Practice Exam Reviews

(18²)(5⁴)/(60³) Reference: MP CAT

15/16 Prime factorization!!! 18² = 2² x 3² x 3² = 2² x 3⁴ 5⁴ 60³ = 2³ x 2³ x 3³ x 5³ = 2⁶ x 3³ x 5³ (2² x 3⁴ x 5⁴) / (2⁶ x 3³ x 5³) = (3 x 5) / 2⁴ = 15/16

RE: Polygons Circles & Cylinders Triangles & Diagonals Reference: MP CAT

54√3 - 27π Shaded area = Area of the hexagon - (area of circle O) - (portion of circles A, B, C, D, E, F that is in the hexagon) perimeter = 26 side = 6 regular hexagon is comprised of 6 identical equilateral triangles A = (√3/4)s² so area of each equilateral triangle is: (36*√3)/4 = 9√3 Area of the hexagon = 6 * (9√3) = 54√3 For circles A, B, C, D, E, and F to have centers on the vertices of the hexagon and to be tangent to one another, the circles must be the same size. Their radii must be equal to half of the side of the hexagon, 3. Area of circle = 9π To find the portion of circles that is inside the hex, need to consider the angles of the regular hexagon. sum of the internal angles = (n-2)(180) = (4)(180) = 720 each of the angle = 720/6 = 120 (since it's a regular hex) 120/360 or 1/3 of circle's area (1/3)(9π) x 6 = 18π Shaded area = Area of the hexagon - (area of circle O) - (portion of circles A, B, C, D, E, F that is in the hexagon) = 54√3 - (9π) - (18π) = 54√3 - 27π

If a, b, and c are integers and abc ≠ 0, is a² - b² a multiple of 4? (1) a = (c - 1)² (2) b = c² - 1 Reference: MP CAT

C a² - b² is a the difference of squares; can be rewritten such as: (a - b)(a + b) for a product to be divisible by 4, either one of the values need to be divisible by 4 or both values need to be divisible by 2 (aka both values are even) S1: no info about b; doesn't specify whether a is even or odd (if c = even, then a = odd; if c = odd, a = even) S2: no info about a, doesn't specify whether b is even or odd S1 + S2: (a + b) = 2c² - 2c = 2(c² - c) = even (a - b) = -2c + 2 = 2(-c + 1) = even (a - b)(a + b) are both even

If 5ˣ - 5ˣ⁻³ = (124)(5ʸ) , what is y in terms of x? A. x B. x - 6 C. x - 3 D. 2x + 3 E. 2x + 6 Reference: GMAT OG 4

C. x - 3 Factor 5ˣ⁻³(5³ - 1) = (124)(5ʸ) 5ˣ⁻³(125 - 1) = (124)(5ʸ) y = x-3

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

$0.09 determine the price at both stores The MSRP is $60. Store A adds 20%: $60 + $12 = $72 Store B adds 30% and then discounts by 10%: - add 30%: $60 + $18 = $78 - discount 10% = $78 - $7.8 = $70.2 The sales price is 5% at both stores. The question asks how much more someone will pay in sales tax at Store A. Don't calculate the full sales tax for the item at each store! The sales tax is identical at the two stores for the first $70.20 spent. Instead, calculate only the sales tax on the difference between the two prices. The difference is $1.80. 10% of that number is $0.18. 5% is half of that figure, or $0.09.

A certain list consists of 21 different numbers. If n is in the list and n is 4 times the average(arithmetic mean) of the other 20 numbers in the list, then n is what fraction of the sum of the 21 numbers in the list? (A) 1/20 (B) 1/6 (C) 1/5 (D) 4/21 (E) 5/21 Reference: GMAT OG 4

(B) 1/6 average₂₀ = sum₂₀ / 20 = n/4 sum₂₀ = 5n sum₂₁ = sum₂₀ + n = 5n + n = 6n n is what fraction of the sum₂₁ n/6n = 1/6 Alternative appropriate: assume values. assume 20 numbers are all 1. Their average is 1 and sum is 20. Then n is 4 and sum of 21 numbers is 24. 4/24 = 1/6

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

(D) 90 rewrite the equations into y = mx + b format Line l: y = 5x + 4 Line w: y = -1/5x - 2 lines l and w are perpendicular to each other since line k does not intersect line l, lines k and l must be parallel. Since line w is perpendicular to line l, it must also be perpendicular to line k. Therefore, lines k and w must form a right angle, and its degree measure is equal to 90 degrees.

If the operation @ is defined for all integers a and b by a@b = a + b - ab, which of the following statements must be true for all integers a, b and c? I. a@b = b@a II. a@0 = a III. (a@b)@c = a@(b@c) (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III Reference: GMAT OG

(E) I, II and III I. we are given that a@b = a + b - ab so if a@b = b@a b@a also equals a + b - ab a + b -ab = a + b - ab :) II. a@0 = a; aka b = 0 a + 0 − (a*0) = a + 0 - 0 a = a :) III. again, we are given that a@b = a + b - ab and we know that one variable @ another variable = first variable + second variable - the product of two variables (a@b)@c (a@b) = a + b - ab = variable 1 c is the second variable product of v1 and v2: ac + bc - abc (a + b - ab) + (c) - (ac + bc - abc) a@(b@c) a = v1 (b@c) = b + c - bc = v2 product of v1 and v2: ab + ac - abc (a) + (b + c - bc) - (ab + ac - abc) (a + b - ab) + (c) - (ac + bc - abc) = (a) + (b + c - bc) - (ab + ac - abc) :)

Set A contains three different positive odd integers and two different positive even integers; set B contains two different positive odd integers and three different positive even integers. If one integer from set A and one integer from set B are chosen at random, what is the probability that the product of the chosen integers is even? Reference: MP CAT

19/25 (odd)(odd) = odd 1 - P(odd) total number of outcomes = 5 x 5 = 25 P(odd) = 3 x 2 = 6 1 - 6/25 = 19/25

John's front lawn is 1/3 the size of his back lawn. If John mows 1/2 of his front lawn and 2/3 of his back lawn, what fraction of his lawn is left unmowed? Reference: MP CAT

3/8 number pick! make size of lawn = 6 size of front lawn = 6(1/3) = 2 1/2 of front lawn = 1 2/3 of back lawn = 4 8/8 - 5/8 = 3/8

A new tower has just been built at the Verbico military hospital; the number of beds available for patients at the hospital is now 3 times the number available before the new tower was built. Currently, 1/3 of the hospital's original beds, as well as 1/5 of the beds in the new tower, are occupied. For the purposes of renovating the hospital's original wing, all of the patients in the hospital's original beds must be transferred to beds in the new tower. If patients are neither admitted nor discharged during the transfer, what fraction of the beds in the new tower will be unoccupied once the transfer is complete? A. 11/30 B. 29/60 C. 17/30 D. 19/30 E. 11/15 Reference: MP CAT

19/30 In problems involving Fractions, it is best to pick numbers that are multiples of the denominators. Therefore, in the problem at hand, pick numbers that are multiples of 3 and 5. Pay attention! The new tower is not itself three times the size of the old wing; the problem states that the capacity of the entire hospital is three times its original value, so the new tower has twice as many beds as the old wing. Therefore, let there be 15 beds in the hospital's original wing and 30 in the new tower. In this case, 5 of the original beds and 6 of the beds in the new tower are occupied. If the 5 patients in the original beds are transferred to the new tower, 11 of the beds in the new tower will be occupied; therefore, 19 of the new tower's 30 beds will be unoccupied.

