GMAT Manhattan Prep CAT Exams + OG Quant

If xy>0, does (x-1)(y-1)=1 (1) x+y=xy (2) x=y

Answer: A tl;dr: expand out equation xy-xy=0 See OG Quant DS #95 for full explanation

Running at their respective constant rates, Machine X takes 2 days longer to produce w widgets than Machine Y. At these rates, if the two machines together produce 5/4 widgets in 3 days, how many days would it take Machine X alone to produce 2w widgets. (A) 4 (B) 6 (C) 8 (D) 10 (E) 12

Answer: E w/x + w/(x-2) = 5/12x 0 = 5x^2 - 34x + 24 x=4/5 or 6 Machine X takes 6 days to produce w widgets and so 12 days to produce 2w widgets

zzz1 The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers? A. 17 B. 16 C. 15 D. 14 E. 13

Answer: E https://www.manhattanprep.com/gmat/forums/the-number-75-can-be-written-as-the-sum-of-the-squares-t2175.html

8^a(1/4)^b = ? (1) b = 1.5*a (2) a = 2

Exponents & Roots: We can simplify the question as follows: 8^a(1/4)^b = ? [Break all non-primes down to primes.] (2^3)a(2^-2)b = ? [Multiply exponents taken on the same base.] (2^3a)(2^-2b) = ? [Add exponents since the two bases are equal.] 2^(3a - 2b) = ? We can rephrase the question as "what is 3a -2b?" (1) SUFFICIENT: b = 1.5a, so 2b = 3a. This means that 3a - 2b = 0. (2) INSUFFICIENT: This statement gives us no information about b. The correct answer is A.

Machine A, working alone at a constant rate, can complete a certain production lot in x hours. Machine B, working alone at a constant rate, can complete 1/5 of the same production lot in y hours. Machines A and B, working together, can complete 1/2 of the same production lot in z hours. What is the value of y in terms of x and z?

First, let's find the individual rates of machines A and B, as well as the combined rate of machines A and B working together, using the following formula: (rate)(time) = (work) Machine A: (rateA) × x = 1 Machine B: (rateB) × y = 1/5 Machines A and B together: (rateTogether) × z = 1/2 Correct answer: (2xz)/(5x-10z)

If $ defines a certain operation, is p $ q less than 20? (1) x $ y = 2x2 - y for all values of x and y (2) p = 4, q = 10

Formulas: (1) INSUFFICIENT: This gives the definition of the $ function, however, it gives us no information about p and q. (2) INSUFFICIENT: This statement gives us no information about the $ function. (1) AND (2) SUFFICIENT: We can use the definition of the $ function given in (1) along with the values of p and q from (2) to solve for the value of p $ q = 2(4)2 - 10 = 22. The correct answer is C.

The useful life of a certain piece of equipment is determined by the following formula: u =(8d)/h2, where u is the useful life of the equipment, in years, d is the density of the underlying material, in g/cm3, and h is the number of hours of daily usage of the equipment. If the density of the underlying material is doubled and the daily usage of the equipment is halved, what will be the percentage increase in the useful life of the equipment? a) 300% b) 400% c) 600% d) 700% e) 800%

Formulas: Before the change: d = 3, h = 2; u = (8)(3)/22 = 24/4 = 6 After the change: d = (2)(3)= 6, h =2/2 =1; u = (8)(6)/12 = 48 Finally, percent increase is found by first calculating the change in value divided by the original value and then multiplying by 100: (48 - 6)/6 = (42/6) = 7 (7)(100) = 700% The correct answer is D.

If a and b are nonzero integers, which of the following must be negative? a) (-a)^(-2b) b) (-a)^(-3b) c) -(a)^(-2b) d) -(a)^(-3b) e) None

Work through the answer choices one by one. Answer: C

zzz4 Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days? A) 2 B) 3 C) 4 D) 6 E) 7

You can solve this problem without any equations. Just to calculate the rate of 1 machine. The rate of 1 machine is 1/(6∗12)=1/72 So, 1 machine in 8 days can do 8/72=1/9. So, to finish the job in 8 days we need 9 machines. Therefore 3 additional machines are required. The answer is B.

Which of the following fractions will terminate when expressed as a decimal? (Choose all that apply.) (A) 1/256 (B) 27/100 (C) 100/27 (D) 231/660 (E) 7/105

Answer: A, B, D See MP FDP Page 127 #5 In order for the decimal version of a fraction to terminate, the fraction's denominator in fully reduced form must have a price factorization that consists of only 2's and/or 5's.

The operation ==> is defined by x ==> y = x + (x+1) + (x+2) + y. For example, 3 ==> 7 = 3 + 4 + 5 + 6 + 7. What is the value of (100 ==> 150) - (125 ==> 150)?

Answer: 2800 112 x 25 = 2800 See MP Word Problem Page 111 #6

On the planet Flarp, 3 floops equal 5 fleeps, 4 fleeps equal 7 flaaps, and 2 flaaps equal 3 fliips. How many floops are equal to 35 fliips?

Answer: 8 floops See MP Word Problem Page 23 #3

If x + 2y = z, what is the value of x? (1) 3y = 4.5 + 1.5z (2) y = 2

Answer: A If we solve the equation x + 2y = z in terms of x, we can rephrase the question. x = z - 2y The question becomes "What is z - 2y ?" (1) SUFFICIENT: We can manipulate this statement to solve for z - 2y: 3y = 4.5 + 1.5z divide both sides by 1.5 2y = 3 + z z - 2y = -3 (2) INSUFFICIENT: We cannot solve for the expression z - 2y, nor can we find the value of z separately.

If x^2 + y^2 = 29, what is the value of (x-y)^2? (1) xy=10 (2) x=5

Answer: A See OG Quant DS #86 for full explanation

Is the average of n consecutive integers equal to 1? (1) n is even (2) if S is the sum of the n consecutive integers, then 0 < S < n.

Answer: D See MP Word Problem Page 141 #4

What is the ratio x:y:z? (1) x + y = 2z (2) 2x + 3y = z

Answer: C See MP FDP Page 103 #4

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip? (A) (180-x)/2 (B) (x+60)/4 (C) (300-x)/5 (D) 600/(115-x) (E) 12,000/(x+200)

Answer: E See Large MP OG PS 182 Total Distance/Total Time

If the average of 4 numbers is 50, how many of the numbers are greater than 50? (1) None of the four numbers is equal to 50. (2) Two of the numbers are equal to 25.

Answer: E See OG Quant DS #107 for full explanation

If m is a positive integer, then m^3 has how many digits? (1) m has 3 digits. (2) m^2 has 5 digits.

Answer: E See OG Quant DS #109 for full explanation

A student committee that must consist of 5 members is to be formed from a pool of 8 candidates. How many different committees are possible? a) 5 b) 8 c) 40 d) 56 e) 336

Combinatorics: The total number of possible five-person committees that can be created from a group of 8 candidates will be equal to the number of possible anagrams that can be formed from the word YYYYYNNN = 8! / (5!3!) = 56. Therefore, there are a total of 56 possible committees. The correct answer is D.

Quentin's income is 60% less than Rex's income, and Sam's income is 25% less than Quentin's income. If Rex gave 60% of his income to Sam and 40% of his income to Quentin, Quentin's new income would be what fraction of Sam's new income?

Everything ultimately depends on Rex's income, and since the information in the problem involves percents, we should assign a value of $100 to Rex's income. 60% less than $100 is $40, so that's Quentin's income. Sam's income, 25% less than $40, is $30. 60% of Rex's income is $60, and 40% of his income is $40. So Sam gains $60 more, for a total of $90. Quentin gains $40, for a total of $80. We're looking what fraction Quentin's new income ($80) is of Sam's new income ($90): 80/90 = 8/9.

If n is an integer, f(n) = f(n - 1) - n, and f(4) = 10. What is the value of f(6)? a) -1 b) 0 c) 1 d) 2 e) 4

In order to evaluate the function f(n), simply substitute the value of n for every instance of n on the right-hand-side of the definition of the function. For example, for n = 6, f(6) = f(5) - 6. Note that the value of f(n) is dependent on the value of f(n - 1). Therefore, in order to find f(6), we must know f(5). However, we do not know f(5) but we do know f(4). Since f(5) = f(4) - 5, we can calculate the value of f(5) from f(4), then f(6) from f(5) as follows: f(5) = f(4) - 5 = 10 - 5 = 5 f(6) = f(5) - 6 = 5 - 6 = -1 The correct answer is A.

The total cost of producing item X is equal to the sum of item X's overhead cost and production cost. If the production cost of producing X decreased by 5% in January, by what percent did the total cost of producing item X change in that same month? (1) The overhead cost of producing item X increased by 13% in January. (2) Before the changes in January, the overhead cost of producing item X was 5 times the production cost of producing item X.

Percents: Answer: C

Two sides of a triangle have lengths x and y and meet at a right angle. If the perimeter of the triangle is 4x, what is the ratio of x to y ?

