GRE Quantitative Study Guide 2, * BEST* GRE: Math Combo 2, GRE Math, GRE Math 2, GRE Math 3, GRE Math 4

In similar hexagons, the ratio of the areas is 16:25. What is the ratio of their corresponding sides?

4:5

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.

Powers of 4 up to 4^4

4^1 = 4 4^2 = 16 4^3 = 64 4^5 = 256

Simplify (a^2 + b)^2 - (a^2 - b)^2

4a^2(b)

How do you find the probability of a series of events in a row? The probability of A and B.

A and B = probability of A x probability of B

Central angle

A central angle has the same measure as the arc it intercepts

What is a central angle?

A central angle is an angle formed by 2 radii.

What is a chord of a circle?

A chord is a line segment joining two points on a circle.

What is a combination?

A group where the order of elements within the group doesn't matter.

Divisibility Rule for 6

A number must be divisible by 2 AND 3 So it must be even and the sum of the digits must be divisible by 3 ex: 267,914,296 = 2+6+7= 15+9+1+4= 29+2+9+6=46 (not divisible by 3 or 6)

Constant

A number or symbol such as pi which doesn't change in value

How do you find the probability of either one event or another? The probability of selecting A or B.

A or B = probability of A + probability of B

What transformation occurs if point C is reflected over the x-axis and then the y-axis?

A reflection about the axis.

What is it called when a point is reflected to the quadrant opposite it (i.e. I to III or II to IV)?

A reflection about the origin.

What is the name for a grouping of the members within a set based on a shared characteristic?

A subset.

What is a tangent?

A tangent is a line that only touches one point on the circumference of a circle.

Define a "term",

A term is a numerical constant or the product (or quotient) of a numerical constant and one or more variables. (3x, 4x^2 and 2a/c)

Standard deviation

Deviation from the mean -How far away the mean are from the individual data points (distance) -How meaningful a separation from the mean is; if this individual is 10 units above the mean and the SD = 2 then this individual is 10/2 = 5 SDs above the mean ex: On a certain test, the score has a mean = 300 and a SD = 25. If John scored 3 SDs sbove the mean, what is his score? John score = 25*3 = 75 + 300 = 375

Motion Question Word Problems

Distance = rate(speed) times time D = RT => R=D/T => T=D/R *If units are inconsistent then we need to change the units using a unit conversion Practice: A car moving at 72 km/hr moves how many meters in one second? (1 km = 1000 m) 1 hr = (60 min/hr)(60 seconds/min) = 3600 seconds R= D/T -> 72 km/hr = 72,000 m/3600 s = 720/36 =20 m/s

How do you find the volume of a cylinder?

Find the area of the circle and multiply that by the height. πr2 x height

Fundamental Counting Principal

General way to approach tasks that can be broken into stage n1 = first stage done in this many ways n2 = second stage N = (n1)*(n2)*(n3)*etc ex: We have 6 books to place on a shelf. How many different order can we place the 6 books? First book = 6 choices second book = 5 choices... N = 6*5*4*3*2*1 = 30*24=720 ex: At a company with 25 employees, they will choose a 3 person committee consisting of a president, secretary and VP. How many different committees can be chosen? N = 25*24*23 = 13,800 different committees

How do you find the probability of an event not happening?

Given event A, A + not A = 1

GEMDAS

Grouping symbols (paraenthesis, brackets, square root sign, long fraction bar, exponent slot 3^x-7) Exponents, Multiplication and Division, Additional and Subtraction

Which quadrant is the upper left hand?

II

Which quadrant is the lower left hand?

III

Which quandrant is the lower right hand?

IV

Compound interest

Interest on interest; the more interest that's accrued, the larger the amount of the next interest payment; no two successive interest payments are ever the same -With large amounts of money and/or long periods of time, the difference between successive interest payments becomes substantial A = P(r^y) P= principal y=years principal will be multiplied by the % increase multiplier A=total amount in account after y years r= multipler (1 + I/100n) -Can change compounding period quarterly n=4, monthly n=12, daily n=365 ex: Suppose the bank pays 5% annual interest, compounding quarterly 5/4 = 1.25% ex: If Susan invests $1000 in an account that yields 5% annual, compounding quarterly, how much does she have in 6 years? Quarterly percent = 5/4 = 1.25% Multiplier = (1+ 1.25/100) = 1.0125 A = 1000*1.0125^6*4

Infinitely many solutions

One equation is a multiple of the other -> will wind up with an always true equation such as 7=7

What is the factored form of x2 - 2xy + y2

(x-y)2

What is the surface area of a cylinder with radius 5 and height 8?

130pi

A triangle is inscribed in a semi circle with legs 5 and 12. What is the circumfermence of the semicircle?

13pi / 2

If the 80th percentile of the measurements is 72degrees, about how many measurments are between 69 degrees and 72 degrees? Round your answer to the nearest tenth

18

If a pair of parallel lines is cut by a transversal that's not perpendicular, the sum of any acute angle and any obtuse angle is

180 Acute Angle an angle that is less than 90° Obtuse Angle:angle that is greater than 90° but less than 180°

The consecutive angles in a parallelogram equal

180°

Properties of Remainders

20 / 6 yields 3 with a remainder of 2 20 = the dividend 6=divisor 3=quotient 2= remainder 0 < or = remainder < divisor D/S = Q + r/s S=divisor *Trick Question that the test loves: What is the smallest positive integer that when divided by 12 has a remainder of 5? It's 5 -> if the divisor is larger than the dividend the integer quotient = 0 and the remainder equals the dividend 5/12 = 0 with r=5 Rebuilding the dividend: D = S x Q + r ex: When positive integer N is divided by positive integer P, the quotient is 18 with a remainder of 7. When is divided by (P +2) the quotient is 15 with a remainder of 1. What is N? N = 18*P + 7 N = 15(P + 2) + 1 = 15P +31 18P + 7 = 15P + 31 3P = 24 P = 8 N = 15(8+2) + 1 = 120 + 30 + 1 = 151

Which is greater? 27^(-4) or 9^(-8)

27^(-4)

If 8 schools are in a conference, how many games are played if each team plays each other exactly once?

28. n = 8, k = 2. n! / k!(n-k)!

Simplify 9^(1/2) X 4^3 X 2^(-6)?

3

Equilateral Triangles

3 equal sides and 3 equal angles - each angle must be 60 degrees -Is also an isosceles because they have at least 2 equal sides; every special fact about isosceles triangles also applies to equilateral triangles

x^2 = 9. What is the value of x?

3, -3

If you have 10 shirts and 3 of them are black, what is the probability of selecting a black shirt from the closet without looking?

3/10 (3 out of 10)

Suppose that the graph of f(x) is the result of stretching y=x + 5 away from the x-axis by a factor of 2. What is the new equation for the graph f(x)?

y = (x + 5)/2

Suppose that the graph of f(x) is the result of sliding the graph of y=2x^2 down 3 units of spaces. What is the new equation?

y = 2x^2 - 3

Horizontal line

y = k where k is the is height of the line -the x axis is a horizontal line with a height of zero so y=0

What is the equation of a line?

y = mx + b m= the slope b = y-intercept

Slope-intercept form

y = mx + b m = slope b = y-intercept -Horizontal lines have a slope of zero, all rise and no run ex: y = 0*x + 4 => y=4 -Vertical lines have an undefined slope because the slope fraction is #/zero ex: x=k Practice: Find all the points of (a,b) on the line y = -4/3x + 2 such that and b are both integers with absolute values less than or equal to 10 Add 3 to x and -4 to y (0, 2) -> (3, -2) -> (6, -6) -> (9, -10) Subtract 3 from x and -4 from y (0, 2) - > (-3, 6) -> (-6, 10)

What is the formula for finding m, the slope of a line?

y2 - y1 m = x2 - x1

∅ is a multiple of

zero is a multiple of every number, BUT zero is NOT a FACTOR of any number except zero

Intercepts

Points at which the line crosses the x and y axes -Horizontal lines only have a y intercept -Vertical lines only have an x intercept -Plug in y=0 to find x intercept and x=0 to find y intercept ex: 2x - 6y = 3 y=0 -> x= 3/2 x=0 -> -3/6 or -1/2

∅ Is neither

Positive or Negative

The normal distribution

Practice: For adult males, heights are normally distributed with a mean of 175 cm and a SD of 10 cm. What % of males have a height less than 185 cm? 185-175 = 10 cm => mean + 1 SD Would be between M & (M + S) as well as everyone below M. 50 +34 = 84%

Positive integers that have exactly 2 positive divisors are

Prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23)

Graphs of Quadratics

Standard form for a quadratic equations y=ax^2 + bx +c -When a>0, the parabola opens upward -When a<0, the parabola opens downward -When |a|>1 the parabola is skinny -When |a|<1 the parabola is wide -When y=0, those are the x intercepts 0=ax^2+bx+c Can have 2, 1 or no solutions

No solutions

Started with the system of equations and led us to a NEVER TRUE equation ex: x - 2y = 5 => x=2y + 5 & 3x - 6y = 8 3(2y + 5) - 6y = 8 => 6y + 15 - 6y = 8 => 15 = 8 *Same expression equals 2 different things means that there are parallel lines x - 2y = 5 -> (Mult. by 3) 3x - 6y =15 & 3x - 6y=8

Complements

The complement of A = "not A" P(not A) = 1 - P(A)

What is the range

The difference between the biggest and smallest number in the set. In the set {2, 6, 13, 3, 15, 4, 9}, 15 is the biggest and 2 is the smallest, so 15-2 = 13 (the range)

Absolute Value

The distance of the number from the origin |6| = 6 |-14| = 14 |0| = 0 |x - 1| > 4 x > 5 OR x < -3

Which is greater? 200x^295 or 10x^294?

Relationship cannot be determined (what if x is negative?)

Stranger Operators

Remember to find the numerical value of the expression in side parenthesis first

How to square a number ending in 5; e.g 75^2

Remove the five, add one to the remaining digit, find the product of these two #'s, put the number in front of 25; e.g 7x8 = 56 75^2=5625

What are proportions?

The equivalent relationship between two fractions or ratios

What is the absolute value of a number equal to?

The number's distance away from) on the number line.

What are the members or elements of a set?

The objects within a set.

What is the "union" of A and B?

The set of elements which can be found in either A or B.

What is the "domain" of a function?

The set of input values for a function.

What is the "range" of a function?

The set of output values for a function.

How to recognize a # as a multiple of 3

The sum of the digits is a multiple of 3 (i.e. 45 ... 4 + 5 = 9 so the whole thing is a multiple of 3)

How to recognize a # as a multiple of 9

The sum of the digits is a multiple of 9.

How to recognize if a # is a multiple of 12

The sum of the digits it a multiple of 3 and the last two digits is a multiple of 4. (i.e 144: 1+4+4=9 which is a multiple of 3, and 44 is a multiple of 4, so 144 is a multiple of 12.)

What is the mode?

They number in a set that occurs the most frequently

If ETS asks for the value of √16, do they want a positive or negative root?

They only want the positive root.

What are congruent triangles?

Triangles with same measure and same side lengths.

T or F? Given d,e &f =/ 0, [(d^3)e(f^5)] / 2d(e^3) / [3(d^2)(e^3)(f^7)] / [6(e^5)(f^2)]?

True

True or false? 4.809 X 10^7 = .0004809 X 10^11

True

What are "supplementary angles?"

Two angles whose sum is 180.

What are complementary angles?

Two angles whose sum is 90.

When you plug-in for a Quantitative Comparison, what should you plug in the second time?

Use ZONE F

Age Question Word Problems

Usually easier to pick variable to represent the age right now, then use addition or subtraction to create expressions for ages at other times ex: Right now Steve's age is 1/2 Tom's age. In 8 years, twice Tom's age will be 5 more than 3 times Steve's age. How old is Tom now? S=1/2T -> T=2S 2(T+8) = 3(S +8) + 5 2((2S) + 8) = 3S + 29 4S + 16 = 3S + 29 S = 13 T = 2(13) = 26

What is a subset?

a grouping of the members within a set based on a shared characteristic.

How to solve Quadratic Equations

ax^2 + bx + c = 0 *can have 2 solutions, 1 solution or no solution 1) get everything on one side of equation and set equation = to zero 2) Divide by any GCF 3) Factor 3) Use zero product property to create 2 linear equations and solve ex: 3x^2 - 9x -70 = 14 -> 3x^2 - 9x - 84 = 0 divide by 3 x^2 - 3x - 28 = 0 -> (x-7)(x+4) = 0 -> x=7 or x=-4

If a is negative and n is even then aⁿ is (positive or negative?)

aⁿ is positive

How do you find b in the equation of the line?

b is where the line crosses the y-axis

Negative Exponents

b^-n = 1/b^n ex: 13^4 / 13^7 = 13^-3 = 1/13^3 -A negative exponent on a fraction will be the reciprocal to the positive power (p/q)^-n = (q/p)^n -A negative power in the numerator of a fraction can be moved to the denominator as positive power and vice verse ex: b^5*d^-8 / h^-4*k^7 = b^5*h^4 / d^8*k^7 Practice: (1/3) ^ -8 = 3^8 3^-3 = (1/3)^3

To increase a number by x%

multiply by 1+x% i.e. 100 x (1+50%)=100x1.5=150

To decrease a number by x%

multiply by 1-x% i.e. 100 x (1-50%)=100x.5=50

What is 80% of 200?

