GRE Math Foundations and Formulas 2022

Cube

A number raised to the 3rd power. 4^3=64; 64 is the cube of 4

Average

Average = Sum of Terms/Number of Terms

Circumference of Circle

C = 2πr C = πd

Counting Number of Possibilities

Use multiplication to find the number of possibilities when items can be arranged in various ways. EX: How many three-digit numbers can be formed with the digits 1, 3, and 5 each only use once? hundreds place: 3 possibilities tens place: 2 possibilities ones place: 1 possibility 3 x 2 x 1 = 6 possibilities or numbers can be formed

Weighted Average

WA: (Sum of Weighted terms) / Number of Terms TIP: give each term the appropriate "weight" EX: the girls' average score is 30. Boys' is 24. If there are twice as many boys as girls, what is the overall average? WA= ((1 x 30) + (2 x 24)) / 3 = 78/3 = 26 OR (30 + (2)(24)) / 3 OR (30 + 24 + 24) / 3

PEMDAS

Parentheses Exponents Multiplication, Division (simultaneously from left to right) Addition, Subtraction (simultaneously from left to right) "Please Excuse My Dear Aunt Sally" EX: 66(3-2)÷11 = 66(1)÷11 = 66÷11= 6

Probability

Probability = (Number of desired outcomes) / (Number of total possible outcomes) EX: What is the probability of throwing a 5 on a fair six-sided die? # of desired: 1 ; # of total: 6 -> Probability= 1/6

Combined Percent Increase and Decrease with no specified original value

Start with 100 as the starting value EX: A price rises by 10% one year and by 20% the next year. What is the combined percent increase? Year 1: 100 + (10% of 100) = 100 + 10 = 110 Year 2: 110 + (20% of 110) = 110 + 22 = 132 That is a combined 32% increase.

Greatest Common Factor (GCF)

The largest factor a group of integers share

Multiple

The product of a specified number and an integer. EX: 3, 12, and 90 are multiples of 3: 3 = (3)(1); 12 = (3)(6); 90 = (3)(30)

Sum

The result of addition.

Overlapping Sets problem involving Both/Neither

Group 1 + Group 2 + Neither - Both = Total EX: Of the 120 students at a certain language school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither language. How many are studying both French and Spanish? 65 + 51 + 53 - Both = 120 169 - Both + 120 Both = 120

Even

Integers that are evenly divisible by 2

Isosceles, Equilateral, and Similar Triangles

Isosceles: have at lease 2 equal sides and angles Equilateral: all sides are equal and all angles are 60º Similar: corresponding angles are equal, corresponding angles are proportional

Element

One of the members of a set

Pythagorean Theorem

a^2 + b^2 = c^2 Used to find the Hypotenuse (c) of a RIGHT triangle

Associative Laws of Addition and Multiplication

regrouping the numbers does not affect the result (a+b) + c = a + (b+c) ; (ab)c = a(bc)

Predicting weather a sum, diff., or product will be EVEN or ODD

save time: DON'T MEMORIZE THE RULES plug 2 in for the "even" #, plug 3 in for the "odd" # and see what the result is EX: If m in even and n is odd, is the product mn even or odd? m=2, n=3; 2 x 3 = 6 -> EVEN

Rules of Divisibility & How to recognize MULTIPLES of a number

2: (divisible) if its last digit is divisible by 2, (multiple) last digit is even 3: (both) if it's digits add up to a multiple of 3 4: (both) if its last two digits are a multiple of 4 5: (both) if its last digit is 0 or 5 6: (divisible) if it is divisible by both 2 and 3, (multiple) sum of digits is a multiple of 3 and the last digit is an even number 9: (both) if its digits add up to a multiple of 9 10: (multiple) last digit is 10 12: (multiple) sum of digits is a multiple of 3, and last two digits are a multiple of 4

Special RIGHT Triangles

3:4:5 and 5:12:13 -> represent the SIDE lengths 30º-60º-90º and 45º-45º-90º In a 30º-60º-90º the side lengths are multiples of 1, √3, and 2 (hypotenuse), respectively. In a 45º-45º-90º the side lengths are multiples of 1, 1, and √2 (hypotenuse), respectively.

SYMBOLS

= is equal to ≠ is equal to < is less than > is greater than ≤ is less than or equal to ≥ is greater or equal to ÷ divided by π pi (the ratio of the circumference of a circle to the diameter) ± plus or minus √ square root

Area of a Triangle

A = (1/2)bh

Area of Trapezoid

A = (average of parallel sides) x (height)

Area of Rectangle

A = LW

Area of Parallelogram

A = bh

Area of Square

A = s^2

Area of Circle

A = πr²

Decimal

A fraction written in decimal system format. 0.6 is a decimal; numerator divided by denominator 5/8=0.625

Operation

A function or process performed on one or more numbers. The four basic arithmetic operations are addition, subtraction, multiplication, and division.

