GRE Review

Cube Root Rules

(Positive)³ = positive (Negative)³ = negative *Therefore can find square root of positive and negative numbers x³=positive = ³√ (positive) = 1 positive solution x³ = negative + ³√ (negative) = 1 negative solution

Law of exponents: multiplying to powers (a^m) * (a^n)

(a^m) * (a^n) = a^(m+n)

Law of exponents: power to a power (a^m)^n

(a^m)^n = a^ (m*n)

Law of Exponents: Exponent distribution (ab)^n (a/b) ^n

(ab)^n = a^n * b^n (a/b) ^n = a^n / b^n *Exponents don't distribute over addition/subtraction*

Negative exponents with fractions (p/q)^-n

(p/q)^-n = (q/p) ^-n

Consecutive Integer Questions- Algebraic Representations 1) If n is an integer less than 50, then the expression (n^2 -2n)(n+1)(n-1) must be divisible by: 8, 12, or 18 2) N = 135 is the lowest number in a set of 41 consecutive multiples of 5. What is the difference between the lowest and highest numbers in the set

*All will be spaced 1 apart* 1) n(n-2)(n+1)(n-1) --> (n-2)(n-1)n(n+1) --> product of 4 consecutive int. = 2 odd / 1 even / 1 multiple of 4 --> multiple of 4 * even = divisible by 8 --> multiple of 3 --> multiple of 4 and 3 = divisible by 12 --> mulitiple of 2*3*3 = unsure for 18 *Can also ask about multiples of N in a consecutive sequence* 2) 1--> 135 2 --> 135 + 5 3 ---> 135 + 5x2 4 --> 135 + 5x3 --> nth = 135 + 5(n-1) --> 41st = 135 + 5(41-1) = 135 + 200 Difference between 135 and 135 + 200 = 200

Right Triangles

- 2 Legs + Hypotenuse --> 2 legs opposite acute angles --> hypotenuse opposite right angle - Pythagorean theorem: a² + b² = c² --> applies only to right triangles --> angle opposite c² will always be right

Isosceles Triangles

- 2 sides are equal and 2 angles are equal --> 2 acute equal angles (or 3 in equilateral) --> 1 right angle / obtuse angle -Line from vertex to opposite side is the altitude, bisects the base, bisects the angle, and is perpendicular --> also true for equilateral bc special isosceles --> Creates 2 congruent right triangles

Equilateral Triangle

- 3 equal sides and 3 equal angles --> angles = 60 each

Special Triangle: Equilateral bisected by altitude

- 30-60-90 triangle - legs = 1-√3-2 --> short L-long L-hypotenuse --> hypotenuse = 2 * (short leg) --> long leg = √3 * (short leg) - all 30-60-90 triangles are similar

Arc Length: Arc measure vs. Arc length

- Arc Measure = Degrees ---> Compares arcs in same circle -Arc Length = Size of 2 arcs in 2 different circles ---> even if they have the same degree measure arcs may not be the same length

Parallel Lines

- Never intersect -Always the same distance apart *CAN'T assume parallel unless explicitly stated

Triangle Facts: True for All 1) Angles 2) Opposite vertex/side facts

-3 sides and 3 vertices 1) Angles = 180 --> Can't have 2 obtuse or two right angles -->Will always have at least 2 acute angles 2) Largest angle is always opposite the longest side --> Smallest angle opposite shortest side

Quadrilateral

-4 line segment sides -Sum of 4 interior angles = 360 - Has exactly 2 diagonals = line connecting opposite vertices

Special Triangles: Isosceles Right 1) Find the legs of 45-45-90 triangle KNJ with hypotenuse JK = 6

-45-45-90 triangle - legs = 1-1-√2 --> ratio --> hypotenuse = leg * √2 - all 45-45-90 triangles are similar 1) JK = KN * √2 --> becasue 1-1-√2 ratio --> KN = JK/√2 --> 6/√2 (√2/√2) = 6√2/ 2 --> 3√2

Comparing Mean to Median

-Look for location of the outliers High value outliers = higher mean than median Low value outliers = lower mean than median

Rationalizing Radical in the Denominator 1) x = 12/√3 2) 4-√6 / 2√3 3)2/ √5 -1

1) 12/√3 * √3/√3 --> 12√3/ 3 --> 4√3 2) 4-√6/ 2√3 *√3/√3 --> (4√3 - √6*√3)/ 2*3 ----> (4√3 - √18)/6 --> (4√3 - 3√2)/6 3) Conjugate creates (P-Q)(P+Q) = P² - Q² 2/ (√5 -1) * (√5+1)/ ( √5+1) --> 2(√5 + 1) / 5 - 1 --> 2(√5 + 1) / 4 --> (√5 +1)/2

2 Variable Equations: Infinite or No solutions 1) 2x-y=5 & 2y-4x=-10 2) x-2y = 5 & 3x-6y = 8

1) 2x-y=5 --> y = 2x-5 --> 2(2x-5) -4x = -10 --> 4x-10-4x = -10 --> -10 = -10 = Always True = INFINITE answers 2) x-2y = 5 --> x = 2y + 5 --> 3 (2y+5) -6y = 8 --> 6y + 15 -6y = 8 --> 15 = 8 --> Never true = NO ANSWER --> Parallel lines which never intersect

Converting Units of Measurement 1) Area --> 5m² to cm² 2) Volume --> 1m³ to cm³

1) 5m² to cm² -Multiply by the unit conversion squared --> 5m² (100cm/1m)² = 5*10,000 = 50,000 cm² 2) 1m³ to cm³ -Multiply by unit conversion cubed ---> 1m³ ( 100cm/1m)³ = 1,000,000 cm³

Cube 1)Properties 2)Volume 3) Surface area 4) Space diagonal

1) 6 congruent faces 8 vertices + 12 edges -Edges and faces meet at perpendicular angle 2) V = s³f 3) SA = 6s² 4) Diagonal² = s² + s² + s² =

Change in SD when adding new numbers to a list

1) Add Outliers - Increase/decrease the mean - Increase SD 2) Add values that don't change mean - Values are equal distance from the mean - Distance from mean > SD = Increase SD - Distance from mean = SD = Same SD - Distance from mean < SD = Decrease SD 3) Add values equal to mean - Decreases the SD the most because distance from the mean = 0

Find altitude of rhombus/parallelogram JKLM is rhombus. Line KN is the altitude of the rhombus. If JN =1 and NM = 2 find the area of the rhombus

1) Add in any known information --> JN + NM = JN --> 1 + 2 = 3 --> All sides are equal --> Sides = 3 2) Find the sides of the small triangle (JKN) using Pythagorean theorem --> KN² = JK² - JN² --> 3² - 1² --> 9-1 = 8 --> KN² = 8 --> KN = √8 --> 2√2 = height 3) Find the area with Base * Height --> 3* 2√2 = 6√2

Parallelogram 1) 4 properties of all parallelograms 2) Area of a parallelogram

1) All parallelograms are quadrilaterals 1) Opposite sides are parallel 2)Opposite sides are equal 3) Opposite angles are equal 4) Diagonals bisect each other --> intersect at midpoint of each line *Must have ALL 4 of the properties to be a parallelogram* --> If any one of these properties is true then the other 3 must be true as well 2) Area = Base * height ---> height = altitude perpendicular to base

Rectangular Solids 1)Properties 2)Volume 3) Surface area 4) Diagonals

1) All sides are rectangles - 4 congruent faces - long sides - 2 congruent faces - short sides - 8 vertices + 12 edges 2) V = h * w * d 3) SA = 2hw + 2hd + 2wd 4) Diagonals - Face diagonal = diagonal on one face of solid --> Standard Pythagorean theorem - Space diagonal = diagonal through interior between opposite vertices --> Square each individual leg lengths to find length of space diagonal --> Diagonal² = h² + w² + d² = √3 *s

