Tarpon

Caluclation Percent change

% change = ((New Value-Original Value)/(Original value))*100 ex. If a $5 investment grows to $6, by what percent does it increase? ((6-5)/5)*100 20%

Midpoint formula

((x1+x2)/2),(y1+y2)/2)=midpoint

Area of a triangle

(base * height)/2

(x+y)^2 = ?

(x+y)^2 = (x+y)(x+y) = x^2+2xy+y^2 x^2+8x+16=0 -> (x+4)^2=0, and x=-4

(x-y)^2= ?

(x-y)^2=(x-y)(x-y)=x^2-2xy+y^2 x^2-18x+81=0->(x-9)^2=0, and x = 9

Height of an equilater triangle

(√3)/2 times the side of the triangle

(√a/b) = ?

(√a/b)=√a/√b

Greatest possible distance in a rectangular box (Formula)

(√l^2+w^2+h^2)

Language of Mathematics - "Increases by"

+

Language of Mathematics - "Decreases by"

-

1/11

.0909 -> x/11 follows patter of .xyxy

1/10

.1

1/9

.11111111

1/8

.125

1/7

.143

1/6

.166666667

1/5

.2

1/3

.3333333

1/99

.xyxy (1/99=.0101), 15/99 = .1515, 66/99 = .66666

0!

1

Venn Diagrams

1) Label each set on the outside with the totals for each set and the overall total above the two sets 2)Only write on the inside of the circles those numbers that are not double counted (or triple counted in 3 set problems) 3) Do not forget to consider the value for "neither" if you are not told that every component is a member of at least one of the two sets Total = set 1 + set 2 - (overlap of 1 and 2) + neither ( which is 0 here)

Name the places 1,876.534

1,876.534 1-thousands place 8-hundreds place 7-tens place 6-units place . 5-tenths place 3-hundreths place 4-thousandths place

Least Common Multiple (LCM)

1. Determine the prime factors of each number 2. Take each prime factor to the highest power in which it appears 3. Multiple together the results of the previous step to form the least common multiple ex. What is the LCM of 6 and 9? prime factors of 6: 2*3 prime factors of 9: 3*3 Highest power of 2: 2 Highest power of 3: 3*3 LCM of 6 and 9 is 2*3*3=18

Language of Mathematics - "Cent"

100

11^2

121

5^3

125

12^2

144

2^4

16

14^2

196

Perimeter of a rectangle

2(Length)+2(Width)

Is the number divisible by 2?

2-> If the dividend is even, it is divisible by 2. 6->(6/2)=3

15^2

225

Percentages must be taken "of" a value

25 increases by 20% 25+20/100.... The 20% must be a fraction of a value; in this cases, it's a fraction of itself. 25 increases by 20% (of itself) 25+(20/100)*25 25+(1/5)*25 25+(5)=30

4^4

256

3^3

27

Perimeter of a parallelogram

2a+2b

2√2+√8

2√2 + √8 =2√2+ √4*2 = 2√2+√4√2 = √2√2 + 2√2=4√2

√2+√2+√8

2√2+√8

Special Right Triangles based on sides

3-4-5 5-12-13 7-24-25 8-15-17 Note: Largest side is always hypotenuse Two legs 3 and 5 -> hypotenuse is not 4

2^5

32

32 is what percent of 40?

32=(x/100)*40 32 out of 40 is what fraction of 100? (32/40)=(x/100)

20^2

400

Perimeter of a square

4L

√2+√2+√2+√2 = ?

4√2

25^2

625

5^4

625

2^6

64

4^3

64

3^4

81

Tangent of a circle

A line that touches at only one point on a circle. A tangent is perpendicular to the radius at the point of tangency.

Multiple

A multiple of an integer N is any integer that is the result of multiplying N with another integer. Ex. 3, the following are multiples -3, 0, 3, 6, 9, 12, 30, 99, 300, 627, 4503 The following are not multiples: 4, 13, 31, 103, 200 Multiples and factors are related: If N is dvisible by integer d, then N is a multiple of d.

