important math formulas (upcat)

The Associative Property

(a+b)+c=a+(b+c)

(ab)c=a(bc)

The Distributive Property

a*(b+c)=ab+ac

a*(b-c)=ab-ac

(a+b)/c=(a/b)+(b/c)

(a-b)/c=(a/b)-(b/c)

Fast Fractions

{1/x}+{1/y}={x+y}/{xy}

For example: {1/2}+{1/5}={2+5}/{2*5}=7/10

Laws for Combining

{x^a}*{x^b}=x^(a+b)

{x^a}/{x^b}=x^(a-b)

(x^a)^b=x^(a*b)

1 and 0 as bases

1 raised to any power is 1. 0 raised to any nonzero power is 0

Any nonzero number to the power of 0 is 1, e.g. 7^0=1

Fractions as exponents

x^(1/2) = sqrt{x}

Negative exponents

x^(-y) = 1/{x^y}

Negative bases

With an even exponent: positive result

With an odd exponent: negative result

Estimating roots

To estimate square roots of numbers that aren't perfect squares, just examine the nearby perfect squares. For example, to find sqrt{50}, you know that sqrt{49}=7 and sqrt{64}=8, so find sqrt{50}must be between 7 and 8.

Cube roots

root{3}{n}=a number that, when cubed, equals n.

E.g.:

root{3}{-8}=-2

Simplifying roots

Separate the number into its prime factors, and take out matching pairs. E.g.:

sqrt{20}=sqrt{225}=2{sqrt{5}}

sqrt{72}=sqrt{33222}=3{sqrt{222}}=3*2{sqrt{2}}=6{sqrt{2}}

Adding roots

Roots can be added like variables. E.g.:

2{sqrt{7}}+9{sqrt{7}}=11{sqrt{7}}

1

is not a prime.

2

is the smallest prime and the only even prime.

integer

is any counting number including negative numbers (e.g. -3, -1, 2, 7...but not 2.5)

Ratios

let us compare the proportions of two quantities.

Ratios are given by x:y, x to y, or x/y. If a question says "for every x there is/are a y," you are most likely dealing with a ratio question. Ratios can also be x:y:z.

Ratios can be simplified like fractions. 3:6 is the same as 1:2.

The Meaning of "Percent"

x% = x/100

Calculating Percentages

% = {part / whole} * 100

Percent Change

% change = {change / originalvalue} * 100

3 divisibility

sum of digits divisible by 3

4 divisibility

the last two digits of number are divisible by 4

5 divisibility

the last digit is either a 5 or zero

6 divisibility

even number and sum of digits is divisible by 3

8 divisibility

if the last three digits are divisible by 8

9 divisibility

sum of digits is divisible by 9

FOIL

First, Outer, Inner, Last:

(x+2)(x+7) = {xx}+{7x}+{2x}+{27} = {x^2}+{9x}+{14}

Common patterns to memorize

(a+b)^2={a^2}+{2ab}+{b^2}

(a-b)^2={a^2}-{2ab}+{b^2}

{a^2}-{b^2}=(a+b)(a-b)

Cross-Multiplication

a/b=c/d right ad=bc

Quadratic equations

For ax^2+bx+c, where a is not 0, if you can factor it to (x+y)(x-z), then the solutions are -y and z. For example:

{x^2}-7x=-10

{x^2}-7x+10=0

(x-2)(x-5)=0

x=2 or x=5

right angle

an angle that measures 90 degrees

straight line

180 degrees

If two lines intersect

then their intersection is exactly one point

resulting angles = 360degrees

Circle Area

=pi{r}^2

circle circumference

2pi{r}

circle Arc length

{x/360}2{pi}r

circle Area of sector

={x/360}{pi}r^2

General triangle area

Area={1/2}b*h

Side A - Side B < Side C

Side A + Side B > Side C

Right Triangles

a^2 + b^2 = c^2 where c is the hypotenuse.

45˚-45˚-90˚ triangle

triangle has sides in a ratio of x : x : x√2, with x√2 as the hypotenuse

30˚-60˚-90˚ triangle

triangle has sides in a ratio of x : x√3 : 2x, with the 1x side opposite the 30 degree angle and 2x as the hypotenuse

Certain right triangles

triangles have sides with all integer lengths. These sets of numbers are called Pythagorean triples,

Pythagorean triple

sets of numbers and you should memorize some of them: 3-4-5, 5-12-13, and 8-15-17. A multiple of a Pythagorean triple is also a Pythagorean triple (e.g., 6-8-10).

Square Perimeter

4s, where s = side

square Area

s^2

Rectangles Area

l*w, where l = length and w = width

rectanglr Perimeter

2l+2w

Trapezoid

{{Base1+Base2}/2}*height

Polygons

Total degrees = 180(n-2), where n = # of sides

Average degrees per side or angle measurements of regular polygon = 180(n-2)/n

Slope-Intercept Form

y=mx+b, where m is the slope and b is the y-intercept.

Slope

m=({y_2}-{y_1})/({x_2}-{y_1})

The Distance Formula

sqrt{({x_2}-{x_1})^2+({y_2}-{y_1})^2}

Sequences

1+2+3+ . . . +n={n(n+1)}/2

Probability

Probability of event = {number of ways that fit the requirement}/{number of total ways}

Mean

Mean of n numbers = {sum}/{n}

Median

the middlemost value when numbers are arranged in ascending order; for an even count of numbers, take the average of the middle two

Mode

The number that occurs most frequently in a list or set

Distance, Rate, and Time

d=rt, Distance=Rate*Time

Combination

nCr={n!}/{r!(n-r)!}

When the order does not matter—for example, picking any 3 friends from a group of 5.

Permutation

nPr={n!}/{(n-r)!}

When the order does matter—for example, how many ways you could order 3 letters from the word PARTY?