The ratio of the number of boys to the number of girls in a certain school club is 3 to 4. If there are 5 more girls than boys in the club, how many girls are in the club? Reference: GMAT OG 3

20 3/4 = x/x+5 3x + 15 = 4x x = 15 G = 15 + 5 = 20

If 5²¹ * 4¹¹ = 2 x 10ⁿ, what is the value of n? Reference: GMAT OG

21 get a common base!! 5²¹ * 4¹¹ = 2 * (2*5)ⁿ 5²¹ * 4¹¹ = 2 * 2ⁿ * 5ⁿ power of 5 on both sides must equal; don't need to worry about the 2 n = 21

(a + b)² - (a - b)² = ?

4ab

In a sequence 1, 2, 4, 8, 16, 32, ... each term after the first is twice the previous term. What is the sum of the 16th, 17th and 18th terms in the sequence ? A. 2¹⁸ B. 3(2¹⁷) C. 7(2¹⁶) D. 3(2¹⁶) E. 7(2¹⁵) Reference: GMAT OG 3

7(2¹⁵) a1: 2⁰: 1 a2: 2¹: 2 a3: 2²: 4 ... a16 + a17 + a18 = (2¹⁵)(2¹⁶)(2¹⁷) 2¹⁵ (1+2+4) = 7(2¹⁵)

The average (arithmetic mean) height of five of Peter's friends is 70 inches. What will be the new average height if Peter is included in the group? (1) Peter's height is 1 foot greater than the average height of his five friends (1 foot = 12 inches). (2) Peter is 5 inches taller than the tallest of his five friends. Reference: MP CAT

A It is often times easier to attack an average question by focusing on the sum. If the average height of the group of five friends is 70 inches, we can find the sum of their heights. 5(70) = 350 inches To find the new average of the group with Peter, we must know Peter's height, so we can add this to the 350 inches and then divide by six. We can rephrase the question as "What is Peter's height?" S1: Peter is 70 + 12 or 82 inches avg = (350 + 82)/6 S2: obviously not sufficient

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. Reference: MP CAT

A S1: let prod = product of all integers except median given, prod * median = median median * (prod - 1) = 0 median = 0 or prod = 1 but product of all integers except median cannot be 1, as all are distinct => median = 0 => not positive => sufficient conceptually: The only way to multiply a bunch of integers together and arrive at the same starting point is to multiply by 0 or 1 In order to use 1, every number on the list except one number would have to equal 1; for example, 1 × 1 × 1 × 3 = 3. In order to use 0, though, only one number has to equal zero; for example, 1 × 18 × -3 × 0 = 0. S2: Sum of all nine integers = (Median) + (Sum of other eight integers) Median = (Median) + (Sum of other eight integers) 0 = Sum of other eight integers This isn't enough, though, to determine the median; in fact, any number can still be the median of the list, as long as the surrounding numbers sum to zero. For example, the list -10, -9, -8, -7, 5, 7, 8, 9, 10 satisfies the criterion and has a median value of 5; the list -10, -9, -8, -7, -1, 6, 8, 9, 11 also satisfies the criterion and has a median value of -1.

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet? A. 19,200 B. 19,600 C. 20,000 D. 20,400 E. 20,800 Reference: GMAT OG 4

A. 19,200 perimeter: 2x + 2y = 560 x + y = 280 diagonal: x² + y² = 200² we are asked to find xy x² + 2xy + y² =280² x² + y² = 200² 2xy = (280 - 200)(280 + 200) 2xy = (8)(480) xy = 19200

If a car traveled from Townsend to Smallville at an average speed of 40 mph and then returned to Townsend along the same route later that evening, what was the average speed for the entire trip? (1) The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend. (2) The route between Townsend and Smallville is 165 miles long. Reference: MP CAT

A To determine the average speed, we need to know the average speed in each direction. Because the distance in each direction is the same, if we have the average speed in each direction we will be able to find the average speed of the entire trip by taking the total distance and dividing it by the total time. S1: SUFFICIENT: This allows us to figure out the average speed for the return trip. If the return time was 2/3 the outgoing time, the return speed must have been 3/2 that of the outgoing. Whenever the distance is fixed, the ratio of the times will be the inverse of the ratio of the speeds. number pick!!! Let's say the distance between the two towns was 120 miles. going time = 120/40 = 3hrs since going time took 50% longer than the return trip, the returning time is 2 hours avg rate for the return trip = 120/2 = 60mph avg speed for the whole trip = total distance / total time = (2*120)/(3+2) = 240/5 = 48 mph S2: INSUFFICIENT: we only know the distance; no indication of how fast the car traveled on the way back and therefore no way of calculating the average overall speed.

If 2xy + z = 9, what is the value of the positive integer z? (1) xyz - z² = 0 (2) x + y - 3z = -5 Reference: MP CAT

A given: z = -2xy + 9 S1: z (xy - z) = 0 xy - z = 0 z = xy z = -2z + 9 3z = 9 z = 3 S2: equation cannot be manipulated and combined with the original equation to solve for any of the variables

If n is a positive integer, is n/18 an integer? S1: 5n/18 is an integer S2: 3n/18 is an integer Reference: MP CAT

A prime factorization!!! 18 = 2 x 3 x 3 S1: since 5 does not contain 2, 3, 3 n must contain ALL of them; n has 2, 3, 3 as prime factors so n is divisible by 18! S2: using the same logic in order for 3n to be divisible by 18, 3n must contain all the prime factorizations (2, 3, 3) However, 3n contains one of the prime factors (3), n may only contain a 3 and a 2 cannot determine whether n has (2, 3, 3) or just (2, 3)

If r > 0, is rs < 0 ? (1) s < 0 (2) s < r Reference: GMAT OG 4

A r is positive, is rs < 0? S1: s is negative (+)(-) = (-) rs < 0 S2: s < r s is less than r r can be 5, s can be 3 or -2 not sufficient

If there are x men and y women in a choir, and there are z more men than there are women in that choir, what is z? (1) x² - 2xy + y² - 9 = 0 (2) x² + 2xy + y² - 225 = 0 Reference: MP CAT

A z = x - y S1: DIFFERENCE OF SQUARES!! x² - 2xy + y² = 9 (x - y)² = 9 x - y = 3 S2: SUM OF SQUARES!! x² + 2xy + y² = 225 (x + y)² = 225 x + y = 15 or -15 can't figure out what x - y is!