4 Lines inside a triangle 1)Altitude 2) Perpendicular Bisector 3) Median Line 5) Angle Bisector

1) Altitude --> vertex to opposite side with perpendicular line --> Always makes a right angle --> typically not midpoint of opposite side 2) Perpendicular Bisector --> Bisects one side at midpoint --> Always creates right angle --> May not go through a vertex 3) Median Line --> vertex to midpoint of the opposite side --> divides side in half but not necessarily angle 4) Angle bisector --> divides angle into 2 equal parts --> may not evenly divide opposite side

GRE Geometry Assumptions 1) Always True - CAN be assumed 2)Always False - CAN'T be assumed

1) CAN ASSUME - All lines that look straight are straight - relative position --> Item to left or right --> Order of points on a line 2) CAN'T ASSUME - Unless stated - Diagram is to scale - Equal lengths - Parallel/Perpendicular lines - relative angle size - right angles DO NOT TRUST THE DIAGRAMS - aside from stated facts - only trust geometric deductions

SD as a unit of measurement -Find location of a specific value compared to the rest of the set 1) mean = 50, SD = 2 , value = 60 2) mean = 50, SD = 20, value = 60

1) Compare specific value to the mean Mean = 50 & Value = 60 --> 10 from the mean 2) Divide distance from mean by SD 1) 10/20 = 1/2 deviation from the mean 2) 10/2 = 5 deviations from the mean Number of deviations = |Mean - Value|/SD

Counting: Eliminating Repetition -Don't care about order of occurrence 1) Suppose there is a room of 20 people, and each one will shake hands with the other person. How many handshakes occur? 2) From a set of ten different identical items, Lisa is going to select three as a gift for someone. How many different sets can she pick?

1) Count total number of arrangements 2) Divide by number of un-ordered positions/ repeated placements 1) 20 people -> shake 19 hands = 20*19 --> but counts each handshake twice --> i.e: Elliot->Lisa = Lisa->Elliot 20*19/2 =190 2) 10 * 9 * 8 = total choices --> counts arrangements of items 3! = number of repeats (i.e. moving gift to new place) --> 10 * 9 * 8/ 3! = 120

Set/Venn Diagram Questions At a school of 200 students, they can take French, Spanish, both or neither. As many students study both as neither. One quarter who study Spanish also study French. The total number of students taking french is 10 less than those who study Spanish only. How many study french only.

1) Draw a venn diagram of the info given - 4 distinct areas presented - regions all add to the total number of items F + S + B + N = 200 B = N 1/4 (S+B) = B --> S + B = 4B --> S = 3B F + B = S - 10 --? F + B = 3B -10 --> F = 2B -10 200 = 2B-10 + 3B + B + B -- 210 = 7B --> B = 30 F = 2(30) -10 = 50 students 2) For people in 3 overlapping categories use a 3 circle venn diagram

Special Triangle: Equilateral Triangle Area

1) Draw the altitude --> 2 (30-60-90) triangles --> Hypotenuse = S = all sides of triangle --> Short side = S/2 ---> Long leg/Altitude = S√3/2 2) Use full base length (S) as base in formula ---> 1/2 (S) (S√3/2) --> (S²√3)/4

When does Mean = Median

1) Evenly spaced list of numbers - i.e: consecutive integers 2) Symmetrical list -numbers on either side of the median are an equal distance away from the median -i.e.: 13, 23, 25, 27, 37 --> 23 = 25 - 2 & 27 = 25 + 2 --> 13 = 25 - 12 & 37 = 25 + 12

Law of Exponents: sums and differences of powers 17^30 + 17^20

1) Factor out the common elements on the exponents --> (17^20)(17^10) + (17^20)(1) --> factor 17^20 --> (17^20) (17^10 + 1)

Polygons 1) Properties required to make a polygon 2) Properties of all polygons 3) Regular polygons

1) Figure must close --> All sides connected -Sides cannot cross -All sides must be straight 2) Any segment connecting 2 non-adjacent vertices = a diagonal 3) All angles and sides are equal - Diagonal that splits shape in equal halves will split vertices (angles) in equal halves as well

Find Distance between 2 points Ex: distance between (7,5) and (3,2)

1) Find vertical and horizontal distances of the points --> horizontal = 7-3 = 4 --> vertical = 5-2 = 3 2) Find hypotenuse length with Pythagorean theorem --> 2 points can be connected by a right triangle --> 4² + 3² = c² = 25 --> length/distance = 5

Line y = -x -General -Reflections over y=-x i.e. (2,5) or (2,-4)

1) General info Slope = -1 & Y-int = 0 45 degree angle with y and x axes All points of x are equal and opposite of y 2) Reflections -x and y coordinates are switched and given the opposite signs i.e. (2,5) --> (-5,-2) or (2,-4) --> (4, -2)

Line y=x -General -Reflections over y=x i.e: (5,4) or (-2,3)

1) General info Slope = 1 & Y-intercept = 0 45 degree angle with x and y axes All points of x and y are identical 2) Reflections -coordinates for reflection are switched from the original i.e: (5,4) --> (4,5) or (-2,3) --> (3,-2)

Probability and Counting 1)A committee of three will be selected from eight employees, including Alice and Bob. What is the probability that the chosen committee of three includes Alice and not Bob?

1) Groups of 3 from 8 employees = 8C3 ---> 8*7*6/3! = 56 possible outcomes A __ __ --> 7 employees No Bob --> 6 employees Successes = 6C2 = 6*5/2 = 15 Probability = 15/56

Consecutive Integer Questions - 2 important rules

1) N consecutive integers will always have 1 number divisible by N 2) Sum of the set/ number of integers = mean/median of the set

Circles in the x-y plane 1) general 2) equation of a circle at origin (0,0)

1) Orientation of the circle -Radius = hypotenuse of a slope triangle -Center --> y-coord = height of slope triangle --> x -coord = distance from origin 2) r² = x² + y² Slope triangle of each radius --> horizontal = |x| --> vertical = |y|

Rectangles 1)Properties of rectangles 2)Area of a rectangle

1) Parallelogram with 4 90 degree angles - has big 4 parallelogram properties - all angles = 90 - diagonals are of equal length *Diagonal property is the only separable property - Big 4 + 90 degree angles must be true 2) Area = base * height

Rhombus 1) Properties of all rhombuses 2) Area of a rhombus

1) Parallelogram with 4 equal sides - has big 4 properties of parallelogram - All 4 sides are equal - Diagonals are perpendicular *Perpendicular diagonals is the only separable property of rhombuses* - Big 4 properties + equal sides must be true at all times 2) Area = Base * height ---> height = altitude perpendicular to base

Square 1)Properties of a Square 2)Area of a Square

1) Parallelogram with 4 equal sides and angles -Has all big 4 parallelogram properties -Has all rhombus special properties --> equal sides/perpendicular diagonals -Has all rectangle special properties --> 90 degree angles/equal length diagonals *VERY HARD to prove something is a square unless specifically told - even if drawn to scale and looks like a square 2) Area = s² (side x side)

Reflection in the x-y plane -General Rules of reflection -Reflection over x-axis -Reflection over y-axis

1) Reflect a point over the mirror line - Original and reflection are equidistant from mirror line - Mirror line is perpendicular bisector of the segment created between original + reflection - every point on the mirror line is equidistant from original and reflection 2) X-axis reflection - X-coordinate is the same in original and reflection -Y-coordinate is the same value with opposite sign 3) Y-axis reflection -Y-coordinate is the same in original and reflection -X-coordinate is the same value with opposite sign

Normal Distribution 1) Standard deviations 2) Normal Mean

1) Shows percent of population within each deviation -1 SD = Between M and (M+S) = 34% -2SD = Between (M+S) and (M+2S) = 13.5% -3SD = Between (M+2S) and (M+3S) =2.5% 2) Mean = Median --> 50% above and below M

1) Percentile meaning - 40th percentile -0th percentile -99th percentile 2)Percentile and Quartiles 3)Percentile and Normal Distribution

1) Shows the position of an individual score in the larger population - 40th percentile = score is higher than 40% of total distribution -0th percentile = minimum score received -99th percentile = maximum score received 2) Min = 0th percentile Q1 ~ 25th percentile Median ~ 50th percentile Q3 ~ 75th percentile Max = 99th percentile 3) M = 50th percentile M-S = 50th - 34% = 16th percentile M-2S = 16th - 13.5% = 2.5% < 5th percentile M+S = 50th + 34% = 84th percentile M+2S = 84th + 13.5% = 97.5% > 95th percentile

FCP: Restrictions 1) So we have seven children with alphabetical names (A-G). We're going to sit them in adjacent seats. Dahlia must be in the middle chair and George must be next to Dahlia. How many different seating arrangements are possible?

