GRE Math

If x+y = -3 and X^2 + y^2 = 12, what is the value of 2xy? Ch 8

-3

If x^2 - 2xy = 84 and x - y = -10, what is the value of absolute value y? Ch 8

4

-1 < a < 0 < absolute a < b < 1 Quantity a: [(a^2 * sqrt b) / sqrt a] Quantity b: (ab^5) / (sqrt b)^4 Chapter 9, #28

A

Ch 7 #12 (5 lb book)

A

b does not equal 0 absolute value a / b > 1 a + b < 0 Quantity A: a Quantity B: 0 Chapter 9, #23

B

Ch 7, #13 (5 lb book)

D

Chapter 9, #20

D

a=2b=4c and a, b, and c are integers Quantity A: a + b Quantity B: a + c Ch 8

D

When would you find the probability that something does not happen?

If "success" has multiple possibilities Especially if wording contains "at least" and "at most" e.g. What is the probability that, on three rolls of a single fair die, at least one of the rolls will be a 6? Probability that it will not be a 6 is 5/6 --> 5/6 * 5/6 * 5/6 = 125/216

A farmer has two rectangular fields. Field N has 6 times the length and 23 times the width of the field M. If field M has area A, then the area of field N is greater than the area of field M by what amount?

Let the length of field M be l, and let the width of field M be w. Then the length of field N is 6l and the width of field N is 2w3. The area of a rectangle is length times width. The area of field M is lw=A. The area of field N is 6l×2w3=4lw=4A. The area of field N is greater than the area of field M by 4A - A = 3A.

Major vs. minor arcs

Minor arcs go from the center of a circle to two points on the circle. Major arcs have three points on the circle. The angle of the minor arc is double the angle of the major arc

How may different positive integer factors does (64)(81)(125) have?

Start by finding the prime factorization. 64 = 2^6 81 = e^4 125 = 5^3 Any positive integer factor of (64)(81)(125) must be in the form of (2^a)(3^b)(5^c). a can be between 0 and 6 inclusive. b can be between 0 and 4 inclusive. c can be between 0 and 3 inclusive. There are 7 possibilities for a, 5 for b, 6 for c. Therefore, the answer is 7*5*6 = 140 different positive integer factors

fair die with sides numbered 1, 2, 3, 4, 5, and 6 is to be rolled until a number greater than 4 first appears. What is the probability that a number greater than 4 will first appear on the third or fourth roll?

The probability that a number greater than 4 will first appear on the third or fourth roll is equal to the probability that a number greater than 4 will first appear on the third roll plus the probability that a number greater than 4 will first appear on the fourth roll. Appear on third roll: 4/6 * 4/6 * 2/6 = 4/27 Appear on fourth roll: 4/6 * 4/6 * 4/6 * 2/6 = 8/81 4/27 + 8/81 = 20/81

Formula for overlapping sets. (e.g. something that would require making a venn diagram)

Total = Option 1 + Option 2 - both + neither https://www.youtube.com/watch?v=WuXSfQnUNRshttps://www.youtube.com/watch?v=WuXSfQnUNRs

If 3 of 7 standby passengers are selected for a flight, how many different combinations of standby passengers can be selected?

Unique items or people on top, repeated labels on bottom 7 unique people on top, 3 seated and 4 no seats on bottom 7!/(3!*4!)

Weighted average formula

[Weight1 * value1 + weight2 * value2] / [value 1 + value 2]

How many ways can you arrange n distinct (no repetition or identical) objects (i.e. chairs)?

n! e.g. How many ways can you arrange 4 chairs? 4*3*2*1 = 24

If x+y=-3 and x^2 + y^2 =12, what is the value of 2xy?

x^2 + 2xy + y^2 (x+y)^2 x^2 + y^2 + 2xy = (x+y)^2 12 + 2xy = (-3)^2 12 + 2xy = 9 2xy = -3

How many committees of 7 people can be made from a total of 10 people?

10C3 10! / [7! * 3!]

How do you calculate the probability that X or Y will occur if the events are mutually exclusive (i.e. two events cannot both occur, "either" problem and are mutually exclusive)?

Add the probabilities

How do you calculate the probability that X or Y will occur if the events are not mutually exclusive?

Add the probabilities of each individual event and subtract out the probability that both events occur. P(X or Y) = P(X) + P(Y) - P(X and Y)

A lab technician needs a 40% solution of a certain chemical for an experiment, but the lab that she works in only has a 20% solution and a 50% solution in stock. In what ratio should she mix the 20% solution with the 50% solution in order to obtain the 40% solution that she needs?

Alternatively, you could set up an equation of weighted averages to solve for the ratio. Let x be the number of portions of the 20% solution that the lab technician uses, and let y be the number of portions of the 50% solution that the lab technician uses. The total amount of the solution, then, would be 20x + 50y. The number of portions would be x + y. And, the average, as given by the question stem is 40. Putting these three pieces together gives the equation: 20x+50yx+y Multiply both sides by x + y to get 20x + 50y = 40(x + y). Distribute the 40 and consolidate: 20x + 50y = 40x + 40y 10y = 20x y = 2x Remember that x represents the portions of the 20% solution and y represents the portions of the 50% solution. Because the question asks for the ratio of the 20% solution to the 50% solution, rearrange the reduced equation in this order: 2x = y. This means that y is twice the value of x, or in other words, the ratio x:y is 1:2. Once again, (B) is correct.