Prime Numbers

A prime number is an integer that is dvisible by exactly two factors: itself and 1 Prime numbers must be positive, and neither 0 nor 1 is a prime number. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 Note: 2 is the smallest prime number, and the only even prime number (all other even numbers are, by definition, divisible by 2)

Properties of a rectangle

A rectangle is a parallelogram that has all four angles equal to 90∘. A summary of the properties of a rectangle is: Both pairs of opposite sides are parallel. Both pairs of opposite sides are of equal length. Both diagonals bisect each other. Diagonals are equal in length. All angles at the corners are right angles.

Properties of a rhombus

A rhombus is a parallelogram that has all four sides of equal length. A summary of the properties of a rhombus is: 1) Both pairs of opposite sides are parallel. 2) All sides are equal in length. 3) Both pairs of opposite angles are equal. 4) Both diagonals bisect each other at 90∘. 5) Diagonals of a rhombus bisect both pairs of opposite angles.

Properties of a square

A square is a rhombus that has all four angles equal to 90∘. A summary of the properties of a square is: Both pairs of opposite sides are parallel. All sides are equal in length. All angles are equal to 90∘. Both pairs of opposite angles are equal. Both diagonals bisect each other at 90∘. Diagonals are equal in length. Diagonals bisect both pairs of opposite angles (ie. all 45∘).

Properties of a parallelogram

A trapezium with both sets of opposite sides parallel is called a parallelogram. A summary of the properties of a parallelogram is: Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. Both pairs of opposite angles are equal. Both diagonals bisect each other (i.e. they cut each other in half).

Area of a circle

A=╥r^2

Adding or subtracting effect on standard deviation

Adding or subtracting a constant from each element in the set has no effect on the standard deviation. The number of terms is very important in determining standard deviation. If two sets have the same number of terms and the same spacing between terms, their standard deviation will be the same. No concrete relationship b/w range and standard deviation Standard deviation can never be negative

Real numbers

All numbers except for imaginary numbers (the square roots of negative numbers). Irrational numbers (such as Pi and the square root of 2) have an infinite number of decimal places but are real numbers.

Factors all positive

All positive integers other than 1 have an even number of factors unless the number is a perfect square. -If perfect square of a prime, we know that the number of factors will be three -Consider 16- a perfect square of 4 (not prime) has an odd number of factors: 1, 16, 2, 8, 4. The number that is being squared is only counted once so you end up with an odd number

Integers

An integer is any "whole" number, i.e. not a fraction or decimal. Positive integers are integers greater then 0. Negative integers are integers less than 0. 0 is also an integer. ex. -47, -12,-2,-1,0,1,2,3,8,12,17

Central Angle

Any angle whose vertex (point of origin) is at the center of a circle.

Inscribed Angle

Any angle whose vertex (point of origin) is on the circumfernce of a circle.

Even Integers

Any integer that, when divided by 2, results in another integer. ex. -4,-2,-0,2,4....244,246,248

Odd Integers

Any integer that, when dvidied by 2, does not result in another integer.

Properties of an even exponent

Any number or variable raised to an even exponent will always be positive. x^2, (-4)^6, y^20

Consecutive Even Integers

Any set of even integers where each integer is equal to the previous integer plus 2

Consecutive Integers

Any set of integers where each integer is equal to the previous integer plus 1. ex. -3,-2,-1,0,1,2,3,4,5,6,7,8

Consecutive Odd Integers

Any set of odd integers where each integer is equal to the previous integer plus 2 ex. -3,-1,1,3,5,7,9

Properties of a Fraction

As the numerator increases, the fraction increases As the denominator increases, the fraction decreases A>B>0 (A/B)>1

Standard Deviation

Assess how closely terms in the set are spread around its mean-> *can never be negative. Only 0 when range of set is 0. To date, no GMAT question has asked to calculate-> should be able to analyze and compare the magnitudes. To find: 1) find the mean 2)Compute the differences between the mean and each number in a set 3) Square these differences and add them together 4) Divide the sum of the squared differences by the number of terms 5) Take the square root of the rsult.

Circular permutation order

Circular arrangements of N items = (N-1)!

(x+y)(x-y)= ?

