How do you multiply fractions?

what's a trick for large numbers?

just multiply the numerators and denominators.

you can cancel common factors numerators and denominators 4/5 *15/12 = 1/1 *3/3

must have common denominators. If you have mixed you can use the carry trick 4 5/8 - 2 7/8 = borrow 8 from the 4 which makes it 3 13/8

ALWAYS REDUCE!!!

3/5 *100 

3*20/5

60%

.66666

66 2/3 %

.33333

33 1/3 %

What if the percent is 60 and the part is 12?

12 is 60% of what number?

70-50 =20

20/50 = 2/5 *100%

2*20%/1

40%

The percent increase is the difference from 100% so 350%-100% = 250% increase

percent increase is just the difference in percent from the whole(which is always equal to 100%)

7*4=28

28-(3+5+8) = 12

The probability of two independent events both occurring is the product of the individual probabilities.

There are six possible outcomes for each roll of a die, 6*6=36 possible outcomes. Three outcomes that yield 4. 

1/12

How do you multiply powers with the same base?

x³ * x⁴

keep the base the same and add the exponents

x³ * x⁴ = x⁷

how do you divide powers with the same base?

y¹³/y⁸

keep the base the same and subtract the exponents

y¹³ / y⁸ = y⁵

keep the base the same and multiply the exponents

(x³)⁴ = x¹²

How do you multiply powers with the same exponent?

3^x * 4^x

multiply the bases and keep the exponent

3^x * 4^x = 12^x

divide the bases and keep the exponent

6^x/2^x = 3^x

use long division. Don't mix up (-) minus (-)

Always try to remove the first term

How do you simplify a factor similar to all terms?

How do you factor difference of squares?

How do you factor square of binomial?

ax + ay = a(x+y)

a² - b² = (a-b)(a+b)

a² + 2ab + b² = (a+b)²

simplify 3x² -11x +6/ 9-x²  x does not = -+3

If the answer choices are algebraic expressions you can use the trick of plugging in but you have to check all answers

you can solve this 2 ways 

1. factor out terms and simplify

2. plug 0 to the equation, then plug it into the answers. See what matches. If more than one match plug in 1 and see

MAKE SURE TO CHECK ALL ANSWER CHOICES IF DONE THIS WAY

What if you have an unknown in the denominator when trying to solve an equation?

1 + 1/x = 2 - 1/x

or

5/x+3 = 1/x + 1/2x

you can manipulate the equation, just do the same thing to both sides.

Times both side by x in the first one = 2

cross multiply on the second one after to come terms on the right = 9/7

What if you have an unknown in the exponent when trying to solve for x?

8^x = 16^x-1 then x = ?

Look for common powers, for example they are both powers of 2. (8 = 2³) (16= 2⁴)

Now that both sides have the same base, you can set the exponents equal to each other and solve for x

x = 4

How do you solve quadratic equations?

x² + 12 = 7x

x + 4x + 2 = 0 

put in form ax² + bx + c = 0

then factor

x = 3 or 4

use quadratic formula

x = (-b +- √(b²-4ac))/2a

x = -2 +- √2

What if there are 2 equations with 2 variables?

2x - 9y = 11 

x + 12y = -8

what is the value of x + y?

Quickly see if adding them together as is will simplify quickly...if not, adjust them and add them together to eliminate.

x + y = 1

what if you have an absolute value equation?

absolute value sign = []

[x-12] = 3

think of two equations

x-12=3

x-12=-3

x = 15 or 9

How do you solve 

absolute value = []

[2x-3] < 7

-7 < 2x-3 < 7

-2 < x < 5

a = 1/2 b * h

The height is the perpendicular distance between the side that's chosen as the base and the opposite vertex

-two equal sides

-two equal angles

-the two equal angles are call base angles

-three equal sides

-three equal angles

-all angles measure 60^o

-area of equilateral triangle = (s²*√3)/4

-or find area by dividing into 2 30,60,90 triangles and find the area of each right triangle

-has a right angle 90^o

-two sides that form the right angle are called legs

-you can use legs as base and height to find the area of right triangles

-area of right triangle = 1/2 (leg₁)(leg₂)

