TTP Combined Flash Cards

Slope of a line

"Rise over run" - (y2 - y1) / (x2 - x1), where y is the y coordinate and x is the x coordinate.

Total # of Unique Elements when number of unique items is unknown

# in (Group A) + # in (Group B) + # in (Group C) - # in (Groups of Exactly Two) - 2[#in (Group of Exactly Three)] + # in (Neither)

Total # of unique members in a venn diagram problem

# in group A only + # in group B only + # in group C only + # in double overlaps (i.e. A,B + B,C + C,A) + # shared between A,B,C + # is neither A,B,C

Fundamental counting principle and coin flips

# of coin flips TIMES 2 outcomes (i.e. either heads or tails) 2^number of flips

%Profit equation

%Profit is equal to Profit/Cost x 100 where the profit is equal to Total revenue - total cost. Example use: If you know someone has a profit of 30% of the cost, and that the cost is equal to 10 then you can set up an equation to be equal to [Revenue - 10]/10 = 3/10, and from here can you cross multiply and solve for revenue.

Permutation where two items must be next to each other

(1) Account for the items as if they were combined. So if 5 total items, and two must be next to each other calculate as 4x3x2x1 (2) If the two items that must be next to each other can be in opposite orders, then multiply by step (1) by 2 to account for situation where one is ahead of the other and vice versa.

What is the average of all three digit numbers

(1) All three digit numbers can be represents as a consecutive set of integers from 100 to 999 (2) For a consecutive set, to find the average take the first and last divided by 2 (3) [100 + 999]/2 = 549.5

Permutation where two items must NOT BE next to each other

(1) Calculate the TOTAL POSSIBLE PERMUTATIONS with NO RESTRICTIONS (2) Calculate the permutations where two items MUST be together Calculate (1) - (2) to determine permutations that are NOT equal to each other. Note: must be collectively exhaustive and mutually exclusive to utilize this approach.

Inferring the SD without calculating all of it

(1) Calculate the mean of the set (2) Determine the absolute difference from values within the set (3) Sum the differences The resulting sums could be compared to different sets to determine which has a larger or smaller SD. Note: The set that has the greatest sum has the greatest SD.

Sum of odd integers from 5 to 55 inclusive

(1) Count of integers is equal to (55-5/ 2)+1 (+1 because of inclusive, and divided by two because asking for only odds) (2) Average of integers is 55+5/2 (3) 30*26 = 780

Successive percent changes

(1) DO NOT ASSUME YOU CAN NET OR SUM OR DIFFERENCE THE PERCENTS!!! (2) To handle these type of questions take your initial value, and multiply by each individual percent greater/less factor. Ex: Initial = 100, and two successive 20% increases is equal to 100(120/100)(120/100) Ex: Initial = 100 and two sucessive 20% decrease is equal to 100(80/100)(80/100)

Given you can choose from 5 rubies 4 emrealds and 3 diamons. If you select a total of 5 jewels, and 3 will be rubies, how many total combinations?

(1) Determine how many possible combinations of rubies you can select, which will be 5 choose 3. (5!)/(2!)(3!) (2) Determine the number of non-rubies, which will be 7 choose 2. (7 because 4 emrealds + 3 diamons, and 2 because 5-3 is 2). NOTE: Since you already accounted for the number of combinations of ruby in step one you don't consider rubys at all in the rest of the problem.

Probability of NOT selecting some items from a group (using combinations)

(1) Determine the total outcomes/denominator by simply taking total population and sample in the form of a combination of populationCsample. (2) Determine the numerator by taking population MINUS the not selection number, and perform population-not selectedCsample. Example: 5 students from 8 students, and what is the probability 2 of the eight not selected? (1) Denominator = 8c5 (2) Numerator = 6c5 - the concept on why this works is because by excluding some amount, you are removing them from possible combinations. Answer = 3/28

Determining the number of LEADING ZEROES in a decimal

(1) Express the fraction in the form 1/x (2) If between 1<x<=10, no leading zeroes, 10<x<=100 then ONE leading zeroes, etc Essentially if x is an integer with k digits, then the trailing zeroes are (k-1), UNLESS X is a perfect power of 10, in which case it will be k-2 leading zeroes. Ex: 1/123123 = 5 leading zeroes ( because len(123,123) -1) ) Ex: 1/10000 = 3 leading zeroes ( because perfect power of 10 so len(10,000) - 2 )

How to find LCM(a,b)

(1) Find the prime factorization of a and b (2) Take the HIGHEST power from each factorized number (3) multiple together Example: 24 = 3^1 x 2^3 and 32 = 2^5 3^1 x 2^5 is equal to the LCM

Finding the LCM

(1) Find the prime factorization of each integer. That is, prime factorize each integer and put the prime factors of each integer in exponent form. (2) Of any REPEATED prime factors, take only those with the LARGEST exponent. For example, given a 3^2 in one # and a 3^3 in another, take the 3^3. If left with the same of two numbers, take one of the duplication. (3) Of what is left, take all NON-REPEATED (4) Multiply together what is found in 2 and 3 LCM of 24 and 60 (1) 24 = 2^3 x 3^1 and 60 = = 2^2 x 3 x 5 (2) 2^3 x 3 (3) 5 (4) 2^3 x 3 x 5 = 120

Procedure for finding GCF

(1) Find the prime factorization of each number (2) Identify repeated prime factors among the numbers (3) Take the SMALLEST power of each REPEATED prime (4) Take the product of the numbers in 3 NOTE: If there is no common primes among the numbers the GCF is 1.

Finding the GCF

(1) Find the prime factorization of each number. (2) Of any REPEATED prime factors, take only those with the SMALLEST exponent. (Note: if no repeated factors found, then GCF = 1) (3) Multiply together the numbers that you found in step 2; this product is the GCF. GCF 24 and 60 (1) 24 = 2^3 x 3^1 and 60 = = 2^2 x 3 x 5 (2) 2^2 and 3^1 (3) 2^2 x 3^1 = 12

Finding count/number of factors in a number

(1) Find the prime factorization of the number (2) Add 1 to the value of each exponent. Then multiply these results and the product will be the total number of factors for that number. Note: for odd take only the odd bases exponents, and for even take total factors minus odd factors.

What is the greatest integer value of n such that 9^n is a factor of (81!)^3

(1) First need to conceptually realize that we need to determine how many 9^n's evely divide into (81!)^3, in other words 81!^3/9^N (2) A second extremely important concept, and it's easy to forget, is that before determining the number of N's you need to first prime factor the base of the denominator to be 3^2n. (3) Determine number of 3's in 81/3^N, which can be done by 81/3^1, 81/3^2, etc procedure, and which equals 40. (4) Since there's 40 3s in 81!, then that means there's 120 3s in (81!)^3. (5) Finally you can determine the max N's by setting up that 2n <= 120, so maximum n is equal to 60.

Finding the count of factors

(1) Get prime factorization (2) Take each exponent of the prime factorization and add 1 (3) Multiply each number in (2) together, and this is all even and odd factors Note: To get odd factors, perform above for only odd primes, and to get even take total - odd.

Combination restriction example: In a treasure chest are contained five different flawed diamonds and three different perfect diamonds.If a pirate selects four diamonds from the chest, in how many ways can four diamonds be selected, if exactly two of them must be perfect diamonds?

(1) Given that you need EXACTLLY TWO perfect diamons, you have 3c2 (2) Now that you have the exact two perfect diamond combination, you need to take the rest, so you'll select from the imperfect, so 5c2 Multiply 3c2 and 5c2 together to get the answer of 30.

What two things do a ratio represent?

(1) How one part of a ratio (i.e. cats) relates to the other part (i.e. dogs) (2) How each individual part relates to the whole (i.e. cats to sum of cats and dogs)

LCD approach to comparing fractions

(1) If given a set of different fractions, multiply top and bottom of each to get all with the same common denominator (2) The larger the numerator, the larger the fraction after all fractions have the same denominator Note: A similar approach works to getting all fractions with the same numerator, but instead for step two the smaller denominator the larger the fraction

Multiplication of decimals

(1) Ignore the decimals points and treat the #s as whole numbers (2) Count the total decimal points for EACH number (ex: 0.02 = 2) (3) In the product of 1, move the decimal places to the sum of numbers counted in (2) Ex: .02 x .02 (1) 2x2 = 4 (2) 4 decimal places because .02 is 2 and .02 is 2 (i.e. the 0 and 2 after the decimal place (3) .0004 because counted 4 in step 2, so need 4 total places after the decimal point NOTE!!!!: A shortcut to answer questions can be to simply count the decimal places instead of having to actually perform the multiplication. In counting places in 2, the answer MUST have that amount of decimal places, so if there is only one answer with that then that is the answer.

Properties of one

(1) One is a factor of all numbers, and all numbers are multiples of 1 (2) One raised to any power is equal to 1 (3) Multiplying or dividing any number by 1 does not change the number (4) One is an ODD number (not evenly divisible by 2) (5) One is the only number with exactly 1 factor (1 is only a factor of 1) (6) One is NOT a prime number (The first prime number is 2)

Properties when abs(a+b) = abs(a) + abs(b)

(1) One or both quantities are 0 OR: (2) Both quantities have the SAME sign

Steps for LCM

(1) Prime factorize each number in the set (2) For each repeated prime factor, take the one with the highest power (3) Add in any non-repeated primes (4) Multiply (2) and (3) together to get LCM Note: When dealing with easier numbers, another easy way is to take the largest integer of the set, and take multiples of that number until the other smaller numbers in the set divide evenly into the multiple of the largest (Ex: 5 and 12 [12, 24, 36, 48, !60!] Note: Another property of the LCM is that it will contain AT LEAST one of every unique prime factor of the set of numbers. Note: LCM can also be used to solve repeating pattern questions (refer to blinking light example, where you know when they first start, when is the next time they repeat at the same time) Note on step (2) that if you have more than two numbers, repeats in ANY PAIR count for step two, doesn't have to be repeated/duplication in all of the numbers, just at least two.

How to determine the number of digits in a number

(1) Prime factorize the number (2) Court the number of 5x2 pairs, noting that each pair adds a trailing zero/digit (3) Collect the number of unapired 5 and 2s, along with nonzero primes if any, and multiply together. Count the digits in this product (4) Sum the digits of steps 2 and 3 together Remember that exponents distribute over multiplication, example: 50^8 = [5x2x5]^8 = [5^16x2^8]

How to determine if a fraction has a decimal that terminates?

(1) Reduce the fraction into its lowest form (2) If the denominator of the lowest form fraction prime factorizes to either 2's, 5's or both, then it terminates, if not then it DOESN'T terminate.

Compound inequality rules

(1) Similar as in regular equations you can do the same thing to EVERY PART of the equation (2) If multiplied or divided by a negative, every sign must change direction

How to solve Which of the following values for x will hold true for the inequality |-4x + 2| > 2x?

(1) Solve for the positive option where -4x+2 >2x (2) Solve for the negative option where -(-4x+2)>2x The combination of (1) and (2) provides the possible solutions to x. So in general, remember for ha absolute value inequality you solve one equation as is, and another equation where one side is multiplied by -1 (be consistent and use the absolute value side)

How do you handle questions where you are given one equation, and they ask by what value does one value need to change, in order for the RESULT of the equation to equal something else.

(1) Solve for the variable that needs to change. For example if given E=MC^2, and you need to change M, then solve for M. (2) Change the variable that needs to change, for example, if you are looking for E to double in E=MC^2, then (2E)/C^2 = M (3) Divided the new equation by the old after it's transformed. (I.e. (2E)/C^2 DIVIDED BY E/C^2, which equals 2). So in the example given the M would need to change by 2 so that E is doubled.

Adding and subtracting remainders with multiple terms

(1) Take the division of each term separately (2) Add/subtract the whole numbers together (3) Add/subtract the remainders together, remembering that a remainder can't be over the divisor Note: If performing subtraction, if you have a negative, add the divisor take the net for the remainder.

How to determine x^2 = 4

(1) Take the square root of both sides (2) Take abs(x) = 2 (3) Realize that the aboslute value of x is asking what values are two units from 0 on the number line, which means that the final form of the equation is x = +/- 2 Note that the square root of a single number would simply be positive (i.e. sqrt(144) = 12). This concept holds true for ALL EVEN ROOTS.

Properties when abs(a-b) = abs(a) - abs(b)

(1) The SECOND quantity is 0, OR (2) Both quantities have the same sign and the absolute value of abs(a-b) is greater than or equal to the absolute value of abs(a) - abs(b) example1: abs(5-0) = abs(5) - abs(0) example2: abs(5-4) = abs(5) - abs(4)

If given an odd list of numbers, which includes one variable, what are the possible median values of that list?

(1) The median where the variable is LESS than the half way point (i.e. N,1,2,3,4) (2) The median where the variable is GREATER than the half way point (i.e. 1,2,3,4,N) (3) The median IS the half way point (i.e. 1,2,N,3,4) Relevant question: "If the average (arithmetic mean) of 2, 3, 7, 9 and n is equal to the median of 2, 3, 7, 9 and n, which of the following COULD be the value of n?"

Even odd pattern of addition/subtraction

(1) The sum or difference will be an even number only if BOTH of the numbers are even OR BOTH are odd (2) otherwise, the sum or difference will be equal to odd Ex: 2+2 = 4 (E), and 3+3 = 6 (E), 2+3 = 5 (O)

Catch up questions

(1) The time it takes for one object to catch up to another is a function of the DIFFERENCE in the speeds of the two objects (2) Once the two objects meet each other, they will have traveled EQUAL DISTANCES In these questions you're often adjust the time variable based on when the second object leaves vs. when the first object leaves

Three ways triangles can be similar

(1) Three angles are the same in both triangles (2) RATIOS of the three sides are the same in both triangles (example multiples 2x each side of the smaller) (3) One angles is the same in both triangles, and the two closest sides are of the same ratio

Reflections over y=x, y=-x, y=b, and x=a

(1) When something is reflected over y=x, the points "switch places", (x,y) becomes (y,x) (Ex: (2,3) -> (3,2) (2) When something is reflected over y = -x, the points "switch places" and take on negative signs, (x,y) becomes (-y, -x) (Ex: (-1, 4) -> (-4, 1) (3) When something is reflected over y = b the original (x,y) becomes (x, 2b-y) (Ex: (5,-2) -> (5, 4) when reflected over y = 1 (4) The line x = a: When a point (x, y) is reflected over the line x = a, its image is (2a - x, y). That is, (x, y) → (2a - x, y). For example, (5, -2) becomes (-1, -2) when it's reflected over the line x = 2. (5) The point (a, b): When a point (x, y) is reflected over some point (a, b), its image is (2a - x, 2b - y). For example, (5, -2) becomes (-1, 4) when it's reflected over the point (2, 1).

Diverging rate questions

(1) When two objects start at the same point and move away from each other, the distance traveled is a function of the sum of their speeds. (2) Total distance is equal to the sum of the distances of the individual objects

Things to consider when comparing whether a new number to a power or to a root is higher or lower

(1) Whether the base is a proper fraction or a whole number (2) Whether the base is positive, negative, or 0 (3) Whether the exponent is even or odd (4) Whether the exponent is positive or negative (5) Whether the exponent is a whole number or a proper fraction

Round trip questions

(1) With round trip problems you know each leg of the trip will equal the same distance (2) You CAN NOT assume that the TIME TRAVELED is the same, you can instead represent one leg as time t, and the other as total traveled - t. To solve these set up each leg of the trip in a DRT table as two separate rows and solve for the related variables.

Simplest way to solve two similar triangles where one is "extended" from the other

(1) Write down two separate triangles, one that has sides of x, y and 16 and the other that has X+10, y+12 and 24 (2) Take the proportion of 16/24=x/x+10 and 16/24 = y/y+12 (3) Solve for each proportion and that provides the values of x and y Note: Part one the sides are x+10 and y+12 respectively because for the larger triangle you are given the bottom piece of the whole, but are not given the whole parts of the sides so you need to build them

How to explain abs(x)<y

(1) X must be less than y, and (2) X must be greater than negative y -y<x<y Example abs(x) < 8 means that x is less than 8, and greater than negative 8 (i.e. within a distance less than 8 from 0 on the number line <---(-8)-------0-------(8)------->

Calculating the reminder from multiple terms that are MULTIPLIED

(1) You can divide by each term, and take each terms remainder fraction (2) Multiply the remainder fraction NUMERATORS together (3) Divide step 2 into the original divisor Ex: 500x600x700/8 = 62 + 4/8, 75 + 0, 87 + 4/8 = 4x0x4 = 0/8 = no remainder

Properties of 0

(1) Zero multiplies by any number is zero (2) Zero divided by any number other than 0 is 0 (3) Any number divided by 0 is undefined (4) The square root of 0 is 0 (5) Zero raised to any positive power is 0 (6) Zero is the only number that is neither negative nor positive (7) Zero is the only number that is equal to its opposite (0 = -0) (8) Zero is a multiple of ALL numbers (9) Zero is the only number that is equal to all of its multiples (because 0 times any number = 0, so 0 x 1 = 0, 0x2 = 0, etc) (10) Zero is NOT A FACTOR OF ANY NUMBER EXCEPT ITSELF (11) ANY NUMBER (except 0) RAISED TO THE ZERO POWER IS EQUAL TO 1 (12) Zero is an even number (because it is evenly divisible by 2 (13) Zero can be added or subtracted to any number without changing its value.

Three approaches to percent change calculation

(1) [New - old]/ old X 100 Note: the times 100 is required to get it into percent form!!!!!! (2) If it's an increase you can use Final value = initial value(1 + x/100) where x is the percent change (3) If it's an decrease you can use Final value = initial value(1 - x/100) where x is the percent change

How to explain abs(x)>y

(1) x must be greater than y, and (2) x must be less than negative y -y>x>y Example: abs(x) > 20 means that x is less than -20 and greater than 20 (i.e. greater than 20 distance from 0 on the number line.)

What is 100% greater than z, and 800% greater than z

(1) z(1 + 100/100) = z x 200/100 = 200 percent of z (2) z(1+800/100) = z x 900/100 = 900 percent of z This is why it's important to procedural walk through the percent problems. The smaller percents are easier to work with and logically think through, but the larger percents like 800 you should just follow the formula to be sure you've got it right. Remember the formula is initial value (1 + "Percent greater" value/100) = Final value

"What percent less than z" in equation form

(1-X/100) times z Critical to realize when you have what percent LESS THAN you must set up the formula to be 1-x/100, not just x/100. Similar idea for what percent greater than except with a 1+ instead of 1-.

