of n > 6 that satisfy the statement. The value n = 7 satisfies the statement, since 72 - 1 = 48 is even. The remainder of 7/6 is 1. The value n = 8 does not satisfy the statement, since 82 - 1 = 63 is not even. The value n = 9 satisfies the statement, since 92 - 1 = 80 is even. The remainder of 9/6 is 3. At least two different remainders are possible, so the statement is insufficient. (1) AND (2) SUFFICIENT: Statement (1) indicates that n is a multiple of 3 and statement (2) indicates that n is odd. From this point, you can either use algebra or test numbers. Algebra According to the two statements, n must be 3 times some odd integer, so it can be written as n = 3(2k + 1), where k is an integer. Distributing gives n = 6k + 3, which is 3 more than a multiple of 6. Therefore, the remainder upon dividing n by six must be 3. Testing Cases Test numbers that are multiples of 3, odd, and greater than 6: If n = 9, then 9 / 6 = 1 remainder 3. If n = 15, then 15 / 6 = 2 remainder 3. If n = 21, then 21 / 6 = 3 remainder 3. Will this always return a remainder of 3? Yes! An odd multiple of 3 will always have a remainder when divided by 6 (because 6 is even). Further, an even multiple of 3 will always have a remainder of zero (because an even multiple of 3 is divisible by both 2 and 3). Therefore, the odd multiple of 3, which is always 3 more than an even multiple of 3, will always have a remainder of 3. The correct answer is (C).

If n is a positive integer greater than 6, what is the remainder when n is divided by 6? (1) n^2 - 1 is not divisible by 3. (2) n^2 - 1 is even.

The problem can be solved algebraically or using smart numbers (testing cases). Both methods are shown below. (1) NOT SUFFICIENT: Algebra The expression n2 - 1 = (n - 1)(n + 1). Together with n, the three expressions represent three consecutive integers: n - 1, n, n + 1. Statement 2 indicates that the product (n - 1)(n + 1) is not divisible by the prime number 3; therefore, neither n - 1 nor n + 1 is a multiple of 3. Since every third integer is a multiple of 3, it follows that n must be a multiple of 3. This fact alone is not sufficient, though, since different multiples of 3 can give different remainders upon division by 6. For example, 9/6 gives a remainder of 3, and 12/6 gives a remainder of 0. Testing Cases Find values of n > 6 that satisfy statement 1. The value n = 7 does not satisfy the statement, since 72 - 1 = 48 is divisible by 3. The value n = 8 does not satisfy the statement, since 82 - 1 = 63 is divisible by 3. The value n = 9 satisfies the statement, since 92 - 1 = 80 is not divisible by 3. The remainder of 9/6 is 3. The value n = 10 does not satisfy the statement, since 102 - 1 = 99 is divisible by 3. The value n = 11 does not satisfy the statement, since 112 - 1 = 120 is divisible by 3. The value n = 12 satisfies the statement, since 122 - 1 = 143 is not divisible by 3. The remainder of 12/6 is 0. At least two different remainders are possible, so the statement is insufficient. (2) NOT SUFFICIENT: Algebra If n2 - 1 is even, then n2 must be odd, so n itself is odd. This fact alone is not sufficient, though, since different odd numbers can give different remainders upon division by 6. For example, 7/6 gives a remainder of 1, and 9/6 gives a remainder of 3. Testing Cases Find values of n > 6 that satisfy the statement. The value n = 7 satisfies the statement, since 72 - 1 = 48 is even. The remainder of 7/6 is 1. The value n = 8 does not satisfy the statement, since 82 - 1 = 63 is not even. The value n = 9 satisfies the statement, since 92 - 1 = 80 is even. The remainder of 9/6 is 3. At least two different remainders are possible, so the statement is insufficient. (1) AND (2) SUFFICIENT: Statement (1) indicates that n is a multiple of 3 and statement (2) indicates that n is odd. From this point, you can either use algebra or test numbers. Algebra According to the two statements, n must be 3 times some odd integer, so it can be written as n = 3(2k + 1), where k is an integer. Distributing gives n = 6k + 3, which is 3 more than a multiple of 6. Therefore, the remainder upon dividing n by six must be 3. Testing Cases Test numbers that are multiples of 3, odd, and greater than 6: If n = 9, then 9 / 6 = 1 remainder 3. If n = 15, then 15 / 6 = 2 remainder 3. If n = 21, then 21 / 6 = 3 remainder 3. Will this always return a remainder of 3? Yes! An odd multiple of 3 will always have a remainder when divided by 6 (because 6 is even). Further, an even multiple of 3 will always have a remainder of zero (because an even multiple of 3 is divisible by both 2 and 3). Therefore, the odd multiple of 3, which is always 3 more than an even multiple of 3, will always have a remainder of 3. The correct answer is (C).

The temperature inside a certain industrial machine at time t seconds after startup, for 0 < t < 10, is given by h(t) = 42t + 1 - 4t + 2 degrees Celsius. How many seconds after startup is the temperature inside the machine equal to 128 degrees Celsius? 3/2 2 5/2 3 7/2