Perhaps the most straightforward way to solve this problem is to "plug" answer choices back into the problem and check the truth of the final condition (i.e., the perimeter of the triangle must work out to 4x). Given a ratio x : y, we can select two specific values x and y having that ratio, use the Pythagorean theorem to find the hypotenuse of the triangle, add up all three sides to find the perimeter, and, finally, check whether the perimeter is indeed equal to 4x as required. (A) Let x = 2 and y = 3. Using the Pythagorean theorem,9 These values yield a perimeter which is not equal to 4x = 8. (B) Let x = 3 and y = 4. Using the Pythagorean theorem, these values yield a perimeter of 12, which is equal to 4x. Using the answer choices is the most efficient method on this problem, but there is an algebraic solution. The two legs of the triangle are x and y. The perimeter is 4x, so we can find an expression for the length of the third side by subtraction: 4x - (x + y) = 3x - y. The triangle is a right triangle, so the Pythagorean theorem applies: x^2 + y^2 = (3x − y)^2 x^2 + y^2 = 9x^2 - 6xy + y^2 0 = 8x^2 - 6xy 0 = 2x(4x - 3y) This equation yields x = 0 or 4x - 3y = 0. The question asks for the ratio of x to y, or x/y. rearrange the equation to find the value of x/y: 4x - 3y = 0 4x = 3y x/y = 3/4 The correct answer is B.

A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

Polygons Circles & Cylinders: Answer: 5(Root3 - 1). Find diagonal of face using Pythagorean Theorem, and then again to find the diagonal of cube. The length of diagonal minus the diameter divided by 2 gives you one of the excess distances from the corner to the edge of sphere.

Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A and B work together at their respective rates to complete the job, which of the following represents the fraction of the job that B will not have to complete? a) (x-y)/(x+y) b) x/(y-x) c) (x+y)/xy d) y/(x-y) e) y/(x+y)

Rates & Work: We can solve this problem as a VIC (Variable In Answer Choice) and plug in values for the two variables, x and y. Let's say x = 2 and y = 3. Machine A can complete one job in 2 hours. Thus, the rate of Machine A is 1/2. Machine B can complete one job in 3 hours. Thus, the rate of Machine B is 1/3. The combined rate for Machine A and Machine B working together is: 1/2 + 1/3 = 5/6. Using the equation (Rate)(Time) = Work, we can plug 5/6 in for the combined rate, plug 1 in for the total work (since they work together to complete 1 job), and calculate the total time as 6/5 hours. The question asks us what fraction of the job machine B will NOT have to complete because of A's help. In other words we need to know what portion of the job machine A alone completes in that 6/5 hours. A's rate is 1/2, and it spends 6/5 hours working. By plugging these into the RT=W formula, we calculate that, A completes (1/2)(6/5) = 3/5 of the job. Thus, machine B is saved from having to complete 3/5 of the job. If we plug our values of x = 2 and y = 3 into the answer choices, we see that only answer choice E yields the correct value of 3/5.

In the diagram to the right, triangle PQR has a right angle at Q and line segment QS is perpendicular to PR. If line segment PS has a length of 16 and line segment SR has a length of 9, what is the area of triangle PQR? (See diagram in MP CAT 5 Q7) a) 72 b) 96 c) 108 d) 150 e) 200

Since QSP, PQR, and QSR are all right angles, we know that: w + x = 90 x + y = 90 Therefore, w = y x + y = 90 y + z = 90 Therefore, x = z Analyzing the smaller triangles in the diagram (PQS and SQR), we can see that they must be similar triangles. Since these sub-triangles are similar triangles, we know that the ratios of the sides must be equal: QS2 = 144, meaning QS = 12. The area of triangle PQR is: (1/2)*(12)(16+9) = (6)(25) = 150. The answer is D.

Given the ascending set of positive integers { a, b, c, d, e, f}, is the median greater than the mean? (1) a + e = (3/4)( c + d) (2) b + f = (4/3)( c + d)

Since the set {a, b, c, d, e, f} has an even number of terms, there is no one middle term, and thus the median is the average of the two middle terms, c and d. Therefore the question can be rephrased in the following manner: Is (c + d)/2 > (a + b + c + d + e + f)/6 ? Is 3(c + d) > a + b + c + d + e + f ? Is 3c + 3d > a + b + c + d + e + f ? Is 2c + 2d > a + b + e + f ? (1) INSUFFICIENT: We can substitute the statement into the question and continue rephrasing as follows: Is 2c + 2d > (3/4)(c + d) + b + f ? Is (5/4)(c + d) > b + f ? From the question stem, we know c > b and d < f; however, since these inequalities do not point the same way as in the question (and since we have a coefficient of 5/4 on the left side of the question), we cannot answer the question. We can make the answer to the question "Yes" by relatively picking small b and f (compared to c and d) -- for instance, b = 2, c = 7, d = 9 and f = 12 (still leaving room for a and e, which in this case would equal 1 and 11, respectively). On the other hand, we can make the answer "No" by changing f to a very large number, such as 1000. (2) INSUFFICIENT: Going through the same argument as above, we can substitute the statement into the question: Is 2c + 2d > a + e + (4/3)(c + d) ? Is (2/3)(c + d) > a + e ? This is also insufficient. It is true that we know that a + e < (4/3)(c + d). The reason we know this is that the set of integers is ascending, so a < b and e < f. Therefore a + e < b + f, and b + f = (4/3)(c + d) according to this statement. However, we don't know whether a + e < (2/3)(c + d). (1) AND (2) SUFFICIENT: If we substitute both statements into the rephrased inequality, we get a definitive answer. Is 2c + 2d > a + b + e + f ? Is 2(c + d) > (3/4)(c + d) + (4/3)(c + d)? Is 2(c + d) > (25/12)(c + d)? Now, we can divide by c + d, a quantity we know to be positive, so the direction of the inequality symbol does not change. Is 2 > 25/12 ? 2 is NOT greater than 25/12, so the answer is a definite "No." Recall that a definite "No" is sufficient. The correct answer is C.

If x, n, and m are positive integers and x/ n = m, is x divisible by 3? (1) m is divisible by 6. (2) n is divisible by 15.

Since x/ n = m, we can multiply both sides of the equation by n to get x = mn. Thus we can rephrase the question as follows: "Is mn divisible by 3?" (1) SUFFICIENT: 6 can be broken down into the prime factors of 2 and 3. If m is divisible by 6, it must be divisible by 3. Therefore, the product mn must be divisible by 3. (2) SUFFICIENT: 15 can be broken down into the prime factors of 3 and 5. If n is divisible by 15, it must be divisible by 3. Therefore, the product mn must be divisible by 3. The correct answer is D.

A retail item is offered at a discount of p percent (where p > 10), with a 5% state sales tax assessed on the discounted purchase price. If the state sales tax were not assessed, what percent discount from the item's original retail price, in terms of p, would result in the same final price? a) (p+5)/1.05 b) (p/1.05) +5 c) (1.05*p)-5 d) (p-5)/1.05 e) 1.05/(p-5)

The original retail price and the value of p are undetermined, so both can be selected arbitrarily. Let the original retail price be $100, and let p = 20. Then the discounted price is 20% off $100, or $80. Five percent of 80 is 4, so adding the 5% sales tax gives a final price of $84. If no sales tax is assessed, then a final price of $84 is equivalent to a discount of $16, or 16% off the original retail price. Check the answer choices: (A) 25/1.05 = slightly less than 25. (B) 20/1.05 + 5 = (slightly less than 20) + 5 = slightly less than 25. (C) 1.05(20) - 5 = 21 - 5 = 16. (D) 15/1.05 = less than 15. (E) 1.05(15) = 15.75. Only choice (C) works, so the correct answer is (C).

The number of antelope in a certain herd increases every year by a constant factor. If there are 500 antelope in the herd today, how many years will it take for the number of antelope to double? (1) Ten years from now, there will be more than ten times the current number of antelope in the herd. (2) If the herd were to grow in number at twice its current rate, there would be 980 antelope in the group in two years.

The question is asking us to find the minimum number of years it will take for the herd to double in number. In other words, we need to find the minimum value of n that would yield a population of 1000 or more. We can represent this as an inequality: 500x^n > 1000 x^n > 2 In other words, we need to find what integer value of n would cause x^n to be greater than 2. To solve this, we need to know the value of x. Therefore, we can rephrase this question as: "What is x, the annual growth factor of the herd?" (1) INSUFFICIENT: This tells us that in ten years the following inequality will hold: 500x^10 > 5000 x^10 > 10 There are an infinite number of growth factors, x, that satisfy this inequality. For example, x = 1.5 and x = 2 both satisfy this inequality. If x = 2, the herd of antelope doubles after one year. If x = 1.5, the herd of antelope will be more than double after two years 500(1.5)(1.5) = 500(2.25). (2) SUFFICIENT: This will allow us to find the growth factor of the herd. We can represent the growth factor from the statement as y. (NOTE y does not necessarily equal 2x because x is a growth factor. For example, if the herd actually grows at a rate of 10% each year, x = 1.1, but y = 1.2, i.e. 20%) Time: Population Now: 500 in 1 year: 500^y in 2 years: 500y^2 According to the statement, 500y^2 = 980 y^2 = 980/500 y^2 = 49/25 y = 7/5 OR 1.4 (y can't be negative because we know the herd is growing) This means that the hypothetical double rate from the statement represents an annual growth rate of 40%. The actual growth rate is therefore 20%, so x = 1.2. The correct answer is B.