1) Start with the most restrictive stage and continue only only unrestricted stages are left 1) D = 1 choice G = 2 choices A, B, C, E, F = 5 x 4 x 3 x 2 x 1 All = 1 x 2 x 5! = 240

1)Triangle inequality theorem 2)Generalized Formula: Find 3rd side --> Lengths of 2 sides of triangle = P and Q

1) Sum of any two sides must be greater than the 3rd side --> side A + side B > side C --> Sum CAN'T be equal to 3rd side either 2) Generalized Formula (P-Q) < 3rd side < (P+Q)

Circle Properties: Angles 1)Triangles in a circle 2)Arc Angles

1) Triangles in a circle - Triangle with 2 radii as sides = isosceles - Triangle where 2 radii sides and 1 chord side are equal = equilateral 2) Arc Angles - Central angle = opposite arc angle - Diameter = 180 central angle --> 180 arc --> 180 arc = semicircle -2 angles are the same = 2 arcs are the same -2 chords are the same = 2 arcs are the same

Cylinder 1) Properties 2) Volume 3) Surface Area

1) Two circles with a rectangle between 2) V = πr² * h 3) SA = 2 circles + Rectangle --> 2(πr²) + (h* w) --> (2πr²) + (2πr *h)

Writing Equations of a Line 1) Given Slope and a Point 2) Given two points on a line

1) Use slope intercept form --> plug in x, y, and slope to find b 2) Find slope using 2 points Then plug in one of the point coordinates to solve for b

Factoring Quadratics 1)x² + 8x + 15 2) x² + 4x - 21 3) x² - 2x - 35 4) x² - 16x + 48

1) find 2 numbers who's product is 15 and sum is 8 --> 5 and 3 (x +3) (x+5) 2) 7 and -3 (x+7) (x-3) 3) -7 and 5 (x-7) (x+5) 4) -12 and - 4 (x-12) (x-4)

Comparing the size of different roots 1) b>1 2) 0<b<1

1) if b > 1 then ---> 1 < ³√b < √b < b Larger root = smaller number but must be bigger than 1 2) if 0<b<1 then --> 0 < √b < ³√b < 1 Larger root = larger fraction but never bigger than 1

Determine when to include the negative square root answer

1) if the root sign is written by the test maker (i.e. printed on the test) consider only the POSITIVE root 2) If problem contains a variable squared or calculations lead to a variable squared always consider the POSITIVE + NEGATIVE roots

Patterns of Exponents - base > 1 - 0 < base < 1 - base < -1 - 0> base > -1

1) positive base greater than one will get very big very quickly 2) positive base between 0 and 1 will get very small very fast 3) negative base less than -1 will have alternating +/- signs but the absolute value gets very big 4) negative base between -1 and 0 will have alternating +/- signs but absolute value gets much smaller

Interpreting Equations of a line 1) Slope-Intercept Form 2) Linear equation 3) Slope-Intercept --> Linear i.e: y = -2/5x - 2 --> ?

1) y=mx+b --> solving equation for y --> m = slope & b = y-intercept 2) Ax + By = C --> solve equation for y to get to slope-intercept form 3) (5)y = 5(-2/5x - 2) --> 5y = -2x -10 --> 5y + 2x = -10

Rules of Square Roots 1)B > 1 2) 0 < B < 1

1) √B < B --> makes numbers smaller 2) √B > B --> makes fractions bigger

Area of a Triangle -Formula -Obtuse -Right

1/2 b*h -->Base = Any line segment of a triangle -->Height = Perpendicular line from the base to the opposite vertex * Altitude always forms 2 smaller right triangles * Obtuse triangles -must extend the line to create a perpendicular altitude outside of the triangle Right Triangles - B and H will be the two sides that meet at the right angle

Finding Common Denominator 1/3 + 1/7 1/12 + 7/24

1/3 * 7/7 + 1/7 *3/3 --> 7/21 + 3/21 = 10/21 1/12 *2/2 + 7/24 --> 2/24 + 7/24 = 9/24 =3/8

Number Sense with Percents What is 80% of 200 360 is 30% of what # 49 is what % of 700

10% of 200 = 20 *8 = 160 10% = 360/3 = 120 *10 = 1200 10% of 700 = 70 --> 1% of 700 = 7 7*7 = 49 --> 7%

Probability: The AND Rule -Simple

2 Completely Independent Events Occurring = Simple Rule -->independent events can't be mutually exclusive - P(A) AND P(B) = P(A) * P*(B)

Supplementary Angles

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

Congruent Shapes

2 shapes that have an equal size and shape - orientation doesn't matter

Similar Triangles

2 triangles that are the same shape but different sizes Angles --> Both triangles have equal angles --> Parallel line within a triangle creates a smaller similar triangle --> Prove two triangles are similar if 2 of the angles are the same in each Sides --> proportional to one another --> can set ratio of any 2 sides equal to corresponding sides of 2nd triangle i.e. triangle ABC is similar to DEF ---> AB/BC = DE/EF

Pythagorean Triplets

3, 4, 5 5,12, 13 8, 15, 17 - Can use these as ratios which can be multiplied by any number to create another right triangle

Approximations of Pi

3.14 OR 22/7

Multiplying/Dividing with Radicals 3√ 5 * 7√ 2

3√ 5 * 7√ 2 ---> (3*7)(√ 5 *√ 2) = 21 √10

Comparing Distance between percentile to distance between scores 660 = 70th percentile & 760 = 80th percentile. What can we say about 75th percentile

70th-80th = 10% of population --> Based on normal distribution the area between these two values is not even when cut in half 75th percentile = lower than the mean of the upper and lower scores --> due to decreasing amount of population closer to the tail

Work Questions 1) Proportion A machine, working at a constant rate, manufactures 36 staplers in 28 minutes. How many staplers does it make in one hour and 45 minutes? 2) Total work done at different rates When Amelia and Brad detail a car together, 1 car takes 3 hours to detail. When Amelia details the car alone 1 car takes 4 hours. How long does it take Brad working alone to detail one car?

A = RT ( Amount = Rate*Time) 1) A/T = R --> 36/28 min = ? / 105 min --> 9/7 = ?/105 ---> 9/1 = ?/15 = 135 staples 2) The combined rate is the SUM of the individual rates Am+B = 1/3 hrs -- Am = 1/4hrs -- B =1/? B = (Am+B) - Am --> 1/3 - 1/4--> 4/12 - 3/12 B = 1/12

Perpendicular Bisector

A line that bisects and is perpendicular to the segment --> every point on the perp. bisector is equidistant from the 2 end points of the segment ---> perp. bisector is the set of all points equidistant from 2 segment end points

Circles: Tangent Lines

A line that touches the circle at ONE Point - radius to point of tangency = perpendicular line ---> Allows for creation of right triangles

Divisor

A number which divides evenly into another number - with no remainders - A *B = C --> A and B are divisors of C -EVERY FACTOR IS A DIVISOR

Circles: Radius

A segment from the center of the circle to any point on the circle - all radii are equal length in the same circle

Circles: Chord

A segment with both endpoints on the circle - can all have different lengths and angles within the same circle - no shortest chord --> anything down to 0 - longest chord = diameter

Transversal Line

A third non-parallel line that cuts across 2 parallel lines -->8 angles made by the transveral --> Each angle is equal to 3 other angles --> 2 angles on top will equal 2 angles on bottom

Fundamental Counting Principle (FCP) 1) For a formal dinner, guests have the choices of one of 4 salads, one of 5 appetizers, one of 12 entrees and one of 4 desserts. How many meals are possible 2)Suppose we have six different books that we will place on a shelf. In how many different orders can we place these books?