A list of measurements in increasing order is 4, 5, 6, 8, 10 and x. If the median of these measurements is 6/7 times their arithmetic mean, what is the value of x? A. 16 B. 15 C. 14 D. 13 E. 12 Reference: GMAT OG

A. 16 bc the list is in increasing order, we know that the median is (6+8)/2 = 7 mean = (33 + x)/6 median = (6/7)(mean) 7 = (6/7)[(33 + x)/6] NOTE: the 6s cancel out!! 7 = (33 + x)/7 49 = 33 + x x = 16

Tom is on a certain diet that requires him to limit the number of calories he takes in each day. He is allowed to take in 2400 calories each day from three square meals, and 200 calories each day from snacks and dessert combined. On some days, he splurges by taking in three times the recommended number of calories from snacks and dessert. The rest of the days, he follows the calorie guidelines precisely. If his average calorie intake for a 10 day period was 2720, on how many days did he splurge? A. 3 B. 4 C. 5 D. 6 E. 7 Reference: MP CAT

A. 3 WEIGHTED AVG!! LOVE THIS PROBLEM!!! avg of 10 days = 2,720 regular diet = 2,600 calories splurges = 3,000 calories 2,720 is closer to 2,600 than 3,000 eliminates C, D, E immediately! 120 : 400 OR 3: 10 splurge 3 of the 10 days!

Are at least 10 percent of the people in Country X who are 65 years old or older employed? (1) In Country X, 11.3 percent of the population is 65 years old or older. (2) In Country X, of the population 65 years old or older, 20 percent of the men and 10 percent of the women are employed. Reference: GMAT OG

B (1) In Country X, 11.3 percent of the population is 65 years old or older. This particular group composes 11.3% of total population. Clearly insufficient, as no info about employment rates in this group. (2) In Country X, of the population 65 years old or older, 20 percent of the men and 10 percent of the women are employed. 20% of the men in this group and 10% of the women in this group are employed. No matter how many men and women are in this group, more than 10% will be employed. This is because the weighted average of 2 individual averages (10% and 20%) must lie between these individual averages, so percent of employed people in this group is between 10% and 20%. Sufficient.

If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Reference: GMAT OG 4

B 1. K is odd. clearly insufficient, for k=1 answer is No. for k=3, answer is yes. 2. When k is divided by 3, the remainder is 2. Remainder is 2, so K can never be divisible by 3, Hence n will not be divisible by 6. So Sufficient.

If n = 10¹⁰ and nⁿ = 10ᵈ, what is the value of d? A. 10³ B. 10¹⁰ C. 10¹¹ D. 10²⁰ E. 10¹⁰⁰ Reference: MP CAT

C. 10¹¹ nⁿ = 10¹⁰^(10¹⁰) = 10¹⁰*¹⁰^¹⁰ = 10¹⁰^¹¹ 10¹⁰^¹¹ = 10ᵈ d = 10¹¹

Is the integer k divisible by 4 ? (1) 8k is divisible by 16. (2) 9k is divisible by 12. Reference: GMAT OG

B S1: 8k is divisible by 16 8k/16 = k/2 k is a multiple of 2, but it may or may not be a multiple of 4; In this case, k should have a power of 2, hence it has to be divisible by 2, but we are not sure if it will be divisible by 4 Not sufficient. S2: 9k is divisible by 12 9k/12 = 3k/4 in order for 3k/4 to be an integer k must be a multiple of 4. We know that k must be divisible by 4 because the constant 3 is reduced to its lowest factor (prime factorization) Sufficient.

What is the minimal value of function f(x)? (1) f(0) = 16 (2) f(x) = (x - 4)2 Reference: MP CAT

B S1: INSUFFICIENT: This statement tells us the value of function f(x) for just one value of x. We have no information about other possible values of this function. S2: This statement provides the equation of the function, which enables us to compute its smallest value. Note that the value of (x - 4)2 will always be non-negative (i.e., positive or zero), since the square of any number is always greater than or equal to zero. Therefore, the minimal value of this function is equal to zero.

Reference: MP CAT

B S1: Insufficient multiply both sides by xy the value of xy depends on knowing the value of x and y S2: x³ - (2/y)³ = 0 x³ = 8/y³ x³y³ = 8 cube rt both sides xy = 2

2/3 of all the people who attended a certain party arrived by 8:00 PM. How many people arrived by 8:00 PM? (1) 9 people arrived after 9:00 PM. (2) 21 people arrived between 8:00 PM and 9:00 PM, and by 9:00 PM 90% of all the people who attended the party had arrived. Reference: MP CAT

B S1: insufficient S2: 2/3 of the total arrived by 8:00 PM and 90%, or 9/10, of the total arrived by 9:00 PM, we know that 9/10 - 2/3 must have arrived between 8:00 PM and 9:00 PM. 9/10 - 2/3 = 27/30 - 20/30 = 7/30 of the total arrived between 8:00 PM and 9:00 PM 21 = 7/30 30/30 or the total = 90

Is positive integer n - 1 a multiple of 3? (1) n³ - n is a multiple of 3 (2) n³ + 2n² + n is a multiple of 3 Reference: MP CAT

B S1: n(n² - 1) = n(n-1)(n+1) n-1, n, n+1 are 3 consecutive numbers! Any three consecutive positive integers include exactly one multiple of 3.. but we don't know which of the 3 is the multiple of 3 S2: n(n+1)(n+1) 2 consecutive numbers! Any three consecutive positive integers include exactly one multiple of 3.. since n or n+1 is a multiple of 3 n - 1 will not be !

In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR ? (1) The area of region NOPR is 30. (2) The area of the shaded region is 5. Reference: GMAT OG 4

B Since NOPR is a parallelogram, angle OPQ and angle NRM are equal in measure. Furthermore, since triangle MNR is isosceles, the height drawn from vertex N to base MR divides the triangle into two congruent right triangles and each of these right triangles will have the same area as triangle OPQ. Therefore, if we know the area of triangle OPQ, then we can determine the area of triangle MNR. S1: The area of NOPR is 30. No info on the height or base; NS S2: The area of the shaded region is the area of triangle OPQ. Therefore, the area triangle MNR is twice as much, or 10. Statement two alone is sufficient.