The problem indicates that the given formula is equal to 128 degrees: 42t + 1 - 4t + 2 = 128 The challenge lies in solving this equation. It can be solved algebraically, but not easily. Instead, test the answers! (The algebraic solution is also shown below.) Test the Answers In general, start with answer (B) or (D) when testing answers. (This is particular true on this problem; answers B and D are both integers while the others are not!) (B) t = 2. Plug this number into the equation: 42(2) + 1 - 4(2) + 2 = 128 45 - 44 = 128 Yikes! Don't multiply out 45 and 44. Rather, factor out a 44 term: 44 (4 - 1) = 128 44 (3) = 128 128 is not a multiple of 3, so the equation has to be false. In order to know whether you want to try a larger or smaller answer next, estimate out the left-hand side of the equation until you can tell whether it's too big or too small. 42 = 16, 43 = 64, 44 = 256. And you still haven't multiplied by 3! Try answer (A) next (or just pick it, if you're feeling confident!). (A) t = 3/2. Plug this number into the equation: 42(3/2) + 1 - 4(3/2) + 2 = 128 44 - 47/2 = 128 You know already that 44 = 256. What about 47/2? A fractional exponent indicates that you raise to the power of the numerator (in this case, 7) and also take the root of the denominator (in this case, 2). Take the root first: the square root of 4 is 2. Then raise 2 to the power of 7. 27 = 128. 256 - 128 = 128 The correct answer is (A). Algebraic solution 42t + 1 - 4t + 2 = 128 You could try factoring 4t + 2 out of the left-hand side. This action gives (4t + 2)(4t - 1 - 1) = 128. That's even uglier than the original equation. Trying other common factors doesn't do much better. But wait! This equation vaguely resembles a quadratic. Move 128 over to the left side: 42t + 1 - 4t + 2 - 128 = 0 The trick is to get the left-hand term to involve the square of the middle term. (Factoring a "normal" quadratic, such as x2 - 3x - 10, depends on the fact that x2 is the square of x.) Write those ugly terms more simply by pulling out the portion that doesn't include a variable: 41(42t) - 42(4t) - 128 = 0 4(42t) - 16(4t) - 128 = 0 This is genuinely a quadratic equation, since 42t is the square of 4t. Divide by 4 and factor: 42t - 4(4t) - 32 = 0 (4t - 8)(4t + 4) = 0 4t = 8 or 4t = -4 The second solution is impossible; powers of positive numbers are always positive. Therefore, 4t = 8. (22)t = 23 22t = 23 Drop the bases and set the exponents equal: 2t = 3 t = 3/2

The temperature inside a certain industrial machine at time t seconds after startup, for 0 < t < 10, is given by h(t) = 4^(2t + 1) - 4^(t + 2) degrees Celsius. How many seconds after startup is the temperature inside the machine equal to 128 degrees Celsius? 3/2 2 5/2 3 7/2

The problem indicates that the given formula is equal to 128 degrees: 42t + 1 - 4t + 2 = 128 The challenge lies in solving this equation. It can be solved algebraically, but not easily. Instead, test the answers! (The algebraic solution is also shown below.) Test the Answers In general, start with answer (B) or (D) when testing answers. (This is particular true on this problem; answers B and D are both integers while the others are not!) (B) t = 2. Plug this number into the equation: 42(2) + 1 - 4(2) + 2 = 128 45 - 44 = 128 Yikes! Don't multiply out 45 and 44. Rather, factor out a 44 term: 44 (4 - 1) = 128 44 (3) = 128 128 is not a multiple of 3, so the equation has to be false. In order to know whether you want to try a larger or smaller answer next, estimate out the left-hand side of the equation until you can tell whether it's too big or too small. 42 = 16, 43 = 64, 44 = 256. And you still haven't multiplied by 3! Try answer (A) next (or just pick it, if you're feeling confident!). (A) t = 3/2. Plug this number into the equation: 42(3/2) + 1 - 4(3/2) + 2 = 128 44 - 47/2 = 128 You know already that 44 = 256. What about 47/2? A fractional exponent indicates that you raise to the power of the numerator (in this case, 7) and also take the root of the denominator (in this case, 2). Take the root first: the square root of 4 is 2. Then raise 2 to the power of 7. 27 = 128. 256 - 128 = 128 The correct answer is (A). Algebraic solution 42t + 1 - 4t + 2 = 128 You could try factoring 4t + 2 out of the left-hand side. This action gives (4t + 2)(4t - 1 - 1) = 128. That's even uglier than the original equation. Trying other common factors doesn't do much better. But wait! This equation vaguely resembles a quadratic. Move 128 over to the left side: 42t + 1 - 4t + 2 - 128 = 0 The trick is to get the left-hand term to involve the square of the middle term. (Factoring a "normal" quadratic, such as x2 - 3x - 10, depends on the fact that x2 is the square of x.) Write those ugly terms more simply by pulling out the portion that doesn't include a variable: 41(42t) - 42(4t) - 128 = 0 4(42t) - 16(4t) - 128 = 0 This is genuinely a quadratic equation, since 42t is the square of 4t. Divide by 4 and factor: 42t - 4(4t) - 32 = 0 (4t - 8)(4t + 4) = 0 4t = 8 or 4t = -4 The second solution is impossible; powers of positive numbers are always positive. Therefore, 4t = 8. (22)t = 23 22t = 23 Drop the bases and set the exponents equal: 2t = 3 t = 3/2

If x is a positive integer, is x prime? (1) x has the same number of factors as y2, where y is a positive integer greater than 2. (2) x has the same number of factors as z, where z is a positive integer greater than 2

The question stem tells us that x is a positive integer. Then we are asked whether x is prime; it is helpful to remember that all prime numbers have exactly two factors. Since we cannot rephrase the question, we must go straight to the statements. (1) SUFFICIENT: If x has the same number of factors as y2, then x cannot be prime. A prime number is a number that has only itself and 1 as factors. But a square has at least 3 distinct factors. For example, if y is prime, y = 2, then y2 = 4, which has 1, 2, and 4 as factors. If the root (in this case y) is not prime, then the square will have more than 3 factors. For example, if y = 4, then y2 = 16, which has 1, 2, 4, 8, and 16 as factors. In either case, x will have at least 3 factors, establishing it as nonprime. (2) INSUFFICIENT: If z is prime, then x will have only two factors, making it prime. But if z is nonprime, it will have more than two factors, which means x will have more than two factors, making x nonprime. Since we do not know which case we have, we cannot tell whether x is prime. The correct answer is A.