A list contains twenty integers, not necessarily distinct. Does the list contain at least two consecutive integers? (1) If any single value in the list is increased by 1, the number of different values in the list does not change. (2) At least one value occurs more than once in the list.

This problem is annoying because of the number of terms in the list; it's hard to wrap your head around 20 integers. Check the statements to see whether you can think through the problem using a smaller list, or whether it really is necessary to stick with a list of 20. In the case of both statements 1 and 2, the full size of the list doesn't matter; you can think the problem through using an easier list (say, 10 or even 5 numbers) that still represents the basic principles in question. (1) NOT SUFFICIENT: If the list consists of the numbers 2, 4, 6, 8, 10, then all of the values are different. If any value is increased by 1, the list will still consistent of five different values, so this scenario satisfies the statement. This list does not contain two consecutive integers, so the answer to the question is no. If, on the other hand, the list consists of four "1"s and a "2", then there are only 2 different values (1 and 2). If any of the 1's is increased, the result is 2, which is already in the list, so there are still two different values. If the 2 is increased, then the list will contain four 1's and a 3, and so the list will still contain only two distinct values. In this case, the original list does contain two consecutive integers (1 and 2), so the answer to the question is yes. Because there are two conflicting answers to the question (no and yes), this statement is not sufficient. (2) NOT SUFFICIENT: A list containing four 1's and a 2 contains two consecutive integers (1 and 2). If the list contains four 1's and a 3, then it doesn't contain any consecutive integers. Because there are two conflicting answers to the question, this statement is not sufficient. (1) AND (2) SUFFICIENT: Statement 2 indicates that at least one value occurs twice; call that value a. Statement 1 indicates that increasing any value in the list by 1 will not change the number of distinct values in the list. In this case, then, increasing one of the a values by one, to a + 1, will still leave you with a in the list (since there are at least two a values) as well as a + 1. The value of a + 1, then, must already have been in the original list; if it wasn't, then you would have just added a new value without getting rid of an old value, and statement 1 forbids this. For example, if the original list is {1, 1, 2, 4, 6}, then a = 1 and there are four distinct values in the list. Changing one of the 1's to 2 makes the list {1, 2, 2, 4, 6} and there are still four distinct values. This list does contain two consecutive integers (1 and 2). If the original list were {1, 1, 3, 5, 7}, then a = 1 and there are four distinct values in the list. Changing one of the 1's to 2 makes the list {1, 2, 3, 5, 7}, but now there are 5 distinct values in the list! This is not allowed, according to statement 1. As a result, whatever a is, a + 1 must also be in the original list. The original list must contain at least one pair of consecutive integers. The correct answer is (C).

If the box pictured to the right is a cube, then the difference in length between line segment BC and line segment AB is approximately what fraction of the distance from A to C? (AC = edge, AB = diagonal face, BC = diagonal cube) a) 10% b) 20% c) 30% d) 40% e) 50%

Triangles and Diagonals: AB = xroot2, BC = xroot3 Now, we need to subtract AB from BC to find their difference in length. The approximate values are 1.7x and 1.4x, so the difference is 0.3x. Dividing by the length of AC (which is an edge of the cube, so its length is x), we find that the answer is 0.3, or 30%. The correct answer is C.

If ab ≠ 0, is ab > a/b ? (1) |b| > 1 (2) ab + a/b > 0

(1) NOT SUFFICIENT: According to this statement, the magnitude of b is greater than 1; that is, either b > 1 or b < -1. On the other hand, a can have any value whatsoever (except zero). With that freedom, it's possible to choose values of a and b that yield contrasting answers to the question. If a = 1 and b = 2, then the answer to the question is "yes." (Is 2 > 1/2 ?) If a = -1 and b = 2, then the answer is "no." (Is -2 > -1/2 ?) (2) NOT SUFFICIENT: In interpreting this statement, note the influence of signs. The values ab and a/b have to have the same sign (either positive or negative). In order for their sum to be positive, they must both be positive. Therefore, the two variables a and b are either both positive or both negative. Within these constraints, it's possible to choose values of a and b that yield contrasting answers to the question. If a = 1 and b = 2, then the answer to the question is "yes." (Is 2 > 1/2 ?) If a = 1 and b = 1/2, then the answer is "no." (Is 1/2 > 2 ?) (1) AND (2) SUFFICIENT: Statement 1 indicates that b > 1 or b < -1. Statement 2 indicates that a and b are either both positive or both negative. First, consider the case in which both a and b are positive. In that case, b > 1. Consider a the "starting" number. Multiplying a by a number greater than 1 will make a larger. Dividing a by a number greater than 1 will make a smaller. In other words, ab must be greater than a/b. Now consider the remaining case, in which both a and b are negative. In this case, b < -1. Since both numbers are negative, the negative signs cancel when they are multiplied or divided—in other words, the outcome is exactly the same as in the case where both a and b are positive. Again, ab must be greater than a/b. Therefore, ab > a/b in all cases. Both statements together are sufficient. 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? a) 8/5 b) 5/2 c) 3 d) 11/3 e) 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 Answer: D

Guests at a recent party ate a total of fifteen hamburgers. Each guest who was neither a student nor a vegetarian ate exactly one hamburger; no other guests ate hamburgers. If half of the guests were vegetarians, how many guests attended the party? (1) The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians. (2) 30% of the guests were vegetarian non-students.

Answer: A V NON-V TOTAL S NON-S 15 TOTAL x x 2x = ? Each non-vegetarian non-student ate exactly one of the 15 hamburgers, and nobody else ate any of the 15 hamburgers. Therefore, there are exactly 15 people in the non-vegetarian non-student category. The question stem also indicates that half of the group is vegetarian, so the other half must be non-vegetarian. The two groups are equal in size, then! Put an x in both boxes in the table. The second statement is easier than the first statement; start there. (2) INSUFFICIENT: This statement provides information about one group: vegetarian non-students. The figure given, though, is a percentage of another unknown figure; a real number can't be calculated from this. The statement is insufficient. (1) SUFFICIENT: This statement provides two pieces of information. First, for every 2 vegetarian students in attendance, there were 3 vegetarian non-students. Next, this 2:3 rate is half the rate for non-vegetarians. To double a rate, double the first number (4:3). The rate is 4 non-vegetarian students to 3 non-vegetarian non-students. The problem provides a real number for one of these categories! There are 15 non-vegetarian non-students. If that's the case, then there must be 20 non-vegetarian students. Add this info to the table. Therefore, there were 20 + 15 = 35 total non-vegetarians (see the chart below). Since the same number of vegetarians and non-vegetarians attended the party, there were also 35 vegetarians, for a total of 70 guests.

The sequence s1, s2, s3, ..., sn, ... is such that sn = 1/n - 1/(n+1) for all integers n>= 1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10? (1) k > 10 (2) k < 19

Answer: A tl;dr: 1 - 1/(n+1) See OG Quant DS #111 for full explanation

The sequence a1, a2,..., an is such that an = 2a(n-1) - x for all positive integers n>= 2 and for a certain number x. If a5 = 99 and a3 = 2, what is the value of x? (A) 3 (B) 9 (C) 18 (D) 36 (E) 45

Answer: A OG PS 144

In the sequence S of numbers, each term after the first two terms is the sum of the two immediately preceding terms. What is the 5th term of S? (1) The 6th term of S minus the 4th term equals 5 (2) The 6th term of S plus the 7th term equals 21

Answer: A See OG Quant DS #113 for full explanation

Car B starts at point X and moves clockwise around a circular track at a constant rate of 2 mph. Ten hours later, Car A leaves from point X and travels counter-clockwise around the same circular track at a constant rate of 3 mph. If the radius of the track is 10 miles, for how many hours will Car B have been traveling when the cars have passed each other for the first time and put another 12 miles between them (measured around the curve of the track)? (A) 4pi - 1.6 (B) 4pi + 8.4 (C) 4pi + 10.4 (D) 2pi - 1.6 (E) 2 pi - 0.8

Answer: B

The integer 6 is the product of two consecutive integers (6 = 2 × 3) and the product of three consecutive integers (6 = 1 × 2 × 3). What is the next integer greater than 6 that is both the product of two consecutive integers and the product of three consecutive integers? a) 153 b) 210 c) 272 d) 336 e) 600

Answer: B There's no obvious way to set this problem up algebraically, so work backwards from the answer choices. The problem asks for the next integer that satisfies the requirements—in other words, the smallest of the given choices—so begin by testing the smallest value, 153, and then work upward. First, break the number into its prime components: 153 = (51)(3) = (17)(3)(3). Next, can these primes be combined so that two consecutive integers will produce 153? No. (3)(3) = 9, but this is not consecutive with 17. (17)(3) would be way too big. This can't be the right answer. Try the next smallest answer. First, break 210 into primes: 210 = (10)(21) = (2)(5)(3)(7). (2)(3) = 6, so 210 is the product of 3 consecutive integers (5, 6, and 7). Is there a way to combine the primes into two consecutive integers? Yes! (2)(7) = 14 and (3)(5) = 15. The number 210 fits the requirements, and it is the smallest of the answer choices, so it must be the next largest integer (after 6) that can be written as the product of two consecutive integers and as the product of three consecutive integers.

