Quant - GMAT Final Countdown

4/9 = ?

0.44444... memorize your FDP

Ways to arrange ABCDE where A and C are separated by at least 1 letter? Ways to arrange CIRCLE where two C's are separated by at least 1 letter?

5! - 2*4! = 72 6!/2! - 5! = 360-120= 240

Machines X and Y run at different constant rates, and machine X can complete a certain job in 9 hours. Machine X worked on the job alone for the first 3 hours and the two machines, working together, then completed the job in 4 more hours. How many hours would it have taken machine Y, working alone, to complete the entire job? (A) 18 (B) 13 1/2 (C) 7 1/5 (D) 4 1/2 (E) 3 2/3

A Careful. Make sure you distribute that 4. Gamechanger.

A car traveled 462 miles per tankful of gasoline on the highway and 336 miles per tankful of gasoline in the city. If the car traveled 6 fewer miles per gallon in the city than on the highway, how many miles per gallon did the car travel in the city? (A) 14 (B) 16 (C) 21 (D) 22 (E) 27

B Set 'x' equal to the amt of gallons in a tank. Solve for 'x' and divide 336/x to get the mpg in the city.

If x is not equal to 0, is |x| less than 1? (1) x/|x| < x (2) |x| > x

C First, interpret the statement |x|<1. That's the same as -1<x<1. S1 tells you that x>1 OR -1<x<0 S2 definitely tells you that x is negative, but nothing about |x| < 1. 1+2, you know -1<x<0. That fits the question's intended range.

Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9? I. 3 II. 9*sqrt(3) III. 13.5 I only II only III only II and III I, II and III

C (III only) make sure you multiply out 9*sqrt(3)=9*1.7 = 15.3

When determining if something is divisible by 2 or odd/even or an integer, use janky math operations.

Question: is x/18 an integer? (1) 3x/18 is an integer (2) 5x/18 is an integer Either alone insufficient. But do this: 3x/18 is an integer, so 6x/18 is an integer. 6x/18-5x/18 is an integer (subtraction of two ints). x/18 is an integer

Overlapping Sets

Three groups: Total = A+B+C - sum(exactly2) - 2*all3 + neither Two groups: Total = A + B - Both + Neither

Max/min framework

To find max number in a set, minimize all other numbers. To find min number in a set, max all other numbers.

Summing consecutive integers tricks

○ With an even number of items, sum of all items never a multiple of the number of items ○ With an odd number of items, sum of all items is always a multiple of the number of items

Leila is playing a carnival game in which she is given 4 chances to throw a ball through a hoop. If her chance of success on each throw is 1/5, what is the chance that she will succeed on at least 3 of the throws? a. 1/(5^4) b. 1/(5^3) c. 6/(5^4) d. 13/(5^4) e. 17/(5^4)

E do the math right, remember that you need to do the counting principle on the only 3 successes-> multiply by 4!/3!1!

A cylindrical tank has a base with a circumference of 4*sqrt(3pi) 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 sqrt(2) sqrt(3) sqrt(6)

E (sqrt(6)) Make sure you calculate the area of the circle properly, then remember that the stimulus says isosceles RIGHT triangle. That way it's easy to calculate 1/2bh=3 where b^2=6, b=sqrt(6).

Cardinality of a set is the number of distinct elements

Number of subsets possible for a set of n elements = 2^n (including the empty set & the set itself)

Total number of factors of a number I.e. 1. 9 2. 50 3. 100

Prime factorize and take each exponent, add 1 to each, and product. 1. 3^2 -> 3 2. 50 -> 2*5^2 = 2*3 = 6 3. 100 -> 2^2*5^2 = 3*3 = 9 Total number of factors for a perfect square is ALWAYS odd (implies that all powers of prime factors is always even). Total number of factors for a non-perfect square is ALWAYS even.

A farmer has an apple orchard consisting of Fuji and Gala apple trees. Due to high winds this year 10% of his trees cross pollinated, creating trees that are part Fuji and part Gala. The number of his trees that are pure Fuji plus the number that are part Fuji and part Gala totals 187, while 3/4 of all his trees are pure Fuji. How many of his trees are pure Gala? a. 22 b. 33 c. 55 d. 77 e. 88

b (33) Key here is that they mean to say that 10% of the total trees are mixed (cross-pollinated), not 10% of the previous total cross-pollinated forming an unknown amount of mixed trees. F+M=187 F=0.75T; M=0.1T, T=220 187+G=220, G=33

In a local school district, the high school and middle school each received r dollars toward funding for the student arts program. The high school enrolled 300 students and the middle school enrolled 200 students. Later, the middle school transferred s dollars to the high school so that they would have received the same funding per student. Which of the following is equivalent to s? a. r/2 b. r/3 c. r/4 d. r/5 e. r/6

d Come on, man these are layups...