A way to choose items when order make a difference --> the items are all different and the stages are all different 1) Divide the problem in to different stages of choices 2) Multiply the number of options available in each stage 1) 4 x 5 x 12 x 4 = 960 2) 6 x 5 x 4 x 3 x 2 x 1 = 6! = 720

Area of a Circle

A=πr²

Square root inequalities 1) √41

A> B> C then √A > √B > √C 1) 36 < 41 < 49 --> 6 < √41 < 7 *You can approximate square root answers for the test*

Counting: AND

AND = Multiply

Sequence Questions: Arithmetic Sequences 1) 5, 12, 19, 26 --> Find formula 2) Let S be the set of all positive integers that, when divided by eight, have a remainder of five. What is the 76th number of the set? 3) Let T be a sequence of the form a(n) = a(1) + d*(n-1). If a(3) = 17 and a(19) = 65, find a(10)

Add same constant to get from one term to the next -always an evenly spaced list -find the formula of the list to determine any term 1) 1--> 5 2 -->5 + 7 3 --> 5 + 7 + 7 4 ---> 5 + 7 + 7 + 7 a(n)= 5 + 7* (n-1) General Formula: a(n) = a(1) + (n-1) * d Remainder = a(1) that is added to all terms is the remainder for all terms of the sequence divided by d 2) 5 + 8*(n-1) --? 5+ 8*(75) = 605 3) a(3) = (a1) + d (3-1) --> a1+2d = 17 a(19) = (a1) + d(19-1) --> a1 + 18d = 65 -subtract 2 equations: 16d = 48 --> d = 3 / a1 =11 --> a(10) = 11 + 3(10-1) = 27+11 = 38

Age Questions Now Steve's age is half of Tom's age. In 8 years, twice Tom's age is 5 more than 3 times Steve's Age. How old is Tom now?

Ages change over the course of the problem -associate variables with now or later but not constant S = Steve's Age Now T = Tom's age now Now: T = 2S +8Years: 2(T+8) = 3(S+8) + 5 2(2S + 8) = 3S + 24 + 5 --> 4S + 16 = 3S + 29 ---> S = 13 --> T = 26 Now

Exponential Equations 1) 7^2x = 7^(6-x) 2) 49^x = 7^(6-x) 3) 27 ^ (2x-2) = 81 ^ (x+1)

Always make the bases equal so you can set the exponents equal to each other - or as the power of common smaller number 1) 2x = 6-x --> x = 2 2) (7²)^x = 7^(6-x) --> 2x = 6-x 3) 27 = 3^3 and 81 = 3^4 ---> (3^3)^(2x-2) = (3^4)^(x+1) ---> 3(2x-1) = 4(x+1)

Circle: Inscribed Angle

Angle with vertex on the circle - 2 sides = 2 chords - Inscribed angle = 1/2 arc angle --> Angle = 40 --> Arc Angle = 80 -Inscribed angle that intercepts the end points of the diameter = right angle -2 inscribed angles intercept the same arc or chord --> both angles are equal

Raising Radicals to Powers (5√6)²

Apply the exponent to each part seperately (5√6)² = 5² * (√6)² = 25 *6 = 150

Circumference of a Circle

C = πd OR C = 2πr

Weighted Averages 1) Sum solution 2) Proportion solution

Combined averages of 2 different groups to find average of the whole 1) Find the sum of each group N = number of items in the group Average * number of item = sum of all items (average 1 * N1 ) + (average 2 *N2)/ (N1+N2) 2) Use proportion of each group P = proportion of the whole (P1 + P2 = 1) (Average 1 * P1) + (Average 2 * P2) = A of whole

Probability: Complements

Complement of event A = not A --> i.e A = winning a medal Complement = not winning a medal P(notA) = 1 = P(A) Undesired outcome = something occurring - desired outcome

Growth and Decay Questions When isotope QXW radioactively decays, it loses exactly half its mass in each three-day period. Suppose scientists start with a 96 gram sample of a pure isotope of this on a certain day. What will be the remaining mass in 12 days?

Complete the problem 1 change interval at a time Change interval = 3 days = 1/2Q Start = 96 g --> 3 days = 48 --> 6 days = 24 ---> 9 days = 12 --> 12 days = 6g left

Mixture Questions 1) Suppose we start with 5 liters of a 30% HCl solution. How much water must we add to create a 20% solution? 2) Suppose we start with 8 liters of 60% solution. Add 4 liters of a X% solution, and it results in 12 liters of a 50% solution. What is X 3) Suppose we start with unlimited supplies of a 20% solution and a 50% solution. We combine X liters of the first, with Y liters of the second, to produce 7 liters of a 40% solution, what does X equal?

Concentration = solute/ solution (water+solute) x 100 1)Decrease/Increase Concentration .3 = X/5l --> 5*.3 = 1.5 l solute .2 = 1.5l / X --> .2 * X = 1.5 --> .2 = 1/5 --> 1/5 *X = 1.5 (5)*1/5 *X = 1.5 *5 --> X = 7.5 2) Two solutions with different concentrations Total = .5 * 12 = 6 l total solute 1st = .6 * 8 = 4.8 l solute 2nd solute added = 6l - 4.8l = 1.2 L solute --> X = 1.2/4 * 100 = 30% solution 3) Two solutions with unknown amounts Total = .4 * 7 = 2.8l solute total X + Y = 7 l 0.2 X + 0.5 Y = 2.8l Y = 7-X --> 10*(.2X + .5(7-x) = 2.8 ) --> 2X + 5(7-X) = 28 --> 2X + 35 - 5X = 28 --> -3X = -7 --> X = 7/3 liters

Probability: AND Rule -General 1) A box has five green balls and seven red balls. Assume that all the balls in the box are equally likely and that the balls are picked without replacement. What is the probability that the first two balls picked are both green? 2) From a standard shuffled deck of 52 cards, what's the probability of picking three hearts on the first three cards drawn, if the cards are selected without replacement?

Conditional Probability --> 1 event changes outcome of the other ---> P(A|B): probability of A given B desired probability (A) given condition (B) P(A and B) = P(B) * P(A|B) P(A and B) = P(A) * P (B|A) 1) P (1=G) = 5/12 P (2=G | 1=G) 4/11 --> 5/12 * 4/11 = 5/33 2) P (1 =H) = 13/52 = 1/4 P (2=H | 1=H) = 12/51 P (3=H | 1&2=H) = 11/50 --> 1/4 * 12/51 * 11/50 = 11/850

Slope of Line = 1 or -1

Creates 45-45-90 triangle - 45 degree angle with the axes

Bisector -angle bisector -segment bisector

Cuts something into two congruent (equal) parts - angle bisector = two congruent angles - segment bisector = point/line/line segment --> place where it bisects = midpoint

Motion Questions A lichen advances 4 cm each year across a rock slab. If this rate remains constant over time, how many years will it take to cross 30 meters?

D = R*T 30 m = 30 * 100 = 3,000 cm 3,000 = 4 * T --> D/R = T ---> 3,000/4 = T = 750 years

Average Speed Questions Bob drove 120 miles at 60 mph, then another 120 for 40 mph. What was the average speed for the whole trip

D=RT is true for the 1st and 2nd half of the trip as well as the trip as a whole - Average speed = total distance/total time - Same as R of the whole trip * If not all info provided, pick a value or use variables* 1st Part = R = 60/ D = 120/ T = 2hrs 2nd Part = R = 40/ D = 120/ T= 3hrs Total R = 240 mi/ 5hrs = 48 mph

n-Sided Polygon -Number of diagonals -Number of triangles -Sum of all angles -Find degree of 1 angle

Diagonals = n-3 Triangles = n-2 Angle Sum = (n-2)*180 1 Angle = ((n-2)*180)/ n

Combinations 1) Rules 2) We have a pool of 10 people, and we want to select 4. How many different sets of 4 could we pick?

Don't care about order --> nCr --> n choose r --> n group of individuals and selecting r of them 1) Rules -nC1 = n -10C4 = 10C6 --> nCr = nC(n-r) 2)10C4 --> 10 * 9 * 8 * 7/ 4! = 210 --> total options/un-ordered positions 10C4 = 210 = 10C6

Area in Similar Triangles

Each length increases by scale factor k --> 1/2 (b*k) (h*k) --> Area increases by k²

Multiple Traveler Questions Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 miles an hour, and Paul at a constant speed of 40. When Martha arrived at B, Paul was still 50 miles away. What was the distance between A and B?