ONLY FOR RIGHT TRIANGLES

(leg₁)² + (leg₂)² = (hypotenuse)²

a set of integers that fits the theorem

- 3:4:5 (the simplest) and multiples

-5:12:13 and mutliples

and a lot more

if the leg:leg ratio is 3:4 or leg:hypotenuse is 3:5 of 4:5 just figure out which multiple of 3:4:5 is missing 

the same applies for 5:12:13 triangles

-the sides are in the ratio of 1:1:√2

so if a length is 3 then the other length is 3 and the h is 3*√2

1. four-sided figure with one pair of parallel sides and one pair of nonparallel sides.

2. area of trapezoid = (base₁ + base₂/2) * height

3. bases are the 2 parallel sides, height is the perpendicular altitude

1. four-sided figure with two pairs of parallel sides, opposite sides are equal and opposite angles are equal. Consecutive angles add up to 180^o

2. Area of parallelogram = base * height

HAVE TO HAVE A HEIGHT THAT IS PERPENDICULAR DISTANCE FROM THE BASE TO THE OPPOSITE SIDE

1. four-side figure with four right angles. Opposite sides are equal. Diagonals are equal.

2. perimeter = sum of the four side lengths

3. area = length * width

1. four-sided figure with four equal sides

2. the maximum area for a rhombus is a square, the more it "leans over" the smaller the area

3. area = base * height (height has to be perpendicular to the base and opposite side)

1. four-sided figure with four right angles and four equal sides

2. perimeter = 4x

3. area = x²

it is also a rhombus, rectangle, and parallelogram

total distance around the circle

circumference = 2 π r

or diameter * pi

n = degree measure of the arc's central angle

length of Arc = (n/360)*(2 pi r)

how do you find the area of a sector?

radius of 6 and n = 30^o

area of sector = (n/360) (pi * r²)

3 pi

lateral area of a cone = 1/2*c*l

or area = pi * r* l

does not include the circular base

how do you find the volume of a cone?

r = 3

h = 6

volume of cone = 1/3 * pi * r² * h

18 pi

how do you find the surface area for a sphere?

r = 2

surface area of sphere = 4 * pi * r²

16 pi

volume of a sphere?

if r = 2

volume of a sphere = 4/3*pi*r³

32pi/3

volume of a pyramid?

h = 3

B = 16

volume of a pyramid = 1/3*B(area of base)*h

16

6 rectangular surfaces, 12 edges, and 8 vertices

rectangular solid surface area = 2lw + 2wh + 2lh

what if they ask to find the distance between opposite vertices of a rectangular solid?

B=7

H=4

W=4

use two pythagoreans, find the base diagonal then the other diaganol

or use the distance formula

distance = √(l² + w² + h²)

answer = 9

a uniform solid is what you get when you take a plane and move it in space, without tilting it. The top and bottom faces are parallel and congruent. They are the bases

volume of uniform solid = B(are of base)*h

volume of a rectangular solid?

l = 6

w = 4

h = 5

volume of a rectangular solid = lwh

v = 120

volume of cube?

e = 2

e = edge

volume of cube = e³

v = 8

volume of cylinder?

r = 2

h = 5

volume of cylinder = pi*r²*h

20pi

how do you find the midpoint?

(-4, 4)

(4, -1)

x+x/2, y+y/2

(0,1.5)

what is the distance formula?

(0, 1.5)

(-1, .5)

distance = √((x₁-x₂)² + (y₁-y₂)²)

answer = √2

y = mx + b

m = slope

b = y-intercept

the "steepness" of the line

positive is up to the right

negative is up to the left

- parallel lines have the same slope

-perpendicular lines have the negative-reciprocal

-lines parallel to x-axis have 0 slope and parallel to y-axis have no slope

find out where the graphs intercept the x and y axis by putting 0 in x and y

then pick a point to see where the shaded areas would be

reciprocals 

cotangent = a/o

secant = h/a

cosecant = h/o

sin²x + cos²x = 1

tan²x +1 = sec²x

1 + cot²x = csc²x

2pi = 360^o

1pi = 180^o

put them in a x,y plane in a circle, draw your, 30,60,90 right triangle and use the functions

sinO = b/r

cosO= a/r

tanO= b/a

cotO= a/b

secO= r/a

cscO= r/b

P(a,b) r=radius and hypotenuse

2 - has to be an even number

3 - add the digits of your number, if the sum is divisible by 3, then it is divisible by 3