If given |something| = something

make sure to plug numbers back in to verify solutions (cross verify)

Formula for the sum of n consecutive integers beginning with 1 Otherwise, sum of n consecutive integers in general (not necessarily beginning with 1)

n(n+1)/2 n*average (usually the midpoint)

Probability of getting the same outcome on 3 coin flips

1*1/2*1/2 = 1/4

Perfect cubes 3 4 5 6 7 8 9 10

27 64 125 216 343 512 729 1000

Right triangles

3-4-5 5-12-13

If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y? 5 5(x - y) 20x 20y 35x

C (20x) Because 35x/20x will always be a fraction (7/4). Therefore, it certainly cannot. Others you can find examples for. Btw, for answer B 5(x-y) will be a divisor if both x and y are a multiple of x-y. Easiest way to check then is to do x-y=1.

If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S ? 1. 1/8 2. 1/9 3. 1/10 None I only III only II and III only I, II, and III

C (III only) use bounds analysis. Sum of sequence is less than 10/91 (less than 1/9) and greater than 1/10. That means it is ONLY greater than 1/10 -> C

p and q are different two-digit prime numbers with the same digits, but in reversed order. What is the value of the larger of p and q? (1) p + q = 110 (2) p - q = 36 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D 91 is NOT a prime number! 7*13 = 91. That one got you... Know the variable solving method. It's good practice.

The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ? (1) 2 is the cardinality of exactly 6 subsets of set A. (2) Set A has a total of 16 subsets, including the empty set and set A itself. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D S1: # of 2-element sets is the same as number of (unordered) pairs you can select out of n elements. So nC2 = 6; (n!)/(2!(n-2)!)); n(n-1)/2 = 6; n=4 -> sufficient S2: number of subsets in a list = 2^n; 2^n=16; n=4 -> sufficient

In a certain game, a large bag is filled with blue, green, purple and red chips worth 1, 5, x and 11 points each, respectively. The purple chips are worth more than the green chips, but less than the red chips. A certain number of chips are then selected from the bag. If the product of the point values of the selected chips is 88,000, how many purple chips were selected? a. 1 b. 2 c. 3 d. 4 e. 5

b You got this without looking. Think you were just burnt out... that's stupid. Don't get burnt out. Good to know that p can't be 10 because there are 6 two's and 3 5's, some of which come from the blues.

Number AB (where a and b are digits) can be written as 10A+B →

ex: 21 can be written as 10(2)+1 Original - Reversed = 9(a-b) → AKA difference between AB and BA Original + Reversed = 11(a+b) → AKA sum of AB and BA

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet? 19,200 19,600 20,000 20,400 20,800

A (19,200) Dude, you keep making the same math mistakes. 1/2 of 38,400 is 19,200.

If BE CD, and BC = AB = 3, AE = 4 and CD = 10, what is the area of trapezoid BEDC? A. 12 B. 18 C. 24 D. 30 E. 48

B Yeah... this one is kinda wack. Two triangles are similar, OK that's fair, look at the angles. But you have to realize that because the triangle sides are 6-8-10, it's a right triangle. And can thus calculate areas.

Julie wants to be sure that she has enough pies for each of her 30 guests to have at least one slice. One pie can be divided into eight slices. If ⌈x⌉ represents the least integer greater than x, and x is greater than 0, will ⌈x⌉ pies be enough for each guest to have at least one slice? (1) 5 < 2x < 12 (2) x is a multiple of 3 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

B You misinterpreted the function [x]. It's the least integer GREATER than x. That means if x=3, [x]=4. That's not how you interpreted (I think you thought greater than or equal to). That means that S2 is sufficient x=3->4 is enough pies. S1 is not because x could be 2.8, in which case [x]=3, not enough, but x could be 3.5, in which case [x]=4, is enough.

Stores X, Y, and Z each sell a certain item that has a given list price. Stores X and Y are located in a state with a 5 percent sales tax, and both sell the item at a 5 percent discount off list price, while Store Z is located in a state with no sales tax and gives no discounts. Store X applies its discount first and then charges tax on the discounted price, while Store Y adds the tax first and then applies the discount to the price with tax. If x and y are the prices, with tax and discount, charged by Stores X and Y, respectively, and z is the price charged by Store Z, which of the following statements correctly describes the relationship among x, y, and z ? x = y = z x = y < z x < y < z x < z < y y < z < x

B (x=y<z) Pretty easy in POE but actually know that increasing by 5% and decreasing by 5% is less than the original (makes sense)

In the rectangular coordinate system, line k passes through the point (n,-1). Is the slope of line k greater than zero? (1) Line k passes through the origin. (2) Line k passes through the point (1,n + 2). A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Do the slope math to prove

What is the value of y? (1) 3| x^2 - 4| = y - 2 (2) |3 - y| = 11 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Good logic, but be able to interpret Statement 1.