Weighted average example (Lauren purchased two amusement park tickets for $50 each, six tickets for $240 each, and x tickets for $800 each. What is the value of x if the average ticket price she paid is $260?)

(2x50)+(6x240)+(800x) divided by 2+6+x = 240, and you solve for x. In this example the "weights" are equal to the number of tickets sold. What's critical to realize is that the sum of the weights are equal to the denominator.

circular permutation formula

(K-1)!, where K is equal to number of items to be arranged in a circle.. Example: Arrange 6 people around a table, how many different seating arrangements are possible? (6-1)! = 120

How to calculate the number of degrees in shapes

(N-2) x 180 degrees, where N is the number of sides in the shape Ex: Traingle = 3 sides - 2 times 180 = 180 x 1 = 180

Squares of Fractions

(a/b)^2 = a^2/b^2

Area of a circle

(pi)r^2 OR (pi)(d/2)^2

Common quadratic identities

(x+y)^2 = x^2 + 2xy + y^2 [Square of a sum] (x-y)^2 = x^2 - 2xy + y^2 [Square of a difference] (x+y)(x-y) = x^2 - y^2 [Difference of two squares]

Power to a power rule

(x^a)^b=x^ab (x^5)^3=x^15

Factor properties

- 1 and -1 are divisors of every integer - Ever integer is a divisor of itself - Every integer is a divisor of 0, except 0 itself - Numbers divisble by 2 are even, and not are odd

Midsegment of a triangle

- A triangle has 3 possible midsegments.[think of it from each base] -The midsegment is always half the length of the third side. (i.e. smaller triangle is congruent)

Equilateral triangle

- Area = [s^2 x Sqrt(3)]/4 - Altitude/height = [s x Sqrt(3)] / 2 - For a given perimeter equilateral triangle has the largest area. - For a given area equilateral triangle has the smallest perimeter.

Right triangle inscribed in circle

- If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. - Any angle inscribed from a side of a circle to a semi-circle is a 90 degree angle

Trapezoid Properties

- If both legs are the same length, this is called an isosceles trapezoid, and both base angles are the same. -If the legs are parallel, it now has two pairs of parallel sides, and is a parallelogram. -Area is equal to Height times averge base length [h x a+b/2]

Divisibility rules for 3 and 9

- If the sum of the digits is divisible by 3, the number is also. - If the sum of the digits is divisible by 9, the number is also.

Four different slope types

- Pointing up from left to right through to quadrant 1 is positive - Pointing down from left to right through quadrant 4 is negative - Up and down = undefined (can't divide by 0), and the x coordinate of two points on the line are equal because you can't divide by 0. - Horizontal = 0 slope, and the y coordinates of two points on the line are equal to each other, because with the slope formula the numerator is equal to 0 and any number divided into 0 is equal to 0

Coordinate plane

- Quadrant (1) starts at the top right and goes COUNTER !!!!!! clockwise (it's stupid that it's counter clockwise, remember how stupid it is and you wont forget, who made this shit up) - At the origin/center both x and y are 0 - On the x axis, all values to the right are positive, and all values to the right are negative

45-45-90 Triangle

- Side ratio: 1:1:sqrt(2), where sqrt(2) is the hypotonuse - Area: s^2/2 (beacause two of these triangle make a square you divide area of square by 2)

30-60-90 Triangle

- Side ratio: x, xsqrt(3), 2x (i.e. 1:sqrt(3):2)

Graphing inequalities on the coordinate plane

- When given a line is > or <, etc you are comparing y > mx+b, not any other form such as mx > b + y, etc. Always rewrite the equation into y = mx + b form before working with it. (1) To graph a line inequality you must first select at least two x values to determine at least two ordered pairs, which will allow you to graph the line. (2) If y >, then the region above the line is shaded, if less then then the region below the line is shaded. (3) Compare a given y coordinate, to the mx+b where x is the given x coordinate. If the inequality is correct, then that is a solution. (Note: The idea is if the point falls into the shaded region, then it is a solution.) Ex: In the xy-plane, region G is defined by the inequality: y < 6x + 8. Is point (4,33) a solution? Because 33 (y) < 6(4) + 8 is not a correct inequality (i.e. 32 is not greater than 33) then this point is NOT a solution. Note: if greater than or equal to, remember that a point can fall on the line itself and still be a solution.)

Dealing with square roots on the GMAT with algebra vs. give number

- When the GMAT provides the square root sign for an even root, such as x√x or x√4x4, then the only accepted answer is the positive root. Example: sqrt(25) = 5 (not -5) -HOWEVER, if given x^2 = 25 then x = +5 or -5

Quadrilateral

- a four-sided polygon

Properties of 0

-0 is an EVEN integer. Even because any number evenly divisible by 2 is even, and 0/2 is 0 -0 is NEITHER positive or negative -0 is DIVISBLE by EVERY integer except 0 and itself

Sum of degrees of a triangle

-180 degrees Note: all shapes can be calculated as 180(n-2) where n is the number of sides

360, 180 and 90 degree angle comparison

-A 360 degree angle is one that represents one full rotation around an axis, so if given say a 50 degree angle, and you need to know the larger cooresponding angle of the axis you can take 360-50. -180 degree angle create a straight line. -90 degree angle=right angle

Even and odd powers with negative numbers

-A negative number to an even power is a positive product -A negative number to an odd power is a NEGATIVE product

Parrallelogram relationship to other shapes

-A parallelogram is a quadrilateral with opposite sides parallel and congruent. -A rectangle is a parallelogram but with all angles fixed at 90° -A rhombus is a parallelogram but with all sides equal in length - A square is a parallelogram but with all sides equal in length and all angles fixed at 90°

Define a square in relationship to other polygons

-A square can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal, or a parallelogram with equal diagonals that bisect the angles. -If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square. (Rectangle (four equal angles) + Rhombus (four equal sides) = Square) -a parallelogram but with adjacent sides equal and the angles all 90°

Equilateral Triangle

-All angles are of equal measure -All sides are of equal measure (Note: if either all side, or all angles are equal, then it's given that the other must be all equal)

Sector area, arc length, central length

-Arc length is the portion of the perimeter of the circle associated with the central angle Central angle / 360 = arc length / circumference = area of sector/area of circle

Addition/subtraction of LIKE RADICALS

-Can only add/subtract like radicals. Example: 3sqrt(2) + 4sqrt(2) = 7sqrt(2) -It's similar to how addition/subtraction of a variable x would work. A like radical is something whether the index of the radical, and what's under the radical is the same. Example: sqrt(2) and sqrt(2) is a like radical, but sqrt(2) and cuberoot(2) IS NOT.

equilateral triangle inscribed in a circle

-Center of the triangle is the center of the circle, and creating a triangle from the center point to the middle point on the triangle creates a 30-60-90 -The three vertices of the triangles are associated with three equal arcs. So if you have the arc length of one, you have the arc length of all of them, and also the perimiter.

Polygon Properties

-Closed 2d shape composed entirely of straight line segments. -Included under the definition of polygons are things such as triangles, squares, rectangles, parallelograms

Square properties

-Diagonals are perpendicular -Diagonals bisect -Each diagonal divides into 45-45-90 triangles -Length of the diagonal is equal to sidexsqrt(2) (This is because a 45-45-90 has a ratio of x, x, xsqrt(2))

Verifying the primality procedure

-Dividing n by all integers smaller than sqrt(n) Ex: 161 [sqrt(161) = <13, and from 2 to 13 161 is divisble by 7, hence 161 is NOT prime

Difference between permutation and combination approach to determining two objects that MUST BE together

-For a combination you must minus the objects that must be together from the total, and the choose. -For a permutation you treat the ones that must be together as one entity, and count that as a part of the permutation.

Diagonal of a cube or rectangular solid

-For a rectangle the diagonal is equal to the extended Pythagorean theorem, which is diagonal^2 = length^2 + width^2 + height^2 -For a cube the diagonal is equal to side times sqrt(3)) -Diagonal represents the longest possible point in either the cube or the rectangular solid

Roots of x^n rules for positive/negative

-If an even root is taken on a variable to an even power, it is equal to the ABSOLUTE VALUE of the variable. Example: sqrt(x^2) = abs(x) -If an odd root is taken on a variable to an odd power, it is equal to the value of the variable. Example: cuberoot(-x^3) = -x, and cuberoot(x^3) = x

Consecutive integer properties

-If number of integers is odd, the sum of consecutive integers is always divisible by n. Example: Given {9,10,11}, we have n=3 consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3. -If number of integers is even, the sum of consecutive integers is never divisible by n. -The product of nn consecutive integers is always divisible by n! Example: Given n=4 consecutive integers: {3,4,5,6}. The product of 3*4*5*6 is 360, which is divisible by 4!=24

Squre properties

-If the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are (about 1.414) times the length of a side of the square. -A square has a larger area than any other quadrilateral with the same perimeter. - The diagonals of a square bisect its angles. - The diagonals of a square are perpendicular. - Opposite sides of a square are both parallel and equal. - All four angles of a square are equal. (Each is 360/4 = 90 degrees, so every angle of a square is a right angle.)• -The diagonals of a square are equal.

5 and 10 divisibility rules

-If the last digit is a 5 or a 0, the number is divisible by 5. -If the number ends in 0, it is divisible by 10.

Divisibility rules for 6 and 12

-If the number is divisible by both 3 and 2, it is also divisible by 6. -If the number is divisible by both 3 and 4, it is also divisible by 6.

Evenly spaced set properties

-In any evenly spaced set the arithmetic mean (average) is equal to the median. Mean/Median can be calculated as first and last term divided by 2. -If the evenly spaced set contains odd number of elements, the mean is the middle term, so the sum is middle term multiplied by number of terms. -Sum of n first positive numbers = n^2. Example: N=5 1+3+5+7+9 = 25 = 5^2 -Sum of n first even numbers = n(n+1). Example N = 4 = 2+4+6+8 = 20 = 4(4+1)

Similar Triangles

-It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º. -In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle. -If two similar triangles have sides in the ratio x/y, then their areas are in the ratio x^2/y^2

Addition/Subtraction of like bases/radicals

-Key here is to FACTOR -Ex: 2^10 + 2^11 + 2^12 = 2^10(1+2^1+2^2) Ex: 10sqrt(2) + 5sqrt(2) = sqrt(2)(10+5)

Triangle median

-Midpoint of a line from the vertex to an opposite side -Each median divides the triangle into two smaller triangles, and all the medians divide the triangle into 6 smaller triangles of EQUAL area

Roots operation rules

-Multiplication/division: sqrt(x) times sqrt(y) = sqrt(xy) and sqrt(x)/sqrt(y) = sqrt(x/y) -Exponents: sqrt(x)^n = sqrt(x^n) -Additions/subtraction: NO RULE

Indistingishable items permutation formula

-N represents the total number of objects (INCLUDING THE DUPLICATES -- for example AABC would be equal to 4!, NOT 3!) -Each r represents the FREQUENCY of each indistinguishable object. For example, given AABC, you would have 4!/2!x1!x1!.

Rectangle

-Opposite sides are parallel and congruent -The diagonals bisect each other -The diagonals are congruent -A square is a special case of a rectangle where all four sides are the same length. -Each diagonal divides the rectangle into two congruent right triangles. Because the triangles are congruent, they have the same area, and each triangle has half the area of the rectangle. -Area = width*height -A square is a special case of a rectangle where all four sides are the same length. -It is also a special case of a parallelogram but with extra limitation that the angles are fixed at 90°.

Parallelogram

-Opposite sides of a parallelogram are equal in length. -Opposite angles are equal in measure -Diagonals bisect eachother -Consectutive angles add to 180 degrees - Area = 1/2BH - The area of a parallelogram is twice the area of a triangle created by one of its diagonals.

Multiplying and dividing positive/negative number properties

-P X P = N - N X N = P - P X N = N

Altitude

-Perpendicular from the base to the opposite vertex -As there are three different bases their are three possible altitudes -The three altitude meet at a center point called orthocenter

Square root

-Represents tow numbers multiplied by eachother. Example: 2 x 2, 2^2 x 2^2 -The square root of a number is only for the positive number. (!!!ALONG WITH ALL OTHER EVEN ROOTS!!) Example: Sqrt(4) is equal only to positive 2, not negative

Regular hexagons

-Sum of angles = 720 = 180(6-2) -Six sides, and six angles -Each angle is equal to 120

Converting a fraction to decimal form, and vice versa

-The mixed numeral can be expressed as whole number + fraction in decimal form Note: For the vice versa, going from decimal to whole number and remainder, without additional information you can't determine the EXACT remainder (because the fractional remainder could be a multiple up or down). THIS IS A BIG TRAP, DON'T ASSUME THIS Example: 3.16 = 3+16/100 = 3+32/200 = 3+8/50 = etc infinite possible remainders (but same number, different fractions)

Properties of a perfect square [a^2]

-The number of distinct factors of a perfect square is ALWAYS ODD. -The sum of distinct factors of a perfect square is ALWAYS ODD. -A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. -Perfect square always has even number of powers of prime factors.

Relationship of sides to interior angles in a triangle

-The shortest side is always opposite the smallest interior angle - The longest side is always opposite the largest interior angle

Formula to calculate sum of evenly spaced set

-The sum of the elements in any evenly spaced set is given by first and last term divided by 2 MULTIPLIED by the number of terms -OR [[2 * first number] + [difference(count of set - 1)]/2]*n

X and Y intercepts

-The y intercept is the point where x is equal to 0 (i.e. (0, y)) -The x intercept is the point where y is equal to 0 (i.e. (x, 0)) Calculating the x intercept: (1) y = mx + b (2) y-b = mx (3) y-b/m = x Since we know that the x intercept is where y = 0, then we can say that -b/m = x. If you know the slope and the y intercept, then you can calculate x.

Prime number properties

-Two divisors/factors, 1 and itself -infinite number of positive prime numbers -primes can only be positive -All prime numbers except 2 and 5 end in 1, 3, 7 or 9 -all prime numbers above 3 are of the form 6n-1 or 6n+1

parallel lines intersected by a transversal (i.e. a line that intersects two parallel lines) - what are the properties?

-Vertical angles are equal -Corresponding angles are equal -Any acute + any obtuse will sum to 180

If a is a factor of b and a is a factor of c, then

-a is a factor of [B + C] -a is also a factor of [mB + nC] for all integers m and n example: 2 is a factor of 4 and 6 and 4+6 = 10 which is divisible by 2

Even number

-integer that is "evenly divisible" by 2 -integer of the form n=2kn=2k, where kk is an integer.

Odd number

-integer that is not evenly divisible by 2 -integer of the form n=2k+1n=2k+1, where kk is an integer

Examples of reciprocals

-reciprocal for a number x, denoted by 1/x or x^−1, is a number which when multiplied by x yields 1 1/3 = 3/1 5/4 = 4/5 3^-2 = 1/3^2 = 1/9 = 9/1

Rhombus properties

-special type of parallelogram, and has all the same properties of a parallelogram -The diagonals of a rhombus always bisect each other at 90°. -Two ways to calculate area: (1) - base * altititude, or [diagonal1 * diagonal2 ]/ 2

Taking the square root of a number

-sqrt[x^2] = abs(x) -sqrt[x+y]^2 = abs(x+y)

X and Y axis

-the x axis is the horizontal number line on a coordinate plane, streches from neg to pos infinity -the y axis is the vertical number line on a coordinate plane neg to pos infinity -x and y axis are perpendicular to each other where the intersection = origin = (0,0)

Converting small decimal to scientific notation

.0004 = in the FOURTH decimal place (ten thousandths) so you take the 4 x 10^-4

Converting small decimal to a fraction

.0004 = in the ten thousandths place (10s, 100s, 1000s, 10000s) so that means that you take the 4 divided by 10,000 (4/10,000)

Decimal number digit names and algebraic representation

.ABC A = Tenths B = Hundreths C = Thousandths Note: No units/ones, starts at "Tenths" Ax.1 + Bx.01 + Cx.001 = .ABC Note: Each digit can be of course only from 0 - 9

Any factorial >5! has what as it's last number/units digit

0

What is the only number that is equal to its opposite?

0 Because 0 is equal to 0, so it's equal to its opposite. Examples of other opposites: 4 and -4, 1 and -1, 1/2 and -1/2

0^n is equal to what

0 Note: 0^-n is undefined

Square Roots to Memorize

0 = 0 1 = 1 2 = 4 3 = 9 4 = 16 5 = 25 6 = 36 7 = 49 8 = 64 9 = 81 10 = 100 11 = 121 12 = 144 13 = 169 14 = 196 15 = 225

Cube Roots to Memorize

0 = 0 1 = 1 2 = 8 3 = 27 4 = 64 5 = 125 6 = 216 7 = 343 8 = 512 9 = 729 10 = 1000

What numbers must a perfect square end in?

0, 1, 4, 5, 6, or 9 Note: To verify this simply square all numbers from 0 through 5 and check the units digit, can redo this on the test if I forget.

Perfect squares up to 225

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

perfect cubes (≤ 1000)

0=0, 1=1, 2=8, 3=27, 4=64, 5=125, 6=216, 7=343, 8=512, 9=729, 10=1000

0!

1

Finding the Number of Factors of an Integer procedure

1 - Find the prime factorization of a given number 2 - Take reach POWER of the prime factorization and add 1 3 - Multiply each number from 2 together to find the TOTAL factors -Note: To find the total even factors take the TOTAL factors - the odd factors. Finding the odd factors is the same steps of total factors but ONLY for the ODD factors [i.e. not 2].

Quadrant signs

1 = (+, +) 2 = (+, -) 3 = (-, -) 4 = (-, +) When a question asks about quadrants there's a good chance signs will come into play.

Reciprocal of ANY number

1 divided by ANY number is the reciprocal of that number In general 1/a/b = b/a

Any non-zero base raised to the 0, is equal to what?

1 or -1, depending on the sign of the base Ex: 75^0 = 1, 1^0 = 1, -3^0 = -1 Note: 0^0 is not tested on the GMAT. This is incredibly useful because this means you can set 1 equal to any base^0 in any equation. Example: 3^x = 1, then you could say that 3^x = 3^0, so x must be equal to 0.