Percents as multipliers "is" means equal "of" means multiply x= .8*200 = 160

How do you know if it is a permutation problem?

Permutation problems usually ask for: •Arrangements •Orders •Schedules •Lists

How do you know if it is a combination problem?

Permutation problems usually ask for: •Groups •Teams •Committees

Suppose you have a set of n objects, and you want to select k of them, but the order doesn't matter. What formula do you use to determine the number of combinations of n objects taken k at a time?

n! / (k!)(n-k)!

If you have a set of n objects, but you only want to order k of them, what formula do you use to determine the number of permutations?

n! / (n-k)!

Alternative Methods

n! = total number of arrangements R = number of arrangements that obey the restriction Q = number of arrangements that don't obey n! = R + Q R = n! - Q Ex: 6 children sit in a row of 6 chairs but Jackie and Marilyn can't sit next to each other. How many arrangements are possible? Total = 6! = 720 Not allowed = 4! * 10 = 24*10=240 720-240 = 480

Slope

rise/run = y2 - y1 / x2 - x1 Practice: If a line goes through (2, -1) and has a slope m= 5/3, find all the points (a,b) on the line where a and b are integers whose absolute values are less than or equal to 10 First move right, add 3 to x and 5 to y (2, -1) -> (5, 4) -> (8, 9) Now move left, subtract 3 from x and 5 from y (2,-1) -> (-1, -6)

Perpendicular line

Perpendicular line means one line goes down and one goes up - slopes of perpendicular lines are opposite-signed reciprocals -If original line is m = p/q then the slope of the perpendicular line is m2 = -q/p ex: m = 1/2 m2 = -2

Perpendicular bisector

Perpendicular to the midpoint; in most triangles, it doesn't pass through the opposite vertex

What is the factored form of x2 + 2xy + y2

(x+y)2

What is √1

1

What is a number raised to the power 0 (10000)?

1 no matter what the number is (10000 = 1)

What is √3

1.7

A ratio of 2:1 has how many total parts?

3

In Quantitative Comparison questions, how many times should you plug-in?

Always plug-in at least twice

What are integers?

Counting numbers

Percentage Increase/Decrease Formula

Difference Percentage Change = original x 100

Is zero even or odd?

Even

What is the key to dealing with ratio questions?

Finding the whole or the total

How many angles are formed when 2 lines intersect?

Four angles

What are digits?

Numbers the make up other numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

Can you add or subtract square roots?

Only if the roots are equal √2 + √2 = 2√2 (same root = 1√2 + 1√2) but √2 + √3 ≠ √5

What is the third side rule?

The length of any one side of a triangle must be less than the sum of the other 2 sides and greater than the difference between the other 2 sides. So if a triangle as 2 sides, 3 and 5, then 3 + 5 = 8 and 5 - 3 = 2. Therefore, the third side must be between 2 and 8.

What is the diameter?

The line from one side to the other that goes through the center or 2 times the radius (r) D = 2r

What is the radius of a circle?

The line from the center to the side

What is the median?

The middle value in a set of numbers In the set {1,2,3,4,5,6,7), the median is 4 In the set {1,2,3,4,5,6), the median is the average of 3+4/2 (3.5)

A number raised to the first power (10001) is always _________.

The number itself (10001 = 1000)

What is the hypotenuse?

The side of the triangle opposite the right angle or the longest side.

What is the original number in a percentage INCREASE problem?

The smaller number

How do you find the perimeter of a triangle?

The sum of all three sides

What is the formula for finding the Average?

Total Average = # of things

What is 00?

Undefined

When can you plug-in?

When there are variables in the answer choices

What is the ratio of sides in a right isosceles triangle?

x : x : x√2

What is the ratio of sides on a 30:60:90 triangle?

x : x√3 : 2x

What is the side length of an equilateral triangle with altitude 6?

4sqrt3. The triangle can be divided into two equal 30-60-90 triangles with side 6 as the side in which 6 = xsqrt3. So x =2sqrt3...

1:sqrt3:2 is the ratio of the sides of what kind of triangle?

A 30-60-90 triangle.

How do you find the area of a triangle?

A = ½ bh

Area of a circle

A = πr^2

Define an "expression".

An algebraic expression is a combination of one of more terms. Terms in an expression are separated by either addition or subtraction signs. (3xy, 4ab, -5cd, x^2 + x - 1)

A 30:60:90 triangle is what type of triangle cut in half?

An equilateral triangle

Define a "monomial"

An expression with just one term (-6x, 2a^2)

Median

Goes from the vertex to the midpoint of the opposite side -DOES divide the opposite side in half but does NOT divide the angle in half

Altitude

Goes through vertex and is perpendicular to the opposite side

How do you find the perimeter of a four-sides object?

The sum of the lengths of all the sides

What are the roots of the quadrinomial x^2 + 2x + 1?

The two xes after factoring.

Are there any even prime numbers?

Yes, the number 2

bⁿ

b∧b∧b (where b is used as a factor n times)

How would the SD change when including new members to a list, making the list longer

ex: N = 20 and the mean = 50 and the SD =5 Add 2 more members so N= 22 -If we add 2 numbers that are evenly spaced around the mean, (mean) + k and (mean) - k, or equal to the SD, then the mean won't change so the SD stays the same -If we add 2 numbers that are closer to the mean, or less than the SD, than the SD would decrease (distance decreases) -If we add 2 numbers that are equal to the mean, this would decrease the SD the most (distance greatly decreases) -Add 2 outliers greater than the SD, SD increaes

Parallel lines

have equal slopes (m1 = m2)

Inequalities (ie: 12-6x>0) What happens to the inequality symbol when you multiply and divide?

12 - 6x > 0 -12 -12 (+/- no change in direction) -6x > -12 -6x -6x (x/div. change direction) x < 2

From a box of 12 candles, you are to remove 5. How many different sets of 5 candles could you remove?

12! / 5!7! = 792

1/8 in percent?

12.5%

The perimeter of a square is 48 inches. The length of its diagonal is:

12sqrt2

Legs 5, 12. Hypotenuse?

13

1/6 in percent?

16.6666%

Which is greater? 64^5 or 16^8

16^8 64^5 = (4^3)^5 = 4^15 16^8=(4^2)^8 = 4^16

What is an isosceles triangle?

Two sides are equal in length Two angles are equal

Vertical angles

When 2 lines cross, 4 angles are formed and the pairs of angles opposite each other, sharing only the vertex in common are congruent a=c b=d

Eliminating Repetition

When the same thing is counted more than once, we have to divide the total number of possibilities by the times each item was counted ex: Suppose there's a room of 20 people and each one will shake hands with each other person. How many handshakes will occur? 20*19 -> too big, counts handshakes twice 20*19 / 2! = 190

Exponent Rule: a2 is the same as...

a x a

#1 What is an important property of a 30-60-90 triangle?

• The triangle is a right triangle.

#2 What are the important properties of a 45-45-90 triangle?

• The triangle is isosceles (AC=BC).

How do you solve a permutation problem?

•Figure out how many slots you have •Write down the number options for each •Divide by the factorial of the number of slots

How do you solve a permutation?

•Figure out how many slots you have •Write down the number options for each •Multiply them

∅²

∅

The only number that is equal to its opposite

∅ ∅=∅

Probability of Event all cases

∅≤P(E)≤1

What is foil?

First, Outer, Inner, Last (x + 4)(x + 3) (x · x) + (x · 3) + (x · 4) + (4 · 3) x2 + 3x + 4x + 12 x2 + 7x + 12

First 10 prime #s

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself

√5

2.2

Equilateral triangle with perpendicular line drawn

30-60-90 triangle or the 1-2-√3 triangle hypotenuse = 2*short leg long leg = √3 * short leg Area = (s^2 * √3) / 4

30< all primes<40

31, 37

What is the sum of the angles when 2 lines intersect?

360°

Exponent and Cube Root Rule:

3√a12 = (a12)1/3 = a 12/1 x 1/3 = a12/3 = a4

If r, t, s & u are distinct, consecutive prime numbers, less than 31, which of the following could be an average of them (4, 4.25, 6, 9, 24, 22, 24)

4.25, 6, 22

Evaluate (4^3)^2

4096

40 < all primes<50

41, 43, 47

A company places a 6-symbol code on each product. The code consists of the letter T, followed by 3 numerical digits, and then 2 consonants (Y is a conson). How many codes are possible?

441000 = 1 * 10 * 10 * 10 * 21 * 21

Isosceles Right Triangle

45-45-90 -2 legs of equal length, hypotenuse longer than each leg but less than both legs -If you know it's a right angle with one 45 degree it must be a isosceles right triangle hypotenuse = √2 * leg 1 - 1 - √2 triangle

If 4500 is invested at a simple interest rate of 6%, what is the value of the investment after 10 months?

4725

Employee X is paid 19.50 per hour no matter how many a week. Employee Y earns 18 for the first 40 and 1.5 the hourly wage for every hour after that. If both earned the same amount and worked the same in one week, how many did each work?

48

Factorial Notation

5! = 5*4*3*2*1 Any factorial n! is divisible by all the integers less than n and all factorials less than n! ex: 20! = 20*19! = 20*19*18!

If Madagascar's exports totaled 1.3 billion in 2009, and 4% came from China, what was the value in millions of the country's exports to China?

52

50 < all primes< 60 and mult

53, 59 note 57=3x19

What is the area of a regular hexagon with side 6?

54sqrt3. (divide the hexagon into 6 congruent equilateral triangles.

What percent of 40 is 22?

55%

Powers of 5 up to 5^4

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

10^6 has how many zeroes?

6

Reduce: 4.8 : 0.8 : 1.6

6 : 1 : 2

Arithmetic Sequence

An evenly spaced list Common difference: the fixed amount we add to each term to get the next An = A1 + (n - 1)*d Practice: Let x be the set of all positive integers that when divided by 8, have a remainder of 5. What is the 76th number in the set? An = 5 + 8*(n - 1) A16 = 5 + 8(75) = 605

If a lamp decreases to $80, from $100, what is the decrease in price?

= (actual decrease/Original amount) x100% = 20/100x100% = 20%

How do you find the area of a rectangle or square?

A = length x width

Memorize prime numbers 1-60

A prime number is only a factor of itself and 1 *1 is not a prime number *2 is the only even prime # -If a number less than 100 is not divisible by any prime divisor left than 10, then the number has to be prime (2, 3, 5, 7) Prime numbers : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

Term

A product of constants and variables, including powers of variables ex: 5, x, 6y^2, x^5y^6z^7

What is a Ratio?

A ratio expresses relationships in part to part

What is a finite set?

A set with a number of elements which can be counted.

Divisibility Rule for 2

All even numbers (even in one's place) are divisible by 2

What are the integers?

All numbers multiples of 1.

What is an equilateral triangle?

All three sides are equal in length Each angle is 60°

What is the Pythagorean Theorem?

a2 + b2 = c2 * c = hypotenuse

Exponent Rule: a2·a3 is the same as...

a2+3 = a5

Exponent Rule: (a2)3 is the same as...

a2·3 = a6

Exponent Rule: a6/a2 is the same as...

a6-2 = a4

What is the common monomial factor in the expression 4(c^3)d - (c^2)(d^2) + 2cd?

cd

What is the formula to find the distance or amount in a rate problem?

d = rt

Which quadrant is the upper right hand?

I

What is a set with no members called?

the empty set, denoted by a circle with a diagonal through it.

How many factors does a prime number have?

Only 2 factors, 1 and itself

#3 What is an important property of a 30-60-90 triangle?

• The ratio of the length of the three sides is x:x√3:2x

What is probability?