Divisibility

A number is said to be evenly divisible by another number if the result of the division is an integer with NO remainder. A number that is evenly divisible by a second number is also a multiple of the second number.

Finding an angle formed by a Transversal across Parallel Lines

All acute angles formed are equal All obtuse angles formed are equal Any acute angle plus any obtuse angle equals 180º EX: eº = gº = pº = rº ; fº = hº = qº = sº eº + qº = gº + sº = 180º ......................./ ___________eº/fº__________________ line 1 .............hº/gº ................/ ______pº/qº______________________ line 2 ......sº/rº ......../

The Distributive Law

Allows you to "distribute" a factor over numbers that are added or subtracted; multiply the factor by each number in the group EX: a(b+c) = ab + ac ; (a+b)/c = (a/c) + (b/c) BUT when the sum or difference is in the denominator no distribution is possible! EX: a/(b+c) ≠ (a/b) + (a/c)

Prime Number

An integer greater than 1 that has only two factors: itself and 1 First TEN prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Sequence

An ordered list of terms. The terms of a sequence are often indicated by a letter with a subscript indicating the position of the number in the sequence. EX: a3 is the third number in a sequence, an is the nth term in a sequence

Finding an Angle in Intersecting Lines

Angles across from each other are EQUAL Angles along a LINE add up to 180º EX: aº = cº ; bº = dº ; aº + bº = 180º aº + bº + cº + dº = 360º ...........\ .........../ .............\ bº / ................ \ / ..........aº / \cº ............./ dº\ .........../ ..........\

Average Rate

Average A per B = Total A / Total B -> convert to totals EX: If the first 500 pages have an average of 150 words per page, and the remaining 100 pages have an average of 450 words per page, what is the average number of words per page for the entire 600 pages? Total pages= 500 + 100 = 600 Total words= (500 x 150) + (100 x 450) = 75,000 + 45,000 = 120,000 Average words per page= 120,000/600 = 200

Average Speed

Average Speed= Total Distance / Total Time EX: Rosa drove 120 miles one way at an average speed of 40 miles per hour and returned by the same 120 mile route at an average speed of 60 miles per hour. What was Rosa's average speed for the entire 240 mile road trip? 120 miles at 40mph takes 3 hours. To return at 60mph takes 2 hours. Total time = 5 hours. AS = 240 miles/5 hours = 48mph

Counting Consecutive Numbers

B - A + 1 = Number of Integers EX: How many integers are there from 73 through 419, inclusive? 419 - 73 + 1 = 347

Finding the Common Factor(s) of two numbers

Break both numbers to their prime factors to see what they have in common. Multiply the shared prime factors to find all common factors. EX: what factors greater than 1 do 135 and 225 have in common? 135 = 3 x 3 x 3 x 5; 225 = 3 x 3 x 5 x 5 Common: 3x3x5 ; 3x3= 9, 3x5= 15, 3x3x5= 45 COMMON FACTORS = 3, 5, 9, 15, and 45

Scientific Notation

Examples: 123,000,000,000 is 1.23 x 10^11 0.000000003 is 3 x 10 ^-9

Factorials

Factorial is denoted by ! symbol IF n is a number greater than 1 then n factorial is n! n! = the product of all the integers from 1 to n EX: 2! = 2 x 1 = 2 4! = 4 x 3 x 2 x 1 = 24 0! = 1 6! = 6 x 5! = 6 x 5 x 4! EX: 8! / (6! x 2!) = (8 x 7 x 6!) / (6! x 2 x 1) = 28

Compound Interest

Final balance = Principal x (1 + (interest rate/c))^(time)(c) c= the number of times the interest is compounded annually EX: If $10,000 is invested at 8% annual interest, compounded semiannually, what is the balance after 1 year? = 10,000 x (1 + (0.08/2))^(1)(2) = 10,000 x (0.04)^2 = $10,816

Using Actual Numbers to find Rate

Identify the quantities and keep the units straight. EX: Anders typed 9,450 words in 3.5 hours. what was his rate in words per minute? CONVERT: 3.5 hours = 210 minutes 9,450 words / 210 minutes = 45 words per minute

Simplifying Radicals

If the number inside the radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square. EX: √72 = (√36)√2 = 6√2