Converting decimals to fractions

1. Calculate the total numbers after decimal point 2. Remove the decimal point from the number 3. Put 1 under the denominator and annex it with "0" as many as the total in step 1 4. Reduce the fraction to its lowest terms Example: .56 decimals is 2 then 56/1 = 56/100 = 14/25 Note: For repeating decimals similar procedures but separate the recurring number out (i.e. .39393939 = 39/100=13/33

Converting Improper Fractions to Mixed Fractions:

1. Divide the numerator by the denominator 2. Write down the whole number answer 3. Then write down any remainder above the denominator Example: 11/4 = 2 and 3/4

Determining the last digit of (xyz)^n:

1. Last digit of (xyz)^n is the same as that of z^n 2. Determine the cyclicity number c of z; 3. Find the remainder r when n divided by the cyclisity; 4. When r>0, then last digit of (xyz)^ni s the same as that of z^r and when r=0, then last digit of (xyz)^n is the same as that of z^c, where c is the cyclisity number. Example: What is the last digit of 127^39? - 7^1 = 7, 7^2 = 9, 7^3 = 3, 7^4 = 1, 7^5 = 7 So the cycality is equal to 4 (i.e. 7^1 through 4 repeating), then 39/4 remainder is 3 so the last digit of 127^39 is 3

Converting Mixed Fractions to Improper Fractions:

1. Multiply the whole number part by the fraction's denominator 2. Add that to the numerator 3. Then write the result on top of the denominator Example: 2 and 3/4 = 2*4 + 3 = 11/4

Area of a triangle

1/2 base x height the base of a triangle, and the height of a triangle MUST be perpendicular. Critical to understand the perpendicular to base property especially for triangles such as isoscoleces or equilateral.

Area of a triangle

1/2bh

Trapezoid area

1/2h(b1+b2) (Sum of the bases times height) divided by 2 Where the bases are the two that are parallel, and the height is perpendicular from one base to another.

Representing a three digit integer in algebraic form

100X + 10Y + Z Ex: 312 = 100*3 + 10*1 + 2 This pattern continues

Determined remainders when whole numbers divided by 10, 100, 1000 etc

153 / 10 = 15 and 3/10 (Ten is 1 digit to the left) (i.e. remainder becomes units digit of numerator) 153 / 100 = 1 and 53/100 (100 is 2 digits to the left) 153/1000 = 153/1000 (1000 is 3 digits to the left)

What do three triangles sum up to

180 degrees

Sum of interior angles formula

180(n-2) where n is the number of sides -Determine each angle by 180(n-2)/n

Prime numbers less than 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

first twenty-six prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

Prime factors

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

Consecutive powers of 2 pattern

2-4-8-6

Number 2 units digit powers pattern

2-4-8-6 ex: 2^1 = 1, 2^2 = 4, 2^3 = 8, 2^4 = 16

How many letters in the alphabet

26 Note: This may come up on certain permutation problems.

Bases of 2

2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 2^9 = 512 2^10 = 1,024

Surface area of a rectangle

2lengthXwidth + 2widthXheight + 2lengthXheight Note: May just be easier to calculate each faces area independently without thinking through applying this formula, it's effectively the same approach but doesn't require memorization.

Surface area of a cylinder

2πrh+2πr² 2πrh = diamter x pi x radius, this portion gives the "unraveled" portion, which is equal to the perimeter of one of the circles times the height 2πr² = this gives the top and bottom portions

Distinction between number of prime factors vs. number of DISTINCT prime factors

3 Distinct prime factors = 3*2*5 3 Prime factors = 3*3*3 or 2*2*2 or 3*2*5 Note that distinct prime factors means that they/their bases are different, and says NOTHING about the total number of prime factors a number has. On the contrary, the number of prime factors tells you how many prime factors, but not if their distinct. Example 4 prime factors could be equal to: 2^4, or 2*3*5*7, etc. Be careful of a situation where a question states that x and y have the same DISTINCT prime factors, because that could mean: x = 3^2 x 5^1 y = 3^1 x 5^2 and this would mean they are NOT divisible, but do share the same distinct prime factors

Approximating pi

3 is a reasonable estimate for the GMAT, if for whatever reason answers are very close, than 3.14 can be used as well to be more exact. Note: If pi is in the answer choice, then don't approximate, just work with the pi

Translating the proportion 3/7 = 18/42 to words

3 is to 7, just as 18 is to 42

Two most common Pythagorean triplets

3,4,5 (3^2 + 4^2 = 5^2) 5,12,13 (5^2 + 12^2 = 13^2) Multiples of these two are also Pythagorean triples (i.e. 6, 8, 10). If you know two of them, and it's a triangle with a 90 degree angle, then you know the third side without needing to calculate further. NOTE: THE LARGEST SIDE MUST BE THE HYPOTONEUSE. Don't fall into a trip thinking that if the longest side is 4, and another side is 3, that this is a Pythagorean triplet.

Consecutive powers of 3 pattern

3-9-7-1

Number 3 units digit powers pattern

3-9-7-1

Given a rate of 1 pool in four hours, how many hours will the pool be 75% full?

3/4 hours -Idea here is if you have a work rate problem with 1 in the numerator, you can increment from one to determine how much is completed up to the denominator.

If you are given that 3/5x, and 2/5x and 20 is remaining of 100, how can you best represent this?

3/5x + 2/5x + 20 = 100 Remember that the fractional parts must equal a whole.

sum of exterior angles of any polygon

360 degrees Note: This is true only when you are taking ONE EXTERIOR ANGLE PER VERTEX.

What do the angles created by N lines intersecting through one common point sum up to?

360 degrees. In the example image given the sum of the letters is equal to 360.

Representing a two digit number in algebraic form

37 = 3*10 + 7*1 Any two digit number can be represented as 10x + 1y, where x is the 10's digit and y is the units digit. This pattern continues for all digit (including decimals).

Bases of 3

3^0 = 1 3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243

Area of a hexagon

3sqrt(3)/2 x s^2, where s is equal to the length of any given side 3 square root of 3 divided by 2 times side squared. This can be derived easily by hand by recognizing that a hexagon can be split into six equal equilateral triangles, where you then have s^2sqrt(3)/4 TIMES 6 because six triangles. Estimate = 2.6s^2

Rectangle

4 sided shape (i.e. quadrilateral) with 4 90 degree angles. Area = Length x width Perimiter = 2(length) + 2(width)

Converting large number to scientific notation

4,000 = has four total digits, so you take the 4 x 10^3 (the exponent is # of digits - 1) -Note this is slightly different than for small decimals as you do -1 of the digits

Number 4 units digit powers pattern

4-6 pattern All positive odd powers of 4 end in 4, and all even powers of 4 end in 6.

Consecutive powers of 4 pattern

4-6-4-6

Quadrilaterals

4-sided polygons that encompass: (1) rectangles, (2) parallelograms (3) rhombuses (4) trapezoids

Bases of 4

4^0 = 1 4^1 = 4 4^2 = 16 4^3 = 64 4^4 = 256

Bases of 5

5^0 = 1 5^1 = 5 5^2 = 25 5^3 = 125 5^4 = 625

Surface area of a cube

6s^2 This is because s^2 = one face, and there's six faces

Consecutive powers of 7 pattern

7-9-3-1

Number 7 units digit powers pattern

7-9-3-1

Consecutive powers of 8 pattern

8-4-2-6

Number 8 units digit powers pattern

8-4-2-6

Number 9 units digit powers pattern

9-1 pattern All positive odds end in 9, and all positive evens end in 1

Consecutive powers of 9 pattern

9-1-9-1

Compound interest formula

A = P(1 + r/n)^(n x t), r is the rate, n is the number of times compounded, t is time

A is inversely proportional to the square of B, and cube of C

A = k / [b^2 times cuberoot(C)]

Chord

A line that is on one point on a circle to another, but that is not the diameter (diameter passes through the center of the circle).

How to represent a percent off relative to the original price x (i.e. 30% off of $100)

A percent off of x can be represented [1 - discount%]. So in the case of 30% off of 100, this would be equivalent to paying 70% of 100.

Perfect cubes

A perfect cube, other than 1 and 0, is a number such that all of it's prime factors have exponents that are DIVISIBLE by 3 27 = 3^3

Perfect square and factors

A perfect square, OTHER THAN 0 OR 1, is a number whose PRIME FACTORIZATION contains only EVEN exponents.

Perfect squares

A perfect square, OTHER THAN 1 AND 0, is a number such that all of it's prime factors have even exponents. Ex: 144 = 2^4 x 3^2

Perfect cube

A positive perfect cube, other than 1, is a number whose PRIME FACTORIZATION contains only exponents that are MULTIPLES of 3.

Probability and 0-1 implications

A probability MUST be from 0-1. 0 probability example - Probability of selecting an even prime number >2 (0 because there is no such thing) 1 probability example - Probability of landing on an integer in a six sided die of integers (1 because all sides are integers)

Factoring Quadratics

A quadratic equation in the form ax^2 + bx + c can be rewritten as a product of two factors called the "factored form". This form resembles (x + ?)(x + ?) (What adds to bx, and multiplies to c) Ex: X^2 - 3x - 28 = (x-7)(x+4)

Square

A quadrilateral with 4 sides that are equal and has all 90 degree angles

Splitting a regular hexagon into triangles

A regular hexagon can be split into 6 equilateral triangles.

Evenly Spaced Sets

A set of number where the difference between each subsequent number is equal. (i.e.: 1,2,3,4,5 or 10, 20, 30, 40, 50, or 3, 5, 7, 9, etc) In evenly spaced sets, the mean/average is equal to the median. So if given an evenly spaced sets with odd # terms, the median/mean is equal to the middle number, for even # terms the two middle divided by two is both the mean and the median.

Collectively Exhaustive Events

A situation in which two events represents ALL of the possible events that can occur. Example: A light switch on, and off represents collectively exhaustive events, because the on/off state represent all of the possible events. If two events are both MUTUALLY EXCLUSIVE and COLLECTIVELY EXHAUSTIVE, then total possible events are combinations of A occurring + combinations of B occurring

Symbolism/strange operators as functions

A strange operator or symbol can be used in place of a function such as: @x = x^2 +1, which if you were given @3 that would result in 3^2 + 1 = 10 A strange operator can also be used with more than one variable, such as: a@b = a + b, then that would mean 3@2 = 3 + 2 = 5

LCM short form formula assuming you have a GCF

A x B / GCF = LCM

Area of an equilateral triangle

A=(s²√3)/4

Compound Interest Formula

A=P(1+(r/n))^nt p=principle amount r=rate n=times componded t=years/times compounded annually If $20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 years? A = 20,000*[1+[0.12/4]]^2*4

Whole number digit names and algebraic representation

AB,CDE A = Ten thousands B = One thousand C = Hundreds D = Tens E = units digit/ones Ax10,000 + Bx1,000 + Cx100 + Dx10 + Ex1 = AB,CDE Note: Each digit can be of course only from 0 - 9

What is the square root of x^2

ABSOLUTE VALUE OF X, not just x. This has implications especially on data sufficiency questions.

What is the sign of a square root of a number

ALWAYS POSITIVE, except for 0 which is 0 and 0 is neither negative or positive (but is even!).

Adding or subtracting a constant to the numerator and denominator

Adding the same number to top and bottom moves the # closer to 1 Subtracting the same number to top and bottom moves the number AWAY from 1 (note: for subtraction, this doesn't hold true when the numerator becomes <0 (i.e. negative))

Adding or subtracting a constant to a set, impact on standard deviation

Adding/subtracting some constant to each value in a set does not change the SD of the set. Ex: SD of 1,2,3 is the same as 2,3,4 Note: The mean DOES change, but not the SD.

Operation properties for signed numbers

Addition Positive + Positive = Larger positive number Negative + Negative = Smaller negative number Positive + Negative = A number somewhere BETWEEN the positive and negative number (either positive or negative depending on the two numbers) (Ex: -2 + 5 = 3, and -5 + 2 = -3) Multiplication AND division Positive x Positive = Positive number Negative x Negative = Positive number Positive x Negative = Negative number

When using a venn diagram, what does the "doubles" portion of the circle make up

Adds up to the following: (1) The portion related to ONLY the double (i.e. Salmon and Chicken) (2) The portion related the triple (i.e. Salmon and chicken AND turkey) Remember that this means you're given a total double amount, and a triple amount, then the double - the triple equals the section related to ONLY the double.

When using a venn diagram, what does one whole circle add up to?

Adds up to the following: (1) The portion related to ONLY the label (i.e. Salmon) (2) The TWO portions related to the "doubles" (I.e. Salmon and Chicken + Salmon and Turkey) (3) The ONE portion related to the "triple" (i.e. Salmon and chicken and turkey)

Consecutive powers of 5 pattern

All end in 5

Consecutive powers of 6 pattern

All end in 6

Multiples of the LCM property

All multiples of the LCM of a given set of numbers, would also be evenly divisible by those set of numbers.

Number 5 units digit powers pattern

All positive integer powers of 5 end in 5

Number 6 units digit powers pattern

All positive integer powers of 6 end in 6

[a^x][b^w]=[a^y][b^z], allows you to conclude what?

Allows you to conclude that x=y, and w=z, assuming that the bases are NOT 1, -1, or 0. In the GMAT you'll often need to prime factorize one side to get to this form.

Even roots of numbers and +/- implication

An even root will always be for a positive value under the root, not negative. Conceptually this is because there's no numbers which when multiplied by themselves will generate a positive value. (i.e. can't have sqrt(-16)

Summary of angles in circles

An inscribed angle (i.e. one that starts at one point on the perimeter of the circle to two points on a separate area of a circle) is 1/2 of the arc length degrees

Derangement probability questions (I.e. probability that only 1 out of 4 items are correctly matched

Approach to solving these can be done in the following steps: Example to walk-through: 4 books A-B-C-D are to be matched to four people, A-B-C-D, what is the probability that only one is correctly matched? (1) Create a table of possibilities where only 1 is correctly matched. Example: outcome 1: A-B-C-D to A-C-D-B and outcome 2: A-B-C-D to A-D-B-C So the possible outcomes where one letter is properly matches is two. BUT, note that since there's four slots, there could be four different arrangements, so 4x2 is the number of possible combinations where one is properly matched. (2) Determine the denominator, which will simply be the factorial of possible slots, in this case 4!. Answer: 8/24 = 1/3

verticle/opposing angles

Are EQUAL In the example given the letter c shares the same angle on opposing sides.

Overlapping sets with percents

Assume that the total A, B, A+B, Neither A or B is equal to 100. With this assumption each percent can equal the whole integer. (Ex: 30% would equal 30)

How to solve a combined work problem

Assuming a same start and end, you can create a WRT table with values as follows: Rate = Rate of each individual object on separate rows Time = Time equal to T (or any variable) FOR BOTH. The reason you have the same T variable is because of the fact that both objects are working for the same amount of time. Work = This is equal to the rate x time. After setting up the table add the two work fields together for the rows, and this is the answer to the combined work.

Weighted average and average speed relationship (also average gallon rate)

Average speed is actually a weighted average where the weighting is based off of time. Remember that Sum of Distance / Sum of Time = average speed and that Sum of distance = (RT + RT), and Sum of time = (T+T)

Standard form of the equation of a line

Ax + By = C In putting this into standard slope intercept form (solving for y) you get the following. (1) Ax + By = C (2) By = C - Ax (3) y = -Ax/B + C/B So slope = -A/B and y intercept = C/B

Conceptually, why is any number squared (or to any even power) positive

Because of the multiplication rules, where a negative * negative is equal to positive: Example: -2^2 = -2*-2 = 4, or -2^4 = -2*-2*-2*-2 = 4*-2 = -8*-2 = 16

Circle inscribed in a equilateral triangle

Big concept is to understand that drawing a line from the center down to the center of a base of the triangle creates two 30-60-90 triangles

Triangle base

Can be any three sides -traditionally in an isosoclese the base is the unequal side

Geometric sequence

Can be defined as An = A1 times r^n-1 You can determine the common ratio by dividing the second by the first (or any n / n-1) Ex: 2, 10, 50, 250, thus 10/2 = 5, which means: (1) 2 (2) 5x2 = 10 (3) 5 x 10 = 50 (4) 5 x 50 = 250

Three part circle ratio

Central angle / 360 = arc length / circumference = area of sector/area of circle

How can you validate that a triangle contains a 90 degree angle

Confirm that the Pythagorean theorem works for the 3 sides of a given triangle. This works because only a triangle with a 90 degree angle will the Pythagorean theorem work for the three sides of a triangle.

Calculating profit$ and revenue with a given profit % and a cost

Cost times profit% is equal to the $profit Then $profit + cost is equal to the revenue.

How to handle questions that ask about subsequent percent or fractional deductions of a value. (Ex: In the morning you have 1 cup of coffee c and you drink 1/2, in the middle of the day you drink 1/3 of what remains etc)

Create a table that represents each deductive occurrence as a row, and columns that represent the starting value, amount removed, and amount remaining. Note: The quickest way to solve the remains column is to multiply the starting value by the complement of how much is being taken away. For example, if you drink 3/4 of a half of a cup of coffee, what remains is 1/4 of the 1/2. THIS IS CRITICAL TO ANSWER THESE QUESTIONS EFFICIENTLY. Theoretically if you don't want to do a table you can multiply the variable c by all of the reductive compliments. (I.e. If 1/4 taken in morning, 2/3 in afternoon, and 3/5s at night, then answer = [3/4]x[1/3][2/5]c

Alternative to growth formula for solving growth questions (table approach)

Create a table with columns of start, change/delta, and ending value, with rows as each growth period.

How to solve wet mixture problem

Create a table with columns that are set up as follows: Field1 = % of mixture (Example: acid) Field2 = TOTAL VOLUME (Acid + water) Field3 = Volume of ACID which is calculate as field1 x field2 Note: The sum of rows in Field 2 must equal the total volume of all mixed. Note: Sum of rows in Field 3 must equal the total x % of total mixture. (Example: Mix 3 volumes with different %, and total volume is 45%, then sum of the 3 volumes field 3 must equal the 45% x total of field 2)

What does percent mean?

DIVIDE THAT SHIT BY 100. Whatever it is, it quite literally mean to divide by 100. For example, x percent of y is what? x/100 TIMES y

Determining if a decimal is terminating

Decimal equivalent of a fraction is terminating if and only if the denominator of the REDUCED fraction has a prime factorization of only 2s or 5s, or both.

Sum of n terms of an arithmetic sequence

Defined by the formula given as N/2(a1 + an) where an is the last term. Note that this works because as we learned in other modules an evenly spaced set has a mean that is equal to the median, and to determine the sum of values in an evenly spaced set you take average (first + last divided by 2) times number of terms N. Ex: 2, 5, 8, 11 is an arithmetic sequence, what is the sum of the first 100 terms. (1) We know that the sequence can be defined as An = 2 + 3(n-1), because the first term is 2, and n-1 is the iteration step number (2) To determine the 100th sequence we calculate that as 3(100-1) = 299 (3) First + last = 299 + 2 = 301 (4) 301/2 * 100 because total number of terms is 100 and first and last / 2 is the average of an evenly spaced set.