If a, b, and c are positive numbers such that a is b percent of c, what is the value of c? (1) a is c percent of b. (2) b is c percent of a.

Answer: B (1) NOT SUFFICIENT: This statement can be written as a=(c/100)b = bc/100 or a = (0.01 c) b = 0.01 bc. Either way, this statement is identical to the equation already given in the prompt, so statement (1) does nothing to help determine the value of c. (2) (2) SUFFICIENT: This statement can be written as b=(c/100)a =ac/100 or b = (0.01 c) a = 0.01 ac. Use either fractions or decimals, whichever you prefer. Here's the solution method using fractions: b=(bc/100)c/100 and b=bc^2/1000 . Substitute: The variable b is nonzero (the question stem says b is positive), so divide it out, giving 1=C^2/10000 . Thus c2 = 10,000; since c must be positive, c = 100. Here's the solution method using decimals: a = 0.01bc and b = 0.01 ac. Substitute: b = 0.01(0.01bc) c b = 0.0001bc^2 The variable b is nonzero (the question stem says b is positive), so divide it out, giving 1 = 0.0001c^2, or 10,000 = c^2. Thus c^2 = 10,000; since c must be positive, c = 100.

David used part of $100,000 to purchase a house. Of the remaining portion, he invested 1/3 of it at 4 percent interest and 2/3 of it at 6 percent interest. If after a year the income from the two investments totaled $320, what was the purchase price of the house? (A) $96,000 (B) $94,000 (C) $88,000 (D) $75,000 (E) $40,000

Answer: B 1/3(100,000 - x)(.04) + 2/3(100,000 - x)(.06) =320 OG PS 127

If a and b are positive integers such that a/b = 2.86, which of the following must be a divisor of a? a) 10 b) 13 c) 18 d) 26 e) 50

Answer: B Because 2.86 is a terminating decimal and a and b are integers, it might be easier to convert 2.86 to a fraction: a/b = 286/100, now reduce a/b = 143/50 It might be easier to think through the problem if we cross multiply: 50 × a = 143 × b What does that tell us about a and b? Well, we know that 50, a, 143, and b are all integers. Thus both sides of the equation will be integers (the same integer). For that to be true, both sides of the equation must have IDENTICAL prime factorizations. We know that the left side of the equation has a 2 and 2 5's in its prime factorization (50 = 5×5×2). Therefore, b must have at least a 2, a 5 and another 5 in its prime factorization. So b is divisible by 50. Furthermore, we know that the right side of the equation has an 11 and a 13 in its prime factorization (143 = 11×13). Therefore, a must have at least an 11 and a 13 in its prime factorization. So a is divisible by 11, 13, and 143. The question asks about a. We know that a must be divisible by 13.

If d=1/(2^3+5^7) is expressed as a terminating decimal, how many nonzero digits will d have? (A) 1 (B) 2 (C) 3 (D) 7 (E) 10

Answer: B See Large MP OG PS 188 (1/(2*5))^3 * (1/5)^4 10^(-3) * (.2)^4 = .0000016

For each landscaping job that takes more than 4 hours, a certain contractor charges a total of r dollars for the first 4 hours of work plus 0.2r for each additional hour or fraction of an hour, where r > 100. Did a particular landscaping job take more than 10 hours? (1) The contractor charged a total of $288 for the job. (2) The contractor charged a total of 2.4r dollars for the job.

Answer: B See OG Quant DS #110 for full explanation

100 people are attending a newspaper conference. 45 of them are writers and more than 38 are editors. Of the people at the conference, x are both writers and editors and 2x are neither. What is the largest possible number of people who are both writers and editors? (A) 6 (B) 16 (C) 17 (D) 33 (E) 84

Answer: B This Overlapping Sets problem asks us to determine the maximum value of x, the number of people who are both writers and editors. We can set up a Double Set Matrix using the information given in the question to tackle this problem. From the matrix, we find that the number of people who are not writers is 100 - 45 = 55, and therefore the number of people who are editors and not writers is 55 - 2x. We can continue to populate our matrix: Because the total number of editors must be more than 38, we can set up and solve the following inequality: x + (55 - 2x) > 38 55 - x > 38 x < 17 Since x is less than 17, the maximum number of people who are both editors and writers at the conference is 16. The correct answer is B.

Pascal has 96 miles remaining to complete his cycling trip. If he reduced his current speed by 4 miles per hour, the remainder of the trip would take him 16 hours longer than it would if he increased his speed by 50%. What is his current speed? a) 6 b) 8 c) 10 d) 12 e) 16

Answer: B Work backwards, but algebraic solution possible by setting rates equal to each other and adjusting for the 16 hour difference

If x+y+z > 0, is z>1? (1) z > x+y+1 (2) x+y+1 < 0

Answer: B tl;dr: (1) 2z>1, x>1/2 (2) z> -(x+y) and -(x+y) >1 See OG Quant DS #99 for full explanation

What is the value of x^2 - y^2? (1) x-y = y+2 (2) x-y = 1/(x+y)

Answer: B tl;dr: (x-y)*(x+y)=1 or x^2 - y^2 = 1 See OG Quant DS #67 for full explanation

What is the distance between Harry's home and his office? (1) Harry's average speed on his commute to work this Monday was 30 miles per hour. (2) If Harry's average speed on his commute to work this Monday had been twice as fast, his trip would have been 15 minutes shorter.

Answer: C Distance = Rate × Time, or D = RT. (1) INSUFFICIENT: This statement tells us Harry's rate, 30 mph. This is not enough to calculate the distance from his home to his office, since we don't know anything about the time required for his commute. D = RT = (30 mph) (T) D cannot be calculated because T is unknown. (2) INSUFFICIENT: If Harry had traveled twice as fast, he would have gotten to work in half the time, which according to this statement would have saved him 15 minutes. Therefore, his actual commute took 30 minutes. So we learn his commute time from this statement, but don't know anything about his actual speed. D = RT = (R) (1/2 hour) D cannot be calculated because R is unknown. (1) AND (2) SUFFICIENT: From statement (1) we learned that Harry's rate was 30 mph. From Statement (2) we learned that Harry's commute time was 30 minutes. Therefore, we can use the rate formula to determine the distance Harry traveled. D = RT = (30 mph) (1/2 hour) = 15 miles

At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. At the same time, 70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness. If 80% of the interns receive fewer than 6 hours of sleep, what percent of the interns report no feelings of tiredness during their shifts? (A) 6 (B) 14 (C) 19 (D) 20 (E) 81

Answer: C For an overlapping sets problem we can use a double-set matrix to organize our information and solve. Because the values are in percents, we can assign a value of 100 for the total number of interns at the hospital. Then, carefully fill in the matrix based on the information provided in the problem. The matrix below details this information. Notice that the variable x is used to detail the number of interns who receive 6 or more hours of sleep, 70% of whom reported no feelings of tiredness.

Company Z only sells chairs and tables. What percent of its revenue in 2008 did Company Z derive from its sales of chairs? (1) In 2008, the price of tables sold by Company Z was 10% higher than the price of chairs sold by Company Z. (2) In 2008, Company Z sold 20% fewer tables than chairs.

Answer: C See MP FDP Page 69 #14

Each year, a college admissions committee grants a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. The number of scholarships granted at each level does not vary from year to year, and no student can receive more than one scholarship. This year, how many different ways can the committee distribute the scholarships among the pool of 10 applicants? (1) In total, six scholarships will be granted. (2) An equal number of scholarships will be granted at each scholarship level.

Answer: C This is a combinatorics problem. We can imagine that the admissions committee will choose a "team" of students to receive scholarships. However, because there are three different levels of scholarships, it will not suffice to simply count the number of possible "scholarship teams." In other words, the committee must also place the "scholarship team" members according to scholarship level. So, order matters. If we knew the number of scholarships to be granted at each of the three scholarship levels, we could use the anagram method to count the number of ways the scholarships could be distributed among the 10 applicants. To show that this is true, let's invent a hypothetical case for which there are 3 scholarships to be granted at each of the three levels. If we assign letters to the ten applicants from A to J, and if T represents a $10,000 scholarship, F represents a $5,000 scholarship, O represents a $1,000 scholarship, and N represents no scholarship, then the anagram grid would look like this: ABCDEFGHIJ TTTFFFOOON Our "word" is TTTFFFOOON. To calculate the number of different "spellings" of this "word," we use the following shortcut: 10! (3!)(3!)(3!)(1!) The 10 represents the 10 total applicants, the 3's represent the 3 T's, 3 F's, and 3 O's, and the 1 represents the 1 N. Simplifying this expression would yield the number of ways to distribute the scholarships among the 10 applicants. So, knowing the number of scholarships to be granted at each of the three scholarship levels allows us to calculate the answer to the question. Therefore, the rephrased question is: "How many scholarships are to be granted at each of the three scholarship levels?" (1) INSUFFICIENT: While this tells us the total number of scholarships to be granted, we still don't know how many from each level will be granted. (2) INSUFFICIENT: While this tells us that the number of scholarships from each level will be equal, we still don't know how many from each level will be granted. For example, there could be 1, 2, or 3 scholarships granted at each level without exceeding the number of applicants (10). (1) AND (2) SUFFICIENT: If there are 6 total scholarships to be granted and the same number from each level will be granted, there will be two $10,000 scholarships, two $5,000 scholarships, and two $1,000 scholarships granted.