4 - are the last 2 digits divisible by four? (because 100 is divisible by 4)

5 - is the last digit a 5 or 0

6 - it has to be divisible by 2 and 3 (because that's what 6's multiples are)

9 - have to be divisible by 3 twice, or if you add the digits is the sum dvisible by 9

10 - if it ends in a 0

when you can write a ratio of 2 integers

5/3 is rational

.6666 is rational

but pi or square root of a non-perfect square is irrational

if it continues but doesn't repeat

what is anything to the .5 power?

(.16)^.5

it is the square root of that number

√.16 = .4

If you have a number squared (2.004)²

how can you solve?

you can you foil

(2+.004)(2+.004)

4.016016

notice it is the difference of 2 squares

(18+82)(18-82)

= -6400

How do solve when it asks how many different ways you can arrange things?

How many different ways can you arrange 6 different students if 3 of them always have to be together?

stats and factorial problem!

3! * 4!

= 144

What if you are dividing by a decimal number?

2800/.4?

times top and bottom by 10 and it makes it easier

28000/4 = 7000

What if you have a rate problem with jobs in it?

A can do a job in 10 days B can do a job in 12 days. A does a job for 5 days and them B joins to finish, how long did B work for?

use J(job) = r * t

A rate = 1/10

B rate = 1/12

30/11 days

How do you solve a profit/loss or price/sale word problem?

if you mark the price 10% above your cost and sell it for 10% off the marked price what is your profit/loss?

P_s - P_c = Result

1% loss

what is the formula for simple interest?

if my investment goes from $2000 to $2250 in 3 years how much will I make in 2 years with $3600 at the same rate?

I_simple = P(principle)r(rate)t(time)

$300

d(distance)=r(rate)*t(time)

220 - 200 = r*1hr

20 miles/hr

A/B=1/2

A+10/B+10 = 2/3

B present age = 20

3

if you choose k out of n people your formula is

n/k = n!/k!(n-k)!

when you divide two radicals, you can divide the numbers under one radical sign.

√2/.005 = √400

Average Rate = Total distance/Total time

28 and 4/7

1 and 1/4

you can't separate radicals if you're adding only in division or multiplication.

% increase = amount of increase/original amount

Circumference is directly proportional to the radius

C=2 pi r

answer is 44%

2 * 10^-10

remember scientific notation and try and get it in a decimal form

0<z<1

plug in the extremes of y and those become the range for z

1.44

and 1.21

you can just do 12 * 12 and know there will be 2 decimals

sina/A = sinb/B = sinc/C

use if 3 are given and you need to find 1

cos (a+a)

= cosa*cosa - sina*sina

=cos²a - sin²a

= cos²a - (1 - cos²a)

= 2cos²a - 1

cos²a = 1/2*(1+cos2a)

sin²a = 1/2*(1-cos2a)

1/sin(a)

h/o

h/a

1/cos(a)

a/o

1/tan(a)

cos(a)/sin(a)

csc(a)/sec(a)

pi/4, pi/4, pi/2 - 1,1,sqrt(2)

90,45,45, - 1,1,sqrt(2)

use calculator and write down

if same difference, larger is bigger 5/7 or 7/9?

7/9 is bigger

if low number can find common denominator

.125

12.5%

.05 

5%

.2

20%

16 2/3 %

.1666666

factor a³ + b³

^b2 - 4ac < 0

if true then imaginary roots

I = P(1+rt)

how do you convert to C or F?

23 C to F

or -13 F to C

if in C, * by 1.8 and add 32  = 73.4

if in F, substract 32 and divide by 1.8 = -25