General way to solve catch up problems

Determine each objects distance of terms of other variables, and set the two objects distances equal to each other. This works because at the point the two objects catch up, their distances will be equal.

General way to solve round trip problems

Determine each objects distances in terms of other variables, and set each distance to each other. This works because in a round trip problem the distance traveled on each leg will be the same assuming the same route.

Determining the domain of a function (i.e. possible inputs to the function)

Determining the domain requires evaluating the properties of the function. Ex: If f(x) = x^3/x^2 - sqrt(x), then that means that: (1) x cannot be 0, because if x = 0 then you would have 0 / 0 - sqrt(0) and you can't divide by 0 (2) x must always be non-negative because you can't take the sqrt of a negative number. So the domain is > 0 Primarily for GMAT purposes you want to watch out for the following restrictions: (1) Can't take the square root of a negative (2) We aren't allowed to divide by 0

Rectangles inscribed into a circle

Diagonal of the rectangle is equal to the diameter

Rate time distance formula

Distance = Rate x Time

How to convert a grouped rate to a per unit rate (i.e. group of x people doing something, what does 1 person do)

Divide the rate of the group by the number of units in the group. Ex: 1 hour in 2 hours for 6 people, then one person is [1/2]/6 Note: This is useful in problems where you're given one group rate, and the question wants you to find the rate for another group (i.e. 6 people do x, what do 8 people do?)

Determining number of prime number x, in y!

Divide y by x^n, incrementing n from 1 until x^n > y For each increment take the quotient (IGNORE THE REMAINDER), and sum each of the quotients. This sum represents the number of x's in y! You'll often have to utilize in the context where a question asks something like "find n where 400/5^n" Note: To determine the number of numbers (doesn't have to be prime), you take the denominator number and factorize, and then perform the above procedure on the LARGEST FACTOR (note the largest factors would limit the total number of pairs of primes that could exist divided into the numerator)

Range and domain of a function

Domain = the set of all numbers that a function can use Range = the set of all numbers a function can generate

What is true when abs(a+b) EQUALS abs(a) + abs(b)

Either one of two things: (1) Both a and b are equal to zero, OR (2) Both a and b have the same signs (i.e. +/-) (NOT THE SAME INTEGERS, SAME SIGNS!)

What is true when abs(a-b) EQUALS abs(a) - abs(b)

Either one or two things: (1) The SECOND VARIABLE (in this case b), is equal to ZERO, OR (2) Both a and bare the same sign, AND abs(a) >= abs(b)

How else can you generally represent x^2<b

Either: (1) x<=sqrt(b), or (2) x=>-sqrt(b) Example x^2<81 x<9, or x>-9

Product of two perpendicular lines slopes is equal to what?

Equal to -1, and therefore the slopes are negative reciprocals of each other. Ex: Slope of line 1 = 10, and slope of line 2 = -(1/10), so 10 times -(1/10) = -1. Thus if you know the product of two slopes is equal to -1, then you know they are perpendicular.

Diagonal of a rectangle

Equal to sqrt(length^2 + width^2) -Two diagonals within a rectangle, and the two diagonals bisect each other. The angles formed at this bisections are NOT 90 degrees unless it's a square. -Each diagonal divides the rectangle into two congruent right triangles. -One diagnonal does NOT automatically form two 30-60-90 triangles, you can only verify this type of triangle with the x, xsqrt(3) and 2x ratio

nroot(x^n)

Equal to the ABSOLUTE VALUE OF (x) IF N IS EVEN. -Remember given an absolute value(x) = y, then x = +/-y

nroot([x+y]^n)

Equal to the ABSOLUTE VALUE OF (x+y) IF N IS EVEN. -Remember then you would set abs(x+y) = Q, and then x+y = +/- q

gcd(a, b)*lcm(a, b)

Equal to the original a*b

Dry mixture matrix

Equation is equal to rate x time format. So matrix can be set up as Columns = [Rate, Time] Rows = [X, Y]

Even/odd

Even

How to represent even and odd integers algebraically

Even = 2n, where n is any integer Odd = 2n + 1, where n is any integer

"And" probability for independent events

Events that are INDEPENDENT (i.e. where the probability of one thing has not effect on another) means you simply multiply P(A) x P(B) = P(A and B). This can be extrapolated to more than two probabilities assuming the events are independent. Ex: If P(splitting an apple) in any given arrow shot is 1/4, what is the probability of splitting four apples in four shorts (it's equal to 1/4^4) Note: The example is independent because each shot with a new apple are independent of each other. Which is to stay one shot at the apple, will not affect another.

Comparing decimal sizes methodically

Ex: .9098 vs .907 (1) .9 = .9 (2) .00 = .00 (3) .009 > .007 So the answer is (1) is larger because thousandths place is larger in the first number than second, and all places before it are equal.

Combination formula

Ex: 5 choose 2 is equal to 5!/(5-2)!2! N = total objects you are choosing from K = total objects you will select Used in situations where the order does not matter.

How to handle converting two units in one question?

Ex: Given 24ft/1sec, and you need to convert to yds/min Option 1: CONVERT ONE UNIT AT A TIME Option 2: Convert separately and combined Ex way one: 24ft/1sec x 60sec/1min = 1440ft/1min, and then 1440ft/1min x 1yds/3ft = 480yds/1min Ex way two: 24ft x 1yd/3ft = 8yds and 1sec x 1min/60secs = 1/60min Ex: 8yds divided by 1/60min = 480 yds per min First question for this lesson is an example of how to do this in one question easily while factoring out common factors: https://gmat.targettestprep.com/lesson/546

Problems where you're given one quantity of three different ratios, how to determine how many of the total can be created?

Ex: Given 3x cream, 4x flour and 5x eggs, and given that you have 17 eggs, to determine how much in total you can make you first do 5x=17, and then utilize the x in each part of the equation.

Determining the number of powers of primes in a factorial

Example 30!/4^n = 30!/(2^2)^n = 30/2^2n (1) Factorize the factored BASE (2) Divide the factored base by the number in the numerator (remember not factorial just the numerator) up to the point where the quotient equals 0 (3) Sum all of the quotients up to 2^n Take the sum calculated in 3 and perform 2n<=sum and calculate for n. (note: 2n here would be whatever the factored form is) N is equal to the answer.

Trick to adding same bases and exponents when the value of the base is the same as the count of duplicate items you are adding

Example: 2^2 + 2^2 = 2^3, this is because: (1) 2^2(1 + 1) = 2^2[2^1] = 2^3 This can apply to any base assuming that what you're adding is a duplicate, and the number of duplicates you are adding is equal to the bases.

Probability of selecting 1 item out of a list of items, if in total you are selecting >1 items

Example: 4 girls and 5 boys, what is the probability of selecting 1 girl if you select 3 In these situations you must consider two things: (1) Multiply the and combined probabilities together. For example, P(1 girl) x P(1 boy) x P(1 boy) = 4/9 x 5/8 x 4/7 (Note how the denominator decreases after each successive selection, because after selecting one you need to account for that in the other probabilities.) (2) ADD the "OR" probabilities together. For example, you can select GIRL BOY BOY, or BOY GIRL BOY, or BOY BOY GIRL. The important concept here is that you must consider the order in which the items are selected when performing the probability questions. Each different order represents additional probabilities.

If given total possible combinations, and number of items chosen from a group, how can you determine the population of the group?

Example: Given 2 people are selected, with 66 total combinations, you can set up an equation to be x(x-1)/2 = 66 The idea here is that the numerator X(X-1)(X-2)... etc for the number of selections, and the divisions is the factorial of the items selected, which is set equal to the total combinations given.

Procedure for determining when two objects MUST BE TOGETHER for a COMBINATION question

Example: Given that their are 8 trombonists, and two of them MUST be together, and in total you need to select 4. (1) Take the total MINUS the count of objects that need to be together. (8-2 = 6) (2) Take the number to select MINUS the count of objects that need to be together (4-2 = 2) (3) The answer will be 6c2.

Procedure for determining when two objects MUST BE TOGETHER for a PERMUTATION question

Example: Given there's 6 chairs, and 6 friends, and two of those friends must sit together (1) Treat the ones that must sit together as one entity (2) Permutate based off of step (1) above, so in this case you will have 5!, instead of 6!

If given that some object can complete a work unit in x percent less time than the other object, how do you represent this?

Example: Obj1 25% less time, than obj2 Then that means that ob1 1's rate is equal to Workunit/[3[time]/4] = 4/3t, and the other objects unit is equal to 1/t Critical thing to remember here is to differentiate between less TIME, and less RATE. With a time reference point you need to utilize the W=RT equation and realize that the rate is equal to W/T.

Growth Formula

F = kn + p, where F is the final value, k is the linear growth constant, n is the number of growth periods, and p is the start. Note: this only works for LINEAR GROWTH, not exponential.

What should you do before comparing the relative value of two units?

FIND THE COMMONS UNITS (i.e. what is 20 cents divided by $3, [$3 = 300 cents and then 20 cents/300 cents is the answer])

If y divides evenly into x, we can say y is a factor of x

Factors of 16 are 1, 2, 4, 8, and 16

"Percent greater than" questions

Final value = initial value x (1 + "Percent greater than"/100) Ex: 30% greater is equal to (1+ 30/100) = 1.30

What is the sum of integers from 101 to 202 inclusive?

First it's important to realize that the count * average = sum (as derived from average/mean formula). (1) Count of integers in set is equal to 202 - 101 + 1 (+1 because inclusive counting) (2) Average of integers is equal to 202 + 101 / 2 (3) Average * count = 15,453

If someone does some job j, in 75% less of the time than someone else, how do you represent this rate?

First remember that the rate can be calculated as the work divided by the time. From there can do as follows: t(1 - 3/4) = 1/4, then you can say Job/[1t/4] = 4job/thour This will work for any percent "less" question. The big concept here is to utilize the understand that the OTHER PERSON does T hours, so we're taking 1-3/4, or the complement of 1 - 3/4.

Squaring decimals with zeroes

First remember you multiply decimals you must first determine the number of decimal places, then multiply as if no decimal places, then move to the left the sum of decimals places initially determined in the original numbers. So simply put, [0.02]^2 will be 0.02 x 0.02, which has 4 decimal places, and 2*2 is 4, so the answer will be 3'0s and 4, so 0.0004

Consecutive of numbers greater than 9

Follows the units digits pattern of it's UNITS DIGIT example: 12^1 = 2, 12^2 = 4, etc

Divisibility rule for 6 and 12

For 6, determine if divisible by 2 and 3 For 12, determine if divisible by 3 and 4

Multiplication of radicals

For any positive Real numbers: Q√a × R√b = QR√ab The a and b can be multiplied together under the root, the Q and R are multiplied and kept on the outside. (i.e. outside times outside, and inside times inside, assuming same root index in both.) This is true for any radical, assuming that the radicals are the same for what you are multiplying (i.e. square root,, cube root, tenth root, etc) SIMILAR IDEA FOR DIVISION

Counting "if any of x can go into y"

For example, if any of 7 flyers, can go into any of 5 different mailboxes, how do you determine this: Flyer 1: 5 options Flyer 2: 5 options Flyer 3: 5 options ..etc So the answer is 5^7 The way to think about this is that the SEVEN objects can be placed into FIVE different boxes. So general idea is that the object count is the exponent, and where the object can be placed into is the base (i.e. because it represents the options).

Probability where one object is fixed to an order

For example, if given six names, A B C D E F, that are to speak in a randomized order, what is the probability that B will speak second. To solve this question you can determine the numerator by affixing the B to the second place. For example, 5 X B X 4 X 3 X 2 X 1. Note that you start with 5 because if you affix B to second place then the slots must start with 5 (i.e. instead of six.). The denominator would simply be 6!, as that represents total possible outcomes without restrictions.

Consecutive decrease in percent (i.e. discount) and then increase where you want the same initial value after the increase

For situations such as this it's important to realize that the consecutive INCREASE MUST BE GREATER THAN THE DECREASE. Ex: 100 with 20% discount is $80, to get back to 100 you need 80(1+x/100) = 100 which means x is 25% increase needed to get back to 100

Overlapping sets with fractions

For the overlapping sets questions the best approach is to take the LCM of the denominators of the GIVEN fractions. Ex: Given that 2/5 are passengers, 1/2 like peanuts and 1/8 likes peanuts and are male, then the LCM would be 40 (i.e. LCM(5,2,8) The calculated LCM can be used the inferred total to calculate the rest of the fractions.

Multiplication with consecutive increases/decreased percents

For these situations you start with an original value, x, and you can multiply x by each successive increase/decrease. Ex: increase 1 = 50% and decrease 2 = 50%, then X(3/2)(1/2) is the answer.

Line segment midpoint

Formula = (x1 + x2 / 2, y1 + y2/ 2) Simply the mid point is the average of the x and y of each point respectively.

Regular polygons

Generally means they have equal sides and angles

How to solve work rate questions involving a multiple of objects relative to another multiple (i.e. 10 machines)

Generally there's two things to consider in these types of problems: (1) The per object rate, which can be calculated as rate / count of group (alternatively you can multiply the time by the count of group to get per unit rate) (2) The proportion of object/rate to compared object/compared object rate

How to handle splitting the cost questions

Generally you'll need to set up a formula in the form of: Total cost/# of original people - Total Cost/# of new people = change in cost per person With this formula you can generally solve for total cost and finish the question from there.

How to convert 100 dollars into cents using a conversion factor?

Given a conversion factor of 100cents/1dollar, you would multiply 100 dollars * 100cents/1dollar which would cross out the dollars, keep the cents and 100*100 which equals 10,000 Note: The original unit MUST cancel out, if it doesn't then you are doing something wrong

Logical way to handle conversions

Given a conversion factor, put the unit you want to KEEP on the top, and the one you want to "remove" on the bottom, and multiply by the units of whatever you want to remove. Ex: 1000 basis points x 1%/100 basis points = 10%

Permutations box and fill

Given a total number of objects X, and for example 3 need to be selected you can determine permutations as follows: X(X-1)(X-2) Note with permutations, unlike combinations, you don't divide by the factorial of objects selected.

Absolute value inequalities

Given abs(x+2) < 4 you do two different things: (1) Calculate x+2 < 4 = x<2 (2) Calculated x+2 > -4 = x>-6(Note: you can think of this as starting with -(x+2) <4 and flipping signs because of diving by negative) Note: It's helpful to plot the result on a number line. In this case x is greater than -6 and less than 2

Determining the equation of a line given two points

Given any two points on a line you can find the equation of that line. Ex: (3,3) and (1, -1) (1) Slope = -1-3 / 1-3 = -4/-2 = 2, so y = 2x + b (2) Given y = 2x + b use either of the two given points and input y and x, and then solve for b which is equal to -3 (3) So the formula for the line is y = 2x - 3

Addition of a positive number and negative number

Given one negative, and one positive number (1) Take subtract the one with the SMALLEST absolute value from the LARGEST absolute value number (2) For the result, keep the sign of the largest absolute value Ex: 3 + (-4) is equal to 4-3 and carry the negative. https://gmat.targettestprep.com/lesson/373

perimeter of a square vs a rectangle

Given same areas of a square and a rectangle, the perimeter of a rectangle will always be greater than that of the square. This can be seen as inverse of the square rule for squares and rectangles. Note: The largest possible perimeter for a rectangle will have two sides that are 1, and the other that are maximized.

Area of a square vs a rectangle

Given same perimeters of a square and a rectangle, the area of the square will always be greater than the area of the rectangle.

Using letter representations/anagram to determine the quantity in sets with duplicates

Given something, such as probability of exactlly two wins and one loss this can be represented as follows: (1) WWL (2) WLW (3) LWW Mathmatically, this can represented as a permutations with duplicates where the numerator is 3! and denominator is 2!, so in this anagram there is 3 possible options. This concept can be extended for longer similar anagrams. Represents the numerator is count! and denominator is the product of duplicate counts factorial.

Triangle inequality theorem

Given three lengths of a triangle, A , B , C two things must be true for the relationship between any combination of the three sides: (1) Sum of two individual sides is GREATER than the remaining side (i.e. A + B > C) (2) Difference of two individual sides is LESS than the remaining side (I.e. A-B < C)

A base raised to a negative exponent represents what

Given x^-2, you can express this has 1/x^2.

How to test a units digit pattern when you forget one

Go from x^1 to x^4 taking all of the units digits. All single digit numbers will have a max repeating pattern of 4, some less, but all a max of 4. Note that you don't have to calculate the full number, just the previous units digits times the base. Example: 7^1 = 7 [7x1] 7^2 = 9 [7x7] 7^3 = 3 [7x9] 7^4 = 1 [7x3] The above an extremely important concept for many questions, the idea being that only units digits create units digits

What does it mean when a question talks about growth as a factor (I.e. when it uses the term factor to refer to the growth)

Growth as a factor generally means that the question is talking about EXPONENTIAL GROWTH. Ex: Day1 = 1x Day2 = 1x^2 Day3 = 1x^3 etc

Range (Hint: it's simple)

Highest value in the set MINUS the lowest value in the set. Conceptually, the higher the range, the more the numbers in the set are spread out.

Determining the domain and range using a graph

Identify how many x values are on the graph to determine the domain (possible inputs). Generally if the x's go on into infinity in both directions then the domain is all real numbers. Identify how many y values are on the graph to determine the range (possible outputs). For example, if the graph only goes up in both directions then the range would be all positive real numbers.

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle -The ratio of point AB/2 to BA/2 is equal to AC/BC

Triangles inscribed in circles

If a side of a triangle is on the diameter, then that means that the triangle IS a right triangle, where the 90 degrees is the vertex on the perimeter, thus this also means that the diameter is the hypotenuse.

vertical line test (checking if a graph can be a function)

If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.

Checking whether a point is on a line

If given a point (x, y), and given a slope intercept equation (y = mx + b) then you can: (1) Input the x and y into the y = mx + b equation (2) If the equation is equal to each other, then point (x, y) DOES fall on the line, if not equal then the point does NOT fall on the line

Permutation anchoring strategy

If given a restriction in a permutation, one approach is to "anchor" the restriction, which essentially means to first account for the restriction then account for the rest. Ex: If given 5 dogs, Spot, Lucky, Happy, Jumpy, and Jack will be lined up for baths, how many possible ways can they be lined up if Jack must ALWAYS be first? A permutation can be calculated as follows 1x4x3x2x1. NOTE!!! the first one here represents Jack, because there is only one possible way to select the first spot, which is jack, and the rest of the slots follow a N-1 pattern.

