GRE math
Sum of Consecutive integers
Sum=n*(n+1)/2 Ex: sum of first 200 integers=200∗(201)/2=20100 sum of first 10 integers = 10∗(11)/2=55
(6sqrt3) x (2sqrt5) =
(6 x 2)(sqrt3 x sqrt5) = 12sqrt15
inscribed angle whose triangle base = diameter
*When the base of triangle is the diameter of the circle, triangle must be a right triangle
(a/b)^x
a^x/b^x
Intro paragraph opening issue essay
"the author states that......"
(12sqrt15) / (2sqrt5) =
(12/2) x (sqrt15 / sqrt5) = 6sqrt3
Square root of 2^36 = ?
(2^36)^1/2 = 2^18
Simplify: (3^4)(12^4) = ? (12^7)/(3^7) = ?
(3x12)^4 = 36^4 (12/3)^7 = 4^7
(3/4)^-3 = ?
(4/3)^3 = 64/27 *reciprocal
Determining Slopes with 2 Sets of Coordinates
(y2 - y1) / (x2 - x1)
Isosceles Right Triangle
**A right triangle with the two legs (and their corresponding angles) equal.** (short leg : long leg : hypotenuse) x : x : x square root of 2 has angles 45-90-45
If the sum of 5 consecutive numbers is 180, what is the sum the next 5 consecutive numbers?
*The mean is equal to the median in a consecutive series 180/5 = 36 First set: 34, 35, 36, 37, 38 Second Set: 39, 40, 41, 42, 43 39 + 40 + 41 + 42 + 43 = 205
(ab)^x
(a^x)(b^x)
Simplify the expression [(b^2 - c^2) / (b - c)]
(b + c)
(a^-x)(b^y)
(b^y)/(a^x)
three Pythagorean triples to memorize
(short leg : long leg : hypotenuse) 3 : 4 : 5 6 : 8 : 10 5 : 12 : 13
x^2 - y^2 =
(x + y)(x - y)
x^2 + 2xy + y^2 =
(x+y)(x+y) = (x+y)^2
x^2 - 2xy + y^2 =
(x-y)(x-y) = (x-y)^2
30º, 60º, 90º Triangle
*long leg is opposite 60 degree angle
dividing exponents
*must have same base
multiplying exponents
*must have same base
If a>b then
-a<-b
Finding the vertex of a quadratic equation
-b/2a
1ⁿ
1
a^0
1
Inscribed angle
1/2 of central angle
(x+y)/xy =
1/x + 1/y x,y≠0
(x-y)/xy=
1/y - 1/x x,y≠0
10^5 how many zeros?
100,000 10^5 means 5 zeros
How many more times is 17% than 3%
17/3 or 5 2/3 how many more times is m than n? x = m/n
perfect cubes up to 5
2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125
Ratio of boys to girls is 3/4. If there are 135 boys, how many girls?
3/4 = 135/g *cross multiply and divide
Normal Distribution percentages
34%, 13.5%, 2.5%
3 x 3^4 = ?
3^1 x 3^4 = 3^5 *when you see a bases without an exponent, write in a 1
Triangle third side rule
3rd side of ANY triangle must be greater than the difference between other 2 sides and less than the sum. *Add sides and subtract sides to get a range
For similar triangles, the ratio of their corresponding sides is 2:3. What is the ratio of their areas?
4:9. The ratio of the areas of two similar triangles equals the square of the ratio of the corresponding sides.
Equilateral Triangle
60-60-60 all sides are equal
Ratio of ages of Anna and Emma is 3:5 and of Emma and Nicolas is 3:5. What is the ratio of Anna to Nicholas' ages?
9 : 25
(2a^m)(1/3a^-n)
= (2/3a^m)(a^-n) = 2a^m / 3a^n
If a lamp decreases to $80, from $100, what is the decrease in price?
= (actual decrease/Original amount) x100% = 20/100x100% = 20%
Area of a Hexagon Equation when side length is known
A = (3√3)/2 * known side squared
Combinations
A permutation where you DON'T care about the order nCr = (n!/(n-r)!)/r!
Area of a Parallelogram
A= bh
Area of a Triangle
A= bh/2
surface area of rectangular cube
A=2(wl+hl+hw)
Surface Area of a Cylinder
A=2πrh+2πr2
Integer divisible by 3
Add all the digits, If that digit is a 3,6 or 9, the number is a multiple of 3 Ex: 314159265 3 + 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 = 36 Then, 3 + 6 = 9
Average of Consecutive numbers
Add the two numbers and divide by 2 (ave. of integers from 13 to 77 = 13 + 77/2)
Integer divisible by 9
Add up digits, If the result is divisible by 9, then the original number is divisible by 9. Ex: 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9
if a/b = 1/4, where a is a pos. integer, which of the following is possible for the value a^2/b?: 1/4, 1/2, 1
All · cross multiply 4a=b · substitute b in equation a^2/4a · square root both nominator and denominator a/4 · looking at the options, plug in values to see if they equal any of the possible answers
-4 < -x, then +4 > +x
Always
Percent Increase
Amount of Increase/Original Whole X 100%
Formula to calculate arc length?