# of possible outcomes that satisfy the condition # of total possible outcomes

2³×7³

(2x7)³

Squares

-Most elite quadrilateral, shape with the highest number of special properties -A square is a rectangle, rhombus and a parallelogram so it has all the special properties -Hard to prove something is a square

Fractional Exponents

-Raising a number to the 1/2 is the same as squaring it b^1/2 = √b ex: 2^1/2 = k => (2^1/2)^2 = k^2 => 2 = k^2 => √2 = k The rules: b^1/m = m√b b^m/n = (b^m)^1/n = (b^1/n)^m These are true for all positive numbers. If the denominator of the exponent-fraction is odd then the base can be negative as well ex: 2^3/5 = (2^3)^1/5 = 5√(2^3) = 5√8 OR 2^3/5 = (2^1/5)^3 = (5√2)^3 ex: 8^4/3 = 3√8^4 = (3√8)^4 = 2^4 = 16

Rectangles

-Rectangles are parallelograms and the 4 big properties are true for them 2 special rectangle properties: 1) All 4 angles are equal to 90 degrees 2) Diagonals are congruent QS = PR

Percentiles

-position of an individual score in a large population -More precise than a box plot -If a score is in the 40th percentile then their score is larger than 40% of the distribution -lowest score is 0%, highest score is 99% (no such thing as 100%) -halfway between percentiles is not the same as halfway between scores

Decimal of 1/7

0.143

How to count factors of large numbers

1) Find prime factorization of number ex: 8400 = 84x100= 7x12x10x10 = 7x3x2x2x5x2x5x2= 2^4x3x5^2x7 2) Make a list of the exponents of the prime factors {4,1,2,1} 3) Add 1 to every number on the list {5,2,3,2} 4) Multiply the numbers together 5x2x3x2=60 The number 8400 has 60 factors (including 1 and 8400) *To find the number of odd factors, repeat this procedure but ignore the factors of 2 ex: 21600 = 2^5 x 3^3 x 5^2 {3 x 2} +1 = {4 x 3} = 12 odd factors *To find the even factors, find total factors and subtract odd factors 72 total factors - 12 = 60 even factors

How to find the greatest common factor (GCF)

1) Find the prime factorization ex: GCF of 360 and 800 360 = 36x100 = 6x6x10=3x2x3x2x5x2 = 3^2x2^3x5 800 = 80x10 = 40x2x5x2 = 2x2x5x2x2x5x2 = 2^5x5^2 2) What is the highest power of 2 that they have in common? Each have 3 factors of 2 Highest power of 3? Only one has 3 so it is zero Highest power of 5? 1 Thus GCF = 2^3 x 5 = 8x5 = 40

Perfect Squares 1-15

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

How do you know when you can plug in the answer? (ie: plug-in one of the given answers)

1. When the question begins with: How much..., How many..., What is the value of... 2. Answer choices are in ascending or descending order 3. Your tempted to make an algebraic equation

Write 10,843 X 10^7 in scientific notation

1.0843 X 10^11

How many degrees do the three angles inside a triangle equal?

180°

In a triangle where the two legs are 4 and 3, what is the value of a line directly intersecting the middle coming from the meeting point of the two legs?

2.4. We calculate the area (6) and then turn the triangle on its side and use x as the height to calculate again. (5x)/2=6

A brick with dimensions 10. 15 and 25 weighs 1.5 kg. A second brick (same density) has dimensions 12, 18, and 30. What is the weight of the second brick?

2.592 kg

If a=-1 and b=3, what is the value of (4(a^3)(b^2) - 12(a^2)(b^5)) / (16(a^3)(b^2))?

20.5

20<all primes<30

23, 29

240 is 30% of what number?

240=.3x 240/.3=x 2400/3=x 800=x

2⁵/2³

2²

60 < all primes <70

61, 67

5/8 in percent?

62.5%

What are the smallest three prime numbers greater than 65?

67, 71, 73

Cubes of 1 - 10

6^3 = 216 7^3 = 343 8^3 = 512 9^3 = 729

Convert 0.7% to a fraction.

7 / 1000

When plugging in the answer, what answer should you assume is correct?

C. Use C as a starting point

Definition of rate

Any ratio with different units in the numerator and denominator. Rates are ratios -Must set up an equation of form: ratio=ratio is called a proportion *MUST know there are 360 degrees in a revolution ex: A bumblebee wing flaps 1440 times in 8 seconds. How many times does it flap in a minute? 1440/8 = 720/4 = 360/2 = 180 180 flaps/ second 1 minute = 60 seconds 180*60= 10,800 ex: If gold has a density of 20 grams/cm^3 and the price of gold is $50/gram, then what's the price of the gold in a 2cm*2cm*2cm cube of gold? 8cm^3 * 20 gram/cm^3 * $50/gram = 8*20*50= $8000

Formula to calculate arc length?

Arc length = (n/360) x pi(2r) where n is the number of degrees.

How do you find the area of a circle?

Area = πr2

How do you create a right isosceles triangle?

By cutting a square in half.

Hoe to change fractions to percents

Change to a decimal and move 2 places over for % ex: 3/8 = 0.375 = 37.5% ex: 2/3 = .666667 = 66.67%

Coefficient

Constant factor of a term; ex: 6 is the constant in 6y^2 ex: -1 is the constant in -x

Dividing with decimals

For division, move the decimals to the right until the denominator is a whole number ex: (0.56/.0007) * 10,000 = 5600 / 7 = 800 ex: (0.00013/0.025) * 4 = 0.0052

Divisibility Rule for 5

If last digit is 5 or 0, then divisible by 5

Word problems

If there are 3+ quantities in a word problem, you will always want to relate all of them to a single variable and construct a single equation

P and r are factors of 100. What is greater, pr or 100?

Indeterminable.

How many multiples does a given number have?

Infinite.

Solve: Five people are running in a race. The winner will get a gold medal, the person who comes in second will get a silver medal, and the person who comes in third will get a bronze medal. How many different orders of gold-silver-bronze winners can there be? How many slots, how many options?

Is it a permutation or combination? Permutation because it asks for the order. There are 5 people and three possible prizes so, Slots = 3 (possible prizes) # of options = 5 people (slot 1), 4(slot 2), 3 (slot 1) so 5x4x3 = 60 different orders

What happens to a number raised to a negative power? 2-2

It becomes 1 over the number raised to a positive power (2-2 = 1/22 = 1/4)

Does a negative number raised to an even power become negative or positive? (-22)

It becomes positive -22 = 4

Does a negative number raised to an odd power become negative or positive? (-23)

It stays negative -23 = -8

Is zero positive or negative

It's neither

Laws of Exponents II

Just as multiplication distributes over addition and subtraction (P*(M+N) = PM + PN), exponents distribute over multiplication and division (ab)^n = (a^n)(b^n) (a/b)^n = a^n/b^n ex: 18^8 = (2*3^2)^8 = 2^8*3^16 -ILLEGAL to distribute exponent over addition or subtraction; must solve in parenthesis first (M + N)^p =/= M^p + N^p -Lower power is always a factor of a higher power ex: 3^32 - 3^28 = (3^28)(3^4) - 3(28)(1) = (3^28)(3^4 - 1) = (3^28)(81-1) = 80(3^28) - b^s = b^t => s=t

How do you divide and multiply square roots?

Just like any other number √3 x √12 = √36 √16 = √16 = 4 = 2 4 √4 2

GCD - LCM formula

LCM = PXQ / GCF ex: the LCM of 48x75 48 = 12x4 = 2x2x2x2x3 = 3 x 2^4 75 = 15x5 = 3x5x5 GCF = 3 LCM = 48x75 / 3 = 48x25 = 24x50 = 12x100 =1200

If you raise a number other than 1 to a power greater than 1 with the number become larger or smaller

Larger. 22 = 4

Cirfumference

Length around the circle C = πd or C = 2πr

Arc Length

Length of an arc, part of the whole circumference -Found by setting up a part to whole proportion arc length / 2πr = angle / 360° Ex: If an arc is 120° that is is 1/3 of 360° so the arc must be 1/3 of the circumference

How do you find the volume of a rectangle?

Length x width x height

Divisibility Rule for 4

Look at the last 2 digits, if they form a 2 digit number divisible by 4, then the whole number is divisible by 4

Range

Max - min; difference between highest and lowest value

How to change decimals to percents

Multiply by 100 and move decimal point 2 places to the right ex: 0.68 = 68% ex: 0.075 = 7.5% ex: 2.3 = 230%

Angle bisector

Opposite of the median; divides the angle in half but doesn't usually divide the opposite side in half -The line down the middle of an isosceles triangle from the vertex to the midpoint of the bases plays all 4 roles at once (altitude, median, perpendicular bisector and angle bisector)

One is (a prime or not?)

NOT A PRIME

Are 0 and 1 prime numbers?

No

Are fractions integers?

No

Can you divide a number by 0?

No

Is x2=16 the same as √16?

No because in x2 = 16, x = ±4, but √16 = +4 (NOT -4)

Can the absolute value of a number be negative?

No, it's always positive. |-5| = 5 and |5| = 5

Can you add sqrt 3 and sqrt 5?

No, only like radicals can be added.

Can negative numbers be prime?

No, prime numbers are only positive

Will ETS make you calculate standard deviation?

No, so if ETS is asking for you to calculate standard deviation, then chances are the answer is D (cannot be determined)

Can standard deviation ever be negative?

No, standard deviation can only be positive. So if you see negative numbers in a standard deviation quantitative comparison question, chances are the answer is D (cannot be determined)

7 divided by ∅

Null

What are consecutive integers?

Numbers listed in order (increasing or decreasing). Can count by 1s, 2, 3s, etc.

Probability of an Event

P(E) = number of favorable outcomes/total number of possible outcomes

If Event is impossible

P(E) = ø

What are the orders of operation?

PEMDAS Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction

A quadrilateral where two diagonals bisect each other

Parallelogram

What is 75% of 280?

Percents and Fractions - only use if percent is an easy fraction 3/4 * 280/1 = 3/1 * 70/1 = 210

If a is positive, aⁿ is

Positive

Rules of proportions

Proportion is an equation in the form of fraction=fraction a/b=c/d -> ad=bc ex: 5/7=3/x -> 5x=21 -> x=21/5 *CAN do vertical cancellation in same fraction *CAN do horizontal cancellation *CAN'T do diagonal cancellation -when we multiply two fractions, cross-cancellation is legal (a/b * c/d) but when we are in proportion it is illegal

How to change percents to fractions

Put percent over 100 and simplify ex: 20% = 20/100 = 1/5 ex: 0.02 = 2/10,000 = 1/5,000

Quartiles

Q2 is the median, Q1 is the median of the lower list and Q3 is the median of the upper list ex: Find quartiles {3, 3, 4, 4, 4, 9, 11, 13, 14, 15, 15, 17, 17} Median or Q2= 11 (excluded from upper and lower list) Lower = {3, 3, 4, 4, 4, 9} Q1 = 4 Upper = {12, 14, 15, 15, 17, 17} Q3= 15 *If the whole set is odd then the median is on that list ex: A set of 1203 Exclude median = 1202; on each half list is 601 On lowest list, median is on this list so 600/2 = 300 lowest quarter

In a rectangle, all angles are

Right

Formula for the area of a sector of a circle?

Sector area = (n/360) X (pi)r^2

chord

Segment with 2 endpoints on the circle; a chord that passes through the center (the longest chord) is called the diameter -equal length chords intercept equal arcs

If you raise a fraction between 0 and 1 to a power greater than 1 will the number become larger or smaller?

Smaller (1/2)2 = 1/4

Extraneous solutions

Solutions that result correctly from the math but which won't work in the original equation ex: |2x + 5| = x + 1 2x + 5 = x+ 1 or 2x + 5 = -x - 1 x = -4 x = -2 |2(-4) + 5| = -4 + 1 |2(-2) + 5| = (-2) + 1 |-3| = -3 |1| = -1 3 =/= -3 1 =/= -1 Neither solution works

Substitution Method

Solve one equation, either one, for one of the variables. Get one variable by itself on one side of the equation: x + 2y = 11 - > x = 11 - 2y Now replace x in the other equation with the expression that x equals: 2x + 3y = 15 -> 2(11-2y) + 3y = 15 - > 22 - 4y + 3y = 15 -> 22 - y =15 -> y= 7 Now plug the value of x back into equation to solve for x: x= 11 - 2(7) = 11 - 14 = -3

Equation to find the multipler

Some problems give us the starting and ending values and ask us to find the percent of increase or decrease multiplier = new price/old price ex: Price decreased from $250 to $200, what is the percent decrease? 200/250 = 4/5 = 0.8 1-0.8=.2 -> 20% decrease ex: Price of an item increased from $200 to $800, what's the percent increase? 800/200 = 4; 4-1=3 so there is a 300% increase ex: Price of an item increased from $60 to $102, what's the percent increase? 102/60 = 17/10 = 1.7; 1-1.7 = .7 so there is a 70% increase

When you plug-in for a Quantitative Comparison, what should you start with?

Something nice and happy

An Angle that's 180°

Straight Angle

When dividing exponential #s with the same base, you do this to the exponents...

Subtract them. i.e (5^7)/(5^3)= 5^4

How to recognize a multiple of 6

Sum of digits is a multiple of 3 and the last digit is even.

How do you find the surface area of a rectangle?

Sum of the area of all of its sides

Different ways to present ratios

Test will always give ratios in the simplest form, absolute number of participants will be larger than the number in the given ratio (use scale factor) 1) p to q form: ratio of boys to girls is 3 to 4 2) fraction form: ratio of boys to girls is 3/4 3) colon form: ratio of boys to girls is 3:4 4) idiom form: for every 3 boys, there are 4 girls -To solve ratio problems, set 2 equivalent fractions equal (known as a proportion) ex: In a class, the ratio of boys to girls is 3:7. If there are 32 more girls than boys, how many boys are there? boys=3n girls=7n 7n-3n = 4n 4n = 32 n = 8 Number of boys = 3n = 3(8) = 24 *Each part is part of a whole factor Ex: Purpose concrete is creased using a 1:2:3 ratio of cement to sand to gravel. If we have 150 kgs of sand, how many kgs of concrete can we make? 1+2+3 = 6 total parts sand: concrete = 2:6 or 1:3 1/3 = 150/x -> 1/3x = 150 -> x=150*3 = 450 kgs of concrete

What is the mean?

The average Add all the numbers and divide the total by the number of numbers. ie: 5, 1, 4, 6 = 16/4 = 4 is the average

Describe the relationship between 3x^2 and 3(x - 1)^2

The graph of 3(x - 1)^2 is a translation (shift) of the graph one unit or space to the right.