Kali builds a tower using only red, green, and blue toy bricks in a ratio of 4:3:1. She then removes 1/2 of the green bricks and adds 1/3 more blue bricks, reducing the size of the tower by 14 bricks. How many red bricks will she need to add in order to double the total number of bricks used to build the original tower? (A) 82 (B) 96 (C) 110 (D) 120 (E) 192

Answer: C Use the Unknown Multiplier to set up this ratio. If the ratio of red to green to blue bricks in the original tower is 4:3:1, then the tower contains 4x red bricks, 3x green bricks, and x blue bricks, for a total of 8x bricks. After Kali makes the described changes, the new totals are 4x, 3/2x, and 4/3 x, for a total of 41/6 x bricks in the new tower. The new tower of 41/6 x bricks represents a reduction of 14 bricks from the original total of 8 x: 8x − 14 = 41/6x 7/6x = 14 x = 12 Next, Kali wants to double the size of the original tower, which contained 8x bricks, so she will need a total of 16x bricks. The tower currently consists of 41/6 x bricks, so find the difference. 16x-41/6x = 55/6x = 56/6*12 = 110

K is a set of numbers such that (i) if x is in K, then -x is in K, and (ii) if each of x and y is in K, then xy is in K. Is 12 in K? (1) 2 is in K. (2) 3 is in K.

Answer: C tl;dr: 2 and 3 are in K, so is 6, thus 2*6=12 See OG Quant DS #85 for full explanation

What is the area of rectangular region R? (1) Each diagonal of R has length 5 (2) The perimeter of R is 14

Answer: C tl;dr: 2L + 2W = 14, L + W = 7, plug 7-L = W into Pythagorean theorem See OG Quant DS #64 for full explanation

A palindrome is a number that reads the same forward and backward, such as 121. How many odd, 4-digit numbers are palindromes? a) 40 b) 45 c) 50 d) 90 e) 2500

Answer: C We can use the "slot method" to count all the 4-digit, odd palindromes. __ __ __ __ Since the last digit must be odd, our only choices are 1, 3, 5, 7, or 9 for the first/last digit. There are no restrictions on the inner digits, so we have 10 choices: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Notice that the outer two numbers must match and the inner two numbers must match, creating numbers such as 1221 or 5665. We have 5 choices for the outer two digits and 10 choices for the inner two digits. Our "slot method" diagram looks like this: 5 10 1 1. Once a digit is selected for the left outer digit, there is only one possible choice for the right outer digit, which must match it. Similarly for the two inner digits, the left choice determines the right. Using the counting principle, we have 5 × 10 × 1 × 1 = 50 choices for our 4-digit number. Notice that we do not set the problem up as 5 10 10 5 and multiply, giving 2500. There are really only two choices to be made - number of possibilities for inner digits and number of possibilities for outer digits.

Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days? a) 2 b) 3 c) 4 d) 6 e) 7

Answer: C See MP Word Problem Page 51 #4

If 75 percent of the guests at a certain banquet ordered dessert, what percent of the guests ordered coffee? (1) 60 percent of the guests who ordered dessert also ordered coffee. (2) 90 percent of the guests who ordered coffee also ordered dessert.

Answer: C See OG Quant DS #116 for full explanation

A store sells a certain product at a fixed price per unit. At the product's current price, q units cost a total of exactly $300. If the price were lowered by $5 from its current value, then q + 2n units would cost exactly $300; if the price were raised by $5, then q - n units would cost exactly $300. What is the value of q? a) 10 b) 15 c) 20 d) 25 e) 30

Answer: C The answer choices are fairly "clean" (integers, not ridiculously large), so we can try plugging them into the problem until we find the one that works. Typically, when plugging the answers into the problem, we start with answer B or D and work from there. (Note: it is possible to use an algebraic solution here, but the algebra is so cumbersome that we can't recommend it. In fact, we're not even going to show it!) One other piece of advice: the math is complicated here, even to explain. Try writing out the steps below yourself as you read the explanation. (B) q = 15. If 15 units cost $300, then the fixed price is $20 per unit. If the price were lowered by $5, then the new price would be $15 and $300 would buy 300/15 = 20 units, so q + 2n = 20, or 15 + 2n = 20. This equation gives n = 2.5. If the price were raised by $5, then the new price would be $25 and $300 would buy 300/25 = 12 units. Check this against the final piece of information (q - n = the number of units). If B is the correct answer, we should be able to use the same value (2.5) for n and get the answer 12. 15 - 2.5, however, is 12.5, not 12, so B is not the correct answer. B was not correct but it was very close to correct (12.5 vs. 12), so the correct answer is likely to be either A or C. There is no easy way to tell which to try first; just pick one and try it. (C) q = 20. If 20 units cost $300, then the fixed price is $15 per unit. If the price were lowered by $5, then the new price would be $10 and $300 would buy 300/10 = 30 units, so q + 2n = 30, or 20 + 2n = 30. This equation gives n = 5. If the price were raised by $5, then the new price would be $20 and $300 would buy 300/20 = 15 units. Check this against the final piece of information (q - n = the number of units). If C is the correct answer, we should be able to use the same value (5) for n and get the answer 15. Since q - n = 20 - 5 = 15, we know that q = 20 was the correct starting point.

Al and Barb shared the driving on a certain rip. What fraction of the total distance did Al drive? (1) Al drove for 3/4 as much time as Barb did. (2) Al's average driving speed for the entire trip was 4/5 of Barb's average driving speed for the trip.

Answer: C (8/5)rt See MP Word Problem Page 51 #5

Of the 300 subjects who participated in an experiment using virtual reality therapy to reduce their fear of heights, 40 percent experienced sweaty palms, 30 percent experienced vomiting, and 75 percent experienced dizziness. If all of the subjects experienced at least one of these effects and 35 percent of the subjects experienced exactly two of these effects, how many of the subjects experienced only one of these effects? (A) 105 (B) 125 (C) 130 (D) 180 (E) 195

Answer: D See Large MP OG PS 192

Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store's revenue from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of Newspaper A, which of the following expresses r in terms of p? (A) 100p/(125-p) (B) 150p/(250-p) (C) 300p/(375-p) (D) 400p/(500-p) (E) 500p/(625-p)

Answer: D See Large MP OG PS 205

.99999999/1.0001 - .99999991/1.0003 = (A) 10^-8 (B) 3(10^-8) (C) 3(10^-4) (D) 2(10^-4) (E) 10^-4

Answer: D See Large MP OG PS 206

Bag A contains red, white and blue marbles such that the red to white marble ratio is 1:3 and the white to blue marble ratio is 2:3. Bag B contains red and white marbles in the ratio of 1:4. Together, the two bags contain 30 white marbles. How many red marbles could be in bag A? a) 1 b) 3 c) 4 d) 6 e) 9

Answer: D We are told that bag B contains red and white marbles in the ration 1:4. This implies that WB, the number of white marbles in bag B, must be a multiple of 4. What can we say about WA, the number of white marbles in bag A? We are given two ratios involving the white marbles in bag A. The fact that the ratio of red to white marbles in bag A is 1:3 implies that WA must be a multiple of 3. The fact that the ratio of white to blue marbles in bag A is 2:3 implies that WA must be a multiple of 2. Since WA is both a multiple of 2 and a multiple of 3, it must be a multiple of 6. We are told that WA + WB = 30. We have already figured out that WA must be a multiple of 6 and that WB must be a multiple of 4. So all we need to do now is to test each candidate value of WA (i.e. 6, 12, 18, and 24) to see whether, when plugged into WA + WB = 30, it yields a value for WB that is a multiple of 4. It turns out that WA = 6 and WA = 18 are the only values that meet this criterion. Recall that the ratio of red to white marbles in bag A is 1:3. If there are 6 white marbles in bag A, there are 2 red marbles. If there are 18 white marbles in bag A, there are 6 red marbles. Thus, the number of red marbles in bag A is either 2 or 6. Only one answer choice matches either of these numbers.

A list contains n distinct integers. Are all n integers consecutive? (1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed. (2) The positive difference between any two numbers in the list is always less than n.

Answer: D (1) SUFFICIENT: If the removal of the lowest number causes the average to be exactly 1 higher than does the removal of the highest number, then the difference between the highest and lowest terms must equal 1(number of terms - 1). For example, if the list contains 4 distinct integers, then the difference between the highest and lowest terms is 3. The only way for this to be true is for the 4 distinct integers to be 1 apart; that is, the 4 integers must be consecutive. (2) SUFFICIENT: If all n numbers are consecutive, then the difference between the least and the greatest is n - 1. In that case the list will satisfy statement 2 (the difference is less than n) and the answer to the question is yes. If there are any gaps in the list, then the distance between the least and the greatest values will increase. In that case, the difference between the least and the greatest integer will be n or more, and so the list will not satisfy statement 2. In other words, the only allowable case, according to this statement, is the case in which the numbers are consecutive.