Each traveler and each trip get their own D=RT equation - use system of equation to solve for multiple variables M = D --> D = 60T P = D-50 --> D-50 =40T 60T - 50 = 40T 20T = 50 --> T = 5/2 = 2.5 hours D = 60 * 2.5 = 150 miles

0ⁿ

Equals 0 as long as the exponent is positive - 0 to the power of 0 - NOT TESTED -0 to the power of negative - NOT TESTED

Negative base with exponents

Even exponent = positive answer odd exponent = negative answer

Factors of a Factorial 1) 20!

Factorial n! is divisible by all integers less than n and all factorials less than n 1) 20! = 20 * 19! = 20 * 19 * 18! = etc.

Set/Sequence Questions: Double Matrix - Every item belongs to 2 sets 1) In a company of 300 employees, 120 are females. A total of 200 employees have advanced degrees and the rest have college degree only. If 80 employees are males with college degrees only, how many are females with advanced degrees? 2) In a certain school, there are 80 freshmen, 100 sophomores, and 220 upperclassmen, drawn from three cities: A, B, C -- 60% from A, 30% from B, the rest from C. All the students from C are freshmen. Half the students from B are upperclassmen, and the rest are split evenly between the other two grades. How many sophomores are from A?

Find missing information that can be deduced from the problem 1) Fa = 100 F M T C 20 80 100 A 100 100 200 T 120 180 300 2) Sa = 70 F S U T A 10 70 160 240 B 30 30 60 120 C 40 0 0 40 T 80 100 220 400

Graphs of Quadratic Equations 1) y = ax² + bx + c 2) y = 4 - x²

Graph = parabola - has a line of symmetry through vertex ---> symmetry helps determine points on opposite sides 1) General Equation --> Graph a > 0 = opens up a < 0 = opens down |a| > 1 = skinny parabola |a| < 1 = wide parabola y = 0 = x-intercepts = quadratic solutions @ 0 x= K = line of symmetry = x coord. of vertex 2) y = 4 - x² --> y = (2-x) (2+x) Vertex = (0,4) X-int = (2,0) & (-2,0)

Boxplots - Info -Inter-Quartile Range

Graphical representation of the 5-number summary - 2 ends = min/max -Box area = Q1, Median, Q3 --> middle 50% of pop. falls between Q1 and Q3 --> representative of the pop. without outliers IQR = Q3-Q1 = range of mid 50% of data

Coordinate Plane: Quadrants

I = upper right --> (+, +) II = Upper left --> (-, +) III = Lower Left --> (-, -) IV = Lower right --> (+, -)

Law of exponents: 0 exponent a^0

If a does not equal 0 a^0 = 1

Proportional Reasoning with Right Triangles 1) Right triangle with sides 24 and 45 and hypotenuse X. Find X

If right triangle has very large sides --> Find the scale factor for the 3 sides to reduce numbers 1) 24 and 45 divide by 3 --> 8, 15 --> part of triplet 8, 15, 17 --> X = 17*3 = 51

Two Way Triangle Theorem

If two sides are equal --> 2 opposite angles are equal If two angles are equal --> 2 opposite sides are equal

Probability: At Least 1) Suppose we roll one fair six sided die eight times. What is the probability that we will roll at least one six?

Indicates you can use the complement rule - P(Not A) = 1 - P(A) P(at least 1 success) = 1 - P(0 successes) 1) P(0 success) = 5/6 --> 8 times = (5/6)^8 P(at least 1) = 1-(5/6)^8

Law of exponents: Mistakes

Laws only apply if the base is the same -without the same base none of these laws work No law for sum or difference of powers

Line and Line Segments

Lines go on infinitely -all lines are straight on the test -can't assume horizontal or vertical Line segment has two distinct end points

Mean - Measure of center

Mean = average of all the numbers in the set --> sum of N values / N --> sum of values = N * mean -Mean is impacted by outliers

Slope of a Line 1) Formulas 2) Meaning of m = -2/3

Measure of how steep a line is --> Slope = (y/x) of 2 points on a line --> Slope = (y1 - y2) / (x1 - x2) Can use slope to find other points on the line 2) m = -2/3 - move right 3 and down 2 (3, -2) - move left 3 and up 2 (-3, 2) - move left 3k and up 2k = multiples of slope - move left 1 and up 2/3 (1,2/3) = small distances

Quartiles: Finding Each Quartile

Median -Odd number of values = middle number --> Exclude from upper/lower list -Even number of values = average of 2 middle --> Split between upper/lower list Q1 = median of the lower list Q3 = median of the upper list

Quartiles: General Values

Min = Smallest value Q1 = Bottom 25% (Min --> Q1) Median = Divides upper and lower 50% Q3 = Divides lower 75% from upper 25% Q3 - Max = 25% Max = Largest value *Box plots indicate the location of these 5 values*

Graphing Vertical Lines -Y-axis -X-axis -Slope

Must all have the same x-coordinate 1) X-Axis Equation: x = B --> B = where line passes through x-axis 2) Y-Axis Equation: x = 0 --> Passes through the x axis at 0 3) Slope = undefined slope --> X/0

Graphing Horizontal Lines -Y-axis -X-axis -Slope

Must all have the same y-coordinate 1) Y-Axis Equation: y = B --> B = height of the line & Y-intercept 2) X-Axis Equation: y = 0 --> Passes through the y-axis at 0 3) Slope = 0

Extraneous Roots √x+3 = x-3

Must check for answers that would not work when you are using the square root √x+3 = x-3 --> ( √x+3)² = (x-3)² --> x+3 =(x-3)² --> x+ 3 = x² - 6x + 9 --> 0 = x² -7x + 6 --> (x-6)(x-1) x = 1, 6 * Must check if both answers work in the equation* √x+3 = x-3 --> √1+3 = 1-3 --> 2 = -2 --> NO SOL. √x+3 = x-3 --> √6+3 = 6-3 --> 3 = 3 --> YES

Adding/Subtracting with Radicals √72 -√32=

Must simplify each radical before you can add/sub √72 -√32 --> √ 36*2 + √ 16*2 = 6√ 2 - 4√ 2 = 2√ 2

Probability: The OR Rule -Simple -General 1) Supposing Game M the probability of outcome A is .6, the probability of outcome B is .7, and the probability of A or B is .9. What is the probability of A and B happening at the same time?

Mutually exclusive events = Simple rule --> mutually exclusive can't be independent - P(A) OR P(B) = P(A) + P(B) Events with overlap = General Rule - P(A) or P(B) = P(A) + P(B) - P (A and B) 1) 0.9 = 0.6 + 0.7 - P(A and B) = 0.4

Counting with Identical Items 1) A librarian has seven books to arrange. Four different novels, and three identical copies of the same dictionary. How many different orders could these seven books be put on the shelf? 2) How many different arrangements can be made of the letters in "Mississippi?

N = n!/b! = total items / identical items N = n!/b! * c! * d! = total items/ identical(b) * identical(c) * etc. 1) Total Arrangements = 7! Dictionary Arrangements = 3! --> All same 7! / 3! = 7 * 6 * 5 * 4 = 840 arrangements 2) Total = 11! I's = 4! S's = 4! P's = 2! 11! / 4! * 4! * 2! --> 11*10*9*8*7*6*5/(4*3*2*1)*(2*1) --> = 34,650

Can you take the square root of a negative number?

NO! - positive squared = positive -negative squared = positive -0 squared = 0 i.e. √a = b then a>/=0 and b>/= 0 Anything squared can't yield a negative so you can't take the square root of a negative - since squaring and the square root are opposite/undo each other

Counting: OR

OR = Add

Gap Questions 1) Opposite directions (---> <---) or (<--- --->) City J is 480 miles north of City K. At 10 AM, Car M starts driving south and Car N starts driving north, and it's going twice as fast as M. Both cars maintain constant speeds and they pass each other at 2 PM going in opposite directions. What is the speed of Car N? 2)Same direction (-> ---->) or (----> ->) A car and truck are moving in the same direction on the highway. The truck is moving at 50 miles an hour, and the car is traveling at a constant speed. At 3 PM, the car is 30 miles behind the truck and at 4:30, the car overtakes and passes the truck. What is the speed of the car?