Odd

Integers that are not evenly divisible by 2

Simple Interest

Interest = Principle x Rate x Time EX: if $12,000 is invested at 6% simple annual interest, how much interest is earned after 9 months? 12,000 x 0.06 x (9/12) = 540

Combined Ratio

Multiply one or both ratios by whatever you need to in order to get the terms they have in common to match. EX: The ratio of a to b is 7:3. The ratio of b to c is 2:5. What is the ratio of a:c? multiply a:b by 2 -> 2(7:3) = 14:6 multiply b:c by 3 -> 3(2:5) = 6:15 a:b:c = 14:6:15 -> a:c = 14:15

Exponent Rules

Multiply: same base: x^a • x^b = x^a+b (2^2)(2^3) = 2^2+3 = 2^5 diff base: (3^2)(5^2) = (3x3)(5x5) = (3x5)(3x5) = (3x5)^2= 15^2 Divide: x^a / x^b = x^a-b 4^5 / 4^2 = 4^5-2 = 4^3 Fraction = smaller number (1/2)^2 = (1/2)(1/2)=1/4 x^1/2 = √x Raise a power to another power: (x^a)^b = x^ab (3^2)^4 = 3 ^(2x4) = 3^8 Neg. exponent: x^-a = 1/x^a 5^-3 = 1/5^3 = 1/125 Exponent = 0 5^0 = 1 ; 161^0 =1 ; (-6)^0 = 1 Base = 0 0^3 = 0^4 = 0^12 = 0 Neg. # to even power = Positive number Neg. # to odd power = Neg. number Even # to even power = Even Number Odd # to ANY integer greater than or equal 0 = Odd number

Properties of 1 and -1

Multiplying or dividing a number by 1 does NOT change the number. Multiplying or dividing a nonzero number by -1 changes the number's sign.

Number Added/Deleted Formulas Using Original Average & New Average to find WHAT was added or deleted

Number Added = New Sum - Original sum Number Deleted = Original Sum - New Sum EX: The average of five numbers is 2. After one number is deleted the new average is -3. What number was deleted? Original Sum= 5 x 2 = 10 New Sum = 4 x (-3) = -12 Number DELETED = 10 - (-12) = 22

Consecutive numbers

Numbers of a certain type, following one another without interruption. Numbers may be consecutive in ascending or descending order. Consecutive integers: (-2, -1, 0, 1 ,2, 3) Consecutive even numbers: (-4, -2, 0, 2, 4, 6) Consecutive multiples of 3: (-3, 0, 3, 6, 9) COnsecutive prime numbers: (2, 3, 5, 7, 11)

Rules for Odds and Evens

Odd + Odd = Even Even + Even = Even Odd + Even = Odd Odd x Odd = Odd Even x Even = Even Odd x Even = Even

Digit

One of the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. A number can have several digits 542 -> three digits: 5, 4, and 2 321,321,000 -> nine digits, but four distinct digits: 3, 2, 1, and 0

Addition and Subtraction of Radicals

Only like radicals can be added to or subtracted from one another. EX: 2√3 + 4√2 -√2 - 3√3 = (4√2 -√2) + (2√3 - 3√3) = 3√2 + (-√3) = 3√2 - √3

Overlapping Sets problem involving Either/Or

Organize the info in a grid and fill in the blank boxes from the given information using simple math to find the desired answer

Perimeter of Rectangle

P = 2(L + W)

Perimeter of Square

P = 4s

Square

The product of a number multiplied by itself. A squared number has been raised to the 2nd power. EX: 4^2 = (4)(4) = 16; 16 is the square of 4

Denominator

The quantity in the bottom of a fraction

Numerator

The quantity in the top of a fraction.

Product

The result of multiplication.

Difference

The result of subtraction

Least Common Multiple (LCM)

The smallest multiple (other than zero) that two or more integers have in common. How to find it: - Determine the prime factorization of each integer - Write out each prime number the maximum number of times that it appears in any one of the prime factorizations - Multiply those prime numbers together to get the LCM of the original integers EX: LCM of 6 & 8? 6= 2 x 3 8= 2 x 2 x 2 LCM= 2 x 2 x 2 x 3 = 24

Multiplication and Division of Radicals

To multiply or divide one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs. EX: (6√3)2√5 = (6)(2)(√3)(√5) = 12√15 EX: 12√15 ÷ 2√5 = (12/2)(√15/√5) = 6√(15/5) = 6√3