During a 40-mile trip, Marla traveled at an average speed of x miles per hour for the first y miles of the trip and and at an average speed of 1.25x miles per hour for the last 40 - y miles of the trip. The time that Marla took to travel the 40 miles was what percent of the time it would have taken her if she had traveled at an average speed of x miles per hour for the entire trip? (1) x = 48. (2) y = 20. Reference: GMAT OG

B original total time = y/x + [(40 - y)/(1.25x)] = (0.25y + 40)/1.25x total time if travel at x mph = 40/x original total time / total time if travel at x mph = [(0.25y + 40)/1.25x] / (40/x) = [(0.25y + 40)/1.25x] * (x/40) = [(0.25y + 40)/1.25] * 1/40 S1: x = 48 not sufficient S2: y = 20 sufficient

A jar contains 8 red marbles and y white marbles. If Joan takes 2 random marbles from the jar, is it more likely that she will have 2 red marbles than that she will have one marble of each color? (1) y ≤ 8 (2) y ≥ 4 Reference: MP CAT

B there are a total of 8 + y marbles in the jar Prob(R and R) = (8/8+y)(7/7+y) = 56/[(8+y)(7+y)] Prob(R and W) = (8/8+y)(y/7+y)(2) = 16y/[(8+y)(7+y)] **note the 2 in the second expression; that is because there are 2 different ways Joan could obtain a R & W marble so the question is really asking is: 56/[(8+y)(7+y)] > 16y/[(8+y)(7+y)] or 56 > 16y? or 3.5 > y? Therefore mathematically, if y is less than 3.5 (i.e., 3 or less), the answer to the question will be "YES." Otherwise, the answer is "NO." (1) INSUFFICIENT: All we know is that y is less than or equal to 8. It could be greater than 3 or less than 3. (2) SUFFICIENT: We know that y is NOT less than 3 (in other words, y is at LEAST 4).

(n - 2)⁻¹(2 + n) If n > 2 and 2/n is substituted for all instances of n in the above expression, then the new expression will be equivalent to which of the following: A. (n + 1)(n - 1)⁻¹ B. -(n + 1)(n - 1)⁻¹ C. -(n - 1)(n + 1)⁻¹ D. (2 + n)⁻¹(n - 2) E. (n - 2)⁻¹(2 + n) Reference: MP CAT

B. -(n + 1)(n - 1)⁻¹ number pick! n = 3 (2/3 - 2)⁻¹(2 + 2/3) = (2 + 2/3) / (2/3 - 2) = (8/3) / (-4/3) = -2 plug 3 into the answer choices and see which one gives -2 B: -(3+1)(3-1)⁻¹ = -(4)(2)⁻¹ = -4/2 = -2

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 Reference: MP CAT

B. 13 a/b = 2.86 = 286/100 = 143/50 b = 50a/143 b = 50a/ 11*13 for b to be an integer, a must have all the factors of 143 (50 doesn't have either 11 or 13) hence a must be divisible by 11 and 13 OR 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 the left side has a 2 and 2 5's (PF of 50) b must have at least one 2 and two 5's right side of the equation has an 11 and 13 therefore a must have at least an 11 and a 13 in its PF so a is divisible by 11, 13, 143

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 Reference: MP CAT

B. 2 PRIME FACTORIZATION!!! PF of 88,000 = (11)( 5³)(2⁶) The 11 in the prime box must come from a red chip, since we are told that 5 < x < 11 and therefore x could not have 11 as a factor. In other words, the factor of 11 definitely did not come from the selection of a purple chip The 2's must come from the purple chips, since the other colored chips have odd values and thus no factor of two. Thus, we now know something new about x: it must be even. We already knew that 5 < x < 11, so now we know that x is 6, 8, or 10. However, x cannot be 6: 6 = 2 × 3, and our prime box has no 3's. x seemingly might be 10, because 10 = 2 × 5, and our prime box does have 2's and 5's. However, our prime box for 88,000 only has three 5's, so a maximum of three chips worth 10 points are possible. But that leaves three of the six factors of 2 unaccounted for, and we know those factors of two must have come from the purple chips. So x must be 8, because 8 = 23 and we have six 2's, or two full sets of three 2's, in the prime box. Since x is 8, the chips selected must have been 1 red (one factor of 11), 3 green (three factors of 5), 2 purple (two factors of 8, equivalent to six factors of 2), and an indeterminate number of blue chips.

The charge for a single room at Hotel P is 25 percent less than the charge for a single room at Hotel R and 10 percent less than the charge for a single room at Hotel G. The charge for a single room at Hotel R is what percent greater than the charge for a single room at Hotel G ? A. 15% B. 20% C. 40% D. 50% E. 150% Reference: GMAT OG 4

B. 20% P = 0.75R P = 0.90G 75R = 90G R/G = 90/75 = 1.20 so 20% greater

If y is not equal to 4, x is not equal to 0, and (y² - 16)/(3x) = (y - 4)/6, then in terms of x, y equals: A: (x + 8) / 2 B: (x - 8) / 2 C: (-3x) / 2 D: (-3x + 8) / 2 E: (3x - 8) / 2 Reference: MP CAT

B: (x - 8) / 2 (y² - 16)/(3x) = (y - 4)/6 rewritten as: [(y-4)(y+4)]/(3x) = (y-4)/6 (y+4)/(3x) = 1/6 6y + 24 = 3x 6y = 3x - 24 y = (x-8)/2

If the length of side AB is 17, is triangle ABC a right triangle? (1) The length of side BC is 144. (2) The length of side AC is 145. Reference: MP CAT

C According to the Pythagorean Theorem, in a right triangle a² + b² = c². (1) INSUFFICIENT: With only two sides of the triangle (2) INSUFFICIENT: With only two sides of the triangle (1) AND (2) SUFFICIENT: With all three side lengths, we can determine if a² + b² = c².

If x is a positive number, is x an even integer? (1) 3x is an even integer. (2) 5x is an even integer. Reference: MP CAT

C Careful: the problem does not specify that x is an integer. The problem states only that x is a positive number, not necessarily a positive integer, so we must consider non-integer cases of x that satisfy the statements in the problem. S1+ S2: 3x = even 5x = even Subtract one from another: 5x - 3x = 2x basic rules of number properties, the difference between two even integers is always another even integer; therefore, 2x = even x = even/2 = integer

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? A. 5 B. 7 C. 11 D. 13 E. 17 Reference: MP CAT

C. 11 Even multiple of 15 is of a form 15*2n=30n, hence it's is multiple of 30. First multiple of 30 in the range 295-615 is obviously 300 and the last 600. so we have: 300 + 330 + 360... + 600 OR 3(10 + 11 + 12... + 20) sum of 11 consecutive integers = [(first time + last term) / 2] * number of terms (in this case 11) = [(10 + 20)/2] * 11 = 15 * 11 PF = 3 * 5 * 11