What is the "range" of a series of numbers?

The greatest value minus the smallest.

What is the original number in a percentage DECREASE problem?

The larger number

What is the difference in a percentage increase/decrease problem?

The larger number minus the smaller number

What is the quotient?

The result of division

What is the product?

The result of multiplication

Describe the relationship between the graphs of x^2 and (1/2)x^2

The second graph is less steep.

What is the intersection of A and B?

The set of elements found in both A and B.

Multiplying/Dividing with Tens

When we divide any number by 10 or multiply by 0.1, we move the decimal point one place to the left ex: 0.02 / 10 = 0.002 39.85 * 0.1 = 3.985 64,000 / 0.0001 = 6.4 When we multiply any number by 10 or divide by 0.1, we move the decimal point on place to the right ex: 24 * 10 = 240 2.35/0.01 = 235 4.7/10^-4 = 4.7/0.0001 = 47,000

When does a function automatically have a restricted domain (2)?

When we need to avoid having a zero in the denominator or avoid taking the square root of a number.

Can integers be even or odd?

Yes

Can you simplify sqrt72?

Yes, because you can factor out a perfect square (36). Sqrt(36 x 2) = sqrt36 X sqrt2 = 6sqrt2.

Is a line an angle?

Yes, it is a perfectly flat angle of 180°

Can you subtract 3sqrt4 from sqrt4?

Yes, like radicals can be added/subtracted.

Proportional Reasoning Steps

a) Pick easy numbers to satisfy the equation, can also change any constants to 1 b) Change whatever values need to be changed, leave the quantity in the Q as unknown and solve for it Ex: In V^2 = 2ad, if V triples and a doubles, then d is multiplied by what? 1^2 = (1)(1)d 3^2 = (1)(2)d => 9 = 2d => 9/2 = d ex: In T^2 = KR^3, if T is multiplied by 5, R is multiplied by what? (1)^2 = (1)R^3 => 5^2 = R^3 => 3√25 = R R = 3√25 = 3√(5^2) = 5^2/3

Distributive Property

a*(b+c) = a*b + a*c a*(b-c) = a*b - a*c

If a<b, then

a+c<b+c

Pythagorean theorem and triplets

a^2 + b^2 = c^2 Only for right triangles {3, 4, 5,}, {5, 12, 13}, {8, 15, 17}, {7,24, 24} -If you scale down the sides of a right triangle you could realize they are a Pythagorean triplet -Whenever you have to find the length of a diagonal, chances are VERY good you have to solve with the Pythagorean theorem

Distributive Property: a(b+c) is the same as...

ab + ac

Distributive Property: a(b-c) is the same as...

ab - ac

Find distance when given time and rate

d=rt so r= d/t and t=d/r

formula for distance problems

distance=rate×time or d=rt

In word problems, is means.... of means....

equals multiply ex: What is 3/5 of $400 3/5 * 400/1 = 3/1 * 80 = 340= 240

1 is a divisor of

every number

What is the graph of f(x) shifted left c units or spaces?

f(x + c)

Slope given 2 points

m= (Y1-Y2)/(X1-X2)

The objects in a set are called two names:

members or elements

To multiply a number by 10^x

move the decimal point to the right x places

Circumference of a circle

pi(diameter)

Percentages How do you set-up: 1. 5 is r percent of 25 2. s is 25 percent of 60

r 1. 5 = 100 (25) 25 2. s = 100 (60)

1 is the

smallest positive integer

What is the formula to find the time in a rate problem?

t = d/r

The larger the absolute value of the slope...

the steeper the slope.

0^0

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#3 What are the important properties of a 45-45-90 triangle?

• The ratio of the lengths of the three sides is x:x:x√2.

#1 What are the important properties of a 45-45-90 triangle?

• The triangle is a right triangle.

What is a minor arc?

The shortest arc between points A and B on a circle's diameter.

What is the unfactored form of (x+y)2

x2 + 2xy + y2

x^6 / x^3

x^(6-3) = x^3

Any Horizontal line slope

zero

For what type of triangles can you use the Pythagorean Theorem?

Only right triangles

10<all primes<20

11, 13, 17, 19

What is √2

1.4

-3²

9

Vertical lines

Do not have slopes!

∅ Is

EVEN

binomial product of (x+y)²

(x+y)(x+y)

What is the unfactored form of x2 - y2

(x+y)(x-y)

binomial product of (x-y)²

(x+y)(x-y)

The "At Least" Scenario

"At least one" - the complement of "at least one" is "none" P(at least one success) = 1 - P(zero successes) Practice: We roll one fair six-sided die 8 times. What's the probability that we will roll at least 1 six? -The complement of "at least one six" is "zero sixes" -not getting a six in one roll is 5/6 -not getting a six in eight rolls if (5/6)^8 P(at least one six) = 1 - (5/6)^8

Simplify 4sqrt21 X 5sqrt2 / 10sqrt7

(4x5/10)xsq(21x2/7)=2sqrt6

If 10800 is invested at a simple interest rate of 4%, what is the value of the investment after 18 months?

$11,448

(6sqrt3) x (2sqrt5) =

(6 x 2)(sqrt3 x sqrt5) = 12sqrt15

Equation for a percent decrease

"Y is decreased by 30%" or "x is 30% less than Y" (multiplier for a P% decrease) = 1- (P% as a decimal) ex: After an item was discounted 80% the new price is %150. What was the original price? 1-.8 = .2 x*.2 = 150 x = $750

Equation for a percent increase

"Y is increased by 30%" or "x is 30% greater than Y" (multiplier for a P% increase) = 1 + (P% as a decimal) ex: After a 30% increase, the price of something is $78. What was the original price? x*1.3=78 x=60

Introduction to Counting

"and" means multiply "or" means add ex: If at a dinner you choose 1 of 2 meat entries or one of 2 veggie entrees, how many choices do you have? 2 +3 = 5 ex: If you can choose 1 of 4 entrees and 1 of 2 desserts, how many combinations are there? 2*3 = 6

Solve: A pizza can be ordered with any of eight possible toppings. How many different ways can you order a pizza with three different toppings?

"any of eight possible toppings" = group in no particular order, so this is a combination problem Slots = 3 different toppings # of options = 8 possible toppings so 8x7x6 (possible topping in 3 slots) 3x2x1 (factorial of the number of slots) = 56 different ways

Hector invested $6000. Part was invested in account with 9% simple annual interest, and the rest in account with 7% simple annual interest. If he earned $490 in the first year of these investments, how much did he invest in each account?

$3,500 in the 9% and $2,500 in the 7%.

(12sqrt15) / (2sqrt5) =

(12/2) x (sqrt15 / sqrt5) = 6sqrt3

What are the most common right triangles?

(3, 4, 5), (6, 8, 10), and (5, 12, 13)

Difference of Two Squares

(a + b)(a - b) = a^2 - b^2 ex: 9x^2 - 16 = (3x - 4)(3x + 4) ex: 25x^2 - 64y^2 = (5x + 8y)(5x - 8y) ex: x^2y^2-1 = (xy + 1)(xy - 1) ex: x^7 - 4x^5 = x^5(x^2 - 4) = x^5(x - 2)(x + 2) Practice: If y=5 + x and y=12 - x and if y^2=x^2 + k, what does k equal? y = 5+x -> y-x=5 y=12-x -> y+x=12 y^2=x^2 + k k = y^2 - x^2 = (y+x)(y-x) = (5)(12) = 60

Square of a Difference

(a-b)^2 = (a-b)(a-b) = a^2 - ab -ba + b^2 = a^2 -2ab + b^2

Simplify the expression [(b^2 - c^2) / (b - c)]

(b + c)

Volume of a rectangular solid

(length)(width)(height)

The sum of the measures of the n angles in a polygon with n sides

(n-2) x 180

Simplify the expression (p^2 - q^2)/ -5(q - p)

(p + q)/5

Area of a circle

(pi)r²

Distance

(rate)(time) d=rt

Square of a Sum

*Common mistake = illegal to distribute an exponent across addition or subtraction illegal: (a + b)^2 =/= a^2 + b^2 legal: (a + b)^2 = (a+b)(a+b) = a^2 + ab + ba +b^2 = a^2+2ab+b^2 This is called the Square of a Sum

Average Speed Word Problems

*NOT solved by adding numbers and dividing by 2 Average velocity = total distance / total time *Often need to find the total distance/total by adding the distances and times of the individuals legs *Sometimes the problems won't give you all the #'s Practice: Cassie drove from A to B at constant speed 60 mph. She then returned on the same route, from B to A at constant speed 20 mph. What was her average speed? T1 = D/R = D/60 T2 = D/R = D/20 Tt = T1 + T2 = D/60 + D/20 = multiply by the LCM = D/60 + 3D/60 = 4D/60 = D/15 Vaverage = Dt/Tt = 2D/(D/15) = 2D/1 *15/D = 30 mph

Inequalities

*When you see "number" or see x, you have to think of all numbers - positive, negative, zero, integers, fractions and decimals *Can always add or subtract the same thing from both sides and the inequality remains the same ex: x + 7 > 2 => x > -5 *Multiply or divide both sides by any positive number will preserve the inequality *Multiply or divide both sides by a negative number will reverse the inequality ex: 2/x > 1/3 x=/= 0, x>0 Because all #'s are positive we can cross multiply 6>x ex: -4 < 5 -3x < 17 Subtract 5 from all 3 parts -9 < -3x < 12 Divide -3 from all 3 sides -3 > x > -4

What is the maximum value for the function g(x) = (-2x^2) -1?

-1

Independent Events

-2 events are independent if they have absolutely no effect on each other; the outcome of one has no influence on the outcome of the other ex: 2 rolls of a die, flips of a coin -With replacement: each and every choice comes from a full and shuffled deck;e each choice is independent of previous choice -Without replacement: each new choice is made from a smaller and smaller deck; each choice is NOT independent of the previous choice

Venn Diagram Word Problems

-2 groups and each member can belong to either group, both or none A + B + C + D = total Practice: Of the 100 students in a school, 60 are in the band and 35 are on the baseball team. If 25 students are neither in the band or on the baseball team. How many are in both? A + B= 60 B + C = 35 D = 25 A + B +C = 75 A + 35 = 75 -> A = 40 40 + B = 60 -> B = 20

Isosceles Triangles

-2 sides are equal -Equal angles opposite the equal sides -Can have a right angle and an obtuse angle

-3³

-27

6w^2 - w - 15 = 0

-3/2 , 5/3

For what values should the domain be restricted for the function f(x) = sqrt(x + 8)

-8

Listing vs. Counting vs. Formal Probability Rules

-> Use the formal algebraic rules IF: 1) The problem gives you algebraic expressions P(A) = 0.5, P(B) = 0.6 2) The items concerned are coins, dice, etc 3) The language of the problem uses words like mutually exclusive or independent -> Use listing IF: -The full list is very short (under 10) -Rare to use listing to solve problem, mainly used to find alternate solution -> Use counting IF: -The problem involves a selection of several elements from a set with certain restrictions ->Whether you use formal algebraic rules or counting - may be easier to calculate or count the complement of what you want using The Complement Rule: P(A) = 1 - P(not A)

Tangent Line

-A line that passes by a circle and touches it only at one point -If we draw a radius to the point of tangency, then the radius and the tangent line are perpendicular and form a right angle

Writing Equations of Lines

-A slope and single point are enough to determine the unique line Practice: A line passes through the point (A, 30) with a slope of m = 1/2. What is the value of A? y = 1/2x + 1 => 30 = 1/2x + 1 => 29 = 1/2x => 58=x Practice: Line A has a slope of -5/3 and passes through the point (-2,7). What is the x-intercept of Line A? y = mx + b => 7 = -5/3(-2)+ b => 7 = 10/3 + b => 21/3 - 10/3 = b => 11/3 = b y = -5/3x + 11/3 => 0 = -5/3x + 11/3 => 5/3x = 11/3 => 5x = 11 => x = 11/5 Practice: Line J passes through the points (-3, -2), (1,1), and (7,Q). Find the value of Q m = 1- -2/ 1 - -3 = 3/4 y = 3/4x + b => 1 = 3/4(1) + b => 1 - 3/4 = b => b = 1/4 y = 3/4x + 1/4 => y = 3/4(7) + 1/4 => y = 21/4 + 1/4 => y = 22/4 = 11/2

Geometry Assumption Rules

-Always allowed to assume lines that look straight are straight -CAN'T assume if 2 lengths look the same, are the same -CAN'T assume lines are parallel or perpendicular or right angles -CAN'T assume perfect squares -CAN'T assume equal lengths, horizontal, or veritcal

Fundamental Counting Principal with Restrictions

-Always start with most restrictive stage first -If more than one restriction appear, start with most restrictive then move on to the second most restrictive, etc. Ex: Children A, B, C, D, E, F and G will sit in 7 adjacent chairs. D must be in the middle and G must be next to D. How many different orders can be arranged? D = 1, G = 2, A = 5, B = 4, C= 3, E= 2, F= 1 N = 1*2*5*4*3*2*1 = 10*24 = 240 different orders

Circle properties

-Any triangle in a circle with 2 sides that are radii have to be isosceles -If the "chord" side of a triangle is also equal to the radius, the the triangle would be equilateral