Representing exponential growth in algebra form

If given a start value, and factor/exponential growth of x for a certain number of periods y then you can represent the growth at the end of y as: [start value]x^y Period 1 = start_value[x] Period 2 = start_value[x^2] Period 2 = start_value[x^3] etc

AT LEAST one restriction, and mutually exclusive/collectively exhaustive event approach

If given an AT LEAST ONE restriction, and you know the events are mutually exclusive and collectively exhausting you can do the following: (1) Determine the TOTAL COMBINATIONS of possible events (2) Determine the combination of events where NONE (i.e. the opposite of at least one) Take (1) - (2) and that is equal to the combination of events where AT LEAST ONE is applicable.

Ratio multiplier when given the sum

If given the sum of proportions (i.e. sum of basketballs, golfballs and footballs = 30) and you also have the individual ratios, then you can calculate the ratio multiplier. Ex: Golf: 3x, football: 4x, and basketball: 2x, then 3x+4x+2x = 30

How to represent number of days passed

If one day past the initial day you can write: (X-1), and this pattern continues for # of days as (x-# of days) Example: $40 for first day, 50 for each day after would be 40 + 50(x-1)

Direct variation

If some varies DIRECTLY (remember to look for the direct word in some form), then that makes that what varies directly, varies by some constant k. For example, if a cars mileage M, varies directly with the speed S, you can say that M = SK, where k is the constant, note this means the constant is equal to M/S. Note: The way to solve this is to first determine what the value of K is equal to, from there you'll usually be given a new M or S, and since you have K and a new M or S you can solve for the other variable which would be the answer.

LCM x GCF

If the LCM of x and y is P, and the GCF of x and y is Q, then pq=xy Example for 24 and 60 24 x 60 = 12 x 120

Steepness of a line

If the RUNS in the (rise divided by run formula) is the same in each line, then the greater the ABSOLUTE VALUE of the RISES, the steeper the line.

Determining the range (all possible outputs) of a function in the form of f(x) = kx^n +c

If the k value is positive then the range is >= c If the k value is negative then the range is <= c Ex: f(x) = -5x^6 + 4 has a range <=4 Ex: If x = 0, then f(x) = 4 Ex: If x = 1, then f(x) = -5+1 = -4

Divisibility rules for 2

If the last digit is even, the number is divisible by 2.

Divisibility rules for 8

If the last three digits of a number are divisible by 8, then so is the whole number.

Divisibility rules for 4

If the last two digits form a number divisible by 4, the number is also.

What can you say of the complement of two independent events

If the probability of two events are independent, then that means that their complements are also independent. Ex: Splitting an apple in one shot if 1/4, so that means that P(Split) AND P(Not split) in any given shot, is independent from other shots. So probability of splitting on first attempt, and not splitting on second attempt is (1/4)(1-1/4) = 3/16

SSS (Side-Side-Side)

If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

ASA (Angle-Side-Angle):

If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

SAS (Side angle side)

If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

Inverse variation

If two things x and y varies INVERSELY, then that means that when x goes up, y goes down and vice versa. To represent this x = K/y, where k is some constant. Note that this also means that xy = K. For example, if given that the speed someones runs, is inverse proportional to their weight, you can write that speed = K/weight, and if speed = 10, and weight = 80, then k is equal to 800, and now given a new speed or new weight you can utilize k=800 to solve the other variable.

Factors and LCM relationship

If x and z is a factor of y, then the LCM of x and z is also a factor of y. Example: 24/3x4 then 24/12 is an integer as well Note: This is only true for the LCM itself, not the multiples, which you would need more information to determine Note: This fact is especially helpful when needing to determine other additional factors of a number, given multiple numbers that are known factors

Factors of factors property

If x is a factor of y, then x's factors are also factors of Example: 10/20 means that 1, 2, 5, 10 are also all factors of 20.

Weighted average with percents

If you are given the weights (i.e. 30% of the total) then you simply multiply the percents times the values. Ex: At a certain school, 1,000 students took a standardized exam that was scored between 0 and 80. Thirty percent of the students scored a 40, fifty percent scored a 50, and twenty percent scored a 70. Answer is .3(40) + .5(50) + .2(70) = 51 Note: The reason you don't divide is because the .3+.5+2 is equal to 1 (i.e 100%!!) so you could think of it as divided by the 1, but it's the same idea, where the "weights" are the percents. Note2: If you aren't given all of the percents (i.e. they don't add to 100) then you MUST DIVIDE BY THE SUM OF THE PERCENTS

Sequence data sufficiency implications

If you are only given a couple of outputs of a sequence, but not the formula for the sequence itself, that is NOT sufficient information. You can't assume that a pattern is being followed. Ex: If x is the fifth term of a sequence, what is x? (1) First four terms are 2, 4, 16, 256 (NS) (2) Each term of the sequence from the SECOND term on is obtained by squaring the previous term (NS) Both statements are suffecient

Multiplication of DIFFERENT BASES, with the SAME EXPONENT

If you have difference bases, with the SAME exponent, you can multiply the bases and keep the exponent. Ex: 2^4 times 3^4 = 6^4 Ex: x^a times y^a = [xy]^a

Square diagonal area method

If you know the lengths of the diagonals, the area is half the product of the diagonals. Since both diagonals are congruent (same length), this simplifies to: area=d^2/2, where d is the length of either diagonal

Multiplication/division across an inequality

If you multiply or divide by a negative, flip the sign. If not negative then no flipping of the sign is needed.

Formula way to solving FACTOR formula logic questions (I.e. by what factor would the result change, how many factors would the result change, etc)

If you're asked by what factor does a new formula result change by an original formula, you can simply do: New formula DIVIDED by old formula Ex: What factor does rate change when you multiply distance by 2 (formula original = Distance / Time, and new = 2Distance / Time) 2d/t DIVIDED by d/t is equal to 2d/t TIMES t/d which equals 2. Note: A secondary way to see this is to see that 2 TIMES d/t means that you are DOUBLING the d/t original formula, and if the original equals some number x, then you are doubling x. Conceptually this works because d/t=x and it are equal and replaceable by each other.

f(x) = y properties

In f(x) = y: x is the independent variable y is the dependent variable

Shaded region question

In general, the best way to approach these problems are to perform the following: Area of the ENTIRE figure MINUS the area of the UNSHADED region = the area of the shaded region

A/B approach to "what percent" question

In problems where the question is asking x is what percent of y, you can represent this as a/b x 100, where the x=a, and the y=b. Ex: 105 is what percent of 15, is equal to 105/15 x 100 = 700% Note: The multiplication by 100 is required when the answer choices are in percents.

How to halve a ratio, or double a ratio

In the most simple logical way possible, for halving you just multiply the ratio by 1/2, and for doubling you multiply by 2. Ex: 1/2 of 4x:6x is equal to 4x/12x

Combinations where there are multiple sub groups (ex: Selecting what types of cheese topping, and what type of veggie toppings on a pizza) (I.e. Approach when subgroups separated by "And")

In these situations you approach as follows: (1) Define the subgroups you are working with. In the example given, you have two groups, cheese toppings and veggie toppings (2) Utilize the combination formula on each of the subgroups. (Ex: if 4 cheese and you must choose two, then 4!/(4-2)!2! (3) Multiply the results of each subgroup in step (2) together (note: this is the fundamental counting principle)

"At least" probabilities

In these situations, if you are given a question that asks what is the probability of "at least" something you must add each probability scenario together (I.e. where the probability scenario represents all of the possible selections equal to or greater than). For example: Probability of at least 8 out of 10 P(selecting 8) + P( selecting 9) + P(selecting 10)

Question where you're given two conversion factors that share at least one common unit

In this situation you should calculate amounts that are associated with the COMMON unit. Ex: 1 attometer = 10^-24 meter and 1 yoctometer = 10^-20, and what is 10 att / 1 yoct You would convert attometer into meters by multiplying 10 attometers by 10^-24 meter / 1 attometers to get meter and compare the meter conversions.

How numbers are both perfect squares and perfect cubes

Infinitely many, don't let a question trick you up by thinking that there's a limit on this. NOTE!!! the numbers 1, and 64 however are the only ones between 1-100, so keep in mind that least if dealing with a range. Also conceptually realize what is means to have this property. That means that the number when factorized in some form must have a power that is a multiple of 2 and 3, for a LCM of 6. So it must be a power of 6. (i.e. 1^6 is 1, 2^6 is 64, 3^6 is 729, and that pattern continues)

If given a slope intercept with a slope and y intercept, how to graph a line

Input values of x into the given slope intercept formula. Each ordered pair must line on the line, which will be true for any value of x. Big idea is that if given a slope intercept formula, you can determine ANY point on the line.

Square root of x^2

Is equal to the absolute value of x, can either be x or negative x unless a conidition provides that x is <0 or >0

Dividend

Is equal to the numerator Note: Divisor (i.e. factor) is the denominator

Isosceles right triangles in circles

Isosceles right triangles (45-45-90) inscribed into a circle can have one if its LEGS as the RADIUS of a circle

If known that there are 12 students taking tests, and you are given the average test scores of x students, and y of the students, (where x and y sets are different students). What is true of the sum of the students test scores?

It must be true that the sum of X test scores + sum of Y test scores = Total sum of all test scores. Knowing this concept will help to solve questions where you need to find individual groups/pieces in questions.

Converging rate questions

It's critical to remember that the distanced traveled by object 1, and distance traveled by object2 must equal the total distance of object 1 + 2.

Testing for even/odd with multiple sums/differences

It's not like multiplication where one 2 will give you even, it's more like positive negative where for example: 1 + 3 + 5 3 + 5 = 8 [even], but add in the one now and you get 9[odd]

When is it NOT true that x doesn't equal y in a^x = a^y

It's possiblly not true if the bases of a are either: (1) 1 (2) -1 (3) 0 Ex: 1^15 = 1^2

Combination vs. permutation for job roles

Job roles are a PERMUTATION. For example, of 20 frat member, how many different unique ways can you select a beer pong officer, flip-cup organizer and poker dealer. 20X19X18

Operations involving the same bases:

Keep the base, add or subtract the exponent (add for multiplication, subtract for division) multiplication: a^n * a^m = a^[n+m] division: a^n/a^m = a^[n-m]

Operations involving different bases, and same exponents

Keep the exponent, multiply or divide the bases multiplication: a^n X b^n = [ab]^n [Ex: 2^2 3^2 = 6^2] division: a^n/b^n = [a/b]^n [Ex: 6^2/3^2 = 2^2 ] exponent: [a^m]^n = a^mn exponet^exponent: a^m^n = a^m^n NOT a^mn

LCM and GCF, when y divides evenly into x

LCM(x, y) = x GCF(x, y) = y Example: 100 and 25 LCM = 100 GCF = 25

Determining the last digits of products

Last n digits of a product of integers are last n digits of the product of last n digits of these integers. Example1: The last digit of 85945*89*58307=5*9*7=45*7=35=5? Example2: The last 2 digits of 845*9512*408*613 would be the last 2 digits of 45*12*8*13=540*104=40*4=160=60

What are constonents

Letters that are NOT vowels. (May come up on some permutation problems)

Adding/subtracting decimals

Line up the decimals along the decimal point, and add/subtract as normal

How many exponents must the factorization of a perfect square have?

MUST HAVE EVEN EXPONENTS Example: 4 = 2^2, 16 = 2^4, 36 = [3x2]^2 Note: Similar idea for a perfect cube, where the prime factors must have powers of 3

Standard deviation

Measurement of how far data points in a set are from the mean of such data set. Higher SD = Data points are farther away from the mean Lower SD = Data points are closer to the mean

How to deal with an equation with a square root of two or more terms (i.e. sqrt([x+y])

Move everything, other than the sqrt, on to one side and then square each side. Ex: sqrt(x+y) - 2 = 3, then sqrt(x+y) = 5, then x+y = 25

Dividing a number by 10^x, how many decimal to the left does the number move?

Moves equal to X PLACES TO THE LEFT Ex: 300/100 = 300/10^2 = 3 Ex: 5/100 = 5/10^2 = .05

How to calculate the sum, given an average a number of items in a set?

Multiply the average by the number of items in a set. This is because Sum of Terms/# of terms = Avg, so Avg * terms = Sum

Convert a decimal to a percent

Multiply the decimal by 100 and attach a percent sign. Ex: .25 x 100 = 25%

Converting a fraction to a percent

Multiply the fraction by 100 and attach a percent sign. Ex: 1/4 x 100 = 100/4 = 25% Ex: 1/3 x 100 = 100/3 = 33 and 1/3% Note: If the denominator is a multiply of 100, it's often easy to get the fraction into a form of x/100 and x (i.e. the numerator) is the percent.

Calculating an arc length when given an inscribed angle, but not a central angle

Multiply the inscribed angle by 2, and that will provide the central angle which you can use in a ratio of central angle / 360 degrees to determine the portion of the perimeter that is the arc length.

Rationalizing a two term radical denominator

Multiply the numerator and denominator by it's conjugate pair, which creates a difference of squares pattern in the denominator Ex: 4/a-sqrt(b) = 4(a+sqrt(b))/(a-sqrt(b))(a+sqrt(b)) = 4(a+sqrt(b)/a^2-b Essentially to rationalize a denominator in the form a-sqrt(b) multiply numerator and denominator by a+sqrt(b).

How to calculate a power to a power

Multiply the powers together. Ex: [x^2]^3 = x^6 Ex: [x^a]^b = x^ab

Range of possible remainders

Must be a non-negative integer that is less than the divisor

What is the work of two combined objects equal to?

Must be equal to the work of object1, plus the work of object2 Note: Similarly for opposing work problems, where one objects decreases the output of another, you need to subtract the opposing work from the object creating work.

What is true when abs(a-b), and abs(a) - abs(b)

Must be true that abs(a-b) is GREATER THAN OR EQUAL TO abs(a) - abs(b) Example: A = 6 B = -2 -abs(6-[-2]) >= abs(6) - abs(-2) -8 >= 4

Factorial division properties

N! would be divisible by any number included within N!, or any combination of multiplied numbers with N1 Example 4! is divisible by 4!, 3, 2, 4x3, 2x1, etc.

Difference between a divisor and a factor

NO DIFFERENCE, they are the same example: 3 is a divisor and factor of 9

Does the number of UNIQUE prime factors change when a number is raised to an exponent?

NO! For example: 8 = 2^3 8^2 = 64 64 = [2^3]^2 = 2^6 Another example: 18 = 3^2 x 2 18^3 = 5,832 5,832 = [3^2 x 2^1]^3 = 3^6 x 2^3

When setting up an equation with "what percent", do you have to multiply by 100 to get into percent form?

NO, if you take the x/100 approach (vs. x/y approach), then the x is already in a percent.

Given x<y, and that x and y have the same NUMBER of prime factors, and same distinct prime factors, is y/x an integer?

NO, would never be an integer. Examples: (1) y = 18 [3^2 x 2^1] x = 12 [3^1 x 2^2] Conceptually, you can visualize this by assuming two distinct prime factors, 7 and 5: The numerator MUST be larger, so: y=5^1 x 7^2 and x=5^2 x 7^1 and obviously this wouldn't be an integer, and this same idea would continue for other numbers.

How do you quickly find the POSITION (i.e. where it is, not what it is) of the median in a set of !!!EVEN!!! number?

Need to calculate two different positions to easily find the median: (1) Position n/2, where n is the count of items in the set (2) Position n+2/2, where n is the count of items in the sets. Take the AVERAGE of the VALUES AT THE GIVEN POSITIONS FROM THE FORMULA, and this provides the median.

Equations with two absolute values (both sides)

Need to solve the equation twice, once where both are equal without change, and the other where either side is negative Ex: 16x+14 = 8x+6 AND 16x+14 = -(8x+6)

Converting decimals to percents

Note: perform the inverse for converting a percent to a decimal

A and B ONLY, B and C ONLY, and A and C only in venn diagram

Notice how this portion specifically excludes the overlap of the A AND B AND C. Individually, the total of A AND B minus the A AND B AND C would equal the A and B ONLY.

Regular polygons inscribed into circles

Number of sides of the regular polygon = the number of equal arcs Example: Given a 5 sided polygon, and you need the arcs of three sides, then you can take 3/5 times 360

Mode

Numbers that occurs most frequently in the set Note: If more than one number shares this property, then you can have multiple modes (possibly important concept for data sufficiency) Note: If all numbers are unique OR all numbers occur at the same frequency, then you NO MODE.

A line that is tangent to a circle (i.e. just barely touches) intersects how many times

ONE TIME!!! (not 0 times), the tangent counts as one intersection

Permutations

ORDER DOES MATTER.

Odd/odd

Odd

Average rate question with a round trip and no given distance

Often in these problems you'll be given two rates, but not given a distance, BUT you will know that the distance is the same for the to trip and return trip. So you can either assume an equal easy to use distance or a variable. ( example: [2x / [x/rate1 + x/rate2]] ) The example above can be generalized to be 2D / d/rate1 + d/rate2, and remembering that d/rate = time which means that we're litterally taking here the sum of the distances (2d) and the sum of the times (d/rate1 + d/rate2)

In a converging rate problem, what is true of the distance initially between two objects, and the distance traveled

Once the two objects converge, the SUM OF THE DISTANCES will be equal to the SUM OF THE INITIAL DISTANCE BETWEEN.

Trapezoids

One pair of opposite sides parallel, but the other side is not.

Calculating X standard deviations away from a 50% mean with a 10% standard deviation

One standard deviation is equal to either 40% or 60% (i.e. 50-10, or 50+10) More generally the SD * # of deviations is how much you'll plus or minus. Example question: Given set of 86,79,90,92,76,81,83 how many values fall within a single SD of 5 given that the mean is equal to 87. Answer to this is 4, because 87+-4 is 82-92 and 4 values fall within the 82-92 range.

Prime numbers have how many divisors/factors?

Only ever two factors/divisors: (1) the number 1 (2) and itself

Combinations

Order does NOT matter. For example: Select 3 coffees out of 5, it doesn't matter which order you select, just which ones are selected. (i.e. Selecting coffees 1,2,3 is the same as selecting coffees 3,2,1)

Given a/b = c/d what corresponding ratios can you find?