The integers r, s, and t all have the same remainder when divided by 5. What is the value of t ? (1) r + s = t (2) 20 ≤ t ≤ 24 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Pretty good rule of thumb (know math), but if r, s, and t have same remainder then if r+s=t, then they are all multiples of 5. S2 alone is insufficient, it tells us nothing of 't' besides its range. Careful about bringing in S1 to S2

The measurements obtained for the interior dimensions of a rectangular box are 200 centimeters by 200 centimeters by 300 centimeters. If each of the three measurements has an error of at most 1 centimeter, which of the following is closest to the maximum possible difference, in cubic centimeters, between the actual capacity of the box and the capacity computed using these measurements? 100,000 120,000 160,000 240,000 320,000

C (160,000) Remember error of at most 1 cm, not 2.

There are 11 women and 9 men in a certain club. If the club is to select a committee of 2 women and 2 men, how many different such committees are possible? 120 720 1,060 1,520 1,980

E (1980) 9!/2!7! * 11!/2!9! = 1980 Not sure about the divide by 4 part. I guess you have two distinct lists of 2, where order doesn't matter so you divide by 2!2!

Type 3: |something| = |something| If you have absolute value on both sides (completely), square both sides and remove the absolute value |2x+3| = |x+1| (2x+3)^2 - (x+1)^2 = 0 Now you have difference of squares: (2x+3+x+1)(2x+3-x-1)=0

Ex: If |x+2| = |y+2|, what is the value of x+y? S1) xy < 0 S2) x > 2, y < 2

see picture

C... make sure you realize that the sales tax INCREASES the price, not decreases lol.

|a+b| < |a| + |b| When is it true?

If a and b have different signs (e.g. ab<0)

Reciprocals of inequalities

If same signs, flip (x < y, then 1/x > 1/y) If diff. signs, don't flip (x < y, then 1/x < 1/y)

For positive integer k, is the expression (k + 2)(k2 + 4k + 3) divisible by 4? (1) k is divisible by 8. (2) (k+1)/3 is an odd integer. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

a Absolute tank of a question but nice job.

The Carson family will purchase three used cars. There are two models of cars available, Model A and Model B, each of which is available in four colors: blue, black, red, and green. How many different combinations of three cars can the Carsons select if all the cars are to be different colors? 24 32 48 60 192

b (32) Easiest way to think about it is you can take 4 colors and pick 3 and arrange them in 4!/3!1! ways = 4. Each of the 3 slots has two options. So 4*2^3 = 32. Counting principle: Better yet, do 8*6*4 and divide by 3! = 32. Remember the rule: if we want to avoid double counting we have to divide by 3! In this case, if you purchase B1, R1, G1, that's the same as purchasing R1, G1, B1, so you want to avoid double counting. That's why you divide by 3! Probability: 1*6/7*4/6=4/7; 4/7*56 = 32

For a particular model of moving truck, rental agency A charges a daily fee of m dollars, plus n cents per mile. For the same model of truck, rental agency B charges a daily fee of p dollars, plus q cents per mile. If a driver plans to rent this model of truck for two days, which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal? a. 100(m-p)/(q-n) b. 200(p-m)/(n-q) c. 50(m-p)/(q-n) d. 2(p-m)/(n-q) e. (m-p)/(2(q-n))

b Cents!! Cents! Also remember it says 2 days...

Order matters vs. order doesn't matter

Order matters: AB and BA are distinct outcomes n!/(n-k)! Order doesn't matter: AB is the same as BA n!/k!(n-k)! If reasons are different, order matters. If the reasons are the same, order matters

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? 82 96 110 120 192

c Good ratios question

Test the following types: ■ Negative, Zero, Positive ■ Odd, Even, Proper Fraction, Improper Fraction ○ Every integer and half integer from -2 and 2 would test all of the above cases

-2, -3/2, -1, -½, 0, ½, 1, 3/2, 2

List decimal 1/6 1/7 1/8 1/9 1/11 1/12 1/15

0.16666 0.143 0.125 0.111 0.0909 0.0825 0.0666

5 ways to prove a quadrilateral is a parallelogram

1. Show that both pairs of opposite sides are parallel. 2. Show that both pairs of opposite sides are congruent. 3. Show that one pair of opposite sides are both congruent and parallel. 4. Show that both pairs of opposite angles are congruent. 5. Show that the diagonals bisect each other. (6. It tells us.)

Prime numbers up to 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

If 1/a^2 + a^2 represents the diameter of circle O and 1/a + a =3, what best approximates the circumference? 28 22 20 16 12

22 Square both sides, so you get 1/a^2 + a^2 + 2 = 9; diameter = 7

Po and Shifu had stamps in ratio 5:3. After Po gave 10 stamps to Shifu, the ratio became 7:5. As a result of the gift, how many more stamps does Po have than Shifu?