Rationalizing

-Avoid radicals in the denominator ex: 14/√21 * √21/√21 = (14√21)/21 = (2√21)/3 ex: 4 - √6 / 2√3 *√3/√3 = 4√3 - (√6)(√3) / 2*3 = 4√3 - 3√2 / 6 Multiply by the opposite ex: (4 + 2√5) / (3 + √5) * (3 - √5)/(3 - √5) = (4 + 2√5)(3 - √5) / 9 - 5 = (12 - 4√5 + 6√5 - 10) / 4 = (2 - 4√5 + 6√5) / 4 = (2 - 2√5) / 4 = 1 + √5 = 4

Operations with Roots

-Can combine 2 terms ONLY if they have the same radical factor ex: √72 - √32 = √(36*2) - √16*2) = 6√2 - 4√2 = 2√2 ex: 6√2 - 4√3 Can't be simplified further because the radicals aren't equal -Multiplication is commutative and and associative (can switch the order around). Can multiply the whole by whole and radical by radical ex: (3√5)(2√15) = 6√(5*15) = 6√(5*5*3) = 6*5√3 = 30√3 -Also when we divide, we can divide whole by whole and radicals by radicals ex: 54√35 / 18√5 = 54/18 * √35/√5 = Simplify fractions 3√7 -If we are squaring a radical expression, we square the number and radical separately ex: (2√3)^4 = 2^4 * (√3)^4 = 16 * √3 * √3 * √3 * √3 = 16 * 3 * 3 = 144 OR ((2√3)^2)^2 = (4*3)^2 = 12^2 = 144 -Any even power of a square root can be written as a power of a whole number ex: (√2)^48 = ((√2)^2)^24 = 2^24

Comparing inequalities

-Can combine inequalities (a<b, b<c) in the same direction (a<b<c) -Can add inequalities in the same direction a<b & c<d then (a+c) < (b+d) ex: 5>2 and 11>8 then 5+11 = 16 > 2+8=10 -Can subtract inequalities in opposite direction a>b and d<c then (a-d)>(b-c) ex: 20>15 and 10<12 then 20-10 = 10 > 15-12 = 3 -No general rule for the multiplication or division of inequalities -any positive > any negative -Adding a positive makes a number greater, subtracting a positive makes it less

Rules of fractions

-Can multiply any expression by n/n because multiplying by 1 never changes the value -the product of any fraction with its reciprocal equals one 4/17*17/4 = 1 6*1/6 = 1 -one divided by any fraction equals the reciprocal of that fraction 1/(3/7) = 7/3 -If a number is bigger than 1, then its reciprocal is between 0 and 1 and vice versa

Operations with Fractions

-Can only add or subtract 2 fractions if they have a common denominators ex 1/5 + 3/5 = 4/5 Find a common denominator by multiplying both parts of a fraction by a number to make both denominators equal ex: 3/5 - 1/3 = 3/5(3/3) - 1/3(5/5) = 9/15 - 5/15 = 4/15 -Can multiply fractions by multiplying across the numerators and denominators (can cancel across to make smaller before you multiply) ex: 5/14 * 7/15 = 1/2*1/3 = 1/6 ex: 21/35*30/42=1/7*6/2=6/14=3/7 -Can divide fractions by multiplying by the reciprocal ex: (3/5)/2 = 3/5*1/2=3/10 ex: 6/(3/4) = 6*4/3 = 24/3 = 8

Analyzing Probability Questions

-Can't be mutually exclusive and independent at the same time -Selection processes that are "without replacement" are NEVER independent Practice: In a game, Phase 1 you flip one coin as many as 3 times. If you flip 3 tails, you lose. As soon as you get your 1st head, you advance to phase 2. In phase 2, you roll a six-sided die once. If you roll a 6, you win. Any other roll, you lose. What's the probability of winning? P(^) = 1 - P(TTT) 1/2 * 1/2 * 1/2 = 1/8 P(^) = 1 - 1/8 = 7/8 =7/8 * 1/6 = 7/48

Recursive Sequences

-Can't jump right away to value, have to work out way term by term to the value we want. Some sequences depend on the previous two terms Practice: For a sequence with rate An = An-1 + An-2 for n>2, and starting values of A1 = 1 and An = 3, find value of A6. A1 = 1 and A2 = 3 A3 = A2 + A1 = 3 + 1 = 4 A4 = A3 + A2 = 4 + 3 =7 A5 = A4 + A3 = 7 + 4 = 11 A6 = A5 + A4 = 11 + 7 = 18

Comparing Fractions rules

-Changing numerators, same denominator: If a>b then (a/c) > (b/c) -Changing denominators, same numerator: If p>q, then (s/p) < (s/q) if p,q,s > 0 *Bigger denominators make smaller fractions 7/24>7/36 -If both the numerator gets bigger and the denominator gets smaller then the fraction gets bigger 3/8<4/7 -If we multiply both the numerator and the denominator by the same # then we get an equivalent fraction 3/7 * 12/12 = 36/84 -Can compare fractions by cross multiply ex: 7/11 and 5/8 7*8 and 5*11 56>55 --> 7/11>5/8

Graphing lLines

-Every line in the xy plane has its own equation -For any given line, all the points on the line have x and y coordinates that satisfy the equation of the line -Any linear equation that relate x to y, with no multiplication or division of variables, must be the equation of some line in the xy plane ex: 3y - 4x - 12 If x = 0 then y = 4 (0,4) If y = 0 then x = -3 (-3,0) Practice: The equation of Line M is kx +3ky = 17, for some number k. If Line M passes through the point (2,1) find the value of k. k(2) + 3(1)k = 17 => 2k + 3k = 17 => 5k = 17 => k=17/5

Trapezoids

-Exactly one pair of parallel sides -the 2 parallel sides are called bases and the non-parallel sides are called legs -the 2 angles on a leg are always supplementary ∠ A + ∠ B = 180 and ∠ C + ∠ D = 180 -Some trapezoids have 2 equal legs, called "symmetrical" or "isosceles" trapezoids If KL // JM and if KJ = JM then ∠K = ∠L and ∠J = ∠M and the diagonals have equal length

Factor, divisor and divisible definitions

-Factor and divisor mean the same thing -Every integer is a factor of itself and 1 -If C/A=B then A is a divisor of C because it divides evenly into C Three ways to say the same thing: 1) 8 is a factor of 24 2) 8 is a divisor of 24 3) 24 is divisible by 8 ex: 12 is not divisible by 8 because there is no positive integer

The Units Digit Question

-Focus on the single digit multiplication only -Look for the repeating pattern and determine the period of the pattern (period if often 4) -Extend the pattern using multiples of the period ex: What is the units digit of 57^123? 7^1 = 7 7^2=_9 (9*7=63) 7^3 = __3 (3*7 = 21) 7^4 = __1 (7*1 = 7) 7^5 = __7 Answer: 1

Exponential Growth

-For a positive base greater than 1, the powers get larger at a fast rate ex: 7^5 = 16,801 -For a positive base less than 1, the powers get smaller at a fast rate ex: (1/2)^8 = 1/256 -For a negative base less than -1, the absolute values are getting bigger each time, but the + or - signs are alternating ex: (-3)^5 = -243 (-3)^6 = +729 -For a negative base between -1 and 0, the absolute values are getting smaller but the + or - signs are alternating ex: (-1/2)^7 = -1/128 (-1/2)^8 = +1/256

Using Counting Techniques in Probability

-Fundamental Counting Principle = K*N*D -Permutations = n! -Combinations = 10C3 Practice: A committee of 3 will be selected from a group of 8, including Alice and Bob. What's the probability that the committee includes Alice and not Bob? 8C3 = (8*7*6) / (3*2*1) = 4*7*2 = 56 6C2 = (6*5) / (2*1) = 3*5 = 15 Probability = 15/56

Boxplots

-Gaps from one bar to the next represent the difference in score, not difference in number of people or data points -25% below Q1 and 25% above Q3 -IQR is the middle 50% given by Q3 - Q1

Advanced Numerical Factoring

-Helps find the prime factorization of large numbers ex: 2491 = 2500 - 9 = (50^2) - (3^2) = (50 + 3)(50 - 3) = 47x53 ex: 9975 = 10,000 - 25 = (100^2) - (5^2) = (100 + 5)(100-5) = 105x95 = (5x21)(5x19) = 5x7x3x19 *Can also help with decimals Ex: simplify (0.999856/0.998) - 1 = (1-0.000144/1-0.012)-1 = (1-0.012)(1+0.012)/(1-0.012)-1 =(1 + 0.012) - 1 = 0.012

Comparing the size of different roots

-If b>1 and if n>m then 1 < n√b < m√b < b The higher the root, the smaller the number ex: 19 > √19 > 3√19 > 1 ex: 20√19 > 30√19 -If 0 < b < 1, and if n>m, then 0 < b < m√b < n√b < 1 ex: 2/5 < 50√2/5 < 75√2/5 < 1 The HIGHER the root, the CLOSER to 1

Square Roots continued

-If squaring makes numbers bigger, then taking a square root must make them smaller 1 < b^2, then b < b^2 => √b < b -If squaring makes numbers smaller, then taking a square root must make them bigger 0 < b < 1, then b > b^2 => √b > b

Square Roots

-If test makers write √ consider the positive roots only -If your calculations lead to a variable squared, consider both positive and negative roots √0 = 0 -CAN'T take the square root of a negative -For A > or = to 0, √A > or = to 0 -If the test writes "solve the equation" x^2 = 5, there are 2 solutions, both positive and negative √5 -If A<B<C then √A<√B<√C -Square roots are always positive regardless of whether y is positive or negative √y^2 = |y| -> x^2 = k => x = + or - √k Practice: If (x - 3)^2 = 16 Solve for x √(x - 3)^2 = √16 x - 3 = +4 => x = 7 OR x - 3 = -4 => x = -1

Shrinking and Expanding Gaps Word Problems

-Involve 2 travelers moving to or away from each other -Two travelers moving in opposite directions: *When moving in opposite directions, always ADD the speeds -If the 2 travelers are approaching each other, the sum of the speeds is the speed at which the gap is shrinking -If the 2 travelers are moving away from each other, the sum of the speeds is the speed at which is the gap is expanding -Two travelers moving in the same direction: *When moving in the same direction, we always SUBTRACT the speed -If the faster traveler is in front, then the difference in which the gap is expanding -If the slower traveler is in front, then the difference in speeds is the speed at which the gap is shrinking *In a problem where the gap is obviously shrinking or expanding, sometimes saves time to set up a D=RT for the gap itself Practice: Car X & Y are traveling from A to B on the same route at constant speeds. Car X is initially behind Car Y but Car X's speed is 1.25 times Car Y's speed. Car X passes Car Y at 1:30 pm. At 3:15 pm Car X reaches B and at that moment Car Y is still 35 miles away from B. What is Car X's speed? Time interval: 1:30 to 3:15pm => 1/2 + 1 + 1/4 = 2/4 + 5/4 = 7/4 hr Gap: R = D/T = 35 miles/(7/4 hr) = 35 * 4/7 = 20 mph x = 1.25y x - y = 20mph 1.25y - y = 20 = > 1/4y = 20 => y=80 mph x = 1.25(80) = 100 mph

Work Word Problems

-Machines or workers and how fast they can get certain jobs done A = RT a= amount of work done R= the work rate t=time 2 categories of work problems 1) Using proportions Practice: A machine, working at a constant rate, manufactures 36 staplers in 28 minutes. How many staplers can it make in 1 hour 45 minutes? 1 hour 45 minutes = 60 + 45 = 105 minutes staplers/ time: 36/28 minutes = x/105 minutes *Don't cross multiply 9/7 = x/105 => 9 = x/15 => 135 staplers = x 2) Multiple machines/workers working at different rates and how much they can get done together *the SUM of the individual work rates *Need to match units on each side Practice: When Amelia and Brad detail a car together, 1 car takes 3 hours. When Amelia details a car alone, 1 car takes 4 hours. How long does it take Brad to detail a car alone? Rate = car/hours Rab = Ra + Rb Rab = 1/3 Ra = 1/4 Rb = Rab - Ra = 1/3 - 1/4 = 4/12 - 3/12 = 1/12 It takes Brad 12 hours to detail 1 car

Counting with Identical terms

-arrange sets in which some of the items are identical -In a set of n items with b identical items: N = n! / b! Ex: Librarian has 7 books to arrange 4 different novels and 3 identical copies of the same dictionary. How many different orders could these 7 books be put on the shelf? N = 7! / 3! The 3! on the top and bottom cancel out leaving 7*6*5*4 = 42*20 = 84*10 = 840 -One set of n items with more than one set of identical items: N = n! / (b!)(c!)(d!) Ex: How many different arrangements of the 11 letters of Mississippi? 11 letters, 4 i's, 4 s's, 2 p's N = 11! / 4! *4! * 2! = 11*10*9*8*7*6*5/4*3*2*2 = 11*10*9*7*5 = 99 * 350 = 34,650