A window in the shape of a regular hexagon with each side length 80 centimeters. If a diagonal through the center of the hexagon is w centimeters long, then w = (A) 80 (B) 120 (C) 150 (D) 160 (E) 240

Answer: D 60 degrees, equilateral triangle so double 80 to get diagonal

The function f is defined for each positive three digit integer n by f(n) = 2^x 3^y 5^z, where x, y, and z are the hundreds, thens, and units digits of n, respectively. If m and v ar ethree digit positive integers such that f(m) = 9f(v), then m-v = (A) 8 (B) 9 (C) 18 (D) 20 (E) 80

Answer: D A=a, B=b+2, C=c Therefore, m-v = 20 OG PS 160

There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible selections of 5 books from the shelf contain at least one paperback and at least one hardback. (A) 75 (B) 120 (C) 210 (D) 246 (E) 252

Answer: D T-N, where T is the total number of selections of 5 books and N is the number of selections that do not contain both a paperback and a hardback. The value of T is 10!/5!*5! = 252 To find the value of N, there are only 4 paperback books, so no selection of 5 books can contain all paperbacks. Thus the value of N is equal to the number of selections of 5 books that contain all hardbacks, which is equal to 6 since there are 6 ways a single hardback can be left out. Thus T-N = 252-6=242

Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased? (A) 17.5% (B) 62.5% (C) 87.5% (D) 140% (E) 150%

Answer: D The question asks for the percent increase in the number of biographies in the library Note that the problem never provides a real number of books for any of the steps and the question also asks for a percentage. We can choose Smart Numbers here! If the biographies originally represented 20% of the total books, then let's say that there were 20 biographies originally out of a total of 100 books. The next part is a little more complicated and we need to use a variable to start. When x new biographies are added to the mix, the new fraction is 37.5% or 3/8: (20+x)/(100+x) = 3/8 (Note: don't forget to add x to both the top and the bottom! The total number of books also increases by x.) Now cross multiply: 8(20 + x) = 3(100 + x) 160 + 8x = 300 + 3x 5x = 140 x = 28 If the library started with 20 biographies and added 28 new ones, then the percentage increase is: 28/20=140/100 = 140% Note that the difference equals the 28 added books. Also notice that the question asks for the answer in the form of a percentage, so the answer needs to be "over 100." In this case, multiplying the denominator, 20, by 5 will result in 100 on the bottom. The numerator becomes 28 × 5 = 140.

zzz2 At a certain pet shop, 1/3 of the pets are dogs and 1/5 of the pets are birds. How many of the pets are dogs? (1) There are 30 birds at the pet shop. (2) There are 20 more dogs than birds at the pet shop.

Answer: D http://gmatclub.com/forum/at-a-certain-pet-shop-1-3-of-the-pets-are-dogs-and-1-5-of-t-58028.html

zzz3 If (1/5)^m * (1/4)^18 = 1/(2*(10)^35), then m = ? A. 17 B. 18 C. 34 D. 35 E. 36

Answer: D http://gmatclub.com/forum/if-1-5-m-1-4-18-frac-127321.html

What was the ratio of the number of cars to the number of trucks produced by Company X last year? (1) Last year, if the number of cars produced by Company X had been 8 percent greater, the number of cars produced would have been 150 percent of the number of trucks produced by Company X. (2) Last year Company X produced 565,000 cars and 406,800 trucks.

Answer: D tl;dr: (1) 1.08c/t=1.5, c/t = 1.5/1.08, (2) values given See OG Quant DS #83 for full explanation

Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m? (A) k/m% (B) (k-m)% (C) 100(k-m)/(100+k)% (D) 100(k-m)/(100+m)% (E) 100(k-m)/(100+k+m)%

Answer: D See Large MP OG PS 191

Kim finds a 1-meter tree branch and marks it off in thirds and fifths. She then breaks the branch along all the markings and removes one piece of every distinct length. What fraction of the original branch remains? a) 2/5 b) 7/15 c) 1/2 d) 8/15 e) 3/5

Answer: E In this Fractions problem, a 1-meter branch is marked in thirds and fifths. Because it's difficult to compare, add, and subtract intervals with thirds and fifths, we can write the fractions in terms of the common denominator, 15. We can draw the following diagram to help us visualize each segment of the branch and quickly calculate the lengths of each interval. We see from the figure that the unique lengths of the segments are 3/15, 2/15, 1/15 totaling to 6/15 or 2/5 when simplified. Thus the remaining part of the branch is 9/15 or 3/5.

A sports team played 100 games last season. Did this team win at least half of the games it played last season? (1) The team won 60% of its first 65 games last season. (2) The team won 60% of its last 65 games last season.

Answer: E (1) INSUFFICIENT: The team won 60%, or 39, of its first 65 games last season. What about the last 35 games? Test the extreme cases: winning all of the remaining games and winning none of them. If the team won all 35 remaining games, then it won 74 of 100 total games—more than half of the season total. If, on the other hand, the team lost all 35 remaining games, then it only won 39 of 100 total games—less than half of the season total. (2) INSUFFICIENT: The team won 60%, or 39, of its last 65 games last season. What about the first 35 games? Test the extreme cases: winning all of the games and winning none of them. If the team won the first 35 games, then it won 74 of 100 total games—more than half of the season total. If, on the other hand, the team lost the first 35 games, then it only won 39 of 100 total games—less than half of the season total. (1) AND (2) INSUFFICIENT: The key to treating the statements together is to observe the overlap between "the first 65 games" and "the last 65 games." Sketch a quick number line, with the first game labeled #1 and the last labeled #100. In this case, the first 65 games are #1 to #65, and the last 65 games are #36 to #100. Therefore, games #36 to #65—a total of thirty games—are considered in both statements. Maximize overlapping wins: All 30 of the overlapping games could represent winners. In that case, those games account for 30 of the team's 39 wins in the first 65 games, and for 30 of its wins in the last 65 games. Statement 1 indicates that the team won 39 of the first 65, so the team must also have won an additional 9 games during matches #1 to #35. Likewise, the team must also have won an additional 9 games during matches #66 to #100. The team would win a total of 9 + 30 + 9 = 48 games, less than half the season total.

If x and k are integers and (12^x)(4^(2x+1) = (2^k)(3^2), what is the value of k? (A) 5 (B) 7 (C) 10 (D) 12 (E) 14

Answer: E (12^x)(4^2x+1) = ((3*2^2)x)((2^2)^2x+1) = (3^x)(2^6x+2) x=2 so k = 14 OG PS 147

A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? (A) 4 (B) 6 (C) 8 (D) 10 (E) 12

Answer: E Each side of the square must have a length of 10. If each side of the square were to be a length of 6, 7, 8, or most other numbers, there could only be four possible squares drawn in total, because each side, in order to have integer coordinates, would have to be drawn on the x- or y-axis. A length of 10 is different though, because the side of such a square can be the hypotenuse of a Pythagorean triple, meaning the vertices could have integer coordinates without lying on the x- or y-axis. For example, a square could be drawn with the coordinates (0,0), (6,8), (-2, 14) and (-8, 6). (It is tedious and unnecessary to figure out all four coordinates for each square). If we label the square abcd, with a at the origin and the letters representing points in a clockwise direction, we can get the number of possible squares by figuring out the number of unique ways ab can be drawn. a has coordinates (0,0) and b could have the following coordinates, as shown in the picture:

A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square. The ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width, in inches, of the wooden strip? I. 1 II. 3 III. 4 (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III

Answer: E Let x represent the side length of the entire plaque, let y represent the side length of the brass inlay and w represent the uniform width of the wooden strip around the brass inlay. Since the ratio of the area of the brass inlay to the area of the wooden strip is 25 to 39, the ratio of the area of the brass inlay to the area of the entire plaque is y^2/x^2 = 25/(25+39) = 25/64. Then y/x = 5/8, and y=5/8x. Also, x = y +2w and w = (x-y)/2. Substituting 5/8x for y into this expression gives w for w = (x-5/8x)/2 = 3/16x. I. If the plaque were 16/3 inches on a side, then the width of the wooden strip would be 1 inch, and so 1 inch is possible with for the wooden strip. II. If the plaque were 16 inches on a side, then the width of the wooden strip would be 3 inches and 3 inches is a possible width for the wooden strip. III. If the plaque were 64/3 inches on a side, then the width of the strip would be 4 inches and so 4 inches is a possible width for the wooden strip.

Is the positive integer n a multiple of 24? (1) n is a multiple of 4 (2) n is a multiple of 6

Answer: E See OG Quant DS #115 for full explanation

If 2 different representatives are to be selected at random from a group of 10 employees and if p is the probability that both representatives selected will be women, is p >1/2 (1) More than 1/2 of the 10 employees are women (2) The probability that both representatives selected will be men is less than 1/10

Answer: E See OG Quant DS #119 for full explanation

If m and n are nonzero integers, is m^n an integer? (1) n^m is positive. (2) n^m is an integer.