Arc length = (n/360)(2πr) where n is the number of degrees.
Area of a circle
Area = (Pi)r^2
Area of a Trapezoid
Area = Average of parallel sides X Height OR A=1/2 (b1 + b2) h
Average
Average = Sum of terms/number of terms
Count consecutive numbers (inclusive)
B-A +1 (how many integers from 73-419 = 419-73 + 1
Circumference of a circle
C = 2(pi)r Or C = (Pi)D
Determining slope direction
Coefficient of x (neg. or pos.) shows the direction of the slope. Negative slopes down left to right, while positive slopes up left to right.
As the exponent of a positive fraction increases, what happens to the value of the expression?
Decreases Ex: 3/4^2 = 9/16 3/4 > 9/16 *also applies to decimals
Solving for # of variables
Determine number of unique values per variable and multiply them by each other
When you multiply integers, if ANY of the integers are EVEN, the result will be...
EVEN Ex: 3 x 8 x 9 x 13 = 2808
Integer divisible by 8
Examine the last three digits Ex: 34152: Examine divisibility of just 152: 19 × 8
Pythagorean Theorem
Finding the sides of a right triangle a^2 + b^2 = c^2
Solve for both or neither (venn diagram)
Group1+group2+ neither - both= total
Brainstorming Argument Essay
Identify conclusion, support, and assumptions
Integer divisible by 6
If sum of the digits are divisible by 3 and the last digit is even Ex: 1,458: 1 + 4 + 5 + 8 = 18
Argument Essay Outline
Intro: deconstruct argument by stating conclusion and their supportive info, establish your take (remember they are flawed, you will not agree) Body paragraph 1 Body paragraph 2 Body paragraph 3 *Each body paragraph you will state assumption, provide counter examples, indicate specific evidence Conclusion: restate your position without copying previous statements
Issue essay outline
Intro: summarize issue and state position Body Paragraph A: provide evidence that supports position Body Paragraph B: provide additional evidence for your position Conclusion: summarize your position and acknowledge wrinkle if you haven't already
Median
Middle number
If NONE of the integers are even in a set, when multiplied the result will be...
ODD
Odd and even number operations (addition and multiplication)
Odd + Odd = Even Even + Even = Even Odd +Even = Odd Odd x Odd = Odd Even x Even = even (and divisible by 4) Odd x even = even
Probability
P is probability, m is # of favorable ways, n is total # of ways P = m / n when combining probability of two mutual exclusive events Pa x Pb
Simple probability
Probability = # of desired outcomes/# of total possible outcomes
Revenue Formula
Profit = Revenue - Cost
Rate
R=D/T *identify quantities and units to be compared. Keep units straight. If question gives you a rate in hours, but wants an answer in minutes, then convert the hours in the problem to minutes to solve.
Ratio
Ratio= Of/To 20 oranges to 12 apples = 20/12 = 5/3
Surface area of a cube
SA = 6a^2
Surface area of sphere
SA=4(pi) r^2
Brainstorming Issue Essay
Side 1 pros and cons Side 2 pros and cons *think of ideas to support each side *pick a side
Slope of a Line
Slope = Rise/Run = change in Y/Change in X Example: slope of line with (1, 2) and (4, -5) = (-5-2)/(4-1) = -7/3
Sum of consecutive ODD integers
Sum=(n+1)∗(n+1)/4 = (n+1)2/4 Ex: sum of odd integers from 1 to 199 = 200∗(200)/4=10000 sum of odd integers from 1 to 9 = 10∗(10)/4=25
Sum of consecutive EVEN integers
Sum=n∗(n+2)/4 Ex: sum of even integers from 1 to 200 =200∗(202)/4=10100 sum of even integers from 1 to 10 =10∗(12)/4=30
Integer divisible by 4
The last 2 digits are a multiple of 4. Ex: 144 .... 44 is a multiple of 4, so 144 must also be a multiple of 4
Volume of a sphere
V = (4/3) (pi) r^3
Volume of cylinder
V = (pi) r^2 h
Permutations
When you DO care about the order. For instance, if there are 5 people, how many different ways can they sit in 3 chairs. When a person sits down, it removes the possibility of sitting in any other chair. nPk = n!/(n-k)!