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

C If x > 0, |x| = x the question is: is x less than 1 If x < 0, |x| = -x the question then becomes is -x less than -1? OR x > -1? putting them together we can rephrase the question as: is -1 < x < 1, where x is not equal to 0? (1) NS: if x > 0, then x > 1 if x <0, then x > - 1 (2) NS: if x > 0, then x > x which is not true when x < 0, x < 0 S2 only tells us that x is negative (1 + 2): Sufficient we know that x is negative (S2) and that x > - 1 (S1) which means that x is between -1 and 0 which means that x is definitely between -1 and 1

In the XY plane, does the line with equation y=3x+2 contain the point (r,s)? (1) (3r+2−s)(4r+9−s)=0 (2) (4r−6−s)(3r+2−s)=0 Reference: GMAT OG

C Line with equation y=3x+2 contains the point (r,s) aka s = 3r + 2 OR 3r + 2 − s = 0 So basically we are asked to determine whether 3r + 2 − s = 0 is true or not (1) (3r+2−s)(4r+9−s)=0 --> either 3r+2−s=0 OR 4r+9−s=0 OR both. Not sufficient. (2) (4r−6−s)(3r+2−s)=0 --> either 3r+2−s=0 OR 4r−6−s=0 OR both. Not sufficient. (1)+(2) Both 4r+9−s=0 and 4r−6−s=0 cannot be true (simultaneously), as 4r−s cannot equal to both -9 and 6, hence 3r+2−s=0 must be true. Sufficient.

If x and y are positive integers, what is the value of xy? (1) The greatest common factor of x and y is 10 (2) The least common multiple of x and y is 180 Reference: GMAT OG

C S1: Clearly insufficient as multiple values are possible for xy: for instance, if x = y = 10, GCF(x,y)=10 and xy=100 BUT if x=10 and y=20, GCF(x,y)=10 and xy=200. S2: also insufficient xy can have multiple values! x = 10, y = 180 LCM(x,y) = 180; xy = 1800 x = 1, y = 180 LCM(x,y) = 180; xy = 180 S1 + S2: most important property of LCM and GCF is: for any positive integers x and y, xy = GCF(x,y)∗LCM(x,y) xy = 10 *180

If m, n, and p are integers, is m + n odd? (1) m = p² + 4p + 4 (2) n = p² + 2m + 1 Reference: MP CAT

C S1: INSUFFICIENT: Given that m = p² + 4p + 4, If p is even: m = (even)² + 4(even) + 4 m = even If p is odd: m = (odd)² + 4(odd) + 4 m = odd m can be either odd or even, we don't know anything about n S2: If p is even: n = (even)² + 2(even or odd) + 1 n = even + even + odd n = odd If p is odd: n = (odd)² + 2(even or odd) + 1 n = odd + even + odd n = even don't know whether n is even or odd. Additionally, we know nothing about m. S1 + S2: If p is even, then m will be even and n will be odd. If p is odd, then m will be odd and n will be even. In either scenario, m + n will be odd.

If AD is 6√3 and ADC is a right angle, what is the area of triangular region ABC? (1) Angle ABD = 60° (2) AC = 12 Reference: MP CAT

C formula for the area of a triangle is 1/2(bh) = 1/2(6)(BD + DC) = 3(BD + DC) S1: can help us get BD, don't know about DC S2: we can get DC, but don't know BD S1 + S2: Sufficient

The figure above shows two entries, indicated by m and n, in an addition table. What is the value of n + m ? (1) d + y = -3 (2) e + z = 12 Reference: GMAT OG 4

C m + n = (d + z) + (e + y) rephrase the question: What is the value of d + z + e + y Statement 1 tells us that d + y = -3 Statement 2 tells us that e + z = 12 So, d + z + e + y = (d + y) + (e + z) = (-3) + (12) = 9

In the xy-plane, at what two points does the graph of y = (x + a)(x + b) intersect the x-axis? (1) a + b = -1 (2) The graph intersects the y-axis at (0, -6) Reference: GMAT OG

C to find the x intercepts, set y = 0 (x + a)(x + b) = 0 x² + xb + ax + ab = 0 OR x² + x(a + b) + ab = 0 S1: gives us what a+b is but we don't know what ab is S2: note ab = y intercept to find y intercept, set x = 0 y = (x+a)(x+b) = (0+a) (0+b) = ab = −6 now we have ab, but S2 alone doesn't give us a + b S1 + S2: sufficient

If x² < x and x is written as a terminating decimal, does x have a nonzero hundredths digit? (1) 10x is not an integer. (2) 100x is an integer. Reference: MP CAT

C x² < x really means 0 < x < 1 "terminating decimal" means a decimal that ends at some point; that is, it does not keep going forever. 0.538 is an example of a terminating decimal. The value for pi is not a terminating decimal; nor is the decimal value of 2/3. Both decimals go on forever (1) INSUFFICIENT: If x is multiplied by 10, then the tenths place of x turns into the units digit (an integer), but the other decimal places are still decimal places. For instance, if x = 0.315, then 10x = 3.15. What used to be the hundredths digit (1) has become the tens digit. but if x = 0.305, we get a different answer. (2) same concept as above (1) AND (2) SUFFICIENT: From statement (1), because 10x is NOT an integer, x must have a nonzero digit somewhere to the right of the tenths place—in the hundredths place or further to the right (because there still needs to be a decimal after x is multiplied by 10). From statement (2), because 100x IS an integer, every digit after the hundredths place has to be zero (because there can be no more decimals after x is multiplied by 100). If there must be a nonzero digit somewhere to the right of the tenths place, but all of the digits to the right of the hundredths place must be zero, then x must have a nonzero digit in the hundredths place.

In a certain mixture consisting of sugar and cinnamon, the ratio of the number of ounces of sugar to the number of ounces of cinnamon is 15 to 1. How many ounces of sugar are in 24 ounces of the mixture? A) 11.25 B) 12 C) 22.5 D) 23 E) 24 Reference: GMAT OG 4

C) 22.5 weighted (perform mapping) Sugar : Cinnamon 15 : 1 (15/26)(24) = 22.5

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) D. None of the above Reference: MP CAT

C. -(a⁻²ᵇ) A. (-a)⁻²ᵇ can be rewritten as 1 / (-a)²ᵇ a negative number to an even power, will be a positive number B. may be positive or negative, since the exponent is 3 C. -1/positive = negative D. may be positive or maybe negative, same as B

In a certain region, the number of children who have been vaccinated against rubella is twice the number who have been vaccinated against mumps. The number who have been vaccinated against both is twice the number who have been vaccinated only against mumps. If 5,000 have been vaccinated against both, how many have been vaccinated only against rubella? A. 2,500 B. 7,500 C. 10,000 D. 15,000 E. 17,500 Reference: GMAT OG 4

