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Issa H.

Right triangle: 30-60-90

x, 2x(hypotenuse), x√3

Pythagorean Theorem

a^{2}+b^{2}=c^{2 }

- remember to take square root when looking for "c"

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Right Isosceles triangle

90-45-45

side : side : side √3

Equilateral Triangle

- all sides equal
- all angles equal 60°

Area of a triangle

2

Area of a circle

Πr^{2}

Circumference

2Πr

Circle degrees

- whole: 360°
- 1/4: 90°
- 1/2: 180°
- 3/4: 270°

Area of a rectangle (circle)

b x h

Numbers to know

Π=3

√1= 1

√2= 1.4

√3= 1.7

% change

O

Data Sufficiency

Statement 1: Sufficient- AD

Not Sufficient- BCE

Noun

person, place, thing or idea

Verb

expresses an action

Adjective

modifies a noun

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Adverb

modifies a verb, adjective or another adverb

Preposition

notes relation of a noun to an action or thing.

Phrase

group of words acting as a single part of speech. A phrase is missing either a subject, verb or both.

Prepositional phrase

group of words beginning with a preposition. Like any phrase, does not contain a subject or verb.

Pronoun

takes the place of a noun. (he, she, their, etc.)

Clause

group of words that contains a subject and verb

Two types of pronoun errors:

1. pronoun preference

2. pronoun number

See a pronoun, as yourself...

1. is it clear who/what it's referring to?

2. does it agree in number with noun it's referring to?

Participial phrase

starts with a verb ending in -ing. (supposed to modify noun or pronoun immediately following it)

Spotting a misplaced modifier:

when a sentence begins with a modifying phrase followed by a comma, noun or pronoun right after should be what phrase is referring to.

Identify-the-reasoning questions

ANSWER

ALL QUESTIONS!!

Integers

- can be negative
**or**positive - sometimes called "natural" or "whole" numbers
- do NOT include fractions
- do NOT include decimals
- DOES include
__ZERO.__

Consecutive Integers

Prime numbers

a __positive__ integer that is only divisible by *two* numbers: **1 & itself. ****2, 3, 5, 7, 11, 13**

**0 nor 1 is prime**

Divisible by 3

the sum of its digits is divisible by 3.

Divisible by 4

the number formed by its last two digits is divisible by 4.

7**28: **28--divisible by 4.

Divisible by 6

if it is divisible by **2** AND **3**

Division by zero is undefined

test writers won't ever put a zero as the **denominator. **

- zero in the
*numerator*is fine.

Numbers have a few factors

but an infinite number of multiples.

Absolute value

the distance between a number and zero.

*always*positive.

ZERO

is an integer but NEITHER positive or negative.

Data Sufficiency

"Don't assume there are only two candidates"

- just because the answer to the question is NO doesn't mean it is
*insufficient*.

Yes/no data sufficiency

- yes or no-- sufficient
- maybe-- insufficient

Order of operations

always distribute.

If in distributed form:

Factor them.

if in factored form: distribute.

Adding & subtracting fractions

- fractions must have same denominator.
- multiply the denominators together for a new common one.
- then multiply the numerators by the same number you did the demonimator,

Multiplying fractions

just multiply the numerators and denominators.

Dividing fractions

invert the second fraction and multiply

Converting mixed numbers to improper fractions

- multiply the whole number by the denominator and add the numerator.
- that becomes the new numerator and use the same denominator.

Comparing fractions

cross multiply and whichever product is larger is the larger fractions

Fraction equivalents of decimals

.2= ** 1/5 ** .6=** 3/5**

.25= **1/4 ** .66=** 2/3**

.33= **1/3 ** .75= **3/4**

.4= ** 2/5** .8= **4/5**

.5= ** 1/2**

Dividing decimals

make the divisor a whole number by moving the decimal in the divisor as well as the dividend.

Ratios

part to part relationship.