40 It would be WRONG to say Po has 30*7 and Shifu has 30*5 stamps. Po had 5*30 or 150 stamps, Shifu had 3*30 or 90 stamps originally Don't forget to apply the transaction of 10 stamps: trap answer would be 60 (just saying 150 - 90) Now, Po has 150-10 or 140, and Shifu has 90+10 or 100 stamp

A researcher has determined that she requires a minimum of n responses to a survey for the results to be valid. If p% of the surveyed individuals fail to respond to the survey, how many individuals, in terms of n and p, must the researcher survey to produce twice the minimum required number of responses? a. (200n)/(100-p) b. (2n)/(100-p) c. 200n/p d. 2n(100+p)/100 e. (2n + 2np)/100

A Easy in hindsight. You have to be able to execute on these and NOT BE TIRED. Create dummy variable x and solve for it.

If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive? (1) The product of the nine integers in list S is equal to the median of list S. (2) The sum of all nine integers in list S is equal to the median of list S. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A Good rule of thumb here. If list is composed of only nine distinct integers, only way for the product of all 9 to be one of the numbers is if one of the numbers is 0. Interesting. That means median is 0, so it's not positive->sufficient

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)? 4pi - 1.6 4pi + 8.4 4pi + 10.4 2pi - 1.6 2pi - 0.8

B Make sure you add back the 10 hours to account for the time car B has been traveling...

If a and b are positive integers, what is the remainder when ab is divided by 40? (1) b is 60% greater than a. (2) Each of a^2*b and a*b^2 is divisible by 40. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A Good theory question. So for S1, we know that b=(8/5)a. For b to be an integer, a has to be a multiple of 5. That means b is at least 40 (remember a cannot be 0 since integers are positive). Therefore we know that ab will be a multiple of 40. For S2, you can use the theory, but basically you can make it such that a^b*b and a*b^2 can be divisible by 40 but not ab or the other way around. Say a=10, b=2. Not divisible by 40, but a^b*b and a*b^2 will be. But say a=20,b=2. ab will be divisible by 40 and so will a^b*b and a*b^2. Not sufficient.

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. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A Make sure you realized the last sentence-- "half the guests were vegetarians," you can setup a graph with a lot from that and the 15 neither student nor vegetarian.

A bowl contains pecans, cashews, and almonds in a ratio of 6 : 10 : 15, respectively. If some of the nuts of one of the three types are removed, which of the following could be the ratio of pecans to cashews to almonds remaining in the bowl? i. 1 : 2 : 3 ii. 2 : 3 : 4 iii. 4 : 7 : 10 I only II only III only I and III only II and III only

A That's just a silly mistake. You saw how this works and applied right. Not sure why you thought III worked? Actually issue is that in III, cashews INCREASED. That's not possible.

Barry walks from one end to the other of a 30-meter long moving walkway at a constant rate in 30 seconds, assisted by the walkway. When he reaches the end, he reverses direction and continues walking with the same speed, but this time it takes him 120 seconds because he is traveling against the direction of the moving walkway. If the walkway were to stop moving, how many seconds would it take Barry to walk from one end of the walkway to the other? a. 48 b. 60 c. 72 d. 75 e. 80

A ez layup

If ab ≠ 0 and a + b ≠ 0, is 1/(a+b) < 1/a + 1/b? (1) |a| + |b| = a + b (2) a > b A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A good job, in principle, S1 tells you a and b have to be positive. If that were the case, you know that a+b would always be have a smaller reciprocal than 1/a or 1/b. S2 leaves the possibility of negatives, which presents an alternative trend.

If x and y are non-zero integers and |x| + |y| = 32, what is xy? (1) -4x - 12y = 0 (2) |x| - |y| = 16 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A great absolute value equation. Key is to see that from S1, you know that x=-3y (you got that). But that means they have opposite signs (xy MUST be negative). That also means that |x|=3|y|. You have two equations that you can solve while KNOWING both their signs... that's huge. REDO!

Reiko drove from point A to point B at a constant speed, and then returned to A along the same route at a different constant speed. Did Reiko travel from A to B at a speed greater than 40 miles per hour? (1) Reiko's average speed for the entire round trip, excluding the time spent at point B, was 80 miles per hour. (2) It took Reiko 20 more minutes to drive from A to B than to make the return trip. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

A know the math behind that... Basically, with S1, know that d/(< d/40) is same as d * >40/d = >40

A certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are A. 0 and 13 B. 0 and 14 C. 1 and 10 D. 1 and 9 E. 2 and 8

A PS04711

In a certain sequence, each term, starting with the 3rd term, is found by multiplying the previous two terms. What is the difference between the 6th and 3rd terms in the sequence? (1) The 1st term is equal to 8 times the 2nd term. (2) The 4th term is equal to 1. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

B Plug in an arbitrary number for S2 and you will see the difference between 6th and 3rd.