Difference of squares (x+y)(x-y)= x^2-y^2 x^2-64=0 -> (x+8)(x-8)=0 and x=8 or -8

Distance Formula

Distance between (x1,y1) and (x2,y2)

Language of Mathematics- "Per"

Divided by, represented as /

Language of Mathematics - "Is"

Equality, and can be represented as =

Even +/- Even

Even

Odd +/- Odd

Even 1+1=2 11+11=22

Odd * Even

Even 2*3=6 2*2=4

Even * Even

Even 4*4=16 10*10=100

Mutually exclusive events

Events that can never occur together P(A and B) = 0

Bimonthly

Every other month

Biannual

Every other year

Surface area of a cylinder (formula)

Find area for top and bottom of cylinder πr^2=A -> 2A for both Calculate cylindrical portion of cylinder Length equal to circumference. Width is height.

Pairs probability

First pick does not matter

Median

First put set in ascending order If odd, the middle number is the median If the set has an even number, take average of the two middle terms {-8,-5,3,5,-6,1}->{-8,-6,-5,1,3,5} (-5+1)/2=-2 When you combine sets with the same median, the median of the new set will always be the same as the median of the smaller sets, regardless of how many sets combined or the number of terms in each set.

Divisor

For an integer N, a divisor is a positive integer that can divide N into another integer. Ex. 100, the following are divisors: 100, 50, 25, 20, 10, 5, 4, 2, 1 The following are not divisors: 90, 72, 33.3, 9, 6, 0 Multiples and factors are related: If N is dvisible by integer d, then N is a multiple of d.

Greatest common denominator/factor (GCF/GCD)

GCD of two numbers (negative) can not be greater than the absolute value of the integers individually GCF of 110 and 66? Prime factorization of 110: 2*5*11 Prime factorization of 66: 2*3*11 Pick out exactly what is in common from the sets of prime factors: 2 and 11 2*11=22

Relationship between median and mean

Generally there is no relationship between the median and the mean. However, when sets are evenly spaced (distributed) then the mean will always be equal to the median. {5,6,7,8,9,10) mean and median = 7.5 In evenly spaced sets in ascending order, it is possible to quickly determine the mean and median by averaging the first and last term. set A {2,x,y,z,24) (2+24)/2 = 13 -> median and mean

Determining the equation of a line

If given any of the following, the equation can be found. A) Any two points on the line B) One point on the line and the slope C) One point on the line and the slope or equation of a line perpendicular to that line D) One point on the line and the slope or equation of a line parallel to that line 2 pts - Determine slope = (Y2-Y1)/(X2-X1)=Slope

Complimentary events

If one and only one event must occur. Heads or tails on the flip of a coin P(A or B) = 1

Angle Angle Angle (AAA) similar trianges

If the angle measures in one triangle match the angle measures in the other triangle, then the triangles are similar. Note, you actually only need to match two sets of angles.

Is the number divisible by 10?

If the dividend ends in 0, it is divisible by 10

Is the number divisible by 6?

If the dividend is even and the sum of its digits is a multiple of 3, it is divisible by 6. Note: This simple combines the rules for 2 and 3, as a number divisible by 6 must be divisible by the prime factors of 6, which are 2 and 3

Is the number divisible by 5?

If the last digit of the dividend is a 5 or a 0, it is divisible by 5. ex. 35->ends in 5->write as 7*5 255->ends in 5->write 255 as 51*5

Is the number divisible by 4?

If the last two digits of the dividend are divisible by 4, then it is divisible by 4. ex. 724->24 is divisible by 4->write 724 as 181*4 892->92 is divisible by 4

Dependent Events

If the occurence of one event affects the probability o another (picking balls out of a jar) P(A and B) = P(A) * PA(B) P(A)=probability of event A PA(B) = probability that event B will occur assuming that A has already occurred PA(B) conditional probability

Is the number divisible by 3?

If the sum of the digits of the dividend is a multiple of 3, then it is divisible by 3. Ex. 21-> 2+1 = 3 633->6+3+3=12-> 633/3=211 528->5+2+8=15->528/3=176

Is the number divisible by 9?

If the sum of the digits of the dividend is a multiple of 9, then it is divisible by 9 Ex. 81 ->8+1 = 9 -> write 81 as 9*9 243->2+4+3=90>write 243 as 27*9

Side Side Side (SSS) similar triangles

If there is a constant ratio of corresponding sides for two triangles, then the triangles are similar. That is, if (a/A)=(b/B)=(c/C), then the triangles are similar.

Side Angle Side (SAS) similar triangles

If there is a constant ratio of corresponding sides of 2 sets of sides AND if the angle between those two sides is the same in the first triangle as it is in the second triangle, then the triangles are similar.