Overall: set up D=RT for each traveler OR D=RT for the gap change 1) Add the speeds = speed of gap change T = 4 hrs D = 480 --> R = 480/4 = 120mph N = 2M --> M + 2M = 120 --> 3M = 120 --> M=40 N = 2* 40 = 80 mph 2) Subtract the speeds = speed of gap change T = 1.5 hrs/ D = 30 --R = D/T --> 30 *2/3 = 20mph gap ---> 20 mph = difference in speed if Truck R = 50 --> T-C = 20 --> 50 + 20 = 70=C-rate

Order of Operations

PEMDAS 1) Parenthesis/Groupings 2)Exponents and Roots - must perform under the root first - no operation passes through root 3)Multiplication/Division 4) Addition/Subtraction

Vertical Angles

Pairs of angles that are opposite each other and share only the vertex - always congruent angles

Slope of Parallel Lines and Perpendicular Lines M of Line 1 = 2 M of line 2 = ?

Parallel = Same slope --> M of line 2 = 2 Perpendicular = opposite sign and reciprocal --> M of line 2 = -1/2

Circular Sector: Area of a part of the circle

Part/Whole Proportion Sector Area / Circle Area = Central Angle/360

Arc Length: Find Arc Length

Part/Whole proportion (fraction = fraction) Arc length/circumference = Central angle/360

Permutations

Permutations of n different items = n! -> Put items in as many different orders as possible Same as the fundamental counting principle

Intercepts of a Line -Horizontal -Vertical -Origin -Slanted

Points where the line cross the x or y axis 1) Horizontal = only y-intercept --> y =k 2) Vertical = only x-intercept --> x = k 3) Line through origin = x-int and y-int = 0 4) Slanted = unique x-int and y-int ---> shows 2 points to determine a line

Probability

Probability = ratio = fraction # of successes / # of possible outcomes P(A) = probability of event A occurring Falls between 1-0 --> 0 = impossible --> 1 = guaranteed Random -individual outcome of events = unpredictable --> outcome of coin toss occurring once -overall patterns of events = predictable --> outcome of 10,00 coin tosses ~ 50/50

Range - Measure of Spread

Range = Max value - Min value - Many different lists can have the same mean and range values -Doesn't tell us much about the actual numbers in the list

Ratios of Areas of Similar Figures 1) 2 irregular polygons are similar figures. The base of the smaller = 6 and the base of the larger = 12. The area of the smaller = 13. What is the area of the larger

Related by scale factor k --> Lengths increase by k --> Area increases by k² 1) 12/6 = 2 = k --> K² = 4 --> A = 13 * 4 = 52

Ratios of Volumes of Similar Figures

Related by scale factor k --> Lengths increase by k --> Volume increases by k³

Standard Deviation - Measure of Spread 6 General Facts - 1,2,3 --> Value of SD - 4 --> Comparing SD based on values - 5 --> Impact of Add/Sub on SD - 6 --> Impact of multiplication on SD

SD = space between number = distance between a value and the mean 1) SD can only be positive or 0 -> Distance 2) SD = 0 only if all the numbers are equal 3) If all numbers are same distance from the mean then SD = distance from the mean 4) Set with most values near the extremes with have a higher SD than a set with most values near SD 5) Add/Sub same number to each value on the list--> SD doesn't change 6) Multiply every number on the list by K --> SD is also multiplied by K

Similar Triangle Scale Factor 1) In two triangles EGH and EFJ, FJ and GH are parallel. FJ = 4, GH = JH = 20. What is EJ?

Sides of the bigger triangle written in the numerator = fraction > 1 = Scale factor (k) --> Side of small triangle * k = corresponding side of big triangle -Works for any length in the triangles --> altitude/bisector/etc. --> all related by scale factor 1) EGH and EFJ are similar (EFJ is smaller) --> GH/FJ = 20/4 = 5 =scale factor EJ = X & EH = 5x --> Corresponding sides JH = 20 = EH - EJ = 5x -x = 4x ---> 20 = 4x --> x = 5 = EJ

90 degree angle on coordinate plane

Some point C --> Same x-coordinate as some point A --> Same y-coordinate as some point B ACB = 90 degrees

Circles: Arc

Specific length around the edge of the circle - usually the shortest route --> i.e ABC - can be the longer route --> AC

Square multiples of 10 40^2

Square 1st number Add 2 zeros i.e: 40^2 = 16 + 00 = 1600

Sum of Sequence Questions 1) What is the sum of all the multiples of 20 from 160-840 inclusive 2) What is the sum of all the multiples of 5 that are greater than 100 and less than 200

Sum of first N integers --> N(N+1)/2 Sum of a list --> (N/2)(a(1) + a(n))--> number of terms (first+last)/2 1) 160 = 8 * 20 and 840 = 42 * 20 --> 42 - 8 + 1 = 35 terms --> 35/2 (160 + 840) = 17.5 * 1000 = 17,500 2) 100 = 20 * 5 and 200 = 40*5 --> neither included 5*21 = 105 and 5*39 = 195 --> both included --> 39-21+1 = 19 --> (19/2) ( 105 + 195) = 9.5* 300 = 2850

Median - Measure of Center

The middle number on an ordered list -list must be in ascending order -->Odd number list = find middle number -->Even number list = average 2 middle numbers -Not influenced by outliers --> only accounts for numbers at the center of the list

Angles and Degrees

The movable space where 2 lines meet -->Middle point = vertex --> Vertex must be in middle of name Degree Facts 1) Straight line = 180 2) Right angle = 90 ---> 2 lines meeting at right angles = perpendicular --> CAN'T assume unless told specifically 3) Acute angle < 90 4) Obtuse angle > 90

Mode

The number that appears the most times on a list Single mode = most frequent digit 2 modes = 2 most frequent digits No Mode = all digits appear equally

The Units Digit Question 1)Whats the units digit of 57^123

The units digit of any product will only be influenced by the units digits of the 2 factors - only need to consider single digit products - i.e. 3*6 = 18 --> any length number ending in 3 * any length number ending in 6 will end in 8 *Look for repeating pattern of the units digit (period is often 4) 1) 7^1 = 7 7^2 = ..9 7^3 = ...3 7^4 = ...1 7^ 5 = ...7 7^6 = ...9 etc. ---> repeating pattern 7,9,3,1 --> repeats every 4 numbers --> 7^120 = multiple of 4 = ...1 7^121 = ...7 7^122 = ...9 7^123 = ...3 Units digit = 3

Trapezoids -Properties of a Trapezoid - 1 & 2 -Properties of Isosceles Trapezoid - 3 & 4 -Area of a Trapezoid - 5

Trapezoid 1) Quadrilateral with ONE pair of parallel sides - parallel sides = base - non parallel sides = legs 2) 2 angles on a leg are supplementary (=180) - can have 2 right angles on a leg Isosceles Trapezoid 3) 2 legs are equal - opposite angles are equal 4) Diagonals have equal length Area = (b1 + b2/2) * h ---> h = altitude perpendicular to bases OR --> Divide into 2 right triangles and 1 square Area = A of Square + A of Tri. 1 + A of Tri. 2 ---> (s²) + (1/2bh) + (1/2bh)

Finding Dividend from Divisor and Remainder i.e. D = ? / S = 12/ R = 5

Use any multiple of the divisor (S) and add the remainder (R) to it i.e. 12 +5 = 17/ 24+5 = 29/ etc. Or add S to the first D found i.e 17 + 12 = 29/ 29 + 12 =41/ etc.

Set and Sequence Q's: Inclusive Counting 1)Contract negotiations opened on the morning of March 20th cons, continued every day without break and ended late in the evening of May 10th, or how many calendar days were the contract negotiations in session? 2)How many multiples of 8 are there from 200 to 640, inclusive?