Original a/b = c/d - b/a = d/c - a/c = b/d - [a+b]/b = [c+d]/d - [a-b]/b = [c-d]/d - [a+b]/[a-b] = [c+d]/[c-d]

P(A or B) when A and B are mutually exclusive

P(A or B) = P(A) + P(B) This formula requires the events are mutually exclusive, which means that if one event occurs the other can never occur. For example, flipping a coin and landing on heads is mutually exclusive from landing on tails. Ex: Probability of selecting a cat and dog out of 3 cats, 3 dogs and 4 chickens. P(cat) = 3/13 + P(dog) = 3/13 = 6/13

P(A or B) when A and b are NOT mutually exclusive

P(A or B) = P(A) + P(B) - P(A and B) Example: Probability of selecting an even number or a perfect cube from 1 - 50 inclusive (1) Denominator is 50 because 50-1 + 1 = 50 (2) Determine P(Even) which is equal to (Highest even multiple - smallest even multiple / common difference + 1 = 25, so then P(Even) = 25/50 (3) Determine P(Perfect cube) which you need to consider 1, 8, 27 so 3/50 (4) Determine the P(Even AND perfect cube) and 8 is the only even perfect cube, so P(A and B) = 1/50 (5) Final answer is 25/50 + 3/50 - 1/50 is equal to 27/50 = 54/100

(-1)^0

POSITIVE ONE. Any number to, positive or negative, to the zero positive is equal to positive one.

How do you handle questions that have the word "percent" in them?

Percent on the GMAT means to DIVIDE BY 100. It quite literally means to divide by 100. In general you need to always think of the word percent, or % symbol as meaning divide by 100. Note: The percent % symbol itself also represents dividing by 100. Ex: X percent = x/100 Ex: 500 percent = 500/100

Price per item relationship

Price per item x Number of Items = Total cost of items purchased

Probability as a fraction/ratio

Probabilities effectively also act as fractions/ratios. What this means is that you know the relative amounts, but you don't know the absolute amounts. However if you know any amount individually, like in ratios, you can find the absolute amounts. Ex: probabilities of selecting a blue marble out of red and blue total marbles is 1/4, so that means 1x = blue, 3x = red and blue and red = 4x.

Profit and loss equation

Profit = Total Revenue - Total Cost

When dealing with problems that give you beginning value and ask you to get to an ending value, how do you quick determine the ending value? Ex: Given you start with 100, and 1/5 is taken away, what do you end with?

Quick way: You end with 4/5(100) = 80 Long way: 100 - 1/5(100) = 80 The "quick way" will be critical for solving harder word problems.

Ratios and the impact of increasing or decreasing individual parts of ratios

Ratios act identically to fractions, and as such follow the same rules of the values of changing fractions. Examples: -Adding 1 to each of 1x:3x INCREASES the ratio because the ratio moves closer to 1 -Adding 1 to the numberator of 1x:3x INCREASES the ratio because adding to a numerator with the same denominator results in a larger fraction.

Three common point reflections

Reflecting over the x axis: When a point (x, y) is reflected over the x-axis, the image is (x, -y) (ex: (2,3) reflected over the x axis is (2, -3) Reflecting over the y axis: When a point (x, y) is reflected over the y-axis the image is (-x, y). (Ex: (-1, 4) reflected over the y axis is (1, 4) Reflecting over the origin: When a point (x, y) is reflected over the y-axis the image is (-x, -y). (Ex: (-1, 4) reflected over the y axis is (1, -4) These reflection point approaches hold true for all shapes (triangles, points, line segments, lines). To perform the flection take each point of the shape and perform the associated reflection while connecting the reflected points.

How to represent consecutive integers (even, odd or regular)

Regular = x + x+1 + x+2 Even = x + x+2 + x+4 Odd = x + x+2 + x+4 Multiples of 8 = x + x+8 + x+16 (!!NOT!! x + 8x, this could work if you know x is 1 but you will almost never know this.) NOTE: EVEN AND ODD HAVE THE SAME REPRESENTATION!!!! Critical to remember this.

Fractional parts must add up to a whole

Remember that if parts were split up that must always equal back to the whole.

Pythagorean triplets

Remember these below, there's more but these are common: 3,4,5 5,12,13 8,15,17 7,24,25

"Percent of" word questions

Represents the multiplication of a given percent, by a given number Ex: 5 percent of z = 5/100 X z = 5z/100 = z/20 Ex: m percent of p = m/100 X p = mp/100 Ex: 1 million percent of y = 1,000,000y/100 = 10,000y Ex: 1 one thousandth of z = [1/1000]z/100 = z/100,000

Sample Space (Probability)

Represents the set of ALL possible outcomes, and the sum of the individual outcomes are equal to 1. Ex: 40 red marbles, and 60 blue marbles, thne P(Red) = 4/10 and P(Blue) = 6/10, so P(Blue and Red) = 1 Big idea - the probability of a sample space MUST BE EQUAL TO 1.

Base: negative fraction Exponent: even positive integer

Result is LARGER (-1/2)^2>(-1/2)

Base: negative fraction Exponent: odd positive integer greater than 1

Result is LARGER (-1/5)^3>(-1/5)

Base: positive fraction Exponent: odd positive integer greater than 1

Result is SMALLER

Base: Less than -1 Exponent: Odd positive integer greater than 1

Result is SMALLER (-5)^3<-5

Base: positive fraction Exponent: even positive integer

Result is SMALLER (1/2)^2<1/2

Base: Less than -1 Exponent: even positive interger

Result is larger (-5)^2 > -5

Base greater than 1 Exponent: even positive integer

Result is larger (5^2>5)

Base: Greater than 1 Exponent: odd positive integer

Result is larger 5^3>5

Parallelogram

Shape with two pairs of parallel sides Any two consecutive angles are supplementary (add to 180), for example A+B = 180, B+C = 180, etc Area = base x height Note: Height = perpendicular to the base

Triangle inscribed in a square

Side of the square, and the sides of the triangle are equal length

Catch up and pass questions

Similar to catch up questions, but must be mindful of having to adjust the distance of one object as a function of the other + additional distances. Note: The short way to solve these problems is to calculate the difference between distance divided by difference between time. Note that this short cut DOES NOT WORK IF DISTANCE1 == DISTANCE2

Graphing functions

Similar to graphing lines, where: f(x) = the y value of the ordered pair x = the input (i.e. the x value of the ordered pair)

Trick to get integer remainder when you know both the full decimal quotient, and the denominator of the original fraction

Simply multiply the denominator of original fraction times the DECIMAL portion of the quotient. Example: 9/5 = 1.8, so .80 * 5 = 4, so the remainder here is definitely 4. !!!MUST KNOW THE DENOMINATOR!!!

Rationalizing a SINGLE TERM radical denomiantor

Simply multiply top and bottom of the fraction by the denominator. Ex: 3/sqrt(5) =3 x sqrt(5)/sqrt(5) x sqrt(5) = 3sqrt(5)/5 The quickest approach to this is to realize that give a/sqrt(x) this is equal to a x sqrt(x) / x. (i.e. take full radical up and keep what's under the radical on the bottom) Conceptually this works because you're multiply the fraction by 1 (i.e. denominator/denominator) so actual value of the fraction doesn't change, just the form that it is in.

Counting consecutive numbers inclusive of either only first or last (Ex: # in people between 50th in line and 10th inline, only including 50th in line)

Simply subtract the last from the first. DON'T ADD 1 TO THIS.

Hiding in plain sight - slope intercepts

Slope intercepts may be "hidden" in that they won't be in y = mx + b form, BUT they can be in other forms that you can rearrange. Ex: 3x+5y = 8 is equal to the following in slope intercept form: y = (-3/5)x + 8/5, thus the slope is -3/5 and the y intercept is 8/5 Ex: what is the y intercept of Y = 10 + X (1) place into slope intercept form y = x + 10 (2) y intercept is equal to 10 because 10 is in the place of b in the y = mx + b formula and b is equal to the y intercept

How to solve >=3 overlapping sets

Solving these requires the use of a venn diagram. Note: Outside of the diagram can be a section that doesn't fall into any of the sets.

Splitting an equilateral triangle in half properties

Splitting in half creates two 30-60-90 triangles. Remember a 30-60-90 has a ratio of x to x√3 to 2x This means that the area of an equilateral triangle can be determined with height alone.

Is x>12 (1) sqrt(x^2 - 24x +144) = 1 (2) x^2 - 169 = 0

Statements one: Step 1: sqrt([x-12]^2) = 1 Step 2: abs(x-12) = 1 - remember in this step that the sqrt of a square results in an absolute value. Step 3: x-12 = 1, and x-12 = -1 Step 4: x = 13, and x = 11 Statement two: Step 1: (x+13)(x-13)=0 Step 2: Because of zero product property, set x+13 = 0 and x-13 = 0 Step 3: x = 13 and x = -13 Both statements together allow you to triangulate and confirm that x = 13 so x>12

Box and fill combination approach

Step 1: Draw boxes equal to the number of decisions you'll make (i.e. select 3 coffees out of 7 means you need 3 boxes) Step 2: Within the boxes themselves start box 1 with the largest number from total selections you're making (in example above, = 7), and for the next box highest -1, third box highest -2, etc. Step 3: Multiply the numbers in the boxes together, and divide by the # of boxes FACTORIAL.

How to factor 3^az X 3^bz X 3^cz

Step 1: When you multiply like bases you can add the exponents, so 3^az+bz+cz Step 2: Factor out the common z from the addition of the exponents, so 3^z[a+b+c]

Counting consecutive integers EXCLUSIVE of the first and the last (Ex: # in people between 50th in line and 10th inline, excluding 50th and 10th)

Subtract the last from the first, MINUS one

Triangle perimiter

Sum of all sides -An equilateral triangle has the largest relative area, and smallest perimiter

The average (arithmetic mean) age of a group of 10 workers is 46. If the average (arithmetic mean) age of 6 of the workers is 30, what is the average (arithmetic mean) age of the other 4 workers?

Sum of group of all = 460 Sum of group of 6 = 180 Sum of group of 4 = 4x 180 + 4x = 460 Note: 4x because count times an average is equal to a sum, and 180 + 4x =460 because sum of the parts equals the whole.

Finding number of trailing zeros in n factorial

Sum of rounddown(n/5^k) where 5^k <= n 32! trailing zeroes = 32/5 + 32/5^2 = 6 + 1 = 7 trailing zeroes

Finding the number of powers of a prime number p, in the n!

Sum of roundown (n/p^1)... + n/p^k) Example: Powers of 2 in 25! = 25/2 + 25/4 + 25/8 + 25/16 = 12 + 6 + 3 + 1 = 22 powers of 2 in 25!

Mean/Average Formula

Sum of terms / number of terms

Divisibility rules for 3 and 9

Sum of the numbers divisible by 3/9

what do supplementary angles sum up to?

Sum up to 180 degrees Remember also that supplementary angles effectively create a straight line

Median

THE MIDDLE OF THE SET OF NUMBERS. -If odd quantity in set, then it's simply the middle number. -If even quantity in set, then it's middle two numbers divided by 2.

When given the TOTAL of one of three sections in a venn diagram, how do you find the portion associated with ONLY A (i.e. not A,b or A,B,C) etc

Take the TOTAL - the two doubles (i.e. A,B) and the one triple (i.e. A,B,C)

How do you quickly find the POSITION (i.e. where it is, not what it is) of the median in a set of !!!ODD!!! number?

Take the count of items in the set + 1 and divide that by 2. Ex: 1,2,4,5,8 then this means that 5+1/2 = 3, then position 3 is where the median is Example of where this is useful: "A survey was conducted to determine how much money 95 shoppers spent on holiday gifts. 15 shoppers spent $50 each, 20 shoppers spent $72 each, 10 shoppers spent $80 each, 12 shoppers spent $95 each, and 38 shoppers spent $100 each. Of the 95 shoppers surveyed, what was the median number of dollars spent on holiday gifts?" With using this formula it's easy to see that 48 is the median position, which falls within the $95 range.

Divisability rules for 7

Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7. 28 = 8 X 2 = 2 -16 = 14/7 = 2

How to check whether information given in a problem is useful to a ratio?

Test numbers for the variables that still remain, and determine if the resulting fraction would be different with different numbers. Remember you can use extremes like 1,100, 1000, etc, be careful of using 0 if it doesn't logically makes sense too or if there's an explicit contrainst.

What does a line over a number represent

That the numbers that are UNDER THE LINE repeat infinitely. Remember that it's only the number UNDER THE LINE, not the rest of the number.

What percent questions

The "what percent" in these question types are equivalent to x/100. Ex: 80 is what percent of 50. This equals 80 = x/100[50] Note: When using the x/100 method you don't have to multiply by 100, because the x is already a percent.

Scientific notation

The 10^x power represents the number x places a number moves to the right or left, depending on the sign of x. When expressing to a WHOLE number greater than 1, then the number of exponents represent the number of 0's. Ex: 3 x 10^2 = 300 When expressing a fraction greater than 1, then the number of exponents represents the number of 0's MINUS 1 Ex: 3.3 x 10^2 = 330 When expressing a whole a number less than 1, then the number of exponents represent the number of decimals after the decimal place (i.e. 0's are equal to exponent -1) Ex: 3 x 10^-2 = 0.03

PEMDAS (hint: left to right for certain parts of pemdas)

The GMAT WILL try to trick you on the multiplication/division left to right part of pemdas.

Balance point method for finding average (and median) of an EVENLY SPACED set (i.e. using the middle)

The MIDDLE NUMBER (i.e. the median) is equal to the average. Note: This works because in an evenly spaced set, the mean=median. REQUIRES EVEN SPACING OF NUMBERS (doesn't need to be consecutive)

Simple approach to collectively exhaustive events with restriction

The TOTAL combinations without restriction MINUS the combinations in the OPPOSITE restriction = the combination restrictions you are looking for. Note: This approach requires that you have collectively exhaustive events and mutually exclusive events. Example: Three people are to be selected from a pool of five people to form a writing club. If two of the people can never be together in the club, in how many different ways can the club be formed? (1) Total possible events = 5c3 (2) Opposite restriction, where two ARE together is equal to 3c1 (because two people are already chosen, which leaves 3 remaining to choose from, and 1 slot to fill) (1) - (2) is equal to the answer, so 7.

Area of a 45-45-90 triangle in relation to a square

The area of a 45-45-90 triangle is one half of the area of the square. This means that the diagonal of a square is equal to one of the squares side root 2, because a 45-45-90 has a ratio of x, x, x√2

Consecutive increase in percent (i.e. discount) and then decreasewhere you want the same initial value after the increase

The consecutive DECREASE MUST BE LESS THAN THE INCREASE (this is vice versa from the consecutive decrease with a follow-up increase back to original)

Conversion factors and the DRT questions

The conversion factors are critical to DRT questions. For example, if give a rate of 10 miles per minute, and you want how many miles in 1 hr, you need to first convert 10 miles per minute, to miles per hr, OR convert 1hr to minutes.

Circumfrance

The distance around a circle 2 x pie x R or pie x diamater

General way to solve catch up and pass questions

The distance of the person passing, is equal to the distance of the person NOT passing + the incremental pass units.

What is the range of possible remainders for any division

The divisor/denomiantor - 1 down to 0 Example: a/7, has a range of possible remainders from 0-6. Note: Anything above 6 would just be more added to the quotient

Nonzero base raised to the 0 power

The expression is equal to 1. - x^0 = 1

Relationship between 1, a factor of a number, and the number itself when the number isn't 1 or the factor isn't equal to the number

The factor of a number is going to be between 1 and the number itself. Is y is a factor of x, then 1<=y<=x

Formula for dependent events P(A and B)

The formula for dependent events is P(A and B) = P(A) × P(B|A), where P(A and B) = the probability that both event A and event B will occur, P(A) = the probability that event A will occur, and P(B|A) = the probability that B will occur given that A has already occurred. Example: What is the probability of picking two girls in two attempts, if a girl can only be selected once and you have 14 girls and 14 boys. (1) P(A) = P(Selecting girl on first attempt) = 1/2 (i.e. 14/28) (2) P(B|A) = P(Selecting girl on second attempt) = 13/27 (3) P(A) x P(B|A) = 1/2 X 13/27

Multiply or dividing a constant to a set, impact on standard deviation

The impact to the SD is a FACTOR of the multiplication/division. Ex: SD of a,b,c is 3, then 3a,3b,3c means the new SD is 3 times 3 Ex: Similarly for division if we divided each term by 3, the new SD would be 3 divided by 3 Note: Multiplying or dividing each number by 1 will NOT change the SD, which is something to consider for data sufficiency questions.

Using a triangle to determine the length of an arc

The important theme to understand here is that if given a central angle and an inscribed angle, and if they share the same endpoitns, then the central angle is 2x degrees of the inscribed

Converging rate where two objects leave at the same time

The key to solving these problems is to remember two things: (1) The distance at which one object meets will be X, and the other distance will be Total Distances MINUS x (2) Both of the times will be the same. Option 1 to solve: Ex: D = 240, rate1 = 60mph, rate = 40mph, when will they converge (i.e. T?) Equation 1: 60T/1hr = x Equation 2: 40T/1hr = 240 - x So then you would set x in the secon equation equal to 60T/1hr and solve from there. Option 2: Distance of 1 = 60t, distance of 2 = 40t, then it must be true that 60t + 40t = 240

Relationship between sides and angles in triangles

The largest side cooresponds to the largest angle, smallest angle = smallest side, etc. This means that the length of the middle side must be less than or equal to the side opposing the largest angle, and greater than or equal to the side opposing the smallest angle.

Divisibility rule for 4

The last 2 digits are divisible by 4

Relationship of the Sides of a Triangle

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

Minimum value of f(x) = x^2

The minimum value is 0, as you can never get a value less than 0 from this function. Similarly, f(x) = x^2 + y means that y is the minimum value.

Minor arc vs. minor arc

The minor arc is the shorter portion between the two points, and the major is the longer portion. Be default it will refer to the shorter portion

Trailing zeroes

The number of trailing zeroes of a number is equal to the number of (5 x 2) pairs in the prime factorization of that number. Ex: 100 = 10x10 = [5x2]x[5x2] = two trailing zeroes

How to represent trailing zeros in number form

The number of trailing zeros are equal to the number of 5x2 pairs. Example: 5,000 = [5x2]x[5x2]x[5x2]x5 Note: This also means that any >=5! will have 0 as its trailing zeroes as its units digits

If two absolute values are equal, what must be true?

The numbers are either equal or opposite. To solve a problem like this do: (1) Where both absolute values signs remain unchanged (2) Where ONE SIDE is multiplied by a negative Solve for (1) and (2) and that provides all of the possible solutions.

EXTERIOR angle of a triangle, is equal to the two opposite/remote INTERIOR angles of the same triangle

The opposite remote is equal to the two inners.