Is the number divisible by 11?

If you sum every second digit and then subtract all other digits and the answer is 0 or divisible by 11, it's divisible by 11. Ex. 1364 ((3+4)-(1+6)=0) Yes 3729 ((7+9)-(3+2)=11) Yes 25176 ((5+7)-(2+1+6)=3) NO

Polygons sum of interior angles

In a regular polygon, find each angle by dividing the sum of the interior angles by # of sides. Sum of interior angles for any polygon=(n-2)(180) where n is the number of sides for that polygon Sum interior angles of a pentagon. ex. (5-2)(180)=540

Right triangle identified by measurements of angles - 45-45-90

Isosceles Triangle Whatever legs, hypotenuse is legs times √2

Formula for combinations

K items selected from a pool of N elements is computed from the combinations formula

Area of a square

L^2

Area of a rectangle

Length x width

Slopes of perpendicular lines

Lines that are perpendicular will have slopes that are negative recipricals. If product =-1 then they are perpendicular y=(-1/2x+2) and y=2x+4 Slopes of parallel lines -> same slope

Permutations where k is changing

Look for key language "at least", "at most", "and", "or" K=N and K=N-1 Visit at least 2 cities - visit only London, Paris, Rome or Madrid. N=4 (4 cities to choose from) K=4 (4*3*2*1) = 4! = 24 -> (4!/(4-4)!)=4!/0!=4!=24 K=3 (4*3*2) = 24 -> (4!/(4-3)!)=4!/1!=4!=24 K=2 (4*3)=12 ->(4!/(4-2)!)=(4!/2!)=24/2=12 24+24+12 = 60

Language of Mathematics - "Of"

Multiplied by, represented as *

Whole numbers

Natural numbers ( counting numbers) and zero; 0, 1, 2, 3...

Negative + Negative

Negative

Factor Properties

Negative numbers are never factors and 0 is never a factor. 1 is a factor of all integers

√-16

Not a real number -> undefined

Formula for permutations with repeating elements

Number of arrangements of N elements are repeated is represented by formula. Where N is the number of items being arranged and each factorial in the denominator represents an item that is repeated and the variable represents the number of times the item repeats. A,B,B - > N=3 - two repeats (3!/2!) = 3

Irrational numbers

Numbers that cannot be expressed as terminating or repeating decimals (Pi and the square root of 2)

Odd * Odd

Odd 3*3=9 5*5=25

Odd +/- Even

Odd 5+6=11 5-6=-1

Probability of one event OR another

P(A or B) = P(A) + P(B) - P(A and B)

Profit

Profit = revenue - costs Revenue = price * items sold Costs =fixed costs + variable costs

What do ratios tell us?

Ratios give us proportional relationship between two or more variables but they do not give us the actual value of those variables. Always represent either part/part or part/whole relationships

Ratio

Remember part/part or part/whole. Ratios adding up a whole. Ex. 5:6, 5 parts + 6 parts = 11 parts (the whole). Aren't actual quantities, but they're proportionate to those quantities. If the actual total quantity were 22, divide by total parts (11) to find the multiplier (2). Then multiply each element by the multiplier (2) to find the new ratio.

Right triangle identified by measurements of angles - 30-60-90

Smallest side is opposit smallest angle Equilateral triangle may be divided into two 30-60-90 triangles

A number with only 3 factors

Square of a prime 3^2=9 9,3,1

Integer with only three factors

Square of a prime- Ex. 9 which is the square of prime number 3. Three factors- 1,9 and 3.

Universal Divisibility Strategy

Subtract easy to identify multiples until you reach a recognizable factor or nonfactor. If you reach a factor or get all the way down to 0, the initital number is divisible by that number; if you reach a nonfactor then it is not. ex. Divisible by 7? 952-700=252 252-210=42 42-42=0 -> divisible by 7.