Use inclusive counting when the starting point and ending point are both included in the count -i.e. dates 1) March 20th-31st = 11 days + 1 = 12 April = 30 + May = 10 --> 30 + 10 + 12 = 52 days 2) 8*25 = 200 and 8*80 = 640 --> count integers from 25 to 80 inclusive --> 80-25 + 1 = 56

Fraction Division (1/4) / (3/2)

Use the reciprocal of the 2nd number (1/4) / (3/2) --> (1/4) * (2/3) = 2/12 = 1/6

Distances in XY Plane -Vertical Distance -Horizontal Distance

Vertical Distance: (5,1) and (5,8) --> On same vertical line --> Distance = 8-1 = 7 *same x coordinate = vertically separated Horizontal Distance: (-3,2) and (9,2) --> On same horizontal line --> Distance = 9 - (-3) = 12 *same y coordinate = horizontally separated

Sequence Questions: All positive even numbers

a(n) = 2n

Sequence Questions: All positive odd numbers

a(n) = 2n-1

Sequence Questions: All perfect squares

a(n) = n^2

Sequence Questions: Recursive Sequences 1) If b(n) = ((b(n-1)) -1) ^2 + 3 and b(1) = 1, find the value of b(4)

a(n) is defined in terms of a(n-1) --> the previous # - You can't jump to a term on the list; must start from a(1) 1) b(2) = (1 -1)^2 + 3 --> 3 b(3) = (3-1)^2) + 3 --> 7 b(4) = (7-1)^2 + 3 --> 39

Law of exponents: dividing powers (a^m)/(a^n)

a^m/a^n = a^(m-n)

Fractional Exponents b ^1/n b ^m/n

b ^1/n = ⁿ√b b ^m/n = (ⁿ√b)^m = ⁿ√(b^m)

Negative Exponents b^-n

b^-n = 1/b^n or b^-n = b^(0 - n) = b^0/b^n = 1/b^n

Diameter of a Circle

d = 2r

Counting what you don't want 1) 6 children will sit in 6 chairs, but J can't sit next to M. How many arrangements are possible? 2) Harriet has six novels she wants to read, one of which is Emma by Jane Austen. She plans to create a reading list of four of these novels for an upcoming trip, and different orders count as different lists. How many reading lists are possible if Emma must be on the list?

n! = R + Q -->n! = total # of arrangements -->R = # of arrangements obeying restriction -->Q = # of arrangements not obeying restriction R = n! - Q * look for "NOT" in the question to determine if finding Q might be easier* 1) Total # = 6! = 720 --> 10 arrangements where JM/MJ sit together (draw) --> 4! arrangements of other children 10*4! = 240 = Q 720-240 = R = 480 2) Total = 6 * 5 * 4 * 3 = 360 Q = 5 * 4 * 3 *2 = 120 R = 360-120 = 240

Squaring other numbers 41^2

n^2 + n + (n+1) 40^2 + 40 + 41 = 1600 + 40 + 41 = 1681

Law of Exponents: same base and different powers b^s = b^t

t=s

x² = K Find x

x = +/- √K

Power to a Power Rule x^10

x^10 = x^5*2 = (x^5)²

Find x and y intercept of a line

y = 0 --> solve for x = x-intercept ---> (k , 0) x = 0 --> Solve for y = y-intercept ---> (0, k) * Set equal to 0 the intercept you are not looking for

√y²=

|y| Whether y is positive or negative, once squared and then square rooted you get a positive output

Simplifying Square roots √2800

√2800 = √ 28 * 100 = 10√28 --> 10√4*7 --> 10(2√7) = 20√7

Root Properties: Distribution of roots √PQ √P/Q

√PQ = √P * √Q √P/Q = √P/√Q * Do not distribute over addition/subtraction

Sum of Squared Binomials (a+b) ^2

(a+b) * (a+b) a^2 + 2ab + b^2

Difference of Squared Binomials (a-b)²

(a-b) * (a-b) a² - 2ab + b²

Factoring out Negative Signs -46 -37 = -22 + - 61 = * 23-64 *

-(46 + 37) = -83 - (22 + 61) = -83 * Special case where factoring neg flips order* -(64 - 23) = -41 OR -64 + 23 = - 41

Sequential Percent Changes $100 item increased by 30% then later decreased by 30% Item increased by 30% then decreased by 40%. The new price is what percent below the original

1) 100 * 1.3 = 130 ---> 130 *.7 = 91 2) 1.3X ---> 1.3X * .6 = .78 --> 22% decrease

Multiplying Decimals 6.25 x 0.048

1) Add number of decimal places --> 2 + 3 = 5 - product will have 5 decimal places 2) Multiply as normal integers --> 625*48 = 30,000 3) Place rightmost numbers in X # of decimal places --> 30,000 --> 0.30000

Adding/Subtracting Evens and Odds

1) Add/Sub likes to get even number -E+E = E/ E-E = E /O+O = E /O-O = E 2) Add/Sub opposites to get odd number E+O = O/O-E = O/E-O = O

Multiplying Evens and Odds

1) At least one even in equation will make product even E x E = E E x O = E 2) Must have all odds in the equation to make product odd O x O = O

Find Even number of factors

1) Calculate the number of total factors 2)Subtract the number of odd factors

Solving Quadratic Equations x² + 10x -24 = 0

1) Factor (x+12) (x-2) = 0 2) Set each part of equation to 0 x+12 = 0 x-2 =0 3) Find possible solutions x = -12 OR x = 2 *Quadratics have 2, 1, or 0 solutions 4) Solve to confirm each solution is correct

Simplifying Complex Fractions (x/2 +5/4) / (x/3 + 3/2)

1) Find LCM of the denominators in the smaller fractions (x/2 +5/4) / (x/3 + 3/2) --> LCM = 12 12(x/2 +5/4) --> 6x +15 12(x/3 + 3/2)--> 4x +18 6x+15/4x+18

Find the number of odd factors i.e. Odd factors of 480

1) Find prime factorization - 48x10 --> (6x8) x (5x2) -->(3x2)x(2x2x2)x (5x2) ---> 2^5 x 3 x 5 2) List exponents of odd factors - 1,1 3) Add 1 to each number on list -2,2 4) Multiply together for total odd factors - 4 odd factors of 480

Count Factors of Large Numbers i.e. Factors of 8400

1) Find prime factorization - 84x100--> (7x12)x(10x10) --> (7x3x2x2)x(5x2x5x2) ---> 2^4 x 3 x 5^2 x 7 2) List the exponents of the factors - 4, 1, 2, 1 3) Add 1 to every exponent - 5, 2,3,2 4) Multiply all numbers in new list for the number of total factors - 5x2=10 3x2 = 6 --> 6x10 = 60 60 factors of 8400

2 Variable Equations: Elimination 2x + 3y = 15 x + 2y = 11 Eliminate Y --> Solve for X

1) Make coefficients of variable for elimination equal and opposite - multiply (-2)(2x+3y=15) --> -4x-6y = -30 (3)(x + 2y = 11) ---> 3x+ 6y = 33 2) Add the equations (-4x-6y = -30) + (3x+ 6y = 33) = -x = 3 --> x = -3 3) Solve for 2nd variable 2(-3) + 3y = 15 --> -6+3y=15 --> 3y = 21 --> y=7 *Choose which type of solving is easiest based on the equation*

Combinations of Ratios 1) S: J = 2:3 and J:N = 5:6. S is what fraction of the whole group 2) P:M = 3:8 and C:M = 2:3. If there are 27 P, there are how many C

1) Make common term in the ratios equivalent - 2:3 and 5:6 --> 10:15 and 15:18 - S: J : N = 10:15:18 --> whole = 43 --> S =10/43 2) If given a number, find absolute quantity for each term - 3/8 = 27/M --> 1/8 = 9/M --> M = 72 - 2/3 = C/72 -->2/1 = C/24 --> C = 48

Percent Decreases Y decreases by 30% 170 discounted 30% 80% decrease is 150. What was original?