Percent Decrease
amount of decrease/original whole X 100%
Quadratic Equation
ax^2 + bx + c = 0
Central angle/Arc length/Sector area
central angle/360 = arc length/circle circumference = sector area/circle area
Pi or π
circumference : diameter ratio
distance formula
d= rt
Improper fractions to mixed numbers
e.g. 18/7 = 2 R4 or 2 4/7
even^odd
even always
special right triangle (half of an equilateral triangle)
has angles 30-60-90 (short leg : long leg : hypotenuse) x : x square root of 3 : 2x
Ways in which you can tell standard deviation will be lower
less "spread" = lower standard deviation more terms = lower standard deviation **example: 12, 9, 6, 3 spread = 3 between each term, 4 terms**
odd^odd
odd always
Mode
number that appears most often
Mixed number to improper fractions
numerator = denominator x integer + original numerator denominator remains the same Ex: 2 4/7 7 x 2 + 4 = 18 18/7
Percent
part = percent X whole
common multiple of 2 numbers
prime factor both numbers; multiply each factor the greatest number of times it appears in factorization.
Range
put in order from least to greatest and subtract smallest from largest number in set
exponent is a fraction 25^(1/2)
squar root 25 = 5 or -5
maximum area possible of a rectangle
square
What is the "wrinkle" in the issue essay?
twist in the prompt that they want you to acknowledge ie "However, under certain circumstances...."
0^0
undefined
a/∅
undefined
xy + xz =
x(y+z)
xy - xz =
x(y-z)
y^2 = x
y = ±√x e.g. y^2 = 4 so, y = ±√4 = ±2
find value for x & y for: x - 2y = 2 & 2x +y = 4
· Add Equations · Find common factor for x to cancel it out (2x + y = 4) + -2(x - 2y = 2) · Solve for y 5y = 0 so, y = 0 · Replace y value into either one of the original equations and solve for x x - 2(0) = 2 so, x = 2
value of x for: x + 4y = 7 & x - 4y = 8
· Add Equations · No need for a common factor since 4y & -4y cancel each other out 2x = 15 x = 15/2
Ratio of boys to girls is 2:3. Of 40 students, how many are girls?
· Boy/Girls = 2/3 or 5 students total · There are 8 sets of 5 students in a classroom of 40 students · 8 sets x 3 girls/set = 24
7/12 + 3/5
· Cross multiply & add both products together 7x5 + 12x3 = 71 · Multiply both denominators 12x5 = 60 · So, 71/60, then simplify to a mixed number 1 11/60 or · Find a multiple that makes the denominators equal 60 is the lowest common denominator · Multiply each fraction by their respective multiple 5(7/12) + 12(3/5) = 35/60 + 36/60 · Add the two numerators 71/60 or 1 11/60
Graphing Quadratic Equations
· Factor equation · Set each factor = 0 · Solve for x · xV = (x intersect 1 + x intersect 2) / 2 · To find yV, plug in xV into original equation · To find y intercept, make x=0 into original equation and solve for y
what is c if, 200 = (a+b+c)/2 & 80 = (a+b)/3
· Get rid of the denominator in each equation by multiplying both sides by the denominator value · Now subtract both equations by each other (400 = a+b+c) - (240 = a+b) · Therefore, 160 = c
Quadratic Inequalities
· Remember a quadratic formula = a parabola · You're looking for the two x axis intersections and the direction of the remaining · When you divide both sides by a negative you must switch the direction of the inequality sign
Vertex
· The minimum or maximum of a parabola · Equal distance from the two points intersecting the x-axis.
How many different ways can 5 people sit in 3 chairs?
· Written, 5P3 · Equation for solving 5!/(5-3)! · Because 5x4x3x2x1/2x1, the "2x1" cancel out from both the numerator and denominator, you only solve for: 5 x 4 x 3 = 60 different possibilities
Solve for x, x^2 + 2xy + y^2 = 25, with x + y > 0 & x-y=1
· factor the equation (x+y)(x+y) = 25 or, (x+y)^2 = 25 · Square root both sides x + y = ±5 · Since it is stated that "x + y > 0" we know that 5 can only be positive · add the remaining equations together (x+y=5) + (x-y=1) · Because the y value cancels it's self out, there's no need to multiply by a common factor 2x = 6, x = 3
Simplifying exponential equations
·When comparing expressions to find which is greater find a common root using prime factorization Ex: 64^5 & 16^8 = (4^3)^5 & (4^2)^8 or 4^15 & 4^16
∅/a
∅