Surface area for a rectangular solid

Surface Area = 2(lh) + 2(lw) + 2(wh) Volume = l x h x w where l = length h = height w = width

Principal

The amount of the loan (or deposit)

Average squared difference

The average squared difference is known as the variance. On the GMAT you may need to understand the term and realize that if you know the variance then you automatically know the standard deviation and vice versa. Avg. squared difference: (9+1+1+9)/4 = 5 Standard deviation √5 =~2.2

Mode

The mode is the most frequently occurring number Set A {2,2,3,4,6}->mode 2 B{2,2,3,3,4,4}->mode 2,3,4 C{1,-1,2,3,5,6}->mode->none will not be tested

Independent events

The occurence of one event does not affect the probability of the occurence of another P(A and B) = P(A) * P(B)

Range

The range of a set is the difference between the largest and the smallest value in a set. To find the range -> subtract the smallest value from the largest. set A {-10,-6,-12,14,1,2,5}-> range 14-(-12)=26 1) Always non-negative (positive or 0) 2) If range is 0 -> all numbers must be identical 3) Changing any values other than smallest & largest has no value

Imaginary Numbers

The square roots of negative numbers

Third side rule of triangles

The third side is always greater than the difference of the two sides and less than the sum of the other two sides. 7 and 10 are lengths of two sides of a triangle What is the length of the third side? 2 or 8 or 17? A<X<B (8)

Equations with Multiple Variables

To solve for multiple variables, multiple equations are required

Properties of an even number (factor)

Total number of factors is even

Simple interest rates

Total value of deposit is b(1+in) b=beginning principal i=interest rate for one period n=number of periods

Compound interest rates

Total value of the deposit is b(1+i)^n b=beginning principal i=interest rate for one period (assuming interest compound once per period) n=number of periods

Property of a perfect square (factors)

Uneven number of factors

Work Rate Problems

W=work completed R=rate at which work was completed T=time that the work was being done (W=RT) 1)Always convert times to rates->inverse of time ex. 6hrs for a person to complete a job, then he is completing (1/6) of the job per hr. (Only true if entire job). If you are told that a person takes 6 hours to complete 2/3 of the job, then his rate is not 1/6 of the job per hour and you need to use the W=RT formula to convert to the time to rate. In this example, we plug into W=RT. 2/3=R(6), and solving for R, we find that the rate would be (1/9) of the job per hour. 2)Rates are additive. Once converted time to rates, you can combine numerous rates to calculate the combined rate of numerous machines or people. R-Combined- R1 + R2 + R3 -> if (1/6)+(1/4)+(1/3)=(9/12)=(3/4) 3)Use W=RT to solve. Once rates have been determined- plug in 4)Reset equations if machines are added or taken away

What happens when a variable is raised to an even exponent?

When a variable is raised to an even exponent, we know the result is positive but we do not know the sign of the variable. If x^2=25 then x=±5 (5-x)^4 = 16 -> x=3 and 7

What happens when a variable is raised to an odd exponent?

When a variable is raised to an odd exponent, the result can be positive or negative, depending on the sign of the base. x^3=-125 -> x=-5 x^3=64 -> x=4

Inclusive sets- Multiples of 6 between 30 and 114, inclusive

Whenever you are dealing with inclusive sets, you must add 1 to the final calculation. In this example, 114-30=84-> 84/6 = 14. However, that calculation does not account for each end of the inclusive set. The answer is 14+1, which can be confirmed by writing the multiples out.

Universal Divisibility Strategy

a(b+c) = ab+ac 7*136 = 7(100+30+6) = 700+210+42

Perimeter of a trapezoid

a+b+c+d

Area of a parallelogram

base * height

Circumference of a circle

c=2πr or πd Circumference is the boundary line that encloses that circle Arc-any portion of circumference Major arc is the larger arc Minor arc is the smaller arc

Surface area of a cube (formula)

cube=6s^2 volume of cube=s^3 No direct relationship between volume and surface area for rectangular solids unless it is a cube.

Equation to determine remainder

i=xm+r i =product m=multiple x=multilier r=remainder

Language of Mathematics - "What"

the unknown, and can be represented as a variable, X

Semiannual

twice a year

Volume of a 3D figure

volume=(area of base)x(height)

What is 20% of 15?

x=(20/100)*15 x=(1/5)*15 x=3

x^2<x

x^2<x x is between 0 and 1

x and y intercepts

y=mx+b b is the intersection with the y-axis or y-intercept x intercept- set y=0 x-intercept=(-b/m)

Slope of a line (properties)

y=mx+b higher the m the steeper the line Slope of line = (change in y coordinate/change in x coordinate)=(y2-y1)/(x2-x1)=slope Positive-up to the right Negative-down to the right

√a*b = ?

√a * √b