Answer: E See OG Quant DS #91 for full explanation

Two musicians, Maria and Perry, work at independent constant rates to tune a warehouse full of instruments. If both musicians start at the same time and work at their normal rates, they will complete the job in 45 minutes. However, if Perry were to work at twice Maria's rate, they would take only 20 minutes. How long would it take Perry, working alone at his normal rate, to tune the warehouse full of instruments? (A) 1 hour 20 minutes (B) 1 hour 45 minutes (C) 2 hours (D) 2 hours 20 minutes (E) 3 hours

Answer: E We can use an algebraic approach or a "real-world" approach. Both methods are shown below. Algebraic Approach Organize the given information in an RTW chart, using m and p to represent Maria's and Perry's normal rates, and w to represent work (to tune 1 warehouse full of instruments): R1 = Rate, Time Work R2 = Maria, m, blank, w R3 Perry, p, ?, w R4 Maria & Perry, m+p, 45 minutes, w The question mark denotes the desired value: Perry's time. Note that Perry's time also equals w/p. How do we represent the hypothetical situation in which Perry is twice as fast as Maria? We can create an additional row for this situation: Maria & Perry (hyp), m+2m, 20 minutes, w Note that we would not want to label the combined rate p + (1/2)p. Remember that p represents Perry's normal rate, not the hypothetical "twice as fast as Maria" rate. From the hypothetical situation: RT = W 3m(20) = w m = w/60 Fill this into the original RTW chart. Is there any way to solve for w/p? (w/60 + p)45 = w (3/4)w + 45p = w 45p = (1/4)w 180p = w 180 = w/p Perry's time = w/p = 180 minutes = 3 hours The correct answer is E. "Real-world" approach Start with the hypothetical case. If Perry were working at twice Maria's rate (m), that would be like having three Marias on the job: 2m + m. If the three Marias could finish the job in 20 minutes, then each is doing 1/3 of the job in that 20 minutes. At that rate, a single person working at Maria's rate could finish the entire job in 1 hour. Maria's rate, then, is 1 job per hour. When Perry is working at his real rate, he and Maria together complete the job in 45 minutes. If Maria works for 45 minutes at her rate of 1 job per 60 minutes, then calculate the portion of the work that she completes. RT = W. (45)(1/60) = W. Maria completes 45/60 or 3/4 of the job in that 45 minute timeframe. Perry, then, must complete the other 1/4 of the job in 45 minutes. How long does it take him to complete the entire job? He can do 1/2 of the job in 1.5 hours and the entire job in 3 hours. The correct answer is E.

What is the value of x? (1) 2x - 5y + 6 = 12 (2) 8x - (4x + 10y) + 27 = 39

Answer: E When a data sufficiency question provides equations or algebraic expressions that can be simplified, make sure to do so before drawing any conclusions. (1) INSUFFICIENT: After simplifying the equation, the value of x still cannot be determined without knowing the value of y: 2x - 5y + 6 = 12 2x - 5y = 12 - 6 2x - 5y = 6 2x = 6 + 5y x = (6 + 5y)/2 (2) INSUFFICIENT: After simplifying the equation, the value of x still cannot be determined without knowing the value of y: 8x - (4x + 10y) + 27 = 39 8x - 4x - 10y = 39 - 27 4x - 10y = 12 2x - 5y = 6 2x = 6 + 5y x = (6 + 5y)/2 (1) AND (2) INSUFFICIENT: Note that we ended up with the same equation from both statements. Therefore, we cannot solve for the value of x. On data sufficiency problems involving systems of equations, be aware of duplicates and always simplify the equations before making the final decision.

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? a) 9 b) 17 c) 21 d) 27 e) 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).

If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?

Consecutive Integers: Thus, since there are no multiples of 30 between 295 and 299 and between 601 and 615, finding the sum of all multiples of 30 between 295 and 615, inclusive, is equivalent to finding the sum of all multiples of 30 between 300 and 600, inclusive. Therefore, we can rephrase the question: "What is the greatest prime factor of the sum of all multiples of 30 between 300 and 600, inclusive?" The sum of a set = (the mean of the set) × (the number of terms in the set) Since 300 is the 10th multiple of 30, and 600 is the 20th multiple of 30, we need to count all multiples of 30 between the 10th and the 20th multiples of 30, inclusive. There are 11 terms in the set: 20th - 10th + 1 = 10 + 1 = 11 The mean of the set = (the first term + the last term) divided by 2: (300 + 600) / 2 = 450 k = the sum of this set = 450 × 11 Note, that since we need to find the greatest prime factor of k, we do not need to compute the actual value of k, but can simply break the product of 450 and 11 into its prime factors: k = 450 × 11 = 2 × 3 × 3 × 5 × 5 × 11 Therefore, the largest prime factor of k is 11.

In the number 1.4ab5, a and b represent single positive digits. If x = 1.4ab5, what is the value of 10 - x? (1) If x is rounded to the nearest hundredth, then 10 - x = 8.56. (2) If x is rounded to the nearest thousandth, then 10 - x = 8.564.

Digits and Decimals: Key: Let both options have the light of day, independent of each other. To determine the value of 10 - x, we must determine the exact value of x. To determine the value of x, we must find out what digits a and b represent. Thus, the question can be rephrased: What is a and what is b? (1) INSUFFICIENT: This tells us that x rounded to the nearest hundredth must be 1.44. This means that a, the hundredths digit, might be either 3 (if the hundredths digit was rounded up to 4) or 4 (if the hundredths digit was rounded down to 4). This statement alone is NOT sufficient since it does not give us a definitive value for a and tells us nothing about b. (2) SUFFICIENT: This tells us that x rounded to the nearest thousandth must be 1.436. This means, that a, the hundredths digit, is equal to 3. As for b, the thousandths digit, we know that it is followed by a 5 (the ten-thousandths digit); therefore, if x is rounded to the nearest thousandth, b must rounded UP. Since b is rounded UP to 6, then we know that b must be equal to 5. Statement (2) alone is sufficient because it provides us with definitive values for both a and b. The correct answer is B.

If the square root of p^2 is an integer greater than 1, which of the following must be true? I. p^2 has an odd number of positive factors II. p^2 can be expressed as the product of an even number of positive prime factors III. p has an even number of positive factors a) I b) II c) III d) I and II e) II and III

Divisibility of Primes: Statement I: 36's factors can be listed by considering pairs of factors (1, 36) (2, 18) (3,12) (4, 9) (6, 6). We can see that they are 9 in number. In fact, for any perfect square, the number of factors will always be odd. This stems from the fact that factors can always be listed in pairs, as we have done above. For perfect squares, however, one of the pairs of factors will have an identical pair, such as the (6,6) for 36. The existence of this "identical pair" will always make the number of factors odd for any perfect square. Any number that is not a perfect square will automatically have an even number of factors. Statement I must be true. Statement II: 36 can be expressed as 2 x 2 x 3 x 3, the product of 4 prime numbers. A perfect square will always be able to be expressed as the product of an even number of prime factors because a perfect square is formed by taking some integer, in this case 6, and squaring it. 6 is comprised of one two and one three. What happens when we square this number? (2 x 3)2 = 22 x 32. Notice that each prime element of 6 will show up twice in 62. In this way, the prime factors of a perfect square will always appear in pairs, so there must be an even number of them. Statement II must be true. Statement III: p, the square root of the perfect square p2 will have an odd number of factors if p itself is a perfect square as well and an even number of factors if p is not a perfect square. Statement III is not necessarily true. The correct answer is D, both statements I and II must be true.

If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the value of x? (1) z is prime (2) x is prime

Exponents & Roots: The best way to answer this question is to use the exponential rules to simplify the question stem, then analyze each statement based on the simplified equation. (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14 (3^27)(5^10)(7^10)(z) = (5^8)(7^10)(3^28)(x^y) Divide both sides by common terms 5^8, 7^10, 3^27 (5^2)(z) = 3x^y (1) SUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Statement (1) says that z is prime, so z cannot have another factor besides the 3. Therefore z = 3. Since z = 3, the left side of the equation is 75, so x^y = 25. The only integers greater than 1 that satisfy this equation are x = 5 and y = 2, so statement (1) is sufficient. Put differently, the expression x^y must provide the two fives that we have on the left side of the equation. The only way to get two fives if x and y are integers greater than 1 is if x = 5 and y = 2. (2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5 to balance out the 5^2 on the left side. Since statement (2) says that x is prime, x cannot have any other factors, so x = 5. Therefore statement (2) is sufficient.

Two buses and a van were employed to transport a class of students on a field trip. 3/5 of the class boarded the first bus. 2/3 of the remaining students boarded the second bus, and the rest of the students boarded the van. When the second bus broke down, 1/2 of the students on the second bus boarded the first bus. What fraction of the class was on board the first bus? a) 1/2 b) 2/3 c) 11/15 d) 23/30 e) 4/5

For a fraction question that makes no reference to specific values, it is best to assign a smart number as the "whole value" in the problem. In this case we'll use 30 since that is the least common denominator of all the fractions mentioned in the problem. If there are 30 students in the class, 3/5 or 18 boarded the first bus. This means that 12 students either boarded the second bus or the van. 2/3 of the 12 students who didn't board the first bus, or 8 students, boarded the second bus, so 4 students boarded the van. When the second bus broke down, half of these 8 students (or 4) were able to join the other 18 who had already boarded the first bus. That means that 22 of the 30 students ended up on the first bus. 22/30 reduces to 11/15. The correct answer is C.