Alan's regular hourly wage is 1.5 times Barney's regular hourly wage, but Barney gets paid at twice his regular wage for any hours he works on Saturday. Both men work an integer number of hours on any given day. If Alan and Barney each worked for the same total non-zero number of hours last week, and earned the same total in wages, which of the following must be true? I. Alan worked fewer hours Monday through Friday than did Barney. II. Barney worked at least one hour on Saturday. III. Barney made more money on Saturday than did Alan. I only II only I and II only I and III only II and III only

B Think about this in terms of average hours worked. If they worked the same hours and got paid the same amount, then the average wage was the same!

In the figure to the right, if point C is the center of the circle and DB = 7, what is the length of DE in triangle EDB? (1) x = 60° (2) DE || CA A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

B You had the right idea, but you didn't see it until review. You know you should be looking for similar triangles. S1 doesn't give that to you. S2 does.

If x, y, and z are integers greater than 1, and (3^27)(5^10)(z) = (5^8)(9^14)(x^y), then what is the value of x? (1) y is prime (2) x is prime A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

B You know that (x^y)/z = 5^2/3 from prethinking. That means x^y has to be a multiple of 25 and z has to be a multiple of 3. For S1, test cases. x=10,y=2,z=12 is sufficient, but so is x=5,y=2,z=3. We don't know x. Insufficient For S2, test cases. Only way for x^y a multiple of 5^2 when x is prime is if x=5. Think about it. If 'x' is prime it can only have 1 factor besides 1, so that HAS to be 5 to balance out the 5^2

If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12? (1) x^2 + 2x is a multiple of 3. (2) 3x is a multiple of 2. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

B great divisibility question! 3+ consecutive even integers WILL ALWAYS have a multiple of 3 KNOW THAT RULE therefore, we know that if x=even, then the product will be divisible by 12. S2 tells us that, but S2 does not and can be proved insufficient easily with test cases.

ABC +BCB CDD In the addition shown above, A, B, C, and D represent the nonzero digits of three 3-digit numbers. What is the largest possible value of the product of A and B ? 8 10 12 14 18

B (10) Play around with numbers. Remember no restrictions on A,B,C being same or different. Note it says max of A*B!

If an ≠ 0 and n is a positive integer, is n odd? (1) a^n + a^(n + 1) < 0 (2) a is an integer. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Great gamesmanship on the actual test. You knew that S2 was entirely insufficient by itself, but that probably means it serves a purpose. C is a solid answer. But to be clear, you factor out a^n in S1 to get a^n(1+a)<0 From there you break down the cases. If a^n is positive, then a is positive and 1+a is negative -> a<-1. For a negative a to be positive, then n has to be even. If a^n is negative, then a has to be negative and n has to be odd. Moreover, because 1+a has to be positive, then -1<a<0. So it's NOT an integer. That allows you to bring in S2, if 'a' IS an integer, then a^n must be positive and 'a' is negative. Therefore 'n' is even.

A circle is drawn on a coordinate plane. If a line is drawn through the origin and the center of that circle, is the line's slope less than 1? (1) No point on the circle has a negative x-coordinate. (2) The circle intersects the x-axis at two different positive coordinates. Statement A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C play around with b/a setups compared to the radius of the circle. Basically a rise vs. run comparison here.

If 0 < x < 1, is it possible to write x as a terminating decimal? (1) 24x is an integer. (2) 28x is an integer. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Great question. Easy to find counterexamples for each statement. For S1, if x=1/3, 24x is an integer but doesn't terminate. if x=1/4, 24x is an integer but does terminate. For S2, if x=1/4, 28x is an integer and terminates, but if x=1/7, 28x is an integer but DOESN'T terminate. Now if we combine 24x and 28x, for x to make both integers, x has to have ONLY prime factors that 24 and 28 share. Therefore 'x' can only be 2 or 4, in which case 'x' will be a terminating decimal (any fraction with only 2's and 5's as its denominator will terminate). Easier to think about this as x=a/b where b has to be only 2's and 5's. What a is doesn't really matter as long as the fraction is fully reduced. S1 tells us that b can be 2,3,4,6,8,12,24. S2 tells us that b can be 2,4,7,14,28. Either is insufficient on its own. Combining, we know that b has to be either 2 or 4. Therefore 'b' is only 2's and 4's -> sufficient.