Solving Digits problem

Use logic and trial and error. EX: If A, B, C, and D represent distinct digits in the addition problem below, what is the value of D? AB +BA = CDC C= 1 as AB and BA create a hundreds number. SO, B + A = 1 ; B + A = 11 because a 1 gets carried therefore A & B are any 2 digits that add up to 11 (3&8, 4&7,...). 47 + 74 = 121 ; 83 + 38 = 121 D = 2

Finding the New Average when a number is added or deleted

Use the sum of the terms of the old average to help you find the new average. EX: Micheal's average score after four tests is 80. If he scores 100 on the fifth test, what's his new average? Original Sum = 4 x 80 = 320 New Sum = 320 + 100 = 420 New Average = 420/5 = 64

Remainder

What is "left over" in a division problem EX: 17 ÷ 3 = 5 with a remainder of 2

Standard Deviation

a measure of how spread out a set of numbers is - greater the spread the higher the SD HOW is it calculated: - find the average of the set - find the differences between the mean and each value in the set - square each of the differences - find the average of the squared differences - take the positive square root of the average

Combination Formula

nCk = n! / k!(n - k)! n= number in the larger group k= number you're choosing ORDER DOES NOT MATTER EX: How many different ways are there to choose 3 delegates from 8 possible candidates? 8C3 = 8! / 3!(8-3)! = 8! / 3! x 5! = 8x7x6x5x4x3x2x1 / 3x2x1x5x4x3x2x1 = 8 x 7 = 56 possible combinations

Permutation Formula

nPk = n! / (n-k)! n= number in the larger group k= number you're arranging * used for solving questions about permutations, ex, the number of ways to arrange a elements sequentially* ORDER MATTERS EX: Five runners run in a race. The runners who come in first, second, and third place will win gold, silver, and bronze medals. How many possible outcomes for gold, silver, and bronze medal winners are there? 5P3 = 5! / (5-3)! = 5!/2! = 5 x 4 x 3 x 2 x 1 / 2 x 1 = 5 x 4 x 3 = 60

Distance between Points

Subtract the values that differ. if both values differ make a right triangle and use Pythagorean theorem. EX: (2,3) to (-7, 3) subtract x's as the y's are the same: 2 - (-7) = 9

Finding the SUM using Average

Sum = (Average) x (Number of Terms) EX: 17.5 is the average of 24 numbers, what's the sum? SUM = 17.5 x 24 = 420

SUM of Consecutive Numbers

Sum = (Average) x (Number of terms) EX: What is the sum of the integers from 10 through 50, inclusive? Average: (10 + 50) ÷ 2 = 30; # of integers: 50 - 10 + 1 = 41 SUM= 30 x 41 = 1,230

Solving Remainders problems

TIP: Pick a number given the conditions and see what happens EX: When n is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7? - Find a number that leaves a remainder of 5 when divided by 7. Take any multiple of 7 and add 5 to it. 7 + 5 = 12 -> n=12 -> 2(12) = 24 -> 24 ÷ 7 = 21 r3 ANSWER = 3

Original Whole before percent increase/decrease

TIP: think of a 15% increase over x as 1.15x and set up an equation EX: After decreasing by 5%, the population is now 57,000. What was the original population? 0.95 x (Original Population) = 57,000 Original Population = 57,000/0.95 = 60,000

Average of Consecutive Numbers

The average of the smallest number and the largest number. The average of ALL the integers from 13 to 77 is the same as the average of 13 and 77. EX: (13 + 77) ÷ 2 = (90/2) = 45

Absolute Value

The distance a number is from zero on a number line. ALWAYS POSITIVE |-3| = |+3| = 3 ; -3 and 3 are 3 units from 0 so each have an absolute value of 3. if you are told |x| = 5 then x could be 5 or -5.

Fraction

The division of a part by a whole. Part/Whole = Fraction EX: 3/5

Prime Factorization

The expression of a number as the product of its prime factors (the factors that are prime numbers) There are 2 methods to find this, look in book pg. 189 & 190

Exponent

The number that denotes the power to which another number or variable is raised. 5^3 = (5)(5)(5)

Factors

The positive and negative integers by which a number is evenly divisible AKA: divisors

Range

The positive difference between the highest and lowest values in a distribution EX: Range of 88, 57, 68, 85, 98, 93, 93, 84, and 81 98 - 51 = 41

Finding the 3rd angle of a Triangle (given two angles)

sum of interior angles is equal to 180º EX: 35º+45º+xº = 180º, solve for x

Communicative Laws of Addition and Multiplication

Addition and multiplication are both communicative, which means that switching the order of any two numbers added or multiplied does not affect the result. a+b = b+a ; ab = ba Division and subtraction are NOT communicative