A rectangular wall is covered entirely with two kinds of decorative tiles: regular and jumbo. 1/3 of the tiles are jumbo tiles, which have a length three times that of regular tiles and have the same ratio of length to width as the regular tiles. If regular tiles cover 80 square feet of the wall, and no tiles overlap, what is the area of the entire wall? 160 240 360 440 560

he length and the width of a regular tile can be represented as L and w. The length of a jumbo tile, then, is 3L. If the ratio of length to width is the same for all tiles, and the ratio of the regular tile is L : w then a jumbo tile must have a width of 3w. Represent the area of both types: Area of one regular tile = Lw Area of one jumbo tile = (3L)(3w) = 9Lw If 1/3 of the tiles are jumbo, then the other 2/3 must be regular (since only two kinds of tiles are used). There are twice as many regular tiles as jumbo ones. If J is the number of jumbo tiles, then there are 2J regular tiles. Area of all jumbo tiles = (the area of one jumbo tile)(the number of jumbo tiles) Area of all jumbo tiles = (9Lw)(J)

If, 5 years ago, Jamie was half as old as he is now, how old will he be in x years? x + 10 x + 5 x + 2 x - 5 2x

or questions about ages, it is helpful to draw an age chart featuring one column for each of the times under consideration. We can assign the variable J to represent Jamie's age today. His age five years ago was J - 5, and his age x years from now will be J + x: 5 years ago today x years from now Jamie's age J-5 J J + x The question also tells us that Jamie's age 5 years ago was half of what it is today, which means that J-5 = 1/2 J. Solving for J, we get 1/2 J = 5, and then J=10. Now that we know the value of J, we can say that the Jamie's age x years from now, which our table says is J + x, must be equal to 10 + x. Because there are variables in the answer choices, we can solve the last step of this problem arithmetically by choosing a numerical value for x. Let's set x equal to 6. We are asked how old Jamie will be in 6 years. In 6 more years, he'll be 16. Check the answer choices by plugging in 6 for x; the only answer that equals 16 is A. The correct answer is A.

If N = 4P, where P is a prime number greater than 2, how many different positive even divisors does n have, including n? 2 3 4 6 8

plug in 3 for P n = 12 1,12 2,6 3,4 4 even divisors

A list contains only integers. Are there more positive than negative integers in the list? (1) The median of the numbers in the list is positive. (2) The average (arithmetic mean) of the numbers in the list is positive. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

(1) NOT SUFFICIENT: Test numbers and try to disprove the question. For instance, the list 1, 2, 3, 4, 5 has a positive median (3) and contains more positive than negative integers. On the other hand, consider a list with an even number of entries—the median is halfway between two entries in the list, and not necessarily an entry in the list itself. For instance, the list -3, -2, -1, 101, 102, 103 has a positive median (50), but does not contain more positive than negative integers. (2) NOT SUFFICIENT: Test numbers and try to disprove the question. For instance, the list 1, 2, 3, 4, 5 has a positive median (3) and contains more positive than negative integers. On the other hand, consider a list containing small negative numbers and large positive numbers. For instance, the same list from statement 1 above, -3, -2, -1, 101, 102, 103, has a positive average, but does not contain more positive than negative integers. (In fact, unlike statement 1, this statement can also be satisfied by a list containing fewer positive than negative numbers, e.g., -5, -4, -3, -2, -1, 100). (1) AND (2) NOT SUFFICIENT: As deduced above, the lists 1, 2, 3, 4, 5 does have more positive integers while the list -3, -2, -1, 101, 102, 103 does not have more positive integers. Both lists satisfy both statements, so even when used together, the statements are still insufficient. The correct answer is E.

If x and y are integers and [15^(x) + 15^(x+1)] /4^y = 15^y what is the value of x? 2 3 4 5 Cannot be determined

(15x + 15x+1) = 15y4y [15x + 15x(151)] = 15y4y (15x )(1 + 15) = 15y4y (15x)(16) = 15y4y (3x)(5x)(24) = (3y)(5y)(22y) Since both sides of the equation are broken down to the product of prime bases, the respective exponents of like bases must be equal. 2y = 4 so y = 2. x = y so x = 2. The correct answer is A.

Among the 1,600 students at Hamilton High School, 45% take science courses. If 5/12 of the students taking science courses are taking physics, how many students at Hamilton High School are taking physics?

300

A certain galaxy is known to comprise approximately 4 x 10^11 stars. Of every 50 million of these stars, one is larger in mass than our sun. Approximately how many stars in this galaxy are larger than the sun? 800 1,250 8,000 12,000 80,000

50 million can be represented in scientific notation as 5 x 107. Restating this figure in scientific notation will enable us to simplify the division required to solve the problem. If one out of every 5 x 107 stars is larger than the sun, we must divide the total number of stars by this figure to find the solution: 4 x 1011 5 x 107 = 4/5 x 10(11-7) = 0.8 x 10^4 The final step is to move the decimal point of 0.8 four places to the right, with a result of 8,000. The correct answer is C.

What is the sum of the multiples of 7 from 84 to 140, inclusive? 896 963 1008 1792 2016

84 is the 12th multiple of 7. (12 x 7 = 84) 140 is the 20th multiple of 7. The question is asking us to sum the 12th through the 20th multiples of 7. The sum of a set = (the mean of the set) x (the number of terms in the set) There are 9 terms in the set: 20th - 12th + 1 = 8 + 1 = 9 The mean of the set = (the first term + the last term) divided by 2: (84 + 140)/2 = 112 The sum of this set = 112 x 9 = 1008 Alternatively, one could list all nine terms in this set (84, 91, 98 ... 140) and add them. When adding a number of terms, try to combine terms in a way that makes the addition easier (i.e. 98 + 112 = 210, 119 + 91 = 210, etc). The correct answer is C.