C. 10,000 Number vaccinated only against mumps = x Number vaccinated against both = 2x = 5000 (so x = 2500) Then, number vaccinated against mumps (including both) = x + 2x = 3x Number vaccinated against rubella = 2*3x = 6x to get the number of only rubella: subtract both 6x - 2x = 4x = 4*2500 = 10,000

If the speed of x meters per second is equivalent to the speed of y kilometers per hour, what is y in terms of x ? (1 kilometer = 1,000 meters) A. 15x/18 B. 6x/5 C. 18x/5 D. 60x E. 3600000x Reference: GMAT OG

C. 18x/5 X meters per second --> 3,600X meters per hour (as there are 3,600 seconds in one hour) --> 3,600X/1,000 =18X/5 kilometers per hour (as there are 1,000 meters in one kilometer)

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 Reference: MP CAT

C. 19 Matrix problem! two categories 6+ hours and fewer than 6 hours tire and not tired Need to draw it out

A pump started filling an empty pool with water and continued at a constant rate until the pool was full. At noon the pool was 1/3 full, and 1 + 1/4 hours later it was 3/4 full. What was the total number of hours that it took the pump to fill the pool? A. 2 + 1/3 B. 2 + 2/3 C. 3 D. 3 + 1/2 E. 3 2/3 Reference: GMAT OG 3

C. 3 number pick!! let the total pool be 12 liters (easily divisible by 3 and 4) @noon: 1/3 full = 4 liters 1 + 1/4 hours later = 3/4 full = 9 liters so 5/4 hours = 5 liters 1/4 hour = 1 liter 12 liters = 12/4 = 3 hours

If each term in the sum a1+a2+a3+...+an is either 7 or 77 and the sum equals 350, which of the following could be equal to n? A. 38 B. 39 C. 40 D. 41 E. 42 Reference: GMAT OG

C. 40 7x + 77y = 350 7(x + 11y) = 350 x + 11y = 50 now, if x=39 and y=1 then n = x+y = 40

The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; that is d = 0.123456789101112 What is the 100th digit of d to the right of decimal point ? A. 0 B. 1 C. 5 D. 8 E. 9 Reference: GMAT OG 4

C. 5 1 - 9 -------------------- 9 Digits 10 - 19 ----------------- 20 digits 20 - 29 ----------------- 20 digits 30 - 39 ----------------- 20 digits 40 - 49 ----------------- 20 digits 50 - 54 ----------------- 10 digit 5 and 4 are the 98th and 99th digits respectively hence 100th will be 5 Alternate Approach Assume there are 100 blank spaces to fill. First 9 places will be filled by first 9 single digit numbers i.e. 1 to 9. We are now left with 91 places which will be filled by 2 digit numbers starting from 10. TO fill 91 places we would need 91/2 = 45 numbers starting from 10. which will end on the number 54. Next number, for filling out 91th place, will be 5 from (55).

A circle with center O and radius 5 is shown in the xy-plane. Lines that intersect the circle in 2 points include which of the following ? I. y = -x +1 II. y = 2x + 1 III. y = (1/2)x - 6 A. I only B. II only C. I and II only D. I and III only E. I, II and III Reference: GMAT OG 3

C. I and II only draw it out!

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. Reference: MP CAT

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. example: 2, 3, 4, 5 remove 2 (3 + 4 + 5)/ 3 = 4 remove 5 (2 + 3 + 4)/ 3 = 3 (2) SUFFICIENT The positive difference between any two numbers in the list is always less than n. This holds true for any list of consecutive integers: even the difference between the largest and smallest elements must be less than the number of elements (n). For example, {1, 2, 3} --> 3 - 1 = 2 which is less then n or in this case, 3. Since we are told that the list contains distinct integers, then no other set than the set of consecutive integers can satisfy that.

A satellite is composed of 30 modular units, each of which is equipped with a set of sensors, some of which have been upgraded. Each unit contains the same number of non-upgraded sensors. If the number of non-upgraded sensors on one unit is 1/5 the total number of upgraded sensors on the entire satellite, what fraction of the sensors on the satellite have been upgraded? A. 5/6 B. 1/5 C. 1/6 D. 1/7 E. 1/24 Reference: MP CAT

D. 1/7 Let u be the number of upgraded sensors Let n represent the number of sensors that are not upgraded the fraction of upgraded sensors among all the sensors = u / (30n + u) we are given: n = 1/5 of the total number of upgraded (u) n/u = 1/5 u = 5n substitute into the equation 5n / (30n + 5n) = 1/7

If x is an integer and 4ˣ < 100, what is x? (1) 4ˣ⁺¹ - 4ˣ⁻¹ > 100 (2) 4ˣ⁺¹ + 4ˣ > 100 Reference: MP CAT

D 4¹ = 4 4² = 16 4³ = 64 4⁴ = 256 S1: 4ˣ⁺¹ has to be 4⁴ or greater bc the whole 4ˣ⁺¹ - 4ˣ⁻¹ > 100 but combined with what we are given 4ˣ < 100 x has to be 3 to satisfy both condition S2: same logic ^

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

D 8x > 4 + 6x 2x > 4 x > 2 So, the rephrased question is: "If the integer x is greater than 2, what is the value of x?" S1: 6 - 5 x > -13 -5 x > -19 x < 3.8 Since we know from the question that x > 2, we can conclude that 2 < x < 3.8. The only integer between 2 and 3.8 is 3. Therefore, x = 3. S2: 3 - 2x < -x + 4 3 - 4 < x -1 < x -x + 4 < 7.2 - 2x x < 7.2 - 4 x < 3.2 So, we end up with -1 < x < 3.2. Since we know from the information given in the question that x > 2, we can conclude that 2 < x < 3.2. The only integer between 2 and 3.2 is 3. Therefore, x = 3.

If S is a finite set of consecutive even numbers, is the median of S an odd number? (1) The mean of set S is an even number. (2) The range of set S is divisible by 4. Reference: MP CAT

D If S has an odd number of terms, we know that the median must be the middle number, and thus the median must be even (because it is a set of even integers). If S has an even number of terms, we know the average of two consecutive even integers must be odd, and so therefore the median must be odd. The question can be rephrased: "Are there an even number of terms in the set?" (1) SUFFICIENT: when there is an even number of terms, the mean is odd and when there are an odd number of terms, the mean is even. Hence, since (1) states that the mean is even, it follows that the number of terms must be odd. This is sufficient to answer the question (the answer is "no"). (2) SUFFICIENT: This statement tells us that the last number (call it L) in the set minus the first number (call it F) in the set is divisible by four. Since S is a set of consecutive even numbers, the number of elements in the set is given by: (F - L)/2 + 1. Thus if F - L is divisible by 4, (F - L)/2 is divisible by 2 (even) and (F - L)/2 + 1, the number of elements in the set, is odd. This tells us the median (and mean) of the set is even.