Integer Property Strategies

-Make sure your dealing with integers; must use words "integer", "even", "odd", "prime" -Don't forget about zeros and negative -Question only talks about remainders if all the numbers involved are positive integers -Factors: 13 is a factor/divisor of 78 78 is divisible by/a multiple of 13 13 is part of the prime factorization of 78 -1 is not a prime -2 is the lowest and only even prime number -a negative squared is a positive -LCM = PxQ / GCF -> always cancel with P/GCF or Q/GCF -If test gives variables can always use substitution 1=odd 2=even -For all questions involving a remainder, remember these strategies: 1) Listing possible dividends (when divided by 6 have remainder of 3) 2) Using the rebuilding the dividend formula dividend = divisor * quotient + remainder -For consecutive numbers, often appear in variable form (t^2 - 2t)*(t-1) = t(t-2)*(t-1) = (t-2)*(t-1)*t This expression is the product of 3 consecutive integers only if we know that t is an integer

Mixture Question Word Problems

-Mixing solutions of various concentrations. Many materials dissolve in water which forms a solution. -The dissolved substance is called the solute -Concentration indicates how strong the solution is; how much solute is dissolved in a given quantity of water; always expressed as a percent Concentration = Amt of solute/ amt of solution * 100 -Add water to a solution -> makes a less concentrated solution -Add pure solute to a solution -> makes a more concentrated solution -If the amounts of 2 different solutions are initially unknown, we have to set up simultaneous equations. One equation will be a "total" equation and the other will be the amount of solute. Practice: State with unlimited supplies a 20% H2S04 solution and of a 50% H2S04 solution. We combine x liters of the first with Y liters of the second to produce 7 liters of a 40% H2S02 solution. What does x equal? total equation = x + y = 7 -> y = 7 - x Solute = .4 * 7 = 2.8L Solute #1 = 0.2x Solute #2 = 0.5Y => 0.2X + 0.5Y = 2.8L => 0.2X + 0.5(7 - x) = 2.8 => 2x + 5(7 - x) = 28 => 2x + 35 - 5x = 28 => -3x = -7 => x = 7/3 L

Laws of Exponents

-Multiplying 2 powers of the same base means you can add the exponents (7^m)*(7^n) = 7^m+n -Dividing powers of the same base means you can subtract the exponents a^m / a^n = a^m-n -a^0 = 1 if a=/= 0 -Raising a power to a power results in multiplying the exponents (a^m)^n = a^m*n -CAN'T apply these rules if bases are different ex: 2^3*3^5 -NO LAW for the sum or difference of powers ex: 3^4 + 3^7 or 5^8 - 5^2

Intro to Probability

-Probability is a ratio P = # of successes / total # of outcomes 0 ≤ P ≤ 1 where 0 means success is impossible and 1 means success is certain ex: What's the probability that a month name has an R in it? 8/12 = 2/3 in probability "OR" means add & "AND" means multiply

Regular polygons

-Regular means special and elite when it describes a polygon and means that the polygon must be: 1) equilateral (all sides are equal) 2) equiangular (all angles are equal) "regular triangle" is an equilateral triangle "regular quadrilateral" is a square "regular pentagon" the sum of the angle is 540° so each angle is 540/5 = 108° "regular octagon" is 180*6 = 1080° so each angle is 1080/8 = 135°

Simplifying Roots

-Simplify square roots by factoring out the largest perfect square root ex: √75 = √(25*3) = 5√3 -Find the prime factorization: any pairs of prime factors and any even powers of primes are perfect squares ex: √2800 = √(28*100) = 10√(4*7) = 20√7 ex: √(3^5) = √(3^2 * 3^2 * 3^1) = 3^2√3

The Properties of Roots

-Some properties of exponents are the same as the properties of roots: -Roots also distribute over multiplication and division √PQ = (√P)(√Q) √(P/Q) = √P/√Q ex: Simplify (√12)(√27) = √12*27 = √12*3*9 = √36*9 = √36 * √9 = 6*3 = 18 ex: √(4/50) = Reduce the fraction √(2/25) = (√2)/5 -Roots do NOT distribute over addition and subtraction √P + √Q =/= √(P+Q)

Factors of a perfect square

-The exponents of the prime factors of a square all must be even k = 2^6 x 3^4 x 5^2 just by looking we know its a perfect square k = 360^2 -A perfect square always has an odd number of factors ex: 36 = 6^2

Equations with Square Roots

-Undoing a square root by squaring both sides - unsquaring could be positive or negative -Radical equations can have extraneous roots, must plug back in to make sure they work ex: √(x + 3) = x - 3 => x + 3 = (x - 3)^2 => x+3 = x^2 - 6x + 9 = > 0 = x^2 -7x + 6 => 0 = (x - 6)(x -1) x = {1, 6} Plugged back in √(1 + 3) = 1 - 3 => 2 =/= -2 Doesn't work √(6 + 3) = 6 - 3 => 3 = 3 Works ex: √(2x - 2) = √(x - 4) => 2x - 2 = x - 4 => x = -2 Plugged back in: √(2(-2) - 2) = √(-2 - 4) => √-6 = √-6 Can't take the square root of a negative so no solution -MUST isolate the radical one side so it would make sense to square both sides: Ex: 2+ √(4 - 3x) = x => √(4 - 3x) = x - 2 => 4 - 3x = (x - 2)^2 => 4 - 3x = x^2 - 4x + 4 => 0 = x^2 - x => 0 = x(x - 1) => x = 0, x = 1

When to use Combinations

-Use combinations when order doesn't matter (start with FCP, divide off the repetitions) -if order matters, break problem into stages and count the possibilities in each stage and use FCP ex: Pool of 20 committee members and we need to pick 3 officers: chairperson, treasurer and secretary. Does order matter? Swapping about the 3 roles is a different leadership team, so order matters N = 20*19*28 ex: Chef is making a soup and has 20 vegetables. He can choose any 3, how many different soups can he make? Order doesn't matter => 20C3 = 20*19*18 / 3*2 = 20*19*3 = 1140

Inclusive Sequences

-Used when both the starting value and ending value are included in what we are counting ex: days used for workshop -Perform ordinary subtraction and add one ex: How many multiples of 8 are there from 200 to 640 inclusive? 8*25 = 200 8*80 = 640 200 is the 25th multiple of 8 and 640 is the 80th multiple of 8 and both are included number = 80 - 25 + 1 = 56

X is the opposite of

-X

Rhombuses

-a quadrilateral with 4 equal sides -Rhombuses are parallelograms and the 4 big properties are true for them 2 special rhombus properties: 1) All 4 sides are equal 2) Diagonals are perpendicular (meet at a right angle)

If a>b then

-a<-b

Combinations

-don't care about the order of arrangement; have a group of n individuals and we randomly select r -> the number of combinations is denoted as nCr "n choose r" -If we select a group of r from a pool of n, we automatically create a group of (n - r), people who were left behind, there must be a group of (n-r) for every combo of r nCr = nC(n - r) ex: number of different 4-person combos we could select from a pool of 10 10C4 = 10C6 Practice: An amusement park has 12 different rides. A coupon gives its holder access to any 3 of these rides for free. How many sets of 3 rides are possible? 12C3 = 12*11*10 / 3*2 = 2*11*10 = 220

Multiple Traveler Word Problems

-each traveler of each trip gets its own D=RT equation -More often you will have to use the techniques for solving 2 equations with 2+ unknowns (elimination and substitution) ex: Frank and George started traveling A to B at the same time. G's constant speed was 1.5 times Frank's constant speed. When G arrived at B, he turned back immediately and returned by the same route. He crossed paths with Frank who was coming toward B, when they were 60 miles away from B. How far away are A and B? Frank: D - 60 = RT D + 60 = 1.5R*T D + 60 = 1.5(D - 60) => D + 60 = 1.5D - 90 => 150 = .5D => 300 = D

Distance between 2 points

-find the distance by subtracting x2 from x1 or y2 from y1 -use the Pythagorean theorem to find the distance by creating a triangle Practice: Find the distance between (-5,-1) and (3,3) y2 - y1 = 8 x2 - x1 = 4 8^2 + 4^2 = c^2 => 64 + 16 = c^2 => 80 = c^2 √80 = c => c = √5*2*2*2*2 = 4√5

If a line has a slope of m=1

-rise=run and the slope triangle is a 45-45-90 triangle -lines with slope m = 1 or m = -1, make 45 degrees with the axes

The ratio of the areas of two similar polygons is ...

... the square of the ratios of the corresponding sides.

How many digits are there between the decimal point and the first even digit in the decimal equivalent of 1/[(2^8)(5^3)]

0

Decimal of 1/9 and 7/9

0.11111... and 0.7777...

Decimal of 1/8, 3/8, 5/8 and 7/8

0.123, 0.375, 0.625, 0.875

Decimal of 1/6 and 5/6

0.1666666... and 0.833333...

Decimal of 1/5, 2/5, 3/5 and 4/5

0.2, 0.4, 0.6, 0.8

Decimal of 1/3 and 2/3

0.33333333... and 0.6666666...

Powers and Roots

0^n = 0 -> a negative to any even power is positive and a negative to any odd power is negative -> x^2 = 4 has 2 solutions, x = 2 or x = -2 because either equals +4 -> By contrast x^3 = 8 only has one solution, x = 2 -> Something squared even equals a negative has no solution (x - 1)^2 = -4 no solution -> Something cubed odd equals a negative does have a solution (x - 4)^3 = -1 => x - 4 = -1 => x = 3

1ⁿ

1

A cylinder has a surface area of 22pi. If the cylinder has a height of 10, what is the radius?

1

A cylinder has surface area 22pi. If the cylinder has a height of 10, what is its radius?

1

Evaluate 4/11 + 11/12

1 & 37/132

Probability of E not occurring:

1 - P(E)

Operations with Fractions continued

1) Can cancel any part of the numerator with any part of the denominator ab/cd = a/c*b/d ex: 27(y+5)(2y-2)/6(y-1) = 9(y+5)*2(y-1)/2(y-1) = 9(y+5) 2) Can separate a fraction into 2 fractions by adding or subtracting in the numerator a+b/c = a/c + b/c d-e/f = d/f - e/f *We CAN'T separate a fraction into 2 fractions by adding or subtraction in the denominator a/(b+c) =/= a/b+a/c d/(e-f) =/= d/e - d/f *CAN split up the numerator but the denominator must stay unchanged a+b/c+d = a/c+d + b/c+d 3) Multiplying a fraction by its denominator = numerator ex: 4/7 * 7 = 4 ex: x/5 = 3 -> 5(x/5) = 3(5) -> x=15 4) Simplifying complex fractions ex: (x+1/6)/(x+6/15) Multiply total fraction by 15/15 =(x+1/6)*15/(x+6) = 15x + 5/2/(x+6) Multiply total fraction by 2/2 =30x+5/2x+12

Random

1) Every individual event is absolutely unpredictable 2) The overall pattern of events is completely predictable ex: flip a coin -> in the long run it is 50/50 so completely predictable

How to find the least common multiple (LCM)

1) Find prime factorization and the GCF ex: 24 and 32 24 = 6x4 = 3x2x2x2 = 3 x 2^3 32 = 8x4 = 2x2x2x2x2 = 2^5 GCF = 2^3 = 8 2) Write each number in the form of GCF times another factor 24 = 8x3 32=8x4 The LCM is the product of these 3 factors LCM = 8x3x4 = 8x12 = 96 *LCM helps in adding and subtracting fractions because the LCM is the LCD ex: 1/10 - 1/35 GCF 10=2x5 35=5x7 GCF=5 LCM = 2x5x7= 70 1/10 (7/7) - 1/35 (2/2) = 7/70 - 2/70 = 5/70 = 1/14 *If A is a factor of R, then the LCM of A and R must be R ex: The LCM of 8 and 24 is 24 *If A and B have no factors in common greater than 1 then their LCM would have to be their product AxB ex: 7 and 15 -> LCM = 7x15

Parallelograms

1) Opposite sides are parallel 2) Opposite sides are equal 3) Opposite angles are equal 4) The diagonals bisect each other, Mi is the midpoint of AC and BC AM = MC BM = MD -> if any one of them is true the rest is true

3 Equations with 3 unknowns

1) Pick 2 of 3 equations and using substitution or elimination, eliminate 1 variable 2) pick another pair of original equations and eliminate the same variable 3) Now use 2 equation with 2 unknown technique 4) Plug into any original equation to find the value of the third variable ex: A) w - 2x + 3y = 13 B) 2w + x - 4y = -14 C) 3w -x +2y = 8 B) 2w + x - 4y = -14 + C) 3w - x + 2y = 8 => D) 5w - 2y = -6 => 5w =2y - 6 A) w - 2x + 3y = 13 + B) *2 => 4w + 2x -8y = -28 => 5w -5y = -15 => 5w = 5y - 15 => 2y - 6 = 5y - 15 => 3y = 9 => y=3 => 5w = 2(3) - 6 => 5w = 0 => w=0 => 0 - 2x + 3(3) = 13 => -2x = 4 => x= -2

Reflections

1) Reflect over x-axis -> same x, opposite ± y 2) Reflect over y-axis -> same y, opposite ±x 3) reflect over y = x -> switch x and y 4) reflect over y = -x -> switch x and y and make the opposite ± sign 5) Mirror line is always the perpendicular bisector of the segment between the original point and its reflected image 6) Any point on the mirror is equidistant from the original point and its reflected image