For each month of a given year except December, a worker earned the same monthly salary and donated one-tenth of that salary to charity. In December, the worker earned N times his usual monthly salary and donated one-fifth of his earnings to charity. If the worker's charitable contributions totaled one-eighth of his earnings for the entire year, what is the value of N? 8/5 5/2 3 11/3 4

A number of different approaches can work for this problem. Smart Numbers Solution Let the worker earn $10 per month. (Pick a small number that works easily in the problem. This value can be easily divided by either 10 or 5.) For each month from January to November, the worker earned $10 and donated $1 to charity. Therefore, his total earnings for those 11 months were $110, of which he donated $11 to charity. In December, the worker earned $10N and donated one-fifth of that amount, or $2N, to charity. (Note: we cannot pick our own value for N; the answer choices represent real possible values for N.) Since the worker's charitable contributions totaled one-eighth of his total earnings, write an equation: (total earnings)/8 = total charitable contributions (110 + 10N)/8 = 11 + 2N 110 + 10N = 88 + 16N 22 = 6N N = 22/6 = 11/3 Algebraic Solution Let the worker's average monthly salary be s dollars. Then, for each of the first eleven months of the year, he earned $s, of which he donated $s/10 to charity. In total, over these eleven months, the worker earned $(11s) and donated $(11s/10) to charity. In December, the worker's salary was $Ns, of which he donated $(Ns/5) to charity. Thus, for the entire year, the worker's earnings totaled 11s + Ns, and his charitable donations totaled 11s/10 + Ns/5. These donations totaled 1/8 of the worker's earnings for the year, so write an equation: (11s + Ns)/8 = 11s/10 + Ns/5 Multiply both sides by 40, the least common denominator, to eliminate the denominators: 55s + 5Ns = 44s + 8Ns Since s is positive and appears in every term, divide it out: 55 + 5N = 44 + 8N 11 = 3N N = 11/3 Weighted Average Solution For the first eleven months of the year, the worker donates 1/10, or 10%, of his monthly salary. In December, the worker donates 1/5, or 20%, of his monthly salary. His average monthly contribution for the whole year was 1/8, or 12.5%. Because the actual average was 12.5% (and not a simple average of 15%), we know that this is a weighted average, and the average is weighted more towards the months in which the worker donated 10%. The difference between 10% and 12.5% is 2.5%; this represents the "weight" of the other quantity, the December salary. The difference between 20% and 12.5% is 7.5%; this represents the "weight" of the other quantity, the salary from January through November. The "total weight" is 20% - 10% = 10. Therefore, the December salary is 2.5/10 = ¼ of the total weighting, and the combined Jan-thru-Nov salary is 7.5/10 = ¾ of the total weighting. Divide this value by 11 to get just one month of the 11 months in the Jan-thru-Nov time range: [(¾)]/11 = 3/44. Because we know that the December salary is equal to one month of the normal salary multiplied by N, we can write this equation: One regular month × N = the month of December (3/44)N = ¼ N = (¼)(44/3) = 11/3 The correct answer is D.

In the quadrilateral PQRS, side PS is parallel to side QR. Is PQRS a parallelogram? (1) PS = QR (2) PQ = RS

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. The opposite sides of a parallelogram also have equal length. (1) SUFFICIENT: We know from the question stem that opposite sides PS and QR are parallel, while this statement tells us that they also have equal lengths. The opposite sides PQ and RS must also be parallel and equal in length. This is the definition of a parallelogram, so the answer to the question is "Yes." (2) INSUFFICIENT: We know from the question stem that opposite sides PS and QR are parallel, but have no information about their respective lengths. This statement tells us that the opposite sides PQ and RS are equal in length, but we don't know their respective angles; they might be parallel, or they might not be. According to the information given, PQRS could be a trapezoid with PS not equal to QR. On the other hand, PQRS could be a parallelogram with PS = QR. The answer to the question is uncertain. The correct answer is A.

The population of locusts in a certain swarm doubles every two hours. If 4 hours ago the swarm just doubled to 1,000 locusts, in approximately how many hours will the swarm population exceed 250,000 locusts? 6 8 10 12 14

A population problem on the GMAT is best solved with a population chart that illustrates the swarm population at each unit of time. An example of a population chart is shown below: Time Population 4 hours ago 1,000 2 hours ago 2,000 NOW 4,000 in 2 hours 8,000 in 4 hours 16,000 in 6 hours 32,000 in 8 hours 64,000 in 10 hours 128,000 in 12 hours 256,000 As can be seen from the chart, in 12 hours the swarm population will be equal to 256,000 locusts. Thus, we can infer that the number of locusts will exceed 250,000 in slightly less than 12 hours. Since we are asked for an approximate value, 12 hours provides a sufficiently close approximation and is therefore the correct answer. The correct answer is D.

In the rhombus ABCD, the length of diagonal BD is 6 and the length of diagonal AC is 8. What is the perimeter of ABCD?

A rhombus is a parallelogram with four sides of equal length. Thus, AB = BC = CD = DA. The diagonals of a parallelogram bisect each other, meaning that AC and BD intersect at their midpoints, which we will call E. Thus, AE = EC = 4 and BE = ED = 3. Since ABCD is a rhombus, diagonals AC and BD are also perpendicular to each other. Labeling the figure with the lengths above, we can see that the rhombus is divided by the diagonals into four right triangles, each of which has one side of length 3 and another side of length 4. Remembering the common right triangle with side ratio = 3: 4: 5, we can infer that the unlabeled hypotenuse of each of the four triangles has length 5. Thus, AB = BC = CD = DA = 5, and the perimeter of ABCD is 5 × 4 = 20. The correct answer is C.