Even odd pattern of multiplication

The product of an even number and any integer is equal to even. Note: This is true because any number evenly divisible by 2 is even, and for a number to be even it must contain at least one factor of 2.

If two numbers, x and y DON'T share ANY unique prime factors, what's the LCM of x and y?

The product of x and y is the LCM Note: Any two consecutive numbers DON'T share common factors, so the product is their LCM.

How to quickly determine what remains of a fractional value times a fractional value taken away.

The quickest way to solve the remains column is to multiply the starting value by the complement of how much is being taken away. For example, if you drink 3/4 of a half of a cup of coffee, what remains is 1/4 of the 1/2. THIS IS CRITICAL TO ANSWER THESE QUESTIONS EFFICIENTLY.

What is the remainder of an EVEN number divided by two?

The remainder is always goes to be ZERO

What is the reminder of an odd number divided by two?

The remainder is always going to be ONE

Overlapping sets and mutual exclusivity

The rows and columns must be mutually exclusive. Example: AP Bio and AP History, you could NOT have both Bio and history in the same column or row. But what you can have is Do take bio, and don't take bio as a row, and Do take history, and don't take history as the columns.

"At least" combination example with two different subgroup examples (Subgroup 1 at least 2)

The screenshot example given is a scenario where you are counting number of options where you are choosing 4 slots with AT LEAST two video games from subgroup populations of 6 video games and 6 fruit

Permutation arrangement with a restriction that two items must be next to each other

The simplest approach to these type of questions is to treat the two units as one. NOTE however that after account for the two units as one in the permutation, you'll need to consider if the two units that must be next to each other could be flipped, IF they can be flipped in order then you must multiply by 2 to account for this.

If given that a line has a slope of x/y, and you are told that another line is perpendicular, what is the slope of the other line?

The slope of the other line is equal to -y/x, because (x/y)(-y/x) = -1, and the product of two perpendicular line slopes is equal to -1.

Exterior angles of triangles properties

The sum of the exterior angles (i.e. the angles themselves and their supplements) together sum up to 360 degrees. This rule applies to all polygons.

What is true when you take the even root of an even powered variable where the root and variable are equal (Ex: sqrt(x^2) )

The variable under the sqrt become an absolute value. Ex: sqrt(x^2) = abs(x), and if x^2 = 9, (equals abs(x) = 3 then taking the sqrt means what values are a distance from 0 on the number line (i.e. x=+/-3.

Zero product property and inequalities

The zero product property also applies to inequalities. For example a(b-c) > 0 means that a>0 and b-c>0, or b>c

If a is a factor of b and b is a factor of c

Then a is a factor of c. any integer is divisibile by al of its factors and it's factors factors. I E: 72 is divisible by 8 and 8 is divisible by 2 and 4. Thus 72 is also divisible by 2 and 4. So 2 and 4 are factors of 72

Joint direct variation (Ex: x is directly jointly proportional to y and z)

Then this means that x = k times yz. In simple terms it means that y and z are multiplied to k to be equal to x.

Joint inverse variation (Ex: x is inversely jointly proportional to y and z)

Then this means that x = k/yz In simple terms it means that y and z are divided by k to be equal to x.

How many 0's are there in 10^x

There are X number of zeroes. Example: 10^2 = 100 [i.e. 2 0's] Example: 1,000 = 10^3 [i.e. because there's 3 0's in 1,000) Same idea for 10^x, except you can represent this as 1/10^x Example: 10^-2 = 1/100

Formula way to solving PERCENT formula logic questions (I.e. by what percent would the result change, how much percentwould the result change, etc)

These are almost identical to factor questions, except that you must make sure to use the percent formula. I.e. [New-Old]/Old X 100

"Percent less than" questions

These can be generally represented as Final value = initial value X (1- "Percent less than"/100)

Number of terms in an average formula when you terms such as (x+1), (x+7), etc, how many "terms" do the (x+z)'s count for?

These count for ONE TERMS. Don't trick yourself into thinking that since there's an x and another number in (x+z) that it's two, it is one.

Solving probability questions utilizing qudratic equations

These types of questions will come up most often when you are given a final combined probability, and are asked to find the number of one of the items. For example: 10 total marbles only blue or red, and P(two red marbles, consecutive selections) = 2/15, what is the total of red marbles? (R/10)(R-1/9) = 2/5, and solve for R which wil lend up as R^2-R-12 and finally (R-4)(R+3)

Probability where everything selected is of the same time

These will often come in the form of combinations problems, and can be solved in two steps: (1) Determine the numerator by performing xCy for each subgroup and add the results of each subgroup together, where x is the total for a subgroup, and y is the number of selections. Conceptually you do this because the number of possibilities where they're all the same is directly correlated with the number of subgroups (i.e. all chocolate, all sour, etc). (2) Determine the denominator by performing zCy, where z is the SUM of all of the subgroups, and y is the number of selections. Example: Given 8 Bday cards, 5 holiday cards, and 5 graduation cards, what is the probability of 3 selections all being of the same type. (1) 8c3 + 5c3 + 5c3 is the numerator. (2) 18c3 is the denominator.

What is true regarding the decimals of an integer numerator, and non-zero integer denominator?

They will EITHER: (1) Terminate (i.e. .25) or, (2) Repeat (I.e. .333333) Note: If you have a non-integer in the denominator (i.e. sqrt(3) then it would neither terminate or repeat, another example of irrational number would be pi)

Determining the number of digits in powers of 10

Think of this as taking a 1 and adding as many zeroes as there are powers Example 10^2 = 1 + two zeroes, 10^5 = 1 + 5 zeroes

Divisibility by 4 when the last two digits are 0

This WOULD be divisible by 4, because 0/4 is equal to 0 which is divisible, and 4 is a factor of 0.

Adding two inequality equations together

This can be done ASSUMING the two equation are in the same direction. Note: if they're not already in the same direction you can multiply both sides by negative one, or manually flip the inequality.

How can [x/y]^-z be represented?

This can be represented as [Y/X]^Z. -Flip the fraction, and change the sign to positive.

Direct variation and inverse variation combined (Note: NOT JOINTLY, just combined)

This can be represented as follows: (1) x = k1y (2) x = k2/z Together this means that x = k1y TIMES k2/z, which means that k1k2 times y/z, and since k1 and k2 are both constant you can just say that x = k times y/z. The idea here is that for the combined variations you multiply the individual variations together, and the common constants k can be represented as one k.

What does it mean when abs(x+5) = 6

This equation means "what number, when 5 is added to it is a distance from 6 on the number line. The answer to this is 1, and -11.

What do you know if a right angle is inscribed into a circle?

This means that the hypotenuse is equal to the diameter of the circle

Standard deviation of 0

This occurs when all values in a set are EQUAL TO THE MEAN. (i.e. there's no difference from the mean among any items in the set)

Converting a dollars and b cents into a single unit form

This would be equal to $a + $(b/100). Note the reason you divide b by 100 is to convert the cents into dollars. Ex: Detailed conversion 1$/100cents TIMES b cents = $b/100

If given three things, those who grill, bake, and Fry, how can you determine those who ONLY grill

This would be equal to the total that grill MINUS the two double shared (grill,bake + grill,fry) MINUS the triple shared (grill,bake,fry). The idea is your are taking the total minus the shared parts to get to the "only" part of the venn diagram circle portion.

Extended similar triangle example

To be similar the two bottom lines must be parallel

Multipart ratio and finding a "common" ratio

To compare multipart ratios (ex: given x:y, and y:z) you must transform the fractions so that the COMMON ratios are the same. In the example given you would transform the two fractions so that y=y in both individual ratios. In perform that you can compare all of the ratios together. Note: Utilize the LCM of the two common numbers to determine what you need to multiply each fraction by.

How to solve average rate questions

To determine an average rate you take the sum of the distance, divided by the sum of the time (i.e. could be seen as d1 + d2 / t1 + t2. DO NOT take two rates and divide them by two, this will almost always be incorrect. The only way this would be correct is if the weighting is 50/50, but in general just don't do this. Note: This works because this is a weighted average problem. Where you have (Rate1 x Time1 + Rate2 x Time2) /Time1 + Time2 which means that the rates are WEIGHTED by the times. (Remember Rate x Times = distance)

Determining the HEIGHT of water in a 3d shape after x times passes

To determine the height you need to know ALL of the dimensions (length, width, height) not just volume alone. This is because the height of a liquid in a container will vary based on its dimensions (higher if skinnier, lower if wider) Similar idea to fitting boxes/objects within another larger object, need exact dimensions to determine how many of x could fit within a larger x.

Ratio multiplier when given one quantity

To determine the ratio multiplier when given one quantity, you set the ratio multiplied equal to the quantity and solve for x. Ex: Boys:Girls = 4x:3x, and Boys = 8, then 4x=8 and x = 2 so girls equals 6.

A sequence given in terms of n (i.e. An = n + 1)

To determine the values of these sequences you take the current iteration and put that into the formula. Ex: First = A1 = 1+1 = 2 Note: This notation DOES NOT indicate that you input the prior value. Instead you are entering in the value of the nth iteration. (i.e. think using a for loop and a count variable) With these types of sequences you can determine any value given a step (i.e. you don't need previous values). In comparison, a recursive sequence such as An = An-1 + An-2 you NEED the previous values

Determining the number of individuals who like A OR B

To determine this utilize the formula given as: (A OR B) = #(Total A) + #(Total B) - #(A AND B) Ex: 120 learn english, 140 learn math, and 80 learn both, then 120 + 140 - 80 = 180 (i.e. those who learn either math or english) Conceptually think of a venn diagram where you want only the A section , only the B section , plus the overlap section once. Visually you can take sum of Total A and Total B and minus the cross over portion ONCE which leaves only A + Only B + cross over once.

Determine how many of something must be added or removed from a ratio to equal another ratio

To determine this you must perform the following: (1) Determine the multiplier for the ratio. (2) Multiply each ratio by the multiplier (3) Set the multiplied ratio from (2) +/- a variable for what you're looking for equal to the final ratio. Ex: Given A/P is 5/3 and you want how many minus A to get to 2/3 and A = 45, then x = 9, and 45-x/27 = 2/3, solve for x. NOTE: When utilizing the a ratio of a part over the total, if you need to account for that variable in both the part and the total. For example, if you're looking for A/B = 2/3, then A/Total = 2/3 would be A+X/Total + x = 2/3, NOT JUST A.

Dividing Fractions

To divide fractions, invert the second one and multiply

Counting consecutive multiples of a number in a set (ex: how many multiples of 3 in a set from 1-100)

To do this use the following formula: [Highest # divisible in the set (99) - Lowest # divisible in the set (3) / given number (3)] +1

How to calculate the number of people who want Job A or job B

To do this, you need to use a formula to ensure you don't double count any in the "combined" cells of the table. So you must take Total A + Total B - Those who want both A and B To understand why this works you're essentially taking the sum of only A + overlap, + only B + overlap, and removing the duplicated overlap

Finding the count of values in a set with a common difference (Ex: 8, 16, 24, 32, 40)

To find the number between for example 40 and 8 inclusive, you take [40-8]/8 + 1. More generally, you can say that: [End value - start value] divided by common difference + 1 = the count

Square roots of large perfect squares

To get the square root of some large perfect square (i.e. 10,000) (1) Confirm there is an even number of 0's (2) If there is an even number of 0's, take half of the zeros away (3) Take the square of any of the non-zero portion (4) Combination of 2 and 3 above is the square root Example: sqrt(10,000) = 100 (i.e. 100*100) Example sqrt(81,000,000) = 9,000 Similar idea for cubes but instead of half the zeroes, it will be a THIRD of the zeros

Square roots of small perfect squares

To get the square root of some small perfect square (i.e. 0.16) (1) Confirm there is an even number of DECIMAL PLACES (2) If there is an even number of decimal places, find what half of that would be (3) Take the square of any non-zero portion (4) Combination of 2 and 3 above is the square root Example: sqrt(0.16) = 0.4 Example: sqrt(0.0004) = sqrt(4)/sqrt(10,000) = 2/100 = 0.01 Similar idea for cubes, except for half of the decimals it will be a THIRD of the decimals

Total # of unique members in a venn diagram problem, second version

Total # of Unique Members = # [A Only + B Only + C Only] + # [(A and B) Only + (A and C) Only + (B and C) Only] + # [(A and B and C)] + # [Neither A nor B nor C]

Translate to an equation: "A is what percent of b"

Translating this to an equation is equal to a/b X 100 Conceptually you can thank of this as setting up the equation A = X/100 times B and then solving for X, when you solve for X you get a/b times 100

Isosceles triangle

Two angles of equal size, and two legs of equal size Area = (leg)^2/2, because the base and the height ARE the two legs (because one leg is perpendicular to the other in this triangle)

Two consecutive integers

Two consecutive integers will NEVER share ANY prime factors. Thus, the GCF of two consecutive integers is 1.

Parallel line properties

Two lines are parallel if the following is true The two lines never cross, and therefore have the same slope and different y intercepts

perpendicular lines

Two lines are said to be perpendicular if they meet at right angle. Note: Two liens that are perpendicular actually create 4 right angles, but all you need to know is that one right angle is created to know that two lines are perpendicular

complementary events

Two or more mutually exclusive events that together cover all possible outcomes. The sum of the probabilities of complementary events is 1. Example: 40 red and 60 blue marbles. So 1 - P(Red) = P(Blue) (i.e. 1 - 4/10 = 6/10) The events CANNOT occur together. For example, when you flip a coin, it's either heads or tails, but can't be both. You can represents the compliment of an even with a comma or "Not". For example P(A) + P(A') = 1 (or P(Not A), where the P(A')/P(Not A) is the complementary event of A. -YOU MUST VERIFY THE EVENTS ARE MUTUALLY EXCLUSIVE TO UTILIZE THIS.

Congruence of triangles

Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

Slope intercept form of undefined and 0

Undefined = x = -3 (any number and shifts left to right) 0 slope = y = -1 (any number and shifts up and down) Ex: What is the equation of the x axis? y = 0 Ex: What is the equation of the y axis? x = 0

Permutation with a restriction that two items must NOT BE next to each other

Utilize the approach of mutually exclusive and

Determining the number of numbers within a factorial

Very similar to determining the number of prime numbers, the only difference is the following: (1) You must FIRST factorize the denominator, with the factorization take the HIGHEST prime, and perform the operation of getting quotients for that prime number Note: This works because you are looking for the number of pairs of factorials, and the pairs would be limited by the largest number.

Exponential Equations positive/negative impact

We must consider both positive and negative possibilities for the solutions. Example: a^2 = 25, then a equal 5 or -5 Note: with an ODD exponent it's one solution

How to interpret the meaning of abs(x+5) = 6

What number, when 5 is added to it, is 6 units from 0 on the number line. Which to determine this you would set x+5 = 6, and x+5 = -6, and solve each for x, giving 1 or -11.

Square roots of Large Perfect Squares

When a PERFECT square ends with an even number of zeroes, the square root of such a perfect square will have EXACTLY HALF OF THE NUMBER OF ZEROES to the right of the final nonzero digit as the perfect square Ex: Sqrt(10,000) = 100

Circle inscribed into a square

When a circle is inscribed into a square, the diamater of the circle is equal to the sides of the square

Square inscribed in a circle

When a square is inscribed into a circle, a diagonal of the square is the diameter of the circle

Accounting for a variable in part to total ratio

When adding or subtracting some x to part, that means that x will also be added or subtracted to the total as well. Example: ratio of 3x:4x, then part to total + y is 3x+y/7x+y, NOT JUST 3x+Y. When comparing two individual parts one variable is fine, but logically doesn't make sense when utilizing the total part.

When do you absolutely NEED to check the solution of an absolute value equation

When an absolute value equation is set equal to a variable on both sides Example: abs(x-1) = 2x, or abs(x-1) = x, etc Remember that if you place the solved for x back into the equation, and the abs(equation side) is equal to a negative, then that is NOT a possible solution to the equation, because an absolute value equation can not be equal to any negative number.

Remainder rule when an x units digit is divided by 5

When an x units digit is divided by 5, all of the same x units digits will have the same remainder Ex: 9/5 = 1 and 4/5, 19/5 = 15 and 4/5 Note: All you need to do is find the remainder for one, usually can just start with the lowest, example: 7/5 = 1 and 2/5, now you could find the remainder of 1231231237/5 because it's going to be 2/5

Inclusive counting number of items in sets

When counting the number of items in a consecutive set, you take the end minus the beginning PLUS 1. Ex: set = {1,2,3,4,5}, then 5-1+1 = 5

Rope question strategy

When dealing with a question regarding the length of a rope, need to remember that the sum of the individual pieces sums the the total length of the rope.

"At least ONE" probabilities

When given a question that asks to find the probability of "at least ONE", the simplest way to solve this is to take 1 - P(NONE). This works because there are only two possible outcomes in the sample space, (1) P(at least 1) or (2) P(None), and the sum of these two comes are equal to 1. The reason that these are compliments is because P(at least) means that you must take the addition from 1 to all. Note: For this to work the two events must be collectively exhaustive (i.e. not other possible options). This concept is especially critical when you're working with the opposite of at least, because the complement of none is at least one, and calculating at least one requires you to calculate the probability of each increment greater than and equal to one, and add them together.

"None" probabilities

When given a questions that asks to find the probability of selecting none, then you can utilize 1 - P("At least one"). With this you must be especially careful to consider each increment >= 1 and add them together. Example:

Circular arrangement problems

When given circular arrangements problems there's two things you need to keep in mind: (1) Do you need to select any out of a group before arranging into a circle? If so you need to do a combination in the form of xCy where x is the total, and y is the amount in the circle (2) Multiply the combination result from (1) above and the circle permutation of total circle seats - 1 factorial. Note: You can iterate through the above two steps for any additional/remaining circle. Note that if in a final circle, what's remaining is equal to # of seats then an additional combination of xCy isn't needed.

Mutual exclusivity and two items like vs not like, setting up the table

When given that you have two items that are liked or disliked (or other variation of difference) then one item goes on a row, and the other item goes on a column. What you DON'T DO is set like and not like as a column, and the item one and item two as a row. Ex: Rows = ['Like item one', 'Not like item one'] Columns = ['Like item two', 'Not like item two']

Identifying when to use a venn diagram vs when to use a table

When given three separate topics (i.e. Fish or Chicken, or Beef) use a venn diagram. When given two topics (i.e. College degree/no degree, and Manager/Not a manger) use a table.