For the positive integers a, b, c, and d, a is half of b, which is one-third of c. The value of d is three times the value of c. Quantity A: (a + b) / c Quantity B: (a + b + c) / d Ch 8

C

Quantity A: xy^2z(x^2z + yz^2 - xy^2) Quantity B: x^3y^2z^2 + xy^3z^3 - x^2y^4z Ch 8

C

The GRE may ask you to choose two or more sets of items from separate pools e.g. The frat must choose 3 senior members and 2 junior members for the annual conference. If they have 12 senior members and 11 junior members, how many different combinations are possible?

Count the arrangements separately. Then multiply the number of possibilities for each step. e.g. 12!/(3!*4!) * 11!/(2!*9!)

x > absolute y > z Quantity A: x+y Quantity B: absolute y + z Chapter 9, #30

D

a=2b=4c and a, b, and c are integers. Quantity A: a+b Quantity B: a+c

D (unknown) Chapter 8, #19, Page 375

A bicycle wheel has spokes that go from a center point in the hub to equally spaced points on the rim of the wheel. If there are fewer than six spokes, what is the smallest possible angle between any two spokes? A. 18 B. 30 C. 40 D. 60 E. 72 Chapter 9, #10

E

Fundamental Counting Principle

If you have to make a number of SEPARATE decisions, MULTIPLY the number of ways to make each INDIVIDUAL decision to find the number of ways to make ALL decisions e.g. You want to make a sandwich. You have two types of bread and three types of meat. How many different sandwiches are possibly? 2 breads * 3 meats = 6 sandwiches

How do you calculate the probability that something will occur if the events X and Y are independent of each other?

Multiply the probabilities Majority of GRE probability questions are "and" problems... multiply when in doubt

Compound interest formula

P(1 + r/100)^t P is what you start out with The problem will explicitly say that the interest is compounded. E.g. "8% interest compounded annually"

A jar contains 3 red and 2 white marbles. 2 marbles are picked without replacement. What is the probability of picking exactly one red and one white marble?

Red and then white: 3/5 x 2/4 = 3/10 White and then red: 2/5 x 3/4 = 3/10 Not mutually exclusive: 3/10 + 3/10 = 3/5

The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on the disks selected is 7?

The range of a group of numbers is the greatest number minus the smallest number. If the range is to be 7, then we want 2 of the numbers to differ by 7 and the other 2 numbers to both be between the 2 numbers that differ by 7. Thus, if we select 1 and 8, the other 2 numbers can be 2 different numbers among 2, 3, 4, 5, 6, and 7. If we select 2 and 9, the other 2 numbers can be 2 different numbers among 3, 4, 5, 6, 7, and 8. If we select 3 and 10, the other 2 numbers can be 2 different numbers among 4, 5, 6, 7, 8, and 9. nCk is used to denote the number of different subgroups of k different objects that can be selected from a group of n different objects, where n is a positive integer, k is a nonnegative integer, and 0 ≤ k ≤ n. Let's consider the case of selecting 1, 8, and 2 different integers from the 6 integers 2, 3, 4, 5, 6, and 7. The number of ways to select 2 different integers from 6 different integers is 6C2. So the number of ways to select 1, 8, and 2 different integers from the 6 integers 2, 3, 4, 5, 6, and 7, is 15. Similarly, the number of ways to select 2, 9, and 2 different integers from the 6 integers 3, 4, 5, 6, 7, and 8, is 15. Similarly again, the number of ways to select 3, 10, and 2 different integers from the 6 integers 4, 5, 6, 7, 8, and 9, is 15. So the number of ways that the range of the numbers on the 4 disks will be 7 is 3(15) = 45. The number of ways to select 4 different integers from the first 10 positive integers is 10C4. Thus, the number of ways to have a range of 7, which is the number of desired outcomes, is 45 and the number of ways to select 4 different integers from the 10 different integers which are the first 10 positive integers, which is the number of possible outcomes, is 210. The probability is 45/210 = 3/14

If 7 people board an airport shuttle with only 3 available seats, how many different seating arrangements are possible?

Unique items or people on top, repeated labels on bottom 7 unique people on top, 4 no seats on bottom 7!/4!

At noon of a certain day, when 5 pens and 3 pencils were placed in a drawer, the ratio of the number of pens to the number of pencils in that drawer became 47 to 17. Quantity A: The ratio of the number of pens to the number of pencils in the drawer immediately before noon of that day Quantity B: 3/1

We know that after the additions to the drawer were made, the ratio of the number of pens to the number of pencils in the drawer became 47 to 17. However, we CANNOT assume that the number of pens after the additions is 47 and that the number of pencils after the additions is 17. It's possible that the number of pens after the additions is 47 and that the number of pencils after the additions is 17. However, it is also possible that the number of pens after the additions is 2 × 47 = 94 and that the number of pencils after the additions is 2 × 17 = 34. The ratio of 94 to 34 is equal to the ratio of 47 to 17. If the number of pens after the additions was 47 and the number of pencils after the additions was 17, then immediately before the additions that day, the number of pens in the drawer was 47 - 5, or 42, and the number of pencils in the drawer was 17 - 3, or 14. In this case, Quantity A is 4214, which can be reduced to 31. In this case, the quantities are equal. However, if the number of pens after the additions was 94 and the number of pencils after the additions was 34, then immediately before noon of that day, the number of pens in the drawer was 94 - 5, or 89, and the number of pencils in the drawer immediately before noon of that day was 34 - 3, or 31. In this case, Quantity A is 8931, which is less than 31, or 3. In this case, Quantity B is greater. Since different relationships between the quantities are possible, the relationship between the quantities cannot be determined and choice (D) is correct.