Combined Work

1/r + 1/s = 1/t r & s = number of hours it would take individually t = number of hours it would take together EX: If it takes Joe 4 hours to paint a room and Pete twice as long to paint the same room, how long would it take the two of them, working together, to paint the same room, if each of them works at his respective rate? Joe= 4 hours ; Pete= 2 x 4 = 8 hours 1/4 + 1/8 = 1/t 2/8 + 1/8 = 1/t 3/8 = 1/t t = 1/(3/8) = 8/3 hours OR 2 hours and 40 minutes

Integer

A number without fractional or decimal parts, including positive and negative whole numbers and zero. All integers are multiples of 1 EX: -5, -4, -3, -2, 0, 1, 2, 3, 4, 5

Decimal system

A numbering system base don the powers of 10. ONLY numbering system used on the GRE. 315.246 -> (3)hundreds, (1)tens, (5)ones units. (2)tenths, (4)hundredths, (6)thousandths

Whole

A quantity that is regarded as a complete unit.

Part

A specified number of the equal sections that compose a whole.

Number line

A straight line, extending infinitely in either direction, on which numbers are represented as points. Can contain irrational numbers such as √2 which is between 1 and 2.

Set

A well-defined collection of items, typically numbers, objects, or events. The bracket symbols {} are normally used to define sets of numbers. EX: {2, 4, 6, 8} is a set of numbers

Properties of Zero

Adding or subtracting zero to a number does not change the number. x + 0 = x Subtracting a number from zero changes the number's sign. Subtract 5 from 0: 0 - 5 = -5 The product of zero and any number is zero. 0(z) = 0 Division by zero is undefined. For GRE purposes this translates to " it cannot be done"

Distinct

Different from each other. EX: 12 has three prime factors (2, 2, and 3) but only two distinct prime factors (2 and 3)

Percent Increase/Decrease Formula

Percent Increase = (Amount of increase/Original Whole) x 100% Percent Decrease = (Amount of decrease/ Original Whole) x 100% EX- Increase: Price goes up from $80 to $100, what's the % increase? (20/80) x 100% = .25 x 100% = 25%

Percent Formula

Part = Percent x Whole EX- Finding the Part: 12% of 25? Part = (12/100) x 25 -> Part = 300/100 = 3 EX- Finding Percent: 45 is what % of 9? 45 = (Percent/100) x 9 -> 4,500 = Percent x 9 -> 500 = Percent EX- Finding the Whole: 15 is 3/5% of what #? 15 = (3/5)(1/100) x Whole -> 15 = (3/500) x Whole -> Whole = 15(500/3) = (7,500/3) = 2,500

Probability: problems when probabilities need to be multiplied

Probability= number of desired outcomes/ number of total possible outcomes EX: If 2 students are chosen at random to run an errand from a class with 5 girls and 5 boys, what is the probability that both students chosen will be girls? 1st student girl: 5/10 = 1/2 2nd girl: 4/9 Both girls: 1/2 x 4/9 = 2/9

Using a Ratio to determine an Actual Number

Set up a proportion using the given ratio EX: ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there? (3/4) = (135/g) -> 3 x g = 4 x 135 -> 3g = 540 -> g = 180

Slope of a Line

Slope = Rise/Run = (change in y) / (change in x) EX: slope of a line that contains the points (1,2) & (4,-5) Slope = (-5-2) / (4-1) = -7/3

Classic Polynomials

ab + ac = a(b + c) a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a - b)^2 a^2 - b^2 = (a - b)(a + b)

Dilution or Mixture problem

determine the characteristics of a resulting mixture when different substances are combined OR determine how to combine different substances to produce a desired mixture Straightforward Setup EX: If 5 pounds of raisins that cost $1/pound are mixed with 2 pounds of almonds that cost $2.40/pound, what is the cost per pound of the resulting mixture? ($1)(5) + ($2.40)(2) = $9.80 total cost for 7 pounds of the mixture Cost per Pound= $9.80/7 = $1.40 Balancing Method EX: How many liters of a solution that is 10% alcohol by volume must be added to 2 liters of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume? - make the weaker and stronger substances balance (Percent Difference Between the Weaker solution and the Desired solution) x (Amount of weaker solution) = (percent difference between the stronger solution and the desired solution) x (amount of stronger solution) n = amount in liters of weaker solution n(15 - 10) = 2(50 - 15) -> 5n = 2(35) -> n = 70/5 = 14