The cominations formula,

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The permutations formula,

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x² - y² = (x+y)(x-y)

(x+y)² = (x+y)(x+y) = x² + 2xy + y²

(x-y)² = (x-y)(x-y) = x² - 2xy + y²

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Where [image], [image] and [image] are the times taken by A, B and C respectively to complete the task alone and [image] is the time taken by them to complete the task when they are all working together.

Solving for quadratic equations

ex. x² + x -6 = 0

sum of the two numbers equals the middle constant, and the product of the two number equals the third contant.

(x+3)(x-2)

Whatever the numerator is in the fraction, the outcome will be infinite of itself

ex. ²/₉ = .22222 or ⁸/₉ = .88888

The product of a given integer and another integer.

ex. Multiples of 4 are, 4, 8, 12, 16,....

An integer that divides another integer without leaving a remainder.

ex. 3 is a factor of 12 because ³/₁₂ = 4, without a remainder

xy

x+y

Principal x (1+rt)

r=rate, t=time

Principal x (1+r)^t

r=rate, t=time

PEMDAS

parenthesis, exponents, multiply/divide, addition/subtraction

to solve for all of the variables in a systems of equations, we must have at least as many distinct linear equations n as we have distinct variables n.

Two ways to solve:

1) substitution solve one equation for one fothe variables and substitute that variable into the other

2) combination add or subtract one equation from the other to cancel out one of the variables

1) When we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign

2) Single-variable equations are usually solved for a specific value, whereas inequalities can only be solved for a range of values

average speed = total distance

                         total time

Area = ½ × r² × θ

r=radius

θ = angle of sector

^x/₃₆₀ = ^arc/_C = ^sector/_A

X = angle measurement

C = circumference

A = area of circle

m = rise = y₁ - y₂

        run     x₁ - x₂

y=mx+b

m = slope

b = y-intercept

negative reciprocals

line with the slope -2 has a negative reciprocal of ¹/_2, and makes the line perpendicular

Strategies 

When picking numbers, avoid picking numbers that appear in the question stem.

Strategies

When dealing with variables in exponents, find a common base.

Strategies

When a Data Sufficiency question involves a number's absolute value, remember that the number itself may be negative.

Strategies

Be comfortable with positive/negative rules forwards and backwards. A negative resulf of a product means you multiplied a positive and a negative.

Strategies

Use the Least Common Denominator when picking numbers with fractions.

Strategies

Picking numbers is a very powerful strategy when solving for odds and evens, divisibility, and similar number properties.

Data Suffiency

Insufficient

Data Suffiency

Hard Questions

Skip statements that you do not understand.

Eliminate as much as possible.

Data Suffiency

Hard Questions

On harder DS questions, answer choices tend to be more sufficient than they might appear.

• DON’T CHOOSE (E) if you have to guess.

• Pick between (A) or (C), if you can eliminate (B).

• Historically, (A) is slightly more common as the right answer.

Data Suffiency

Equations

For a system with

n variables:

• If you have as many distinct linear equations as you have variables, you can answer ANY question about the system.

• If you are only asked to solve for part of the system, you don’t necessarily need all

n equations.

• If you are asked to solve for a relationship instead of the value of variables, you don’t necessarily need all

n equations.

A someone must be a person and a something must be a thing, place, or idea. So guess what? The subject is usually a noun because a noun is a person, place, thing, or idea. Although, sometimes the subject is a pronoun such as he, they, it, and so forth.

One way to think about the subject is to say that the subject is the who or what part of the subject-verb pair. The subject-verb pair is the main idea of the sentence, stripped to essentials

Sentence Correction

Present Perfect verb tense

• "have" or "has" followed by past tense verb
• describes an action that began in the past but continues into the present 

Sentence Correction

past perfect

• "had" followed by a verb in the past tense
• used when there are two actions in the sentence, describes an action that stared in the past and ended in the past

Sentence Correction

Modifiers

• part of a sentence that provides more information about a person or thing in the sentence
• should be as close as possible to the noun it is modifying

Sentence Corrections

Challenging Pronouns

• "that" is restrictive; what follows is necessary to the subject of the clause.Ex. The car that is parked next to the fire hydrant, will be ticketed.
• "which" is non-restrictive; using which determines the information that follows is irrelevant to the suject of the clause. Ex. The car which is red, will be ticketed.

Sentence Correction

Parallel Structure

• A sentence maintains parallel structure when its components maintain consistent grammatical structure.
• Generally tested with a list or a 2-part idiom.

Sentence Correction

Passive voice

Passive voice: The cake was bought by Alison.

Active voice: Allison bought the cake.

Sentence Correction

Redundancy

Sentence Correction

Wordiness

Sentence Correction

Idioms

as...as

between...and

regard...as

Sentence Correction

Misused words: less/fewer

Use less to talk about things we cant count

Use fewer to talk about things we can count

Sentence Correction

Misused words: number/amount

Use number to talk about things we can count

Use amount to talk about things we cant count

Sentence Correction

Misused words: between/among

Use between when only two options are available

Use among when more than two options are available

Sentence Correction

Misused words: whether/if 

Whether is correct when we're discussing two options

If is correct for more than two options.

Sentence Correction

Comparisons

Comparisons in sentences need to be like terms, people compared to people.

Common indicators of comparisons: like, unlike, similar to, as, so, in contrast to

Odd + Odd = Even          Odd x Odd = Odd

Even + Even = Even      Even x Even = Even

Odd + Even = Odd        Odd x Even = Even

And mulitplication

involving an even

# creates and even

product