Website W receives orders for its products every day. What is the standard deviation of the numbers of orders that website W received daily for the past 5 days? (1) The average (arithmetic mean) number of orders that website W received per day for the past 5 days is equal to the greatest of the numbers of orders that website W received daily for the past 5 days. (2) The range of the numbers of orders that website W received daily for the past 5 days is equal to 0. Reference: GMAT OG 4

D Statement 1: If the average of the numbers is equal to the greatest of all the numbers then all the numbers have to be that greatest number. Hence there is no deviation. eg. 3,4,5 the greatest is 5 but average is 4. So all the numbers have to be 5 for the average to be 5. Sufficient Statement 2: If the range is 0, then the highest number of delivery - lowest number of delivery is 0. So if the highest number in the sequence is equal to the lowest number in the sequence, all the numbers in that sequence has to be same. Sufficient

If mv < pv < 0, is v > 0? (1) m < p (2) m < 0 Reference: GMAT OG

D if v > 0, and we divide: mv < pv < 0 then m < p < 0 however if v < 0, and we divide: m > p > 0 (flip the sign when we divide by a negative number) S1: m < p then v > 0 S2: m < 0 then v > 0

What is the value of x²- y²? (1) x + y = 2x (2) x - y = 0 Reference: GMAT OG 4

D x²- y = (x-y)(x+y) S1: x + y = 2x subtract 2x from both sides -x + y = 0 rearrange: x - y = 0 0 (x+y) = 0 S1: same info

If x is positive, what is the value of √x (1) x√3 = 2 (2) x² = 64 Reference: GMAT OG 4

D x√3 = 2 is the same as x¹/³ You can't have the square root of a negative number. Irrational numbers are way beyond the scope of GMAT. If a question ask for x√x you can assume x is positive or zero.

During the first 11 months of a recent year, a certain charitable organization received an average of $20,600 per month in donations. How much did the organization receive in donations during December of that year if the average amount of donations per month for the entire year was $21,000? A. $21,400 B. $21,600 C. $24,000 D. $25,400 E. $25,800 Reference: GMAT OG 3

D. $25,400 The average to rise by $400, from $20,600 to $21,000, December donation should compensate those $400 for each of the first 11 months 11*400 + 21,000 = $25,400

At a certain university, the ratio of the number of teaching assistants to the number of students in any course must always be greater than 3:80. At this university , what is the maximum number of students possible in a course that has 5 teaching assistants? A. 130 B. 131 C. 132 D. 133 E. 134 Reference: GMAT OG

D. 133 TA/S = 3/80 5/s = 3/80 3s = 5*80 s = 400/3 = 133.33. Now for ratio to be greater than 3/80 reduce the denominator. So just rounded it to lowest integer as number of student can't be in decimal. The new ratio is 5/133, which is less than 3/80 thus, 133 is the maximum number of students possible.

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 ? A. 5 B. 6 C. 15 D. 16 E. 18 Reference: MP CAT

D. 16 based upon our constraint x + 3y < 1,000 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 = 512 999 - 512 = 487 so the possible value 3y can be is 487 y = 162 remainder 1 2⁷ = 128 2⁸ = 256 largest possible length for y is 7 9 + 7 = 16

If water is leaking from a certain tank at a constant rate of 1,200 milliliters per hour, how many seconds does it take for 1 milliliter of water to leak from the tank? A. 1/3 B. 1/2 C. 2 D. 3 E. 20 Reference: GMAT OG 4

D. 3 (1200mL / hr)(1 hr/ 3600s) = 1200mL / 3600s = 1 mL / 3 s 1 mL takes 3 secs to leak

9 basketball players are trying out to be on a newly formed basketball team. Of these players, 5 will be chosen for the team. If 6 of the players are guards and 3 of the players are forwards, how many different teams of 3 guards and 2 forwards can be chosen? A. 23 B. 30 C. 42 D. 60 E. 126 Reference: MP CAT

D. 60 6C3 * 3C2 [(6!)/(3!)(3!)] * [(3!)/(2!)(1!)] = 20*3 = 60

Boomtown urban planners expect the city's population to increase by 10% per year over the next two years. If that projection were to come true, the population two years from now would be exactly double the population of one year ago. Which of the following is closest to the percent population increase in Boomtown over the last year? A. 20% B. 40% C. 50% D. 65% E. 75%

D. 65% number pick: let's say population now: 100 so... Population one year from now: (100*0.1) = 110 Population two years from now: (110 *0.1) = 121 Since the population two years from now, 121, is exactly double the population one year ago then the population one year ago was 121/2 = 60 (100-60)/60 = 40/60 = 2/3, or roughly 67%

A certain liquid leaks out of a container at the rate of k liters for every x hours. If the liquid costs $6 per liter, what is the cost, in dollars, of the amount of the liquid that will leak out in y hours? A. ky/6x B. 6x/ky C. 6k/xy D. 6ky/x E. 6xy/k Reference: GMAT OG 3

D. 6ky/x For these questions, start with the only value that is NOT a ratio and start canceling units. In this case, that's 'y' hours. (y)(k liters/x hours)(6 / liter) = 6ky/x

For every integer k from 1 to 10, inclusive the "k"th term of a certain sequence is given by (−1)⁽ᵏ⁺¹⁾∗(1/2)ᵏ. If T is the sum of the first 10 terms in the sequence, then T is A. Greater than 2 B. Between 1 and 2 C. Between 1/2 and 1 D. Between 1/4 and 1/2 E. Less than 1/4 Reference: GMAT OG

D. Between 1/4 and 1/2 for infinite geometric progression: sum = b / (1-r) where b is the first term, and r is the common ratio when k = 1 (-1)² x (1/2¹) = 1 x 1/2 = 1/2 when k = 2 (-1)³ x (1/2²) = -1 x 1/4 = -1/4 when k = 3 (-1)⁴ x (1/2)³ = -1 x 1/8 = 1/8 common ratio: -1/2 sum = 1/2 / [1 - (-1/2)] = (1/2) / (3/2) = (1/2)(2/3) = 1/3 This means that no matter how many number (terms) we have their sum will never be more then 1/3.