What is the value of a? (1) -(a + b) = 2a - b + 9 (2) a + b = 6

A) - the "B" variable drops out of the equation

Adam and Brianna plan to install a new tile floor in a classroom. Adam works at a constant rate of 50 tiles per hour, and Brianna works at a constant rate of 55 tiles per hour. If the new floor consists of exactly 1400 tiles, how long will it take Adam and Brianna working together to complete the classroom floor? 26 hrs. 44 mins. 26 hrs. 40 mins. 13 hrs. 20 mins. 13 hrs. 18 mins. 12 hrs. 45 mins.

Because Adam and Brianna are working together, add their individual rates to find their combined rate: 50 + 55 = 105 tiles per hour The question asks how long it will take them to set 1400 tiles. Time = Work / Rate = 1400 tiles / (105 tiles / hour) = 40/3 hours = 13 and 1/3 hours = 13 hours and 20 minutes The correct answer is C.

Narcisse and Aristide have numbers of arcade tokens in the ratio 7 : 3, respectively. Narcisse gives Aristide some of his tokens, and the new ratio is 6 : 5. What is the least number of tokens that Narcisse could have given to Aristide? 9 17 21 27 53

Because Narcisse and Aristide are exchanging tokens (rather than acquiring new ones or spending the ones they have), the total number of tokens must remain the same. Thus, this total must be a multiple of both 10 (7 + 3) and 11 (6 + 5). The lowest common multiple of 10 and 11 is 110, so the least number of tokens they could have is 110. If the two of them have a total of 110 tokens, then Narcisse has 77 to start with, and Aristide has 33. After the exchange, Narcisse will have 60 tokens and Aristide will have 50. 17 tokens will have changed hands. The correct answer is (B).

What is the area of the quadrilateral with vertices A, B, C, and D? (1) The perimeter of ABCD is equal to 16. (2) Quadrilateral ABCD is a rhombus.

Because we do not know the type of quadrilateral, this question cannot be rephrased in a useful manner. (1) INSUFFICIENT: We do not have enough information about the shape of the quadrilateral to solve the problem using Statement (1). For example, ABCD could be a rectangle with side lengths 3 and 5, resulting in an area of 15, or it could be a square with side length 4, resulting in an area of 16. (2) INSUFFICIENT: This statement gives no information about the size of the quadrilateral. (1) AND (2) INSUFFICIENT: The four sides of a rhombus are equal, so the length of one side of the rhombus could be determined by dividing the perimeter by 4. Therefore, each side has a length of 16/4 = 4. However, we do not know the height of the rhombus, so we cannot calculate the area. The diagrams below demonstrate two different heights of the rhombus, leading to different areas for the shape. Example #1: height = 4 In this case, ABCD is a square and the area is 4·4 = 16 Example #2: height = 2root3 In this case, angle D = 60° and the height is 2root3. The area is (4)(2root3) = 8root3 The correct answer is E.

If X and Y are integers is X>Y? (1) X + Y > 0 (2) Y^X < 0

C) Together sufficient 2) tells us Y is negative X must be odd if Y negative and X+Y > 0 then X must be positive

If M is an integer, is m odd? 1) m/2 is not an even integer 2) m-3 is an even integer

D 1) m/2 = odd m = 2*odd m = even 2) m-3 = even m = even + 3 m = odd

A coin collection has 150 pennies and 350 nickels. If 40% of the pennies and 60% of the nickels were minted prior to 1982, what percent of all the coins in the collection were minted prior to 1982? 51% 52% 53% 54% 55%

First, find how many pennies and how many nickels were minted prior to 1982. 150 pennies × 40% = 60 pennies 350 nickels × 60% = 210 nickels The total number of coins minted prior to 1982 is 60 + 210 = 270 coins. As a percent of all the coins (150 + 350 = 500), 270 coins represents 270/500 = 27/50 = 54%. (The easiest way to compute that percentage is to multiply the top and bottom of 27/50 by 2, yielding 54/100.) The correct answer is D.

Is x·|y| > y2? (1) x > y (2) y > 0

It is extremely tempting to divide both sides of this inequality by y or by the |y|, to come up with a rephrased question of "is x > y?" However, we do not know the sign of y, so this cannot be done. (1) INSUFFICIENT: On a yes/no data sufficiency question that deals with number properties (positive/negatives), it is often easier to plug numbers. There are two good reasons why we should try both positive and negative values for y: (1) the question contains the expression |y|, (2) statement 2 hints that the sign of y might be significant. If we do that we come up with both a yes and a no to the question. x y x·|y| > y2 ? -2 -4 -2(4) > (-4)2 N 4 2 4(2) > 22 Y (2) INSUFFICIENT: Using the logic from above, when trying numbers here we should take care to pick x values that are both greater than y and less than y. x y x·|y| > y2 ? 2 4 2(4) > 42 N 4 2 4(2) > 22 Y (1) AND (2) SUFFICIENT: If we combine the two statements, we must choose positive x and y values for which x > y. x y x·|y| > y2 ? 3 1 3(1) > 12 Y 4 2 4(2) > 22 Y 5 3 5(3) > 32 Y Using a more algebraic approach, if we know that y is positive (statement 2), we can divide both sides of the original question by y to come up with "is x > y?" as a new question. Statement 1 tells us that x > y, so both statements together are sufficient to answer the question. The correct answer is C.