A circular rim 28 inches in diameter rotates the same number of inches per second as a circular rim 35 inches in diameter. If the smaller rim makes x revolutions per second, how many revolutions per minute does the larger rim makes in terms of x ? A. 48π/x B. 75x C. 48x D. 24x E. 75x

C Grind this out. Think about it this way. Get the in/minute for 28in rim and divide by 35pi inch rim to get revs/min Or: 28pi*x = 35pi*n where n=rev/s for the 35in rim. n=4/5x rev/s, therefore n*60 = 48x is rev/min

If x and y are integers and xy does not equal 0, is xy < 0? (1) y = x^4 - x^3 (2) x is to the right of 0 on the number line A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C In S1, if x is negative, y is positive, xy negative. If x is positive, y is positive, xy positive. Insufficient. In S2, all we know is that x is positive but nothing of y. With 1+2, we know that both x and y are positive. Therefore, it is sufficient and no the stimulus.

Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array? A. 1 B. 2 C. 3 D. 4 E. 5

C Notice that when you get to (n-k)(n+k)=48, you can do the factor list and then get n-k = smaller; n+k = bigger and realize that for that equation to work, the sum of the two factors have to be even so that 2n=# divides out.

If x, y, and z are positive numbers, is z between x and y ? (1) x < 2z < y (2) 2x < z < 2y A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Pretty easy but be good at prove right AND wrong for test cases to make sure statement is sufficient.

List P contains m numbers; list Q contains n numbers. If the two lists are combined to produce list R, containing m + n numbers, is the median of list R greater than the median of list P ? (1) The smallest number in list Q is greater than the largest number in list P. (2) m = n A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C Test cases to prove each insufficient. Crucial to know actual values in list AND m and n (elements in each list).

A rectangular solid has length, width, and height of L cm, W cm, and H cm, respectively. If these dimensions are increased by x %, y %, and z %, respectively, what is the percentage increase in the total surface area of the solid? L, W, and H are in the ratios of 5:3:4. x = 5, y = 10, z = 20 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C With S1, you can get L, W, and H in terms of each other, butno idea of the increases. With S2, you can get the increases, but no idea what L, W, and H are in terms of each other, you you cannot understand the exact magnitude of the increase. Together, you can get L, W, and H in terms of each other and know exactly how much each increases to understand the total SA % increase.

Each of the five divisions of a certain company sent representatives to a conference. If the numbers of representatives sent by four of the divisions were 3, 4, 5, and 5, was the range of the numbers of representatives sent by the five divisions greater than 2 ? (1) The median of the numbers of representatives sent by the five divisions was greater than the average (arithmetic mean) of these numbers. (2) The median of the numbers of representatives sent by the five divisions was 4. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C good question. Good idea to setup the median waterfall (x<=3, median =4; x=4, median =4; x>=5, median =5) That allows you to test different cases more easily

Let S be a set of outcomes and let A and B be events with outcomes in S. Let ~B denote the set of all outcomes in S that are not in B and let P(A) denote the probability that event A occurs. What is the value of P(A) ? (1) P(A ∪ B) = 0.7 (2) P(A ∪ ~B) = 0.9 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C good way to test set notation -> U = 'or' upside down U = 'and'

The figure above shows the dimensions of a rectangular board that is to be cut into four identical pieces by making cuts at points A, B, and C, as indicated. If x = 45, what is the length AB ? (1 foot = 12 inches) A. 5 ft 6 in B. 5 ft 3√2 in C. 5 ft 3 in D. 5 ft E. 4 ft 9 in

C realize that the trapezoids are symmetrical! they have to be with the same area and same height... (plus question said they were identical)

If x is a positive number, is x an even integer? (1) 3x is an even integer. (2) 5x is an even integer. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C realize that x could be 2/3 for S1 or x could be 2/5 for S2. combined, you can do the algebra trick or test cases. algebra trick is pretty solid since you've seen it more than once (5x-3x = 2x has to be even), 3x-2x has to be even, thus x has to be even (1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

If x and y are integers, is xy + 1 divisible by 3 ? (1) When x is divided by 3, the remainder is 1. (2) When y is divided by 9, the remainder is 8. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

C test cases work, but to be clear. S1 is insufficient alone because we don't know anything about 'y,' there's definitely a situation where it could work (i.e. x=4,y=2). Same for S2 But for 1+2, we do (3q+1)(9r+8) +1 = 27qr+24q+9r+8+1 = 24qr+24q+9r+9 -> that is definitely divisible by 3!