Equations with one absolute value

When solving equations with absolute values, we need to solve the equation twice, first for the condition in which the expression between the absolute value bars is positive and second for negative Example: abs(2x+4) = 12 x=? (1) -(2x+4) = 12 (2) 2x+4 = 12

Calculating the height in the 1/2bh formula using an "altitude"

When triangles don't have an obvious height perpendicular to the base, you can use the "altitude" of the triangle that IS perpendicular to the base to determine the height.

Combinations and the implications of "at least"

When you are told that "at least" some items are being selected out of total, this means you should automatically start thinking in OR/ADDITION terms. Ex: Choose at least 3 video games out of 6 means that (1) 6c3 OR (2) 6c4 OR (3) 6c5 OR (4) 6c6 To determine the total possible options you need to add the OR combination results together.'

with/without replacement

With = If an item is removed from a set, the item is replaced before before any more items are removed. Without = If an item is removed, the item IS NOT replaced, and additional selects are net of the previously removed items.

How to handle exponential decay

With exponential decay the starting value is HALVED for each period. Example $100 in investment account drops 50% each week, what is it after 3 weeks? Wk1 = Start = 100 end = 50 Wk2 = Start = 50 end = 25 Wk3 = Start = 25 End = 12.5

Work rate formula

Work = [Rate x Time]

Given that x, y and m are positive, is it true that x>y if x^m>y^M

YES THIS IS TRUE, assuming the conditions that they are positive is met. Note that this has huge implications for comparing numbers with different radicals. Ex: What's bigger, sqrt(3) or sixroot(7)? In this case you first transform 3^1/2, and 7^1/6, then you raise each of these terms to the LCM of 6 and 2, and the resulting powers provide this answer. Ex: What's bigger, 5^50, or 7^25. In this case you find the GCF of 50 and 25 and raise each term to 1/25, which provides 5^3, and 7^1, which provides the answer.

If given a ratio of quantity weightings, and given the amounts that will go into the average (I.e. number of $dollars), can you determine an answer to the WAVG?

YES, use the ratios as the quantity weightings themselves. Ex: Given that students pay $10, and all others pay $20, and the ratio of Students to others is 9/5, then the WAVG answer is [$10x9 + $20x5]/9+5 Moral of the story here is that a ratio is just as sufficient as the actual quantities to answering the WAVG amount. Important concept for data sufficiency. Note: The same concept works for fractions. For example, in the same example if it was given that 5/14 are students and 9/14 are others, then the formula becomes [10 times 5x/14 + 20 times 9x/14] divided by x. Refer to image as an example

Addition/subtraction under a raidcal

YOU CAN'T SEPARATE OUT INTO UNIQUE RADICALS, the radical acts like brackets, so the addition/subtraction must be performed first. -sqrt(3+2) = sqrt(5) not sqrt(3) + sqrt(2)

Is there always a prime number in a number greater than 1?

Yes, If n is a positive integer greater than 1, then there is always a prime number p within n<p<2n

Multiplication of like bases with exponents

You ADD the exponents, and keep the bases. Ex: x^2 times x^2 = x^4 Ex: 2^2 times 2^3 = 2^5 NEED THE BASES TO BE THE SAME TO APPLY THIS RULE. Note: Same idea for division but instead of addition you subtract.

Overlapping sets with a combination of actuals given and also fractions/percents

You CAN NOT ASSUME 100, or purely LCM approach in the event of a question like this. The goal here is to find a way to determine the total value from the percents/ratios given in the problem.

How to deal with situations where you utilize squaring some variable, or multiples terms that contain a variable in order to solve an equation?

You MUST check if the value(s) for x actually work for the problem. When you start to square variables or terms with variables then you run the risk of creating new extraneous roots. -Remember you need to plug it back into the ORIGINAL EQUATION BEFORE THE SQUARE ROOTS WERE CREATED. If you plug it back into the equation after it was modified you won't be able to see the extraneous roots that were created. Big idea here is that if you need to square any variables to solve a equation you NEED to check the all of possible solutions to the problem.

How to handle an ISOLATED absolute value equation set equal to a negative number (i.e. abs(x+y) = -2

You can conclude that abs(x+y) DOES NOT EQUAL -2, there are no solutions, because an absolute value can not equal a negative. But conceptually understand that if give abs(x-1) = 2x you NEED TO CHECK YOUR SOLUTION. You need to check the solutions because you don't know if x is negative or positive. Thus if the solved for x works for the original equation, then it has solution(s).

Information needed to define a line

You can define a line if given either of the following: (1) The slope of the line, or the slope of a line that is parallel or perpendicular to the line. (2) Any second point on the line. (Could be a y intercept, x intercept, or any other given point) Note: However if given one point on a line that tells you essentially nothing about a line. (There are an infinite number of line possibilities if just given this and they can all rotate 360 degrees around the given point)

If given that everyone takes at least one language out of French and Spanish, what could you say about the matrix cell that equals No French and No Spansih

You can say that this cell would be EQUAL TO ZERO. Important to understand here that just because it isn't explicitly stated to be 0, doesn't mean you shouldn't account for this. If you are told, that of the two options, everyone falls under at least 1, then the neither cell must be zero.

How to determine the product of two numbers when both the GCF and LCM is known

You can simply multiply the GCF * LCM, and that brings you back to the product of the two numbers Note: This formula can be represented as [x][y] = GCFxLCM This formula is important because you can then derive that [x][y]/GCF = LCM and vice versa

probability using combinations

You can utilize the combination formula/approaches to determine total possible outcomes, and "favorable" numerator outcomes. Example: Given 5 flawed and 1 perfect diamond, what is the probability of selecting 1 perfect out of 3 selections? Step 1: Determine the total possible combinations (aka the denominator), which is 6c3 = 20 Step 2: Determine the numerator by first setting aside one of the 3 lots to already be the diamond. You can represent this as 1c1, then to determine the other two slots there are 5c2. So 5c2 X 1c1 = 10 The big concept here is to realize that the first "slot" of 3 is 1c1 because the perfect diamond must be selected.

If the product Z(A+B+C) is an integer, what can you say about Z and (A+B+C) individually?

You can't infer much of anything about the two individually, other than the fact that the product of each other must be an integer. DO NOT fall into the trap in thinking that each individually are integers. For example, Z could equal 4, and (A+B+C) could equal 1/2 leading to an integer of 2.

Given that X is an integer, and [50 times Z]/3 is equal to X, what can you conclude about Z?

You could conclude that Z MUST be a multiple of 3. That is because X must be an integer, so 3 must divide evenly into either the 50 or the Z, and of course it can't divide into the 5, so it must divide into the Z (whether that be 3,6,9,... etc.). Note: This is an extremely important concept for word questions. (I.e. Could a teacher split a class into 3 different study groups?)

If the bases are equal, what can you generally say about the exponents in an equation x^z=x^q

You could, in most cases, say that x is equal to q. Cases where this isn't always true is if the bases are 1, -1 or 0. However if the base is anything other than 1, -1 or 0, then the exponents are equal. This is incredibly important concept for problem solving. Because for example if you're given [2^x][x^y] = 2^2, you could say 2^x+y = 2^2, then x+y=2.

Why do you divide with combination counting

You divide because ORDER DOES NOT MATTER. Since order doesn't matter that means you need to remove duplicate occurrences where the order is different. For example, A and B is the same as B and A, so 2/2 = 1 possible combination of A and B

If given the equation 3r^3, and you are told that r is tripled, do you do (3r)^3, or 3(r^3)?

You do (3r)^3. The way to think this through is that you are entirely replacing r, with 3r, and so that means that with the replacement also comes that the exponent must wrap around the whole term and distribute.

If P/Q > 1, is P>q?

You don't have enough information to conclude this. (1) 2/1, would be yes (2) -2/-1 would be no

Converging rate question where two different departues

You need to adjust one of your t's in the DRT table to account for the difference, so for example, one object will just be time = t, and the other will be time = t + (difference of departure time)

Converging rate questions with relative speed

You need to adjust the RATE of one of the R's in the DRT table. (Example, 10 mph faster, so one rate would be x, and the other would be x + 10) Percent variation: If one is x, and the other is 50% as fast as x, then the r would be x/100 and 50x/100

Given a work rate, how can you determine how many work units of x it takes to generate in a time period

You simply divide the work units divided by the RATE. This will tell you the number of time units. Example. A machine makes 10 widgets per hour, how many hours for 40 widgets? 40/[10widgets/1hr] = 4

X percent less than z, can be express in what way?

Z ( 1 - x/100) Ex: 10 percent less than 100 is equal to (1-10/100) times 100 = 90 Ex: 7 percent less than z is equal to (1-7/100) times z = 93z/100

Bookend method to finding the average with an EVENLY SPACED set (i.e. first and last method)

[First # in set + Last # in set] divided by 2 REQUIRES EVEN SPACING OF NUMBERS (doesn't need to be consecutive)

Percent change Fomrula

[New - old] / old * 100

Distributive Property

[a+c]/b = a/b + c/b

Multiplication of DIFFERENT BASES and SAME exponents

[x^a] multiplied by [y^a] is equal to [xy]^a

Arithmetic sequence

a sequence in which each term is found by adding the same number to the previous term Ex: If A1 = 3 and the common difference = 3, then A2 = 3+3 = 6, and A3 = 6+3 = 9, etc

Adding different denominators

a/b + c/d = [ad + bc]/bd

Comparing fractions bow tie method

a/b > c/d IF ad > bc Ex: 3/4 > 5/7 beacause 21>20

What is a^-n equal to

a^-n = 1/a^n Example: 3^-2 = 1/3^2

Fractions as a power

a^[1/n] = nroot(a) a^[m/n] = nroot(a^m)

Exponents and divisibility:

a^n−b^n is ALWAYS divisible by a−b a^n−b^n is divisible by a+b if n is even. a^n+b^n is divisible by a+b if n is odd, and not divisible by a+b if n is even.

Abs(a+b) ?? abs(a) + abs(b)

abs(a+b) <= abs(a) + abs(b)

What is true of the absolute value of abs(a+b), and abs(a) + abs(b)

abs(a+b) <= abs(a) + abs(b) Example: a = -3, and b = 6 (1) abs(a+b) = 3 (2) abs(a)+abs(b) = 9

Abs(a-b) ?? abs(a) - abs(b)

abs(a-b) >= abs(a) - abs(b)

Evenly spaced set formula

an=a1+d(n−1) given d is equal to the common difference

Multiple square roots

aroot(broot(x)) = (x^1/b)^1/a = x^1/ab

Pythagorean Theorem

a²+b²=c² -Can be used for any triangle that is a right triangle (i.e. has one 90 degree angle

Pythagorean Theorem

a²+b²=c², where c² is the hypotenuse Example: Triangle with 3 sides, where the two legs are 3 and 4, what is the hypotenuse 3^2 + 4^2 = C^2 (remember you need to SQUARE the variable)

Diagonals of a parallelogram

bisect each other

Common non-perfect cube roots to memorize

cuberoot(2) = 1.3 3 = 1.4 4 = 1.6 5 = 1.7 6 = 1.8 7 = 1.9 9 = 2.1

even +/- even

even

even X even

even

odd +/- odd

even

even X odd

even -reason for this is that the even integer as a factor of 2 and the even formula is equal to 2k

Multiplication rules for even and odd numbers

even x even = even even x odd = even odd x odd = odd

mutually exclusive events

events that cannot happen at the same time. For example, the stock market can either go up, or down, or be unchanged -- it can NOT be either of these at the same time. Mutually exclusive events are often signaled by or, and if so will mean you must ADD the combination subgroups together (i.e. not multiply)

Nested functions

f(f(x)) To handle these you must work from the inside out Ex: If f(x) = y + 1, then f(f(x)) where x = 1, then: (1) f(1) = 2 (2) f(2) = 3

What do perpendicular lines form?

four right right angles

Any NONZERO number to the power of 0 equals what

given any NONZERO number n, n^0 is equal to 1

Zero product property

if a x b = 0, then a = 0 or b = 0

Fundamental Counting Principle

if an event can happen in N ways, and another, independent event can happen in M ways, then both events together can happen in N x M ways. For example, assume that we're making breakfast one morning and that we can choose among 5 types of eggs, 3 types of coffees, and 6 types of pastries. If a breakfast consists of 1 egg, 1 pastry, and 1 type of coffee, we can then prepare breakfast in one of a total of 5x 3 x 6 = 90 different ways. Notice that each of the three choices is independent of the others.

Divisibility rule for 8

if the last three digits are divisible by 8

The length of each diagonal of a square is

is side times sqrt(2) where s is the length of any one side.

Volume of a cube or rectangular solid

length x width x height

Sum of n consecutive integers equals

mean multiplied by the number of terms Note: Mean can be calculated by first and last term divided by 2 Note: Number of terms is equal to last term - first term + 1

Permutations formula

nPr = n!/(n-r)! Where n is equal to the total objects, and k represents the number of objects chosen Example: Select a catcher, short stop and pitcher from 7 people, then the formula is 7!/(7-3)!, which is equal to 7x6x5x4

What is a prime number

natural number with exactly two distinct natural number divisors: 1 and itself -A number n>1 is prime if it cannot be written as a product of two factors a and b, both of which are greater than 1: n = ab

Basic probability formula

number of outcomes in which x occurs/total number of possible outcomes in the experiment. Note: This assumes that each outcome has an equal chance of occurring.

even +/- odd

odd

odd X odd

odd

Even/odd rules for addition/subtraction

odd + odd = even even + even = even even + odd = odd odd - odd = even even - even = even

If p is a prime number and p is a factor of ab then

p is a factor of a or p is a factor of b Example: p = 3, and a=4 and b = 3 and ab = 12, then 3 is a factor of 3

Volume of a cylinder

pir^2h

Unit conversion factor

ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit Examples: Converting FROM dollars TO cents = 1 dollar = 100cents, and 100cents/1dollar

Angle-Angle-Angle

says nothing about the size of the two triangles and hence proves only similarity and not congruence.

What is the LCM

smallest positive integer that is a multiple both of a and of b Note: If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero

Non-Perfect Square Roots to Memorize

sqrt(2) = 1.4 sqrt(3) = 1.7 sqrt(5) = 2.2

First 6 non perfect square roots

sqrt(2) = 1.4 sqrt(3) =1.7 sqrt(5) = 2.2 sqrt(6) = 2.4 sqrt(7) = 2.6 sqrt(8) = 2.8

Square root values from 2 to 10

sqrt(2) = 1.41 sqrt(3) = 1.73 sqrt(5) = 2.24 sqrt(6) = 2.45 sqrt(7) = 2.65 sqrt(8) = 2.83 sqrt(10) = 3.16

Rectangle diagonal

sqrt(w^2+h^2) where: w is the width of the rectangle, h is the height of the rectangle.

Radicals express in exponential form

sqrt(x) = x^[1/2] cuberoot(x) = x^[1/3]

Square roots of fractions

sqrt(x/y) = sqrt(x)/sqrt(y)

Terminating non-repeating decimals can be expressed as terminating IF

the denominator is in the form of 2^n X 5^m assuming numerator/denominator is the lowest form. Example 3/30 = 1/10 = 1/5*2

What is the GCF

the largest positive integer that divides the numbers without a remainder -To find the GCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors). Example: GCF(8, 12) 8 = 2^3*1 12 = 2^2*3^1*1 GCF = 2^2*1

If a is a factor of b and b is a factor of a

then a = b, or a = -b

If a is a factor of bc, and gcd(a,b)=1

then a is a factor of c. Example: 1 is a factor of [9*2] = 18, and GCD(1,9) = 1 then 1 is a factor of 9

Division of two integers (even or odd)

there's no strict even/odd pattern for division, don't get tripped up by thinking that

V is jointly proportional to h and the square of r

v = khr^2

Scientific form, how many trailing zeros with whole number bases

whole number base x 10^z, means that you take the base and add z zeros. Note: You can often convert a decimal base to whole number by taking a 10's power from the z and shifting the decimal to the right Ex: 0.2 x 10^3 = 2 x 10^2 The x and zeroes are 1:1 correlated for a whole number base.

Translate x is inversely proportional to the cube of y and directly proportional to z

x = k times sqrt(z)/y^3

Expression set equal to 0

x CAN equal 0 ex: x(x+100) = 0 x = 0, or x+100 = 0

If given a^x, and a^y, when is it NOT TRUE that x = y?

x may not always equal y IF a (i.e the base) is equal to the following: (1) 1, (2) 0, (3) -1 If the base is anything else then x=y when a^x = a^y Ex questions: Is z + x = m (1)a^z+x = a^ m (2) a != 1 Both (1) and (2) ARE NOT sufficient because we can not rule out that a is not equal to either -1 or 0, both of which could mean that z+x != M

Determining the LCM with the GCF

x times y DIVIDED by the GCF(x,y)

Determining the GCF with the LCM

x times y divided by the LCM(x, y)

Ratio of a 30-60-90 triangle

x to x√3 to 2x Note: If you know that one angle is 90 degrees, and if you're given one side which you could use to calculate the other side, and IF that other side is in this ratio, then regardless of the 3rd side you have a 30-60-90.

Ratio of a 45-45-90 triangle

x, x, x√2 Example: Given a length of one leg of 7 meters, the other leg must be 7, and the hypotenuse must be 7sqrt(2) Example2: If given a 45-45-90 with hypotenuse of 6, then you can determine the leg by setting up a proportion of 1/1sqrt(2) = x/6, this works because 1/1sqrt(2) follows the property of x/xsqrt(2) for a 45-45-90

Division Formula

x/y = quotient + remainder / y

Raising a base to a negative exponent

x^-1 = 1/x, and in general x^-y = 1/x^y

aroot of x, can be express in exponent form in what way?

x^1/a Ex: sqrt[4] = [4]^1/2

If 0 < x < 1, then what is the order of x^2, x and sqrt(x)

x^2<x<sqrt(x)

If 0<x<1 then what's the order for x^2, x and sqrt(x)

x^2<x<sqrt(x)

Division of like bases (exponents)

x^a / x^b = x^a-b

Multiplication of like bases (exponents)

x^a multiplied by x^b = x^a+b

[broot[x^a]] can be expressed in exponent form in what way?

x^a/b -You're keep the a in the same position but bringing the index of the root into the denominator.

Division of DIFFERENT BASES and SAME exponents

x^a/y^a = [x/y]^a

Slope intercept

y = mx + b x = the y coordinate in (x,y) x = the x coordinate in (x, y) m = the slope (y2 - y1/x2 - x1) b = the Y INTERCEPT (i.e. the point where the line crosses the y axis) Ex: if given that y = 3/4x + 3, then that means that m = 3/4 (slope), and 3 = b (y intercept)

Dividing radicals

√[a/b] = √a/√b

Multiplying radicals

√ab = √a x √b