A triangle in the xy-coordinate plane has vertices with coordinates (7, 0), (0, 8), and (20, 10). What is the area of this triangle? 72 80 87 96 100

It's not immediately possible to use the standard formula A = bh / 2 for the area of this triangle, because none of the triangle's sides is horizontal or vertical. However, if the triangle is circumscribed by a rectangle, as depicted below, the areas of the three surrounding triangles can readily be found and then subtracted from the rectangle's area to yield the desired result. First, the triangle to the lower left has b = 7 and h = 8, so its area is (7)(8) / 2 = 28. Second, the triangle to the lower right has b = 13 and h = 10, so its area is (13)(10) / 2 = 65. Third, the topmost triangle has b = 20 and h = 2, so its area is (20)(2) / 2 = 20. The total area of the surrounding triangles—that is, the triangles that are not part of the desired area—is 28 + 65 + 20 = 113 square units. The total area of the rectangle is 20 x 10 = 200 square units, so the triangle's area is 200 - 113 = 87 square units. The correct answer is C.

A convenience store currently stocks 48 bottles of mineral water. The bottles have two sizes of either 20 or 40 ounces each. The average volume per bottle the store currently has in stock is 35 ounces. How many 40 ounce bottles are in stock?

Let x = the number of 20 oz. bottles 48 - x = the number of 40 oz. bottles The average volume of the 48 bottles in stock can be calculated as a weighted average: 20x + (40)(48) - 40x = (35)(48) 20x = (40)(48) - (35)(48) 20x = (48)(40 - 35) 20x = (48)5 20x = 240

What is the distance between x and y on the number line? (1) |x| - |y| = 5 (2) |x| + |y| = 11

One tip on this problem is that whenever you are asked about the distance between two points on a number line, it is mathematically equivalent to the absolute value of the difference between those two points. Thus in the problem, the question can be rephrased as: What is |x - y|? Depending upon the clues given in the statements, one of the two interpretations may be easier to use in solving the problem. (1) INSUFFICIENT: This tells us that the difference between the absolute value of x and the absolute value of y is 5. Let's pick some numbers to prove that this is insufficient. Say, for example, x = 6 and y = 1. Then |x| - |y| = 5 and the distance between x and y is 6 - 1 = 5. However, let's pick x = 6 and y = -1. Then |x| - |y| = 5 and the distance between x and y is 6 - (-1) = 7. Since we picked two sets of numbers that fit the criteria and got different answers, the statement is insufficient. (2) INSUFFICIENT: This tells us that the sum of the absolute value of x and the absolute value of y is 5. Once again, let's pick some numbers to prove that this is insufficient. Say, for example, x = 6 and y = 5. Then |x| + |y| = 11 and the distance between x and y is 6 - 5 = 1. However, let's pick x = 8 and y = 3. Then |x| + |y| = 11 and the distance between x and y is 8 - 3 = 5. Since we picked two sets of numbers that fit the criteria and got different answers, the statement is insufficient. (1) and (2) INSUFFICIENT: We know from Statement 1 that |x| - |y| = 5. We know from Statement 2 that |x| + |y| = 11. We can use elimination to produce: |x| - |y| = 5 +(|x| + |y| = 11) 2|x| = 16 |x| = 8 Therefore, x = 8 or -8. Substituting, we get: 8 + |y| = 11 |y| = 3 Therefore, y = 3 or -3. This table shows that we still can't get a consistent value for |x - y| The correct answer is E.

If x and y are positive integers and x = 10y + 5, what is the remainder when x is divided by 5? 0 1 2 3 4

Since both x and y are positive integers, 10y must be a multiple of 10. Also, since any multiple of 10 is also a multiple of 5, 10y is a multiple of 5. Therefore, x represents the sum of a multiple of 5 and 5. Since adding 5 to another multiple of 5 will always result in another multiple of 5, x must be a multiple of 5. As a result, x must be divisible by 5 and will leave no remainder when divided by 5. The correct answer is A. Alternatively, the answer can be derived by selecting a value for y, computing the corresponding value of x and calculating the remainder. For example, if y = 1, then x must be equal to 15 and, when divided by 5, will leave a remainder of 0. The correct answer is 0.

90 students represent x percent of the boys at Jones Elementary School. If the boys at Jones Elementary make up 40% of the total school population of x students, what is x? 125 150 225 250 500

The boys of Jones Elementary make up 40% of the total of x students. Therefore: # of boys = 0.4x x% of the # of boys is 90. Use x/100 as x%: (x/100) × (# of boys) = 90 Substitute for # of boys from the first equation: (x/100) × 0.4x = 90 (0.4x2) / 100 = 90 0.4x2 = 9,000 4x2 = 90,000 x2 = 22,500 x = 150 Alternatively, plug in each answer choice for x. Because the answer choices are ordered in ascending order, start with either (B) or (D). x represents both the percent of boys represented by 90 and the total population. Start with the easier value to plug into: population. From that, find the 40% who are boys and then determine if 90 (given in the problem) is 150% of the calculated number of boys. If answer (B) B is too large, the answer must be (A). If Answer (B) is too small, test answer (D) next. x = Total Pop. #boys = 40% = 10% × 4 Take 10% of x, then multiply by 4 150% of #boys = 90? Multiply # of boys by 1.5 (A) (B) x = 150 (150)(10%) = 15 × 4 = 60 1.5(60) = 90 CORRECT! (C) (D) (E) The correct answer is (B).

A used-book seller purchased 50 copies of a book for a total of m euros. She then sold each book for 25 percent more than her original per-book purchase price. In terms of m, for how many euros did she sell each book? [Algebra] a) M/4 b) 5m/4 c) m/40 d) 125m e) 125/2m

Total Cost = Price × Qty m = total cost p = price per book m = p × 50 Original Price = p = m / 50 Sold at 25% mark-up... New Price = 1.25p = (5/4)P =(5/4)(m/50)=m/40 C

If r, s, and t are all positive integers, what is the remainder when 2 rst is divided by 10? (1) s is even (2) rs = 4

When a number is divided by 10, the remainder is simply the units digit of that number. For example, 256 divided by 10 has a remainder of 6. This question asks for the remainder when an integer power of 2 is divided by 10. If we examine the powers of 2 (2, 4, 8, 16, 32, 64, 128, and 256...), we see that the units digit alternates in a consecutive pattern of 2, 4, 8, 6. To answer this question, we need to know which of the four possible units digits we have with 2 rst. (1) INSUFFICIENT: If s is even, we know that the product rst is even. Knowing that rst is even tells us that 2 rst will have a units digit of either 4 or 6 (2 2 = 4, 2 4 = 16, and the pattern continues). (2) SUFFICIENT: If rs = 4, then rst = 4 t. Because t is an integer, rst must be a multiple of 4. Since every fourth integer power of 2 ends in a 6 (2 4 = 16, 2 8 = 256, etc.), we know that the remainder when 2 rst is divided by 10 is 6. The correct answer is B.