Bill has a set of 6 black cards and a set of 6 red cards. Each card has a number from 1 through 6, such that each of the numbers 1 through 6 appears on 1 black card and 1 red card. Bill likes to play a game in which he shuffles all 12 cards, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value? 8/33 62/165 17/33 103/165 25/33

C (17/33) You gotta get these ones right! In this one, notice how we don't really care about "double counts" that's why we don't have to do the counting principle then divide by c! Counting: 12!/(4!8!) = 495 total arrangements; find 1-P(all different) # all different = 12*10*8*6/4! = 240 1-240/495 = 255/495 = 17/33 Probability: = 1-P(all different) 1*10/11*8/10*6/9 = 1*1/11*8*2/3 = 16/33 1-16/33=17/33

List R contains five numbers that have an average value of 55. If the median of the numbers in the list is equal to the mean and the largest number is equal to 20 more than two times the smallest number, what is the smallest possible value in the list? 35 30 25 20 15

C (25) Easy to see in hindsight. You misread the problem, thinking what is the 'max' like the OG question, but this does the same except asks for the SMALLEST. Therefore you have to make all other numbers as big as possible. That would mean: x, 55, 55, 20+2x, 20+2x=275 would yield the smallest x. x=25.

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT x = w x > w x/y is an integer w/z is an integer x/z is an integer

C (x/y is an integer) For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms. For example, the sum of 1, 2, 3, and 4 (four consecutives -- an even number) is 10, which is not a multiple of 4. The question tells us that y = 2z, which allows us to deduce that y is even. Since y is even, then the sum of y integers, x, cannot be a multiple of y. Therefore, x/y cannot be an integer; choice C is the correct answer. We can verify this by showing that the other choices could indeed be true:

If 6xy = x2y + 9y, what is the value of xy? (1) x = -2 (2) x < 0 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D Factor it out. y(x-3)=0, which means that either y=0 or x=3 or both. From S1, we know that x!=3, so that means y=0, so xy=0. Sufficient. From S2, we know that x<0, so x!=3, so y=0, so xy=0. Sufficient as well.

p and q are different two-digit prime numbers with the same digits, but in reversed order. What is the value of the larger of p and q? (1) p + q = 110 (2) p - q = 36 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D Helpful to see the x+y method, but honestly pretty similar to test cases.

If 2.00X and 3.00Y are 2 numbers in decimal form with thousandths digits X and Y, is 3(2.00X) > 2(3.00Y) ? 3X < 2Y X < Y - 3 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D Make sure you realize that there are bounds on x and y! They both have to be 0<x,y<9 and have to be integers. Therefore for any value in that range, 3x<2y.

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 < 1,000, what is the maximum possible sum of the length of x and the length of y? 5 6 15 16 18

D Nice job, this one is pretty tricky, but you got the logic right. To maximize "length" you want to use smallest prime factor, 2.

A tank is filled with gasoline to a depth of exactly 2 feet. The tank is a cylinder resting horizontally on its side, with its circular ends oriented vertically. The inside of the tank is exactly 6 feet long. What is the volume of the gasoline in the tank? 1. The inside of the tank is exactly 4 feet in diameter. 2. The top surface of the gasoline forms a rectangle that has an area of 24 square feet. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are not sufficient.

D S1 is easy to see. S2 is fine. It gives you defined dimensions of an arc from which you can make a circle (doesn't necessarily have to be the same as the circle on the cylinder), so we can calculate the volume from that arbitrary arc. This formula makes it definitive: (1/2*chord)(1/2*chord) = (height of liquid)(distance from surface to top of cylinder) Also can use the triangle method (https://www.beatthegmat.com/og-16-data-sufficiency-question-86-t294596.html): x = r-2 x^2 + 2^2 = r^2 (r-2)^2+4=r^2 -4r+4+4=0 r-2=0 r=2 (and by consequence x=4)

Each light bulb at Hotel California is either incandescent or fluorescent. At a certain moment, forty percent of the incandescent bulbs are switched on, and ninety percent of the fluorescent bulbs are switched on. If eighty percent of all the bulbs are switched on at this moment, what percent of the bulbs that are switched on are incandescent? A. 22 (2/9)% B. 16 (2/3)% C. 11 (1/9)% D. 10% E. 5%

D Smart numbers works well here, but so does the algebraic approach. I don't remember how you got it, but you did. Good to redo.

Merle's spare change jar has exactly 16 U.S. coins, each of which is a 1-cent coin, a 5-cent coin, a 10-cent coin, a 25-cent coin, or a 50-cent coin. If the total value of the coins in the jar is 288 U.S. cents, how many 1-cent coins are in the jar? (1) The exact numbers of 10-cent, 25-cent, and 50-cent coins among the 16 coins in the jar are, respectively, 6, 5, and 2. (2) Among the 16 coins in the jar there are twice as many 10-cent coins as 1-cent coins. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D do this one out. S2 is unique, you have to do some inequality type math such that only one number can make the equation work

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. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D good intuition. if you know that the difference between the highest and lowest (H-L) is less than the number of integers in the series, then the series MUST be consecutive integers. both statements get at this principle. solve S1 for H-L

Ines has $15 to purchase fruit. All apples are priced equally and all bananas are priced equally. If 5 apples and 5 bananas cost exactly $15, could she afford 4 apples and 6 bananas with the money she has? (1) The cost of one apple exceeds the cost of one banana by more than $0.20. (2) Ines cannot afford 6 apples and 4 bananas with the money she has. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D good job noticing that if A>B, then she can afford it. If not, then she can't. Pretty straightforward when you frame it that way.