A candy wholesaler needs to quickly sell some candy bars that are nearing their expiration date, so he reduced the price of the candy bars. By what percent did he reduce the price of the candy bars? (1) The price of a candy bar was reduced by 36 cents. (2) If a candy retailer purchases a case of 144 candy bars from the wholesaler, she will save $51.84 as a result of the price reduction. Reference: MP CAT

E The formula for percent change is (Price Change/Original Price) × 100% = [(New Price - Original Price)/Original Price] × 100%. Thus we need to know any two of the following three values: the original price, the new price, or the price change. (1) INSUFFICIENT: Just knowing the price change is insufficient, because we still don't know anything about the original price. For example, if the original price was $1.00, then the percent reduction was 36%: % change = (-$0.36/$1.00) × 100% = (-0.36) × 100% = -36% However, if the original price was $2.00, then the percent reduction was only 18%: % change = (-$0.36/$2.00) × 100% = (-0.18) × 100% = -18% (2) INSUFFICIENT: This statement tells us that the price reduction for a case of 144 candy bars is $51.84. To calculate the cost reduction for each candy bar, divide the savings by 144: $51.84/case) ÷ (144 candy bars/case) = $0.36/candy bar. Thus, statement (2) gives us information identical to statement (1), and is insufficient for the same reason given above. (1) and (2): INSUFFICIENT since they give the same info

If #p# = ap³ + bp - 1 where a and b are constants, and #-5# = 3, what is the value of #5#? A) 5 B) 0 C) -2 D) -3 E) -5 Reference: MP CAT

E) -5 p = -5, #p# = 3 sub in original equation: (-5)³a + (-5)b - 1 = 3 -125a - 5b = 4 asked to solve for #5#. If we plug 5 into our formula, we get: (5)³a + (5)b - 1 = ? 125a + (5)b - 1 = ? first equation we know that -125a - 5b = 4. By multiplying both sides by negative one, we see that 125a + 5b = -4. 125a + 5b - 1 = ? -4 - 1 = -5

An investment of d dollars at k percent simple annual interest yields $600 over a 2 year period. In terms of d, what dollar amount invested at the same rate will yield $2,400 over a 3 year period? A. (2d)/3 B. (3d)/4 C. (4d)/3 D. (3d)/2 E. (8d)/3 Reference: GMAT OG

E. (8d)/3 Simple interest = principal * interest rate * time An investment of d dollars at k percent simple annual interest yields $600 over a 2 year period = d * (k/100) * 2 = 600 k = 30,000/d What (x) dollar amount invested at the same rate will yield $2,400 over a 3 year period x * 30,000/d * 3 = 2400 x = 8d/.3 OR interest is to be 4 times more in 1.5 times longer time period then investment must be 4d/1.5 = 8d/3

If a rectangle with length √a units and width √b units is inscribed in a circle of radius 5 units, what is the value of a + b? A. 10 B. 20 C. 25 D. 50 E. 100 Reference: MP CAT

E. 100 lets say that √a = x, √b = y the question is asking what is a + b? a + b = (√a)² + (√b)² a + b = x² + y² can rephrase question as what is x² + y²? The problem doesn't indicate exactly how to inscribe the length and the width in the circle. Whatever way this is done, though, it will still be the case that the radius of 5 will represent half of the diagonal of the rectangle. The full diagonal of the rectangle, then, must be 10. This diagonal also creates a right triangle (all rectangles have angles of 90°). The values of the sides x and y, then, can be represented using the Pythagorean Theorem: so x² + y² = 100. The question asks for that exact value! The answer is 100.

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 Reference: GMAT OG

E. 13 Write down all perfect squares less than 75: 1, 4, 9, 16, 25, 36, 49, 64. Now, 75 should be the sum of 3 of those 8 numbers. Also to simplify a little bit trial and error, we can notice that as 75 is an odd numbers then either all three numbers must be odd (odd + odd + odd=odd) OR two must be even and one odd (even + even + odd = odd) We can find that 75 equals to 1+25+49 = 1² + 5² + 7² = 75 1 + 5 + 7 = 13

Two ants, Arthur and Amy, have discovered a picnic and are bringing crumbs back to the anthill. Amy makes twice as many trips and carries one and a half times as many crumbs per trip as Arthur. If Arthur carries a total of x crumbs to the anthill, how many crumbs will Amy bring to the anthill, in terms of x? A. x/2 B. x C. 3x/2 D. 2x E. 3x Reference: MP CAT

E. 3x number pick! Arthur carries a total of x crumbs x = 3 Amy makes twice as many trips but carries 1.5 as many crumbs 2(3*1.5) = 9 plug 3 into the answers and see which one gives you 9!

A piece of wire is bent so as to form the boundary of a square with area A. If the wire is then bent into the shape of an equilateral triangle, what will be the area of the triangle thus bounded in terms of A ? A. √3/64(A²) B. √3/4(A) C. 9/16(A) D. 3/4(A) E. 4√3/9(A) Reference: MP CAT

E. 4√3/9(A) Number pic!! area = 36 a perfect square that is both divisible by 3 and 4 (since the problem involves both a square and a triangle) side of the square would be 6 so the total length of the wire is 24 if the wire is bent into the shape of the equilateral triangle, then each side of the triangle = 8 bc 24/3 = 8 area of the equilateral triangle = (√3/4)s² A = (64√3)/4 = 16√3 check the answer choices, subbing A = 36 and see which gives us 16√3

It takes the high-speed train x hours to travel the z miles from Town A to Town B at a constant rate, while it takes the regular train y hours to travel the same distance at a constant rate. If the high-speed train leaves Town A for Town B at the same time that the regular train leaves Town B for Town A, how many more miles will the high-speed train have traveled than the regular train when the two trains pass each other? Reference: MP CAT

[z(y-x)] / (x+y) number pick!! Let's say that the distance between the two towns is z = 30 miles. If the high-speed train takes x = 2 hours to travel that distance, then it travels at a rate of 15 miles per hour. If the regular train takes y = 3 hours to travel that distance, then it travels at a rate of 10 miles per hour. (Note that the regular train should be slower than the high-speed train.) Both trains leave their respective stations at the same time. The two trains have traveled a total of 25 miles in one hour; in other words, together they travel at a rate of 25 miles per hour. They need to travel another 5 miles collectively in order to cross paths. Notice that, in the first hour, the high-speed train covered 15 out of the 25 collective miles, or 3/5 of the miles covered by the two trains. The regular train covered 10 out of the 25 miles, or 2/5 of the total miles covered. The same proportion must hold true for the remaining 5 miles. The high-speed train, then, covers 3 miles and the regular train covers 2 miles. The high-speed train has traveled 15 + 3 = 18 miles. The regular train has traveled 10 + 2 = 12 miles. The high-speed train has traveled 6 miles more than the regular train. Plug back into the answer conceptually tho: rate(hs): distance / time = z/x rate(normal): distance / time = z/y time in which they meet = distance / combined rate aka z / (z/x + z/y) = xy / (x+y) difference in distances = [time * rate(hs)] - [time * rate(normal) = [[xy / (x+y)] * (z/x)] - [[xy / (x+y)] * (z/y)] = [z(y-x)] / (x+y)

If x and y are positive and x²y² = 18−3xy, then x²= ? Reference: MP CAT

x² = 9/y² x²y² = 18−3xy x²y² + 3xy - 18 = 0 (xy + 6)(xy - 3) - 0 xy = -6 or 3 since we are given that x and y are positive, xy = 3 x = 3/y x² = 9/y²