The ratio of boys to girls in a class is 3 to 5. If 2 boys are added to the class and 6 girls leave the class, the new ratio of boys to girls is 5 to 6. How many girls were in the class to begin with? (A) 15 (B) 20 (C) 25 (D) 30 (E) 35

b = numberofboys = 3x g = number of girls = 5x Unknown Multiplier (3 x + 2)/(5x-6) = 5/6 6(3x + 2) = 5(5x - 6) 18x + 12 = 25x - 30 42 = 7x 6=x Therefore 6 * 5 = 30

If positive integer x is a multiple of 6 and positive integer y is a multiple of 14 is xy a multiple of 105? 1) X is a multiple of 9 2) Y is a multiple of 25

b) Factors of x are 2 and 3 Factors of Y are 2 and 7 factors of 105 are 3,5,7 1) not sufficient we already have a 3 2) Sufficient, gives us the 5 we were missing

A circle with a radius of 4 feet is cut from a piece of sheet metal with uniform thickness. The circle weighs 20 pounds. If another circle is cut from the same sheet and weighs 60 pounds, then its radius is closest to which of the following?

If the new circle weighs three times as much as the old circle, the area of the new circle needs to be three times the area of the old circle. (Because the sheet has uniform thickness, weight will be proportional to area.) The old circle has an area of 16π, so the new circle should have an area 3 times that size, or 48π. This would require a radius of around 7, as 72 = 49. The correct answer is (B).

Six mobsters have arrived at the theater for the premiere of the film "Goodbuddies." One of the mobsters, Frankie, is an informer, and he's afraid that another member of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists upon standing behind Joey in line at the concession stand, though not necessarily right behind him. How many ways can the six arrange themselves in line such that Frankie's requirement is satisfied? 6 24 120 360 720

Ignoring Frankie's requirement for a moment, observe that the six mobsters can be arranged 6! or 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways in the concession stand line. In each of those 720 arrangements, Frankie must be either ahead of or behind Joey. Logically, since the combinations favor neither Frankie nor Joey, each would be behind the other in precisely half of the arrangements. Therefore, in order to satisfy Frankie's requirement, the six mobsters could be arranged in 720/2 = 360 different ways. The correct answer is D.

A triangle in the xy-coordinate plane has vertices with coordinates (7, 0), (0, 8), and (20, 10). What is the area of this triangle? 72 80 87 96 100

It's not immediately possible to use the standard formula A = bh / 2 for the area of this triangle, because none of the triangle's sides is horizontal or vertical. However, if the triangle is circumscribed by a rectangle, as depicted below, the areas of the three surrounding triangles can readily be found and then subtracted from the rectangle's area to yield the desired result. First, the triangle to the lower left has b = 7 and h = 8, so its area is (7)(8) / 2 = 28. Second, the triangle to the lower right has b = 13 and h = 10, so its area is (13)(10) / 2 = 65. Third, the topmost triangle has b = 20 and h = 2, so its area is (20)(2) / 2 = 20. The total area of the surrounding triangles—that is, the triangles that are not part of the desired area—is 28 + 65 + 20 = 113 square units. The total area of the rectangle is 20 x 10 = 200 square units, so the triangle's area is 200 - 113 = 87 square units. The correct answer is C.

Of the birds in a particular aviary, 30 are both spotted and crested and 10 are neither spotted nor crested. What is the total number of birds in the aviary? (1) The number of birds that are spotted but not crested is 10 greater than the number of birds that are crested but not spotted. (2) More than 3/5 of the birds in the aviary are spotted.

The information presented can be set up in a Double-Set Matrix: C No C Total S 30 No S 10 Total ? (1) INSUFFICIENT: If x represents the number of birds that are crested but not spotted, this statement yields the following: C No C Total S 30 x + 10 x + 40 No S x 10 x + 10 Total x + 30 x + 20 2x + 50 = ? Without a value for x, it is not possible to find the total. (2) INSUFFICIENT: By itself, this statement does not provide values for any unknowns in the problem. C No C Total S 30 > 3/5 Total No S 10 < 2/5 Total Total Total = ? (1) and (2) INSUFFICIENT: The two statements together provide a filled-in matrix and a relationship between the total number of spotted birds and the total number of birds in the aviary. However, this only provides a range for the total: C No C Total S 30 x + 10 x + 40 No S x 10 x + 10 Total x + 30 x + 20 2x + 50 = ? Total S > 3/5 of Total Birds x + 40 > 3/5 (2x + 50) x + 40 > 6x/5 + 30 10 > x/5 50 > x If x < 50, the total number of birds, 2x + 50, must be less than 150, but this does not provide a definitive value for the total. The correct answer is (E).

x, y, a, and b are positive integers. When x is divided by y, the remainder is 6. When a is divided by b, the remainder is 9. Which of the following is NOT a possible value for y + b? 24 21 20 17 15

The problem states that when x is divided by y the remainder is 6. In general, the divisor ( y in this case) will always be greater than the remainder. To illustrate this concept, let's look at a few examples: 15/4 gives 3 remainder 3 (the divisor 4 is greater than the remainder 3) 25/3 gives 8 remainder 1 (the divisor 3 is greater than the remainder 1) 46/7 gives 6 remainder 4 (the divisor 7 is greater than the remainder 4) In the case at hand, we can therefore conclude that y must be greater than 6. The problem also states that when a is divided by b the remainder is 9. Therefore, we can conclude that b must be greater than 9. If y > 6 and b > 9, then y + b > 6 + 9 > 15. Thus, 15 cannot be the sum of y and b. The correct answer is E.