A lemonade stand sold only small and large cups of lemonade on Tuesday. 3/5 of the cups sold were small and the rest were large. If the large cups were sold for 7/6 as much as the small cups, what fraction of Tuesday's total revenue was from the sale of large cups? a) 7/16 b) 7/15 c) 10/21 d) 17/35 e) 1/2

For a fraction word problem with no actual values for the total, it is best to plug numbers to solve. Since 3/5 of the total cups sold were small and 2/5 were large, we can arbitrarily assign 5 as the number of cups sold. Total cups sold = 5 Small cups sold = 3 Large cups sold = 2 Since the large cups were sold at 7/6 as much per cup as the small cups, we know: Pricelarge = (7/6)Pricesmall Let's assign a price of 6 cents per cup to the small cup. Price of small cup = 6 cents Price of large cup = 7 cents Now we can calculate revenue per cup type: Large cup sales = quantity × cost = 2 × 7 = 14 cents Small cup sales = quantity × cost = 3 × 6 = 18 cents Total sales = 32 cents The fraction of total revenue from large cup sales = 14/32 = 7/16. The correct answer is A.

In 2003 the price for each of Acme Computer's computers was five times the price for each of its printers. What was the ratio of its gross revenue from computers to its gross revenue from printers in 2003? (1) In the first half of 2003, Acme sold computers and printers in a ratio of 3:2; in the second half of 2003, Acme sold computers and printers in the ratio of 2:1. (2) Acme's 2003 price for each of its computers was $1,000.

Ratios: (1) INSUFFICIENT: Statement (1) says that the ratio of computers to printers sold in the first half of 2003 was 3 to 2; one set of values satisfying this statement is 3 computers and 2 printers. Using example prices of $5 per computer and $1 per printer, we arrive at a gross revenue of $15 from computers and $2 from printers. During the second half of 2003, the ratio of computers to printers sold was 2 to 1; one set of values satisfying this statement is 2 computers and 1 printer, grossing $10 and $1 respectively if we use the sample prices chosen above. With these numbers, then, the full-year gross revenues are $25 from computers and $3 from printers. Alternatively, for the second half of 2003 Acme may have sold 4 computers and 2 printers, still in the required ratio of 2 to 1. In this case, Acme would have grossed $20 and $2 from computers and printers, respectively; with these numbers, then, the full-year gross revenues are $35 from computers and $4 from printers, yielding a different overall ratio. Therefore, statement (1) is insufficient to give us a definitive answer. (2) INSUFFICIENT: Statement (2) tells us that one of Acme's computers cost $1,000 in 2003, but it tells us nothing about the ratio or numbers of computers and printers sold. (1) AND (2) INSUFFICIENT: Statement (2) fixes the price of a computer at $1,000, but, if we inflate the sample prices from $5 and $1 to $1,000 and $200, both of the solutions given in the explanation of statement (1) will still be possible. (The arithmetic will be identical; the prices will be 200 times as great.) Therefore, statements (1) and (2) together are still insufficient. The correct answer is E.

For any integer k > 1, the term "length of an integer" refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y? a) 5 b) 6 c) 15 d) 16 e) 18

The problem asks us to find the greatest possible value of (length of x + length of y), such that x and y are integers and x + 3y < 1,000 (note that x and y are the numbers themselves, not the lengths of the numbers - lengths are always indicated as "length of x" or "length of y," respectively). Consider the extreme scenarios to determine our possible values for integers x and y based upon our constraint x + 3y < 1,000 and the fact that both x and y have to be greater than 1. If y = 2, then x ≤ 993. If x = 2, then y ≤ 332. Of course, x and y could also be somewhere between these extremes. Since we want the maximum possible sum of the lengths, we want to maximize the length of our x value, since this variable can have the largest possible value (up to 993). The greatest number of factors is calculated by using the smallest prime number, 2, as a factor as many times as possible. 29 = 512 and 210 = 1,024, so our largest possible length for x is 9. If x itself is equal to 512, that leaves 487 as the highest possible value for 3y (since x + 3y < 1,000). The largest possible value for integer y, therefore, is 162 (since 487 / 3 = 162 remainder 1). If y < 162, then we again use the smallest prime number, 2, as a factor as many times as possible for a number less than 162. Since 27 = 128 and 28 = 256, our largest possible length for y is 7. If our largest possible length for x is 9 and our largest possible length for y is 7, our largest sum of the two lengths is 9 + 7 = 16. What if we try to maximize the length of the y value rather than that of the x value? Our maximum y value is 332, and the greatest number of prime factors of a number smaller than 332 is 28 = 256, giving us a length of 8 for y. That leaves us a maximum possible value of 231 for x (since x + 3y < 1,000). The greatest number of prime factors of a number smaller than 231 is 27 = 128, giving us a length of 7 for x. The sum of these lengths is 7 + 8 = 15, which is smaller than the sum of 16 that we obtained when we maximized the x value. Thus 16, not 15, is the maximum value of (length of x + length of y). The correct answer is D.

Which of the following integers is NOT a divisor of x if x = (21)(37) - (112)? (A) 7 (B) 11 (C) 15 (D) 17 (E) 35

The problem asks which of the answer choices is NOT a factor of x if x = (21)(37) - (112). Four of the answers choices must be factors of x and the other will not be a factor. In order to find the factors of this number, manipulate the expression to discover the divisors. First, strip the terms down to prime numbers: x = (3×7)(37) - (7×24) Pull out the common 7: x = 7[(3)(37) - (16)] 7 must be a factor of x, so eliminate answer choice (A). Next, combine the exponents in the first term: x = 7(38 - 16) Notice that both 38 and 16 are perfect squares, so the term between the parentheses is an example of a2 - b2! x = 7(34 + 4)(34 - 4). The second term (34 - 4) is once again an example of a2 - b2: x = 7(34 + 4)(32 - 2)(32 + 2) Determine the actual numbers in the parentheses: x = 7(81 + 4)(9 - 2)(9 + 2) x = 7(85)(7)(11) x = 7(17)(5)(7)(11) These numbers all represent factors of x. Answers (A), (B), and (D) are all represented among those factors. Answer (E) can be made by multiplying 5 and 7. Only answer (C), 15, cannot be made using the given factors because there is no 3. The correct answer is (C).

Is x a multiple of 4? (1) x + 2 is divisible by 2 (2) 6 is a factor of 3x

We can rephrase the question as "is x divisible by 2 twice?" since a number that is divisible by four must have two 2's in its prime make-up (prime box). (1) INSUFFICIENT: if x + 2 is divisible by 2, then x itself must be divisible by 2, but not necessarily 4. (RULE: for x + y to be divisible by y, x itself must be divisible by y) We could also plug numbers here to see that the statement is insufficient. According to the statement, x could be 2 (2 + 2 = 4, which is divisible by 2) and x could be 4 (4 + 2 = 6, which is divisible by 2). One of these values is a multiple of 4 and one is not. (2) INSUFFICIENT: If 6 is a factor of 3x, than x must be divisible by 2. For 6 to be a factor of a number, that number's prime make-up (prime box) must consist of a 3 and a 2, the prime components of six. 3x naturally has a 3 in its prime box because of the 3 coefficient. x therefore must be the source of the 2 and so x is divisible by 2. This does not, however, guarantee us that x is divisible by 4. Again we plug numbers here to illustrate the lack of sufficiency. According to the statement x could be 2 (3 × 2 is divisible by 6) and x could be 4 (3 × 4 is divisible by 6). (1) AND (2) INSUFFICIENT: Statements 1 and 2 lead to the same conclusion, namely that x is even, but not necessarily divisible by 4. The correct answer is E.

Maryam's wallet contains a combination of $5 and $10 bills. What is the ratio of $5 bills to $10 bills? (1) The ratio of the number of bills to the total dollar value is 1/9. (2) The total dollar value of the bills is $360.

We have been asked for the ratio of $5 bills to $10 bills. Let's call the number of five dollar bills f and the number of ten dollar bills t. The desired ratio, then, is f:t or f/t. (1) SUFFICIENT: The total number of bills is f + t and the total dollar value is 5f + 10t. Using these values, translate the statement as follows: (f+t)/(5f+10t)=1/9 cross multiply to get: 9(f + t) = 5f + 10t 9f + 9t = 5f + 10t 4f = t f/t= 1/4 The ratio f/t, or f:t, is 1/4. (2) INSUFFICIENT: Translate this statement into math: 5f + 10t = 360. Are there multiple possible values for f and t that will make this equation true, or just one set? Try some numbers. If f = 2, then t = 35, in which case the ratio of f:t would be 2:35. If f = 4, then t = 34, in which case the ratio would be 4:34 or 2:17. Because there are multiple possible values for the ratio of f to t, this statement is not sufficient. The correct answer is A.