Standard deviation facts

1) SD can be positive or zero (never negative) 2) Only way SD can equal zero is if all the number's on a list are identical to each other 3) If all the numbers on a list are exactly the same distance from the mean, that distance is the SD ex: list = {2, 2, 2, 8, 8, 8} Mean = 5 Every number is exactly 3 away from the mean so SD =3 4) A set with most numbers clustered toward the extremes will have a higher SD than a list with most values equal or close to the mean ex: A = {15, 25, 25, 25, 25, 25, 25, 25, 25, 35} B = {15, 15, 15, 15, 15, 35, 35, 35, 35, 35} SD of B = 10 > SD of A 5) If we add or subtract the same number to/from every number on a list then the SD doesn't change 6) If we multiply every number on a list by a positive number K, the SD also gets multiplied by K ex: A = {2, 3, 5, 7, 11, 14} mean = 7 SD = Q mult by 3-> B = {6, 9, 15, 21, 33, 42} new mean = 7*3=21 new SD = 3Q

Probability: The AND Rule

1) Simplified And Rule: And means multiply P(A and B) = P(A)*P(B) -If events A and B are independent ex: What's the probability of tossing 3 coins and getting all heads? 1/2 * 1/2 * 1/2 = 1/8 ex: Suppose events A and B are independent. If P(A)=0.6 and P(B) = 0.8, what does P(A or B) equal? P(A or B) = 0.6 + 0.8 - (0.6*0.8) = 1.4 - 0.48 = 0.92 2) Generalized And Rule: -If event A and B are NOT independent; without replacement -handle with conditional probability P(A|B) means "what is the probability of A, given B?" P(A and B) = P(B) * P(A|B) OR P(A and B) = P(A) * P(B|A) Practice: From a shuffled deck of 52 cards, what's the probability of picking 3 hearts on the first 3 cards drawn, if the cards are selected without replacement? P(1 = H) = 1/4 P(2=H|1=H) = 12/51 = 4/7 P(3=H | 1=H and 2=H) = 11/50 P = 1/4 * 4/7 * 11/50 cross multiply= 1/1 * 1/17 * 11/50 = 11/850

Even and Odd Integer properties

1) Zero is an even number *test loves this* 2) Evens and odds include both positive and negative numbers 3) Evens and odds pertain only to integers, any non-integer is neither even nor odd 4) All even numbers as divisible by 2 - the prime factorization of an even integer always contains 2 5) No odd number is divisible by 3 - prime factorization of a positive odd number will never contain a factor of 2 *Add or subtract "likes" we get EVEN ex: E+E = E E - E = E O+O = E O-O=E *Add or subtract "unlikes" we get ODD ex: E+O = O E - O = O *E*E=E 2X4=8 E*O=E 4X3=12 O*O=O 5X3=15 *At least one even factor in a product = even *Only way to equal odd in a product is if every factor is odd

Steps in the calculation of the SD

1) start with a list of numbers 2) calculate the mean 3) subtract the mean from every number to create a second list (a list of deviations) 4) Square every deviation to produce a third list, a list of squared deviations (all positive) 5) Find the average of the third list, called the variance 6) Take the square root of the variance, this is the SD ex: List = {1, 2, 3, 4, 5, 6, 7, 8, 9} mean = 5 List 2 = {-4, -3, -2, -1, 0, 1, 2, 3, 4} List 3 = { 16, 9, 4, 1, 0, 1, 4, 9, 16} Variance = 60/9 = 20/3 SD = √20/3

√2

1.4

8.84 / 5.2

1.7

√3

1.7

Decimals of 1/20, 1/40, 1/600

1/20 = 1/2 * 1/10 = .5*.1 = 0.05 1/40 = 1/4 * 1/10 = .25*.1 = 0.025 1/600 = 1/6 * 1/100 = 0.166666 * 0.01 = 0.0016666

(a^-1)/a^5

1/a^6

1/p + 1/q = 1/f Solve for q

1/q = 1/f - 1/p 1/q = p/fp - f/fp 1/q = (p - f)/fp q = fp / (p - f)

The negative exponent x⁻ⁿ is equivalent to what?

1/xⁿ i.e. 5^-3 = 1/(5^3) = 1/ 125 = .008

Legs 6, 8. Hypotenuse?

10

There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. In how many ways can the judges award the 3 prizes?

10! / (10-3)! = 720

There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. How many different people can get the three prizes?

10! / 3!(10-3)! = 120

What is the ratio of the sides of an isosceles right triangle?

1:1:sqrt2

What is the ratio of the sides of a 30-60-90 triangle?

1:sqrt3:2

First 15 perfect squares

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

What is the coefficient of the x^2 term in the product of (x + 1)(x + 2)(x -1)?

2

What is √4

2

Evaluate and write as a mixed number: 2/7 - 3/21 + 2 & 4/14

2 & 3/7

Supplementary

2 angles that add up to 180 degrees. 2 angles on a straight line are supplementary

Mutually exclusive

2 events are mutually exclusive if it is impossible for both of them to happen at the same time Event A and B are mutually exclusive means: 1) Possible = A happens alone 2) Possible = B happens alone 3) Possible = neither A nor B happens impossible - A & B both happen P(A and B) = 0 ex: dice, coins, cards

Perpendicular Angle

2 lines or segments meet at a right angle -Don't assume 2 lines are penpendicular

Sequential Percent changes

2 or more percent changes that follow in a sequence *When you have 2 or more changes in a row, NEVER add or subtract the percents. Must take percent of the first and then the percent of the second ex: Anne wants to buy a shirt she saw last week that was $100. When she goes to buy it she notices the price has increased 30%. If she uses her 30% employee discount to buy the shirt, how much will she pay? 100*1.3 = 130 1 - .3 = .7 130* .7 = $91

Circumference of a circle

2(pi)r

How many 3-digit positive integers are even and do not contain the digit 4?

288 (8 * 9 * 4)

Powers of 2 up to 2^9

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

Nine coins are tossed simultaneously. In how many of the outcomes will the fourth coin tossed show heads?

2^9 / 2 = 256

How do you calculate the circumference of a circle?

2πr or πd

(2²)³

2⁶

2⁵*2³

2⁸

Simplify √32

32 has a factor that is a perfect square, 16, so √32 = √16x2 = 4√2

In a Regular Polygon, the measure of each exterior angle

360/n

In any polygon, all external angles equal up to

360°

The sum of all angles around a point

360°

The sum of the angles in a quadrilateral is

360°

3/8 in percent?

37.5%

200 <_ x <_ 300. How many values of x are divisible by 5 & 8?

3: 200, 240, 280

Powers of 3 up to 3^4

3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81

Can you simplify 274?

3x3x3=27, so (3x3x3)4 which equals (3x3x3)(3x3x3)(3x3x3)(3x3x3), so 312

Legs: 3, 4. Hypotenuse?

5

25^(1/2) or sqrt. 25 =

5 OR -5

What is the third quartile of the following data set: 44, 58, 63, 63, 68, 70, 82

70

70 < all primes< 80

71, 73, 79

What is the measure of an exterior angle of a regular pentagon?

72

What number between 70 & 75, inclusive, has the greatest number of factors?

72

What is the ratio of the surface area of a cube with an edge of 10 to the surface area of a rectangular solid with dimensions 2, 4, and 6?

75:11

5/6 in percent?

83.333%

7/8 in percent?

87.5%

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

The four angles around a point measure y, 2y, 35 and 55 respectively. What is the value of y?

90

Find the surface area of a cylinder with radius 3 and height 12.

90pi

The percent decrease of a quantity

= (actual decrease/Original amount) x 100%

If a lamp increases from $80 to $100, what is the percent increase?

= 25%. = (actual increase/original amount) x 100% = 20/80 x 100% = 1/4 x 100% = 25%

Expression

A collection of one or more terms joined by addition or subtraction. *Don't have equal signs ex: y^2 - x^2

Volumes and Surface Area

A cube has 6 faces, 8 vertices and 12 edges Cubes volume: V = s^3 Total surface area of a cube = 6s^2 Volume of a rectangular solid: V = h*w*d Rectangular solid total surface area = 2hd + 2hw + 2wd -A cylinder has a height of h and the radius of each circular base is r Cylinder volume: V = πr^2h Total surface area of a sphere: 2πr^2 + 2πrh

What is the empty set?

A set with no members, denoted by a circle with a diagonal through it.

Circular sector

A slice of the circle To find the area of a sector, we set up another part-to-whole proportion area of sector / πr^2 = angle / 360°

Area of a Parallelogram:

A=(base)(height)

Associative Property

Ability to group numbers in different groupings; addition and multiplication are associative, subtraction and division are generally not a + (b+c) = (a+b) + c a * (b*c) = (a*b) * c

Acute and Obtuse Angle

Acute = less than 90 degrees Obtuse = more than 90 degrees -Triangle can't have 2 right angles or 2 obtuse angles, at least 2 angles must be acute

Simplifying expressions

Add like terms. Multiplication is commutative so orders of factors in multiplication doesn't matter ex: 5xy + 7yx = 12xy *Can't add terms of different variables or powers *When a subtraction sign appears you must change every sign inside the parenthesis to its opposite ex: (x^3 - 3x^2) - (x^3 + 3x^2) = x^3 - 3x^2 - x^3 - 3x^2

When multiplying exponential #s with the same base, you do this to the exponents...

Add them. i.e. (5^7) * (5^3) = 5^10

Divisibility Rule for 3

Add up all digits of the number, if the sum of the digits is divisible by 3 then the number is divisible by 3 ex: 135 = 1+3+5=9 (divisible by 3)

Divisibility Rule for 9

Add up all digits, if the sum is divisible by 9 ex: 1296 = 1+ 2+ 9+ 6 = 18 (divisible by 9)

What are the rational numbers?

All numbers which can be expressed as a ratio of two integers. (All integers and fractions.) (-2, 1, .25, 1/2)

Definition of integer

All positive or negative whole numbers (including zero which is neither positive nor negative)

What are the irrational numbers?

All real numbers which can't be expressed as a ratio of two integers, positive and negative (pi, -sqrt3)

What are the real numbers?

All the numbers on the number line (negative, rational, irrational, decimal, integer). All the numbers on the GRE are real. (-2, 1, .25, 1/2, pi)

Elimination Method

Always allowed to add 2 equations: 7x + 3y = 5 + 2x - 3y = 13 => 9x = 18 => x=2 Plug into either equation to solve for y 2x - 3y = 13 -> 2(2) - 3y = 13 -> -3y = 9 -> y=-3 *Strategy is multiply both sides of 1 equation by 1 number and both sides of another equation by another number so that for 1 of the variables, the coefficients are equal and opposite ex: 2x + 3y = 15 & x + 2y = 11 => -2(x + 2y) = -2(11) 2x + 3y = 15 + -2x - 4y = -22 = -y = -7 -> y=7

Sequence Patterns

An = n is the sequence of all positive integers An = 2n - 1 is the sequence of all positive odd numbers An = 7n is the sequence of all positive multiples of 7 An = n^2 is the sequence of all perfect squares An = 3^n is the sequence of all powers of 3

What is an exterior angle?

An angle which is supplementary to an interior angle.

What is an arc of a circle?

An arc is a portion of a circumference of a circle.

What is a permutation?

An arrangement of things in a particular order ie: how many different ways can you arrange 5 statues on a shelf? 5x4x3x2x1 = 120 different ways

What is the name of set with a number of elements which cannot be counted?

An infinite set.

Prime Factorization

Any integer greater than 1 that's not prime can be expressed as a product of primes ex: 9= 3x3 12 = 2x2x3 96=2x48 = 2x6x8 = 2x2x3x2x2x2 = 2^5x3 1) r is a factor of Q 2) r is a divisor of Q 3) Q is divisible by r 4) Q is a multiple of r 5) every prime factor of r is included the the prime factorization of Q ex: 4680 = 2^3 x 3^2 x 5 x 13 Are the following factors? 25 = 5x5 no 120 = 12x10 = 2x2x3x5x2 yes 45=5x3x3 yes 180=18x10=3x3x2x5x2 yes 65=13x5 yes 85=5x17 no

Definition of number

Any number; positive, negative, zero, whole number, fraction or decimal

Areas of quadrilaterals

Area of a square: A=s^2 Area of other quads: A=b*h -for rhombuses and parallelograms, any side can be a base but the height has to be perpendicular to the base, it won't lie along a side -With trapezoids there are 2 bases with 2 parallel sides so one way to find the area is to find the average of the bases and multiply by the height: A = (b1 + b2 / 2) * h -Sometimes we can find the area of a trapezoid by dividing it into a rectangle and 2 side right triangles

How to find the circumference of a circle which circumscribes a square?

Circumference = Diameter(pi). Use pythagorean theorem to find the diagonal of the square (the diameter).