If m and n are positive integers, is n even? m(m + 2) + 1 = mn m(m + n) is odd. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D solid question. basically each statement forces your hand such that n has to be even for the statement work

A pentagon with 5 sides of equal length and 5 interior angles of equal measure is inscribed in a circle. Is the perimeter of the pentagon greater than 26 centimeters? The area of the circle is 16π square centimeters. The length of each diagonal of the pentagon is less than 8 centimeters. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

D DS75271.01 Statement 2 is tough, but if you get the ratios setup right you can roughly estimate that x < 5. Therefore perimeter < 26.

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. 8

D (6) Great question and great answer. Good job on setting up the ratio properly and looking for the multiples that summed up to 30. You know that Ratio for Bag 1 is 2r:6w:9b and for Bag 2 it is 1r:4w. To get 30w, you can test every multiple of 6 until 30 to see which one yields a multiple of 4 for the whites in bag 2. That leaves us W1=6 (W2=24) and W1=18 (W2=12). If W1=6, R1=2, but there's no answer for that. If W1=18, then R1=6-> that's your answer.

If a jury of 12 people is to be selected randomly from a pool of 15 potential jurors, and the jury pool consists of 2/3 men and 1/3 women, what is the probability that the jury will comprise at least 2/3 men? 24/91 45/91 2/3 67/91 84/91

D (67/91) So first you want to find total number of selections possible = 15!/(12!3!) = 455 Then subtract number of 5W, 7M combos = 5!/5! * 10!/(7!3!) = 1*120 = 120. So 1-(120/455) = 335/455 = 67/91 Driven by counting principle. First find total count of arrangements. Then find all counts with 5F/7M. Subtract those ones out leaves you with all counts of at least 2/3F.

Is x - 1/16 greater than 5/8? (1) Four times the value of x is less than three. (2) One third of two is less than the value of x. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

E Do the math on statement 2. Basically combining you get that x<12/16 and x>10.666/16. That's obviously not sufficient.

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? 4 6 8 10 12

E Don't forget your pythagorean triples (6,8). Then you can count it out or see that for every coordinate, you can make 4 squares out of it by changing around the points of the square in negatives or positives. (0,0), (6,8), (8,6) -> 3*4 = 12

The sum of n consecutive positive integers is 45. What is the value of n? (1) n is even (2) n < 9 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

E Great question. The trick for this question is to realize that 45/n will give you the average, which you can then build a list of consecutive integers from. With this in mind, you can see that you can make a list of 2, around 22.5 (22, 23); a list of 3, around 15 (14, 15, 16); a list of 6 around 7.5 (5, 6, 7, 8, 9, 10), etc. So S1 is insufficient, we have 2 even possibilities for 'n' above. S2 is insufficient we have 3 possibilities for n<9 above. and S1+S2 is insufficient because we have 2 possibilities for n<9 that are even (2, 6) above. E

Bowls X and Y each contained exactly 2 jelly beans, each of which was either red or black. One of the jelly beans in Bowl X was exchanged with one of the jelly beans in Bowl Y. After the exchange, were both of the jelly beans in Bowl X black? (1) Before the exchange, Bowl X contained 2 black jelly beans. (2) After the exchange, Bowl Y contained 1 jelly bean of each color. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient.

E test cases, this one takes a bit of time

If x is a positive integer, then is x prime? 3x + 1 is prime. 5x + 1 is prime. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are not sufficient.

E test even numbers, since in the statements, multiplying even by odd + 1 will yield an odd number, which is typically a prerequisite for being prime. So test 2, 4, 6, etc. You realize that both 2 and 6 work, so 1+2 insufficient.

A certain truck traveling at 55 miles per hour gets 4.5 miles per gallon of diesel fuel consumed. Traveling at 60 miles per hour, the truck gets only 3.5 miles per gallon. On a 500-mile trip, if the truck used a total of 120 gallons of diesel fuel and traveled part of the trip at 55 miles per hour and the rest at 60 miles per hour, how many miles did it travel at 55 miles per hour? A. 140 B. 200 C. 250 D. 300 E. 360

E tough paper math here, but multiply both sides by 63 I think is the best way--cannot get the paper math wrong!

If x > y, x² - 2xy + y² - 9 = 0, and x + y = 15, what is x? A. -3 B. 0 C. 3 D. 6 E. 9

E you can just go with the logic... e is the only answer >7.5, or you can see that (x-y)^2 = 9, so x-y=3 or x-y=-3. Given that x>y, we know that only x-y=3 is possible. Sum up x+y=15 and x-y=3 and you get 2x=18, x=9