If the price of a commodity is directly proportional to m3 and inversely proportional to q2, which of the following values of m and q will result in the highest price for the commodity? m=3, q=2 m=12, q=12 m=20, q=20 m=30, q=36 m=36, q=72

To solve this problem, we need to use our understanding of direct and inverse proportionality. We can start by looking at the generic formulas: If y is directly proportional to x, then y = kx, where k is a constant. In other words, when y gets bigger, so does x. If y is inversely proportional to x, then y = k/x, where k is a constant. In other words, when y gets bigger, x gets smaller (and vice versa). We know that when m3 gets bigger, the price of our commodity gets bigger. However, when q2 gets bigger, that same price gets smaller. We can express this as: Price = km3 / q2 Note that we can combine both pieces of information into one formula because both relate to the price of the same commodity. Using the formula we have derived, we can rephrase the question as follows: What values of m and q will produce the greatest value of m3 / q2? Because the problem asks us for the greatest value, we will need to compare the five answer choices. We don't necessarily need to find all 5 values, though; let's give the choices a quick scan to see which are the most likely to produce the largest values, given our equation. Answer A is the only one in which m is larger than q, but the actual numbers are quite small; this isn't likely to produce the largest value overall. Answers B and C have the same values for m and q; of the two, the 20/20 pair is likely going to result in the larger value - though we have to check to make sure. Answers D and E are large enough to look annoying; let's not start with either of them. Answer C looks like the best place to start. (C) If m and q both equal 20, then 203 / 202 = 20. (Make sure to simplify before you multiply!) Our "starting comparison value" is 20. (B) Because the two values are again identical, this math is going to work the same way as it did in C (where we canceled out two of three 20s); the value will be 12, which is smaller than 20. Cross off B. (A) 33 / 22 = 27/4 = smaller than 20. Cross off A. In both answers D and E, q is greater than m; this might seem to indicate that a smaller value will result. Remember, though, that we are cubing the numerator and only squaring the denominator. Check the math: (D) 303 / 362 = (30)(30)(30) / (36)(36) = (5)(5)(5) / 6 = 125/6. (Again, simplify before you multiply!) 120/6 would be 20, so 125/6 is bigger than 20. This is larger than C, so cross off C. (E) 363 / 722 = 363 / (2×36)2 = 36/22 = smaller than 20. Eliminate answer E. The correct answer is D.

A certain store sells only black shoes and brown shoes. In a certain week, the store sold x black shoes and y brown shoes. If 2/3 of all shoes sold that week were black, which of the following expressions represents the value of y, in terms of x? x/3 x/2 2x/3 3x/2 2x

We can solve this problem algebraically or we can assign values to x and y and use those numbers to determine the correct answer. Let's try picking numbers first. Since the question deals with thirds (2/3, to be precise), let's choose a total number of shoes that is divisible by 3. Let's say 6, for example. If 2/3 of the 6 shoes sold were black, this represents (2/3)(6) = 4 shoes. This leaves 2 brown shoes to make up the difference. The question asks which of the given expressions represents the value of y (the number of brown shoes sold). Since y = 2 in our example, all we have to do is plug 4 (the value of x) into all of the expressions and see which yields 2 as a result: (A) 4/3 = 1.33 Incorrect (B) 4/2 = 2 CORRECT (C) 2(4)/3 = 8/3 = 2.33 Incorrect (D) 3(4)/2 = 12/2 = 6 Incorrect (E) 2(4) = 8 Incorrect

If 8 x > 4 + 6 x, what is the value of the integer x? (1) 6 - 5 x > -13 (2) 3 - 2 x < - x + 4 < 7.2 - 2 x

We can start by solving the given inequality for x: 8 x > 4 + 6 x 2 x > 4 x > 2 So, the rephrased question is: "If the integer x is greater than 2, what is the value of x?" (1) SUFFICIENT: Let's solve this inequality for x as well: 6 - 5 x > -13 -5 x > -19 x < 3.8 Since we know from the question that x > 2, we can conclude that 2 < x < 3.8. The only integer between 2 and 3.8 is 3. Therefore, x = 3. (2) SUFFICIENT: We can break this inequality into two distinct inequalities. Then, we can solve each inequality for x: 3 - 2 x < - x + 4 3 - 4 < x -1 < x - x + 4 < 7.2 - 2 x x < 7.2 - 4 x < 3.2 So, we end up with -1 < x < 3.2. Since we know from the information given in the question that x > 2, we can conclude that 2 < x < 3.2. The only integer between 2 and 3.2 is 3. Therefore, x = 3. The correct answer is D.

If -2 < x < 3, which of the following CANNOT be true? 3x < -3 2x + 2 < 4 3x < 8 3 x > -3 -3x > 8

We must check each of the ranges in the answer choices to see which one does not contain values within the given range -2 < x < 3. (A) 3x < -3 x < -1 Has values within the range -2 < x < 3. (B) 2x + 2 < 4 2x < 2 x < 1 Has values within the range -2 < x < 3. (C) 3x < 8 x < 8/3 Has values within the range -2 < x < 3. (D) 3x > -3 x > -1 Has values within the range -2 < x < 3. (E) -3x > 8 x < -8/3 (must change direction of inequality symbol when dividing by a negative) Does NOT have values within the range -2 < x < 3. The correct answer is E.