- to find the whole part, set up a proportion.
- look at ratio box on page 111

Compound interest problems

- a little bit more than simple
- quarterly= rate is divided into 4 equal parts.
- semiannually= two equal parts

Harder average problems

total number

- plug in numbers to formula
- then divide total sum of remaining numbers by total number of remaining numbers

Mode

number that occurs most

- can be more than one mode

Standard deviation

1. find average

2. subtract each number from average

3. square each difference

4. average those numbers

5 take the square root of those averages

Exponents: multiplying numbers w/ the same base

add their exponents

Exponents: dividing numbers with same base

subtract the bottom exponents from the top exponents

Raising a power to a power

multiply the exponents

(1/3)^{2}

1/3 x 1/3 = 1/9

(-3)^{3}

(-3)(-3)(-3)= -27

*** -27 is smaller than -3

Any number to the first power(x)^{1}:

is itself

Any number to the 0 power is (x)^{0:}

ZERO

√x * √y = √xy

√12 * √3 = √36 = 6

√x/y = √x/√y

√3/16 = √3/√16 = √3/4

The square root of a positive fraction is actually larger than the original fraction:

√1/4 = 1/2

Plugging in

1. pick number for variables in problem

2. using numbers, find answers to problems (target answer)

3. plug numbers into answer choices to see which equals target answer

- avoid using 0 or to plug in
- or numbers in the problem or answer choices
- for problems using percents, use 100
- hours or minutes use 30 or 120

Distance formula

Distance = rate * time

Hidden variable

- plug in a number for total (fractions: multiply denominators)
- write down the totals of each part
- the number you're looking for should reduce into an answer choice

FOIL

first, outer, inner, last

(x+2)(x+5)=x*x, x*5, 2*x, 2*5

x^{2}+5x+2x+10

x^{2}+7x+10

Quadratic equations

- (x+y)
^{2}= x^{2}+2xy+y^{2} - (x+y)(x-y)= x
^{2}-y^{2 } - (x-y)
^{2}= x^{2}-2xy+y^{2}

When two items travel in same line toward each other...

ADD THEIR RATES

Work

rate * time

Functions

the number inside the parentheses you plug into the expression

- don't reduce functions

Probability

Number of outcomes you want / total number of possible outcomes

Probability that A & B will both happen:

A * B

1/6 * 1/6 = 1/36

Probability that A OR B will happen:

A + B

1/8 + 1/8 = 2/8 = 1/4

Odds something won't happen:

add probabilities and subtract from 1.

Odds at least 1 thing will happen:

multiply probabilities and subtract from 1

Permutations

multiply all numbers together

Permutations: single source, order matters

3 teams= 3*2*1

9 teams= 9*8*7*6....

7 teams, 3 rankings= 7*6*5=210

Combination problems (single source, order doesn't matter):

1. find permutation (6*5*4)= 120

2. divide by permutation of # of objects in combination (3*2*1)=6

120/6= 20

Volume

l*w*h

Quadrants

start top right and go counter clockwise

Surface area

Find area of each face and add all together

Slope

x_{1}-x_{2}

Point-slope

↑

slope

Simple interest earned

principle*rate*time

Compound interest earned (4 steps)

1. divide rate by how many times compounded annually.

2. multiply that # by the principle.

3. add that # to the principle. repeat steps 2 & 3 until you reach # of compounding periods.

4. subtract principle from that #

Inference

a statement known to be true because it can be backed up with proof.

First step in evaluating *Multi-source reasoning question*

determine which tab or tabs contain the info you need.

Per capita

must multiply by the population

Math section

- 19 problem-solving, 18 data sufficiency

Verbal section

- 13 reading comprehension
- 17 sentence correction
- 11 critical reasoning

Integrated reasoning section

- table analysis
- graphics interpretation
- multi-source reasoning
- two-part analysis

Early questions

count more than later questions.

--start slowly, check your work

--gradually pick up the pace.

- questions left unanswered will lower your score.

Most obvious answer choices...

are almost ALWAYS wrong!

Output ↓ by 10%, then ↓ again by 20%. How much would new output need to be decreased to restore original level?

1. pick a # & multiply by .10; add that to original output.

2. multiply new output by .20 & add that # to new output.

3. set up a proportion with the total amount added to the outout over the newest output = x/100 to get the percent.

On test day bring:

1. driver's license

2. $28

3. a snack

4. bottle of water

5. resume

MUST

tell computer to keep score

Permutation problems

usually ask for arrangements

Combination problems

usually ask for groups

Regression Lines

Goes down= negative relationship

- negative slope

Goes up=positive relationship

- positive slope

Group equation

Integrated reasoning: pacing yourself

1. do the easier questions first.

2. don't spend too much time on one question-- sometimes its best to guess & move on.

2. don't spend too much time on one question-- sometimes its best to guess & move on.

Tables

Study the column headings first.

Then go straight to the statements

Then go straight to the statements

Square roots

can also be negative!!!!

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