1) Multiplier = Y whole - decrease --> 1-.3 = .7 2) 1-.3 = .7* 170 = 119 3) 1-.8=.2 --> X * 0.2 = 150 --> 150/.2 = 750

2 Variable Equations: Substitution x+2y = 11 2x + 3y = 15

1) Solve 1 equation for a variable x = 11-2y 2) Substitute the variable in the 2nd equation 2(11-2y) + 3y = 15 --> 22-4y + 3y = 15 --> 22-y=15 ---> 7 = y 3) Substitute known variable to find 2nd x = 11-2(7) --> x = -3 *Use if one coefficient of a variable is 1*

Rules for Multiples (6) i.e. P = 36 Q = 12 r = 3

1) every pos. integer is a multiple of one and itself - i.e 12 is a multiple of 1 & 12 2) To find the first X multiples of a # (r=3), multiply the # by each X (i.e. 1-5) -i.e. 1x3 = 3/ 2x3 =6/ 3x3 =9/ 4x3 = 12/ etc. 3) If P is a multiple of r then (P-r) and (P+r) are also multiples of r - i.e. 36-3 = 33 (11*3)/ 36+3 = 39 (13*3) 4) If P and Q are multiples of r, then (P+Q) and (P-Q) is a multiple of r - i.e. 12 + 36 = 48 (16*3)/ 36-12 = 24 (8*3) 5)If P is a multiple of r, then any multiple of P is a multiple of r -i.e. 36*2 = 72 (24*3)/ 36*3 = 108/ etc. 6) If P and Q are multiples of r, then P*Q is a multiple of r - i.e. 36*12 = 432 (144*3)

Number sense to compare fractions 449/150 v. 20/7

1) make numbers closer to a known fraction 449/150 < 450/150 = 3 20/7 < 21/7 = 3 --May be able to stop here knowing approximation -- equal in this case 1)450/150 - 1/150 vs. 21/7 - 1/7 --> 3- (1/150) v.s 3 - (1/7) 1/7 > 1/150 ----> 20/7 is bigger

Dividing Decimals 0.56/0.0007

1) multiply top and bottom by 10 to remove decimals --> 5.6/0.007 -->56/0.07--> 560/0.7--> 5600/7 2) Then divide -->5600/7 = 800

Percent --> Fraction 45%

45% = 45/100 = 9/20

Compare fractions with cross multiplication 7/11 ?? 5/8

7/11 ?? 5/8 --> 7*8 ?? 11*5 --> 56 > 55 7/11 > 5/8

Parts of Ratios Boys to girls is 3:5. Boys are what fraction of the whole?

8 parts in total Girls = 5/8 Boys = 3/8

Prime Factorization of Large Numbers i.e. 96

96 --> 2x48 --> 2x6x8--> (2)x(2x3)x(2x2x2) ---> 2^5 x 3 = Prime factorization of 96

Compound Interest: Example $1,000 with 5% interest yearly, compounding quarterly. How much is there after 6 years

A = 1000 (1.05/4) ^4*6 ----> A= 1000 (1.0125)^24 Can also estimate solution using simple interest

Compound Interest: Formula

A = P(r^ny) A = amount P = principle r = multiplier - r = (1 + In/100) - n = # of compounds per year - I = interest rate per year n = # of compounds per year y = # of years

Questions about Rate Ice melts at 8g/hr. 12:00 there are 30g, when will block fully melt>

A ratio with two different units = rate - set up fraction = fraction - must have SAME UNITS in num and denom 1) 8g/1hr = 30g/X --->4/1 = 15/X -> X= 15/4 or 3 3/4 - melted at 3:45

Divisibility by 9 1) 1296 2) 3072

Add all the digits and see if they are divisible by 9 1) 1+2+9+6 = 18 --> YES 2) 3+7+2 =12 ---> NO

How to rebuild the dividend i.e. N is an integer than when divided by positive integer P gives Q18R7. When N is divided by (P+2) --> Q15R1. What is N?

D = S*Q +R N=(P)*18 + 7 --> 18P + 7 N=(P+2) *15 + 1 --> 15P + 31 18P + 7 = 15P + 31 --> 3P = 24 --> P=8 N = 15*(8) + 31 = 120+31 = 151

Most important rule of integers

DO NOT assume something is an integer if this is not explicitly stated - X is an integer --> 0, Positive, Negative - X is even/odd --> 0, Positive, Negative - X is prime --> Positive - Question only talks about remainder ---> all #s are positive integers

Percent Multipliers What is 55% of 400 240 is 30% of what # 56 is what % of 800 What is 75% of 280

Decimals act as the multiplier of a percent 1) .55 x 400 = 200 2) 240= .3 x A--> 240/.3 = 800 3) 56 = A% x 800 --> 56/800--> 7/100 --> .07--> 7% 4) Easiest to use the fraction because well known - 3/4 x 280/1 -->3*70 --> 210

Doubling and Halving 16x35

Divide 1 number by 2 Multiply 2nd number by 2 i.e: 16 x 35 --> 8*70 = 560

Prime Factorization: Definition

Every integer greater than 1 can be expressed as the product of primes i.e. 3x3 = Prime factorization of 9 2x5 = Prime Factorization of 10 4x3 --> 2x2x3 = Prime factorization of 12

Find which of the following numbers are a prime factor of 4680 = 2^3 x 3^2 x 5 x 13 25, 45, 65, 85, 120, 180

Every prime factor of r is included in the prime factorization of Q 25 = 5 x 5 --> NO 45 = 5 x 3 x 3 --> YES 65 = 5 x 13 --> YES 85 = 5x17 --> NO 120 = 2 x 60 --> 2x (6x10) --> 2x(2x3)x(5x2) --> YES 180 = 2 x 90--> 2 x (9x10)--> 2x(3x3)x(5x2) ---> YES -Shows which numbers are divisors of 4680

Absolute Values |-14| = |x| = Distance of x from 5 = Distance of x from -3

Gives the distance of a number from the origin |-14| = 14 |x| = distance of x from origin x from 5 = |x-5| if x = 7 --> |7-5| = 2 if x = 3 --> |3-5| = |-2| = 2 x from -3 = |x - (-3)| or |x + 3| if x = -4 --> |-4 + 3| = 1 if x = 4 ---> |4+3| = 7

Test if any number < 100 is Prime

If the # is not divisible by any prime less than 10 - 2, 3, 5, 7

Find smallest positive integer when divided by S gives R i.e. Smallest possible when divided by 12 give remainder 5

If the divisor is larger than the dividend, then the integer quotient is 0 Answer = 5 5/12 --> Q = 0 & R = 5

Divisibility by 12

If the number is divisible by both 3 and 4

Divisibility by 3 1) 135 2)431

If the sum of all of the digits is divisible by 3, then the whole number is 1) 1+3+5 =9 --> YES 2) 4 +3+1 =8 --> NO

GCF and LCM Formula

LCM = P x Q / GCF -ALWAYS CANCEL FIRST -Then perform multiplication

Square numbers ending in 5 35^2

Last 2 digits = 25 Find 2 10s digits surrounding number and multiply 10's digits together i.e: 30--35--40 -> 3*4 = 12 + 25 = 1225

Percent --> Decimal 42.5%

Move decimal 2 places left =.425

Decimal --> Percent 0.68 .456

Move decimal 2 places right =68% =45.6%

Dividing by 10 0.02 x 10 39.85 x 0.1

Move decimal place 1 to left - same as multiplying by 0.1 0.02 x 10 = 0.002 39.85 x 0.1 = 3.985

Multiplying by 10 .253 x 10 .00045 x 10

Move decimal to the right 1 .253 x 10 = 2.53 .00045 x 10 = .0045

Sum of 2 Squares a² + b² x² + 9

No way to factor this Only difference of 2 squares can be factored 1) = a² + b² 2) = x² + 9

Fraction Reciprocals 1/ (3/7)

One divided by any number including fractions equals the reciprocal of the denominator 7/3

Multiple v. Factor 91 and 7

Opposite meanings If P is a multiple of r, r is a factor of P i.e. 7 is a factor of 91/ 91 is divisible by 7 i.e. 91 is a multiple of 7 (some number * 7=91)

Fraction Reciprocