GRE Quantitative Reasoning Notes & Formulas
To convert fractions to percentages... 5/8 = ?%
Multiply the numerator by 100% and then divide 5/8 * 100% = 500%/8 = 62.5%
When adding or subtracting radicals... 2√(3) + 4√(2) - √(2) -3√(3) =?
Only add or subtract the outside number of like radicals and do not change the radical (4√(2) - 1√(2)) + (2√(3) - 3√(3)) = 3√(2) - √(3)
For sequence problems (especially if given a sequence table) (pg. 259 #14)
Set up a table so that you have: seq | value | relationship | difference between the value of the sequence # and the relationship mathematical result
Given the ratio of x:y is 7:4 and y:z is 2:1 what is the possible value of x+y+z?
Set up so that all ratios are equal, 7:4:2 Add up, 7+4+2 = 13 Find multiples of 13 (divide all answer choices by 13 and see what gives an integer)
What makes 2 slopes perpendicular to one another? Ex: what line would be perpendicular to 4x+5y=9
The negative reciprocal Ex: the original slope in y=mx+b is -4/5x so the perpendicular slope would be 5/4x
What is a multiple?
The product of a specified, constant, number and all others i.e : the multiples of 3 are --> 3,6,9,12,15,18 RULE: all multiples must be a result of a specific number and an INTEGER. Thus the multiple and the number do not have to be whole, but the other factor does have to be.
What is a common percent increase of decrease pitfall? Ex: compound interest on a loan.
When there are multiply percentage changes, they cannot be simply added up or subtracted, you must find them sequentially. i.e 20% of 200 and then 4% of that is not = 24% of 200
When multiplying/dividing exponential #s with the same base number... Ex: 2^6 * 2^3
You do not multiply or divide the base and add or subtract the exponents. 2^6 * 2^3 = 2^18 2^6 / 2^3 = 2^2
What is the Pythagorean Theorem and what does it measure?
a^2 + b^2 = c^2 Measures the length of any sides of a right triangle where c is the hypotenuse.
What is the equation for a linear equation? When is a good time to use it to solve a problem?
y= mx+b m is the slope (rise/run) and b is the y intercept Useful when trying to find which point will lie on a plotted line 1) find the y intercept value of b and the slope for m 2) plug in all answer choices x and y value for x and y until it is true that y=mx+b
Prime factors
Any factor that does not have any factors other than itself and 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc...
A good way to solve for an unknown variable on the GRE is to...
Backtrack, plug in all answer choices and see what fits
The length of a board is normally distributed, with a mean of 96.00" and a st dev of .05". What is the probability that a board would be longer than 96.1"
2 std devs off of the mean, normally distrubuted so 2.7%
What is the "and" rule as opposed to the "or" law for possibilities?
"And" = multiply all numbers of options "Or" = add up all options for the possibility
Area of: 1) triangle 2) circle 3) polygon
(1/2)*bh πr^2 (divide into 3 triangles) (1/2)bh
What is the sum of the angles in a polygon?
(N-2)(180) where n is the number of sides
PEMDAS "Order of Operations"
(Parenthesis, Exponents, Multiply, Divide, Add, Subtract); order in which an equation is worked out - multiplication and division as well as adding and subtracting can be interchanged, depending on which comes first (from left to right) in the equation
To find the percent profit or discount of a price... Ex: if you buy a car part at $235 and sell it at $132, what was your percent loss?
(Profit or savings margin * 100%) / (original price) = % profit or discount Ex: 235 - 132 = 103 103*100% = 10,300%/235= 43.83% loss
To calculate % change Ex: in 2009, americans ate 400 apples, in 2010 600 were eaten, what is the percent change?
(amnt increase of decreased * 100%) / (original) 600-400 = 200 => (200 * 100%)/(400) => 50% increase
Arrangements of groups formula: If there are 3 squads of 4 people and they must be ordered in a particular way: the first squad can be in any order, the 2nd says that they have a particular way, and the first says that they are either shortest to tallest of tallest to shortest. How many different arrangments can there be?
(factorial of the number of groups) * (factorial of each individual group) ex: 3 squads so 3! 1st squad can be in any order so 4! 2nd squad has no variation so 0! 3rd squad has 2 variations: 2! so 3! * 4! * 2! = 3*2*1*4*3*2*1*2*1 = 288
Combination problem formula (when order doesn't matter) Ex: There can be four finalists, if there are seven entries how many possible finalist groups can exist?
(n!) / (k!(n-k)!) Where ! is a factorial (5! = 1*2*3*4*5) n = number of items in the whole group k= number of items in a subgroup ex: 7! / (4!(3!)) => 7*6*5 / 3*2*1 = 35
What are the percentages of a normally distributed plot and std. devs?
0+-1= 34% +-1+-2= 13% +-2+-3= 2.6% +-3+-4= .1%
1) Vol of a rectangular solid 2) Vol of a cylinder 3) Surface area of a rectangular solid 4) Lateral surface area of a cylinder 5) Total surface area of a cylinder 6) Vol of pyramid
1) lwh 2) π(r^2)h 3) 2lw + 2lh + 2hl 4) 2πrh 5) 2π(r^2) + 2πrh 6) (lwh)/3
How do you solve for an equation when there is a fraction exponent? Ex: x^(5/3) = 32
1.) Take the reciprocal of the fraction and multiply both sides Ex: x = 32^(3/5) 2.) factor out the numerator Ex: x = 32^(3(1/5)) 3.) What is the root of the value for the denominator Ex: what multiplied by itself 5 times is = 32? 2 2*2 = 4*2 = 8 *2 = 16 *2 = 32 so x= 2^(3) 4.) solve the simplified exponent Ex: x= 2^3 = 8 x = 8
To convert a decimal to lowest possible fraction ('lowest term') Ex: 0.875 in lowest term = ?
1.) convert to a fraction dividing by the equivalent 'tens' place (.875 => 875/1000 or .62 => 62/100) 2.) simplify that fraction as low as possible (875/1000 => 35/40 or 62/100 => 31/50)
To convert a ratio to a percent (__ is what % of __) Ex: 4 is what percent of 50?
1.) divide the first number by the second, 2.) multiply by 100 3.) add a percent sign 400/50 = 8%
If x - 2z = 2(y - z) and 2x - 6y + z = 1 and 3x + y - 2z = 4 what is the value of z?
1.) factor in the first equation so that x - 2z = 2y - 2z thus x = 2y 2.) substitute 2y for x in the other 2 equations -2y + z = 1 and 7y - 2z = 4 3.) follow the binomial problem solution by multiplying the first equation by 2 and then adding them together so that the z's cancel and you will find a value for y and then plug and chug.
To find the greatest common factor (GCF) Ex: 32 and 42
1.) find all prime factors (use 'number tree method'), circle them as they appear 2.) Find all commonalities between the two number's prime factors (the amount of each matter, so if one has five 2's and the other has three 2's, then there is only three 2's in common) 3.) Multiply all prime factor commonalities, the product is the GCF Ex: 2, because 32= 2,2,2,2,2 and 42= 2,3,7 so they only share a 2
To find the Least common multiple (LCM) Ex: 6 and 8
1.) find all prime factors (use 'number tree method'), circle them as they appear 2.) For each prime factor, write out how many times it appears (i.e. if there is four 2's in one of the numbers and only three in the other then write out four 2's) 3.) multiply and the product is the LCM Ex: 24 because 8= 2,2,2 and 6= 2,3 so 2*2*2*3=24
How to factor out (solve for multiple values of x) a quadratic equation which has a multiple of x^2 Ex: 6x^2 - x - 12 =0 solve for all values of x
1.) first, you can plug in all answer choices if multiple choice 1.) if you must factor, multiply the leading coefficient by the constant term Ex: 6* -12 = -72 2.) now follow normal factoring steps if the constant value is the product of the previous step and add up to the middle coefficient value. Ex: multiple to -72 but add up to -1 (-9, 8) 3.)Now substitute the possible values for x into the middle term Ex: 6x^2 +8x -9x -12 = 0 4) factor out by grouping, the factor parentheses value should be the same 2x (3x + 4) -3 (3x + 4) = 0 5) now the factor should be one of the equal inside term and the outside terms Ex: (3x +4) (2x-3) =0 6) now solve for each value of x individually Ex: 3x+4 = 0 => x=-4/3 2x-3 = 0 => x=3/2
If x + 4y = 19 and 2x - y = 11 what is y?
1.) multiply one of the equations so that the x value is equivalent 2(x + 4y = 19) 2.) cancel out the x values by adding or subtracting (2x + 8y = 38) - (2x - y = 11) 3.) find y 9y = 27 => y = 3
To solve for multiple unknown variables given multiple equations with them... x -15 =2y and 6y +2x = -10 what is y?
1.) substitute in a value for one variable using another (i.e. if x=2y then anytime x appears use 2y) 2.) find a numerical value for one and then plug & chug add 15 for the first equation to get x= 2y+15 substitute that in for x in the 2nd equation 6y + 2(2y+15) = -10 6y + 4y +30 = -10 10y = -40 y = -4
Solving for an unknown variable in terms of another...
1.) try plugging in 0 for one of the variables and then see what the other variable equals 2.) using that result of the 2nd variable, plug in the numerical value into the original equations and see when the first variable equals 0
What is the probability of rolling a prime number on a 6 sided dice?
1/2 (1 IS NOT A PRIME NUMBER)
If a 6 sided dice has 3 R's, 2 S's, and 1 T on its sides. What is the possibility of rolling an R,S,T in no particular order?
1/2 * 1/3 * 1/6 = 1/36 There is no order so 3! = 6 1/36 * 6 = 1/6 1/6
Combined work formula Ex: John can mow a lawn in 4 hours, Stacy can do it in 6, Bella can do it in 2. If they work together how long will it take?
1/T = 1/a + 1/b + 1/c where T is the total time of all working together and a, b, and c are the times of individuals working alone. 1/T = 1/4 + 1/6 +1/2 => 6/24 + 4/24 + 12/24 = 22/24 = 11/12 = 12/11 hours
Compound Interest Formula Ex: If you deposit $4000 into an account paying 6% annual interest compounded annually, how much money will be in the account after 2 years?
A = P(1 + r/n)^(n x t), r is the rate, n is the number of times compounded, t is time Or, just take the percent interest after every year and make sure not to simply add the percentages together: Ex: $4,000 * 0.06 = $240 + $4,000 = $4,240 $4,240 * 0.06 = $254.4 + $4,240 = $4,494.4
What is a factorial and when do you need to use one to solve a problem?
A factorial (!) is when you take all digits counting down from the initial number and multiply them together: 5! = 5*4*3*2*1 or can be written 5*4*3! Used for finding the amount of possible combinations when order does NOT matter
Formula for the circumference of a circle Formula for arc length? Ex: if one line has a length of 6, the angle is 60 what is the arc length? Formula for area of shaded region of circle?
C=2πr or C=πd Arc = (n/360)*circumference of circle (where n is the angle) Ex: (60/360) * 2π6 = 2π Area of sector: (n/360)*area of circle
What is the sum of all integers so that... ex: sum of all integers x so that -37 < x ≤ 35
Cancel out all negative and corresponding positive integers, the remaining + and - integers is the answer. ex solution: same as saying -36 ≤ x ≤ 35 so -36 is the answer
To convert percentages to fractions... 32% = ?/?
Divide by 100 and then simplify 32% = 32/100 = 16/50 = 8/25
If there is a chart and it has boxes connected by lines, what is a common pitfall (Pg. 374 Example)
Do not look at the difference between the end of once data point and another, the box acts as the top of a histogram bar chart Ex: 2008 customers that switched to company A was 4 mil, not 5-3.5 mil.
To simplify radicals... √(72) =
Find all perfect square factors of the radical and multiply them together √(72) = √(36) * √(2) = 6√(2)
What is the diagonal length of a square give an area of a?
Given area of a, the sides are √a So the diagonal d = √2a
What does a curve with a larger std. dev. look like on a plot. What about a data set with a larger mean? Larger mode?
Larger std dev: more spread out Larger mean: The middle of the plot is move in a positive direction Larger mode: the plot is skewed to the left
To multiply or divide radicals... 2√(3) * 4√(6) =
Multiply or divide the radicals AND the numbers outside of the radical normally 2√(3) * 4√(6) = 8√(18) Simplify if possible: 8√(18) = 8 (√(9) * √(2)) = 8 ( 3 * √(2)) = 24√(2)
To multiply fractions... 2/4 * 5/8 =
Multiply straight across and simplify: 2/4 * 5/8 = 10/32 = 5/16
Convert from Decimals to Fractions Ex: 0.625
Multiply the decimal by any number 1-10, the multiplier is the denominator and the product is the numerator Ex: 0.625 • 8 = 5 —> 5/8 = 0.625
To divide fractions... 1/2 / 2/3 =
Multiply the first fraction by the reciprocal of the fraction following the division sign. 1/2 / 2/3 = 1/2 * 3/2 = 3/4
What is the probability formula? What is the "or" probability formula? Ex: a bag contains 10 marbles: 4 blue and 6 red. If 2 marbles are chosen and not replaced then what is the probability that they are both red? Ex#2: Events A and B are independent, if the probability of A occuring is .6 and either happening is .94, what is the probability of B occuring?
P = (# of desired outcomes) / (# of possible outcomes) Ex 1: 6/10 * 5/9 = 1/3 P(a or b) = P(a) + P(b) - P(a & b) Ex 2: 0.94 = 0.6 + P(B) - 0.6(P(B)) => P(B)= 0.85
Special right triangles
Pythagorean triplets: 3,4,5 5,12,13 8,15,17 7,24,25 45, 45, 90 (x, x, x√(2) for the sides x√(2) is hypotenuse) 30, 60, 90 (x, x√(3), 2x for the sides w/ 2x being the hypothesis and x√(3) being the other long side)
What is the interquartile range (IQR)? Ex: {0, 0, 0, 10, 11, 11, 11, 12, 14, 14, 14, 14}
Q3-Q1 (Used to show the presence of outliers which skew the statistical analysis of data sets) Ex: {0, 0, 0, 10, 11, 11, 11, 12, 14, 14, 14, 14} Q1: 10 Q3: 14 Interquartile range: 4
How can you "pronounce" R/S (other than division)
R to S in terms of a ratio R/S == R:S
What is the range of the set: {-3, -1, 7, -5, 1, 2, ,3 , 10, -7}?
Range: the value of the lowest to the highest number: lowest = -7 highest = 10 so the range of this set would be 17
To find arrangements (permutations) of things in a group (when order does matter) Ex: If there are 7 entries into a competition and 1st place gets gold, 2nd gets silver, 3rd bronze, and 4th a blue ribbon. How many combinations are possible?
Remember that all can be the first place, and then its the remaining n-1 Ex: 7*6*5*4 = 840 possible arrangements
If a zoo is 10% birds and there are a total of 80 birds, how many birds are needed to be added so that at least 20% of the zoo is birds?
Remember that when adding a bird, you not only increase the number of birds but also the total amount of animals Thus, is 80 is 10% there are 800 total animals Set up so that (80+x)/(800+x) > or = 20/100 X=100
How many different appearing arrangements can there be for the letters: AAABBC?
Since all the same letters are considered identical and thus would not make a distinct arrangement, you have to take the factor of the normal 6! by the number of identical letters: formula: (the amount of entries)! / (the number of identical entries 1)! * (number of entries 2)! *etc! 6! / ((3!)(2!)) => 6*5*4*3*2 / ((3*2)*(2)) = 60
List the values of the 1st, 2nd, 3rd, and 4th quartile for the following group of numbers: 13, 8, 11, 0, 0, 7, 14, 12, 1, 8, 10, 7, 0, 9, 1, 3
Steps: 1) arrange in numerical order increasing {0, 0, 0, 1, 1, 3, 7, 7, 8, 8, 9, 10, 11, 12, 13, 14} 2) divide into 4 groups: {0, 0, 0, 1} {1, 3, 7, 7} {8, 8, 9, 10} {11, 12, 13, 14} 3) The largest, or last, number in each group is the value for that quartile: Q1: 1, Q2: 7, Q3: 10, Q4: 14
Find the standard deviation of 1, 3, 8, 11, and 12
Steps: 1) find average: 1+3+8+11+12 / 5 = 7 2) subtract the average from each value: 1-7= -6, 3-7=-4, 8-7=1, 11-7=4, 12-7= 5 3) Find the average of the squares of those differences: -6^2 + -4^ + etc... /5 4) Take the square root of this quotient √(94/5) = 4.34
To find the midpoint of a line given coordinates of its endpoints... Ex: endpoint 1 = (-2, -10) #2 = (-8, -6)
Take the average of the x and y values Ex: (-2+-8)/2 = -5 =x (-10 + -6)/2 = -8 = y (-5,-8)
What is a mathematical "remainder"
The amount between a multiple and the quotient: 7 / 5 = 1 w/ remainder of 2. (the multiple is 7)
Factoring out quadratic equations (x^2 +- __x +- __) Ex: factor x^2 -3x +2 = 0
The first values contain x, if the first value is x^2 then => (x +- __)(x+- __) The middle value is equivalent to adding the other two values together and the last value is their product. ex: x^2 -3x +2 = 0 since the first value is x^2 then (x,)(x,) then find out what adds to -3 and multiplies to a product of +2 ==> -2, -1 (x-2)(x-1)
Digits or weird non-numerical symbols are equal to... ex: if $ = 3 and & = 2 what is $$&$=?
The symbols represent individual numbers which do not interact with one another or multiply ex: $$&$ = 3323
What is the "units place" of a digit?
The tens place
What is the "third side rule" Ex: given side length of a triangle 3,4,x but no angles, what is x?
The third side rule states that the third side of a triangle, given no indication of angle degrees is. 1.) greater than the difference of the other 2 sides Ex: x > 1 2.) and less than the sum of the other 2 sides Ex: x < 7
In a "set" the amount of repeated numbers (does / doesn't) matter and the order of those numbers (does / doesn't).
There is only one occurance of each number and the order doesn't matter {1,2,3,2,3,2,2} => {2,3,1} => {3,1,2}
What is the quadratic formula?
x=(-b±√(b^2-4ac))/2a