Consecutive Integers Properties

Consecutive - in a row, following one another; could be positive, negative or both 1) A set of n consecutive integers will always one number divisible by n 2) If n is odd, then the sum of a set of n consecutive integers will always be divisible by n 3) In a set of 4 consecutive, must have 2 even 2 odd In a set of 3, could have either 2 even, 1 odd or 2 odd, 1 even

Bisector

Cuts something into 2 congruent (equal) pieces -Angle bisector cuts an angle into 2 smaller congruent angles

Multiplying Expressions

Distributive Law allows for multiplication to distribute over addition and subtraction A(B+C) = AB + AC *Does NOT distribute over multiplication 3*(xy) = 3xy -If we multiply a number (a constant) times a monomial with a variable, the constant multiples the coefficient ex: 2*(r^4*s^2*t^3) = 2r^4*s^2*t^3 *Multiplying powers means adding the exponents

How to change from percents to decimals

Divide by 100 and move decimal point 2 places to the left ex: 4% = 0.04 ex: 0.25% = 0.0025

How to eliminate fractions

Eliminate fractions by multiplying by the LCM ex: x/2 + 5/4 = x/3 + 3/2 LCM = 12 12(x/2) + 12(5/4) = 12(x/3) + 12(3/2) = 6x + 15 = 4x +18 -> 2x = 3 -> x= 3/2 *Can simplify a complex fraction by multiplying both the numerator and denominator of a fraction by the LCM of all the denominators of the little fractions

Is 0 even or odd?

Even

2 is the only

Even prime number

∅ is a multiple of

Every number

Combining Ratios Strategy 2 - Solve for the absolute quantity in each term

Ex: One cup of butter is enough for 12 of Kathy's cookies and 1 cup of sugar is enough for 8 of her cookies. If she used 5 more cups of sugar than butter, how many cookies can she make? B:C = 1:12 Multiply by 2 = 2:24 S:C = 1:8 Multiply by 3 3:24 B:S:C = 2:3:24 or B=2n S=3n and C=24n n = 5 -> C=24(5) = 12*10 = 120 cookies

Face and space diagonal

Face diagonal: A diagonal of only one face of the solid - find the length and Pythagorean theorem Space Diagonal: Passes through the interior of the rectangular solid from one vertex to the opposite vertex - find length using a 3D version of the Pythagorean Theorm: (AD)^2 = (AB)^2 + (BC)^2 + (CD)^2 A = length B=width C=height For a cube: (AB)^2 = s^2 + s^2 + s^2 = 3s^2

Factoring Quadratics

Factor : x^2 + 8x + 15 Need to find two numbers whose sum=8 and whose product=15 =(x+3)(x+5)

Whats the difference between factors and multiples?

Factors are few, multiples are many.

Combining Ratios Strategy 1- Find the common element

Find the least common multiplier Ex: On a high school team the ratios of sophmores to juniors is 2:3 and the ratio of juniors to seniors is 5:6. Sophomores are what fraction of the whole team? S:J = 2:3 (multiply by 5) J:Sr = 5:6 (multiply by 3) S:J = 10:15 J:Sr = 15:18 Soph:Junior:Senior = 10:15:18 Whole: 10+15+18=43 Soph:Whole = 10/43

How to square a number not ending in zero or five; e.g 41^2, 69^2, 84^2

Find the square of the closest number ending in five or zero, add the number and add the original number 41^2 = 40^2 + 40 + 41 = 1600 + 81 = 1681 69^2 = 70^2 - 70 - 69 = 4900 - 140 + 1 = 4761 84^2 = 85^2 - 85 - 84 = 7225 - 160 - 9 = 7056

56 is what percent of 800?

Finding the percent 56=800x 56/800=x 7/100=x 0.07=x 7%=x

Multiplying with decimals

For multiplication count the number of digits to the right of the decimal point ex: 6.25 * 0.048 -> 2+3 = 5 decimal points Now ignore the decimal points 625*48 = 1250*24 = 2500*12 = 5000*6 = 30,000 = .3 ex: (.03)^3 = (.03)*(.03)*(.03) -> 2+2+2=6 3^3 = 27 -> 0.000027

Exponential Equations

If two powers with the same base are equal then the exponents must be equal b^x=b^y => x=y ex: 49^x = 7^6-x => (7^2)^x = 7^6-x => 7^2x = 7^6-x => 2x = 6-x => 3x = 6 => x=2 ex: (5√3) ^3x + 7 = 3^2x => (3^1/5)^3x+7 = 3^2x 3^(3/5x + 7/5) = 3^2x => 3/5x + 7/5 = 2x => Multiply each side by 5 => 3x + 7 = 10x => x=1 Can rewrite the bases so they are a power of the other ex: 27^(2x-2) = 81^(x+1) => (3^3)^(2x-2) = (3^4)^(x+1) => 3(2x-2) = 4(x+1) => 6x-6 = 4x+4 => x=5

Triangle Inequality Theorem

In any triangle, the sum of 2 sides must be greater than the third side A + B > C B + C > A A + C > B -The largest angle is always opposite the longest side -The smallest angle is always opposite the shortest side -If we know P and Q then, P - Q < third side < P + Q Ex: Suppose we know 2 sides of a triangle are 8 and 13. What are the possible lengths of the third side x? x + 8 > 13 => x > 5 8 + 13 > x => 21 > x 5 < x < 21

Mean and Median

Means are heavily affected by outliers and the median is entirely unaffected by outliers -If a list consists of evenly spaced numbers, then the mean = median -Mean and median are equal when the list is symmetrical -A distinct outlier pulls the mean away from the median: high value outliers cause the mean to be higher than the median, low value outliers cause the mean to be lower than the median Practice: On test in a class of over 40 students, the score had mean=median=81. Two absent students then took the test and received grades of 83 and 47. What are the new median and mean? One number above and below means the median stays 81; 47 is an outlier which drags down the mean below 81

Ideas of Multiples

Multiple: inverse relationship to factor ex: 91 is a multiple of 7 1) Just as every positive integer is a factor of itself, every positive integer is a multiple of itself 2) If we need the first five multiples of a number, we simply multiple the original number by the numbers {1,2,3,4,5} 3) If P is a multiple of r, then it must be true the (P-r) and (P+r) are also multiples of r ex: 2401 is a multiple of 7 2401+7 = 2408 and 2401 - 7 = 2394 are multiples of 7 4) If P & Q are multiples of r, then (P+Q) and (P-Q) are also multiples of r ex: 700 and 49 are both multiples of 7 700+49 = 749 and 700-49=651 are both multiples of 7 5) If P is a multiple of r, then any multiple of P is a multiple of r ex: any multiple of 52 is a multiple of 13 6) If P and Q are multiples of r, then the product P*Q is a multiple of r ex: 24 and 80 are multiples of 8 24 + 80 = 104 80-24 = 56 24 * 80 = 1920 these are all multiples of 8 CAN'T divide - if we divide 80 by 24 the quotient is not an integer

How to recognize a # as a multiple of 4

The last 2 digits are a multiple of 4. (i.e 144 .... 44 is a multiple of 4, so 144 must also be a multiple of 4.)

What is a major arc?

The longest arc between points A and B on a circle's diameter.

What is the factorial of a number? ie: the factorial for 6!

The number times every positive whole number smaller than that number down to 1 ie: 6! = 6x5x4x3x2x1 = 720

What is the "solution" for a set of inequalities.

The overlapping sections.

What is the "solution" for a system of linear equations?

The point of intersection of the systems.

Sums of Sequences

The sum of a long sequence Sum = N(N + 1) / 2 Sum of a list = N * (a1 + an) / 2 a1= lowest term an = final term Ex: What is the sum of all multiples of 20 from 160 to 840 inclusive? 160 = 8*20 840 = 42*8 Inclusive counting: 42 - 8 + 1 = 35 Number of pairs = 17.5 Sum of list = 17.5 * (160 + 840) = 17.5 * 1000 = 17,500

Angles in polygons

The sum of the angles in a n-sided polygon is: (n -2) * 180° ex: hexagon = 4*180 = 720° Practice: What is the degree of an angle in an 18-sided polygon in which all of the angles are the same? Angle = (16*180)/18 = 16*10 = 160°

Inscribed angle

The vertex is on the circle. Sides of an inscribed angle are always two chords that meet at the vertex -The measure of the inscribed angle is half the measures of the arc it intercepts -Any inscribed angle that intercepts a semi circle is a right angle -If 2 inscribed angles in the same circle intercept the same arc of the same chord then the 2 inscribed angles are equal

Transversal

Third non-parallel line cuts across 2 parallel lines -of the 8 angles formed, the 4 "big" angles are all equal and the 4 "little" angles are all equal a=d=e=h b=c=f=g -A and B are supplementary, if given 1 angle we can find the other 7

Probability: The OR Rule

This rule has 2 versions: 1) the simple rule -> in probability, the word "OR" means add -If events A and B are mutually exclusive events then: P(A or B) = P(A) + P(B) 2) The generalized rule -> If events are NOT mutually exclusive and there is some overlap; rule works for any events: P(A or B) = P(A) + P(B) - P(A and B) ex: In Game M, the probability of outcome A is 0.6, the probability of outcome B is 0.7 and the probability of A or B is 0.9. What's the probability of A and B happening at the same time? 0.9 = 0.6 + 0.7 - P(A and B) 0.4 = P(A and B)

Unit conversion

To change one unit to another, multiply by the unit conversion fraction that has the given unit in the denominator and the desired unit in the numerator ex: 6 feet to inches => 6 ft (12 in/1 ft) = 72 in ex: 5 m^2 to cm^2 => 5 m^2 * (100 cm/1 m)^2 = 50,000 cm^2 -To get an area must raise unit to power of 2 -To get a volume must raise unit to power of 3

Can you plug-in for fraction and percentage problems?

Yes. Plugging in for the missing value will make your life much easier!

How do you figure out what number to multiply by?

You take the total number of coins (24) and divide it by the total of the ration (2:1 = 3), so 24 divide by 3 = 8. Then you multiply each item (pennies and nickels) by 8 to figure out the total number of each coin in the pocket.

What is ZONE F

Zero One Negative Extremely small or big Fractions

5x^2 - 35x -55 = 0

[(7+ sqrt93) /2], [(7 - sqrt93) / 2]

Absolute Value Equations

ex: |x| = 5 has 2 solutions, could be +5 or -5 ex: |2x + 3| = 5 2x + 3 = 5 -> x= 1 or 2x+3 = -5 -> x = -4 *If the absolute value is not by itself on one side then we will have to isolate it ex: |3x + 2| +1 = 5 -> |3x + 2| = 4 3x+2 = 4 -> x = 2/3 or 3x+2 = -4 -> -2

Function notation

f(x) "x" is the input to the function "f" ex: Given the function f(x) = x^2 + 4x - 21 Find the values of x that would satisfy f(x) = 24 f(x) = x^2 + 4x - 21 = 24 x^2 + 4x - 45 = 0 -> (x+9)(x-5) = 0 -> x=-9, x= 5 ex: Given the function f(x) = x^2 +kx + 4 Find the value of k if f(2) = 18 (2)^2 + k(2) + 4 = 18 -> 8 + 2k = 18 -> k=5

What is the graph of f(x) shifted upward c units or spaces?

f(x) + c

What is the graph of f(x) shifted downward c units or spaces?

f(x) - c

What is the graph of f(x) shifted right c units or spaces?

f(x-c)

Solve the quadratic equation ax^2 + bx + c= 0

x = [(-b)+/- (sqrt b^2 - 4ac)]/2a

Vertical line

x = k -the equation of y-axis is a vertical line that intersects the x-axis at zero so x=0

Factor x^2 - xy + x.

x(x - y + 1)

Consecutive integers

x, x+1, x+2

What is the unfactored form of (x-y)2

x2 - 2xy + y2

What is the factored form of (x+y)(x-y)

x2 - y2

Other Roots

x^2 = positive has 2 solutions x^2 = negative has no solution (positive)^3 = positive; x^3 has one positive solution (negative)^3 = negative; x^3 has one negative solution -With square roots, can only find square roots of positives, NOT negatives -Can take the cube root of any number on the number line, positive negative or zero ex: 3√8 = 2 3√0 = 0 3√-8 = -2 -Same positive and negative rules extend to even and odd roots -> n√ a n > or = 2 then for all n, n√0 = 0 and n√1 = 1 -> All roots preserve the order of inequality: If 0 < a < b < c then, 0 < n√a < n√b < n√c

(x+y)²

x²+2xy+y²

factored binomial product of (x+y)²

x²+2xy+y²

(x-y)²

x²-2xy+y²

factored binomial product of (x-y)²

x²-2xy+y²

(x-y)(x+y)

x²-y²

binomial product of (x+y)(x-y)

x²-y²

Absolute Value Inequalities

|x| is the distance of x from zero |x - 5| is the distance of x from +5 |x + 3| is the distance of x from -3 Ex: Express |x-7| < 3 as an ordinary inequality 7 - 3 = 4 7+3 =10 4<x<10 ex: Express the region -3<x<11 as an absolute value inequality *Middle of average endpoints (11+-3)/2 = 4 X can be as far as 7 above 4 or 7 below 4 |x - 4| < 7

#3 What is the relationship between lengths of the sides of a triangle and the measure of the angles of the triangle?

• Sides with the same lengths are opposite angles with the same measure.

#2 What is an important property of a 30-60-90 triangle?

• The hypotenuse is twice the length of the shorter leg.

Estimate √32

√32 is in between √25 and √36, so we can guess that √32 is somewhere between 5 and 6

What is the absolute value of √a2

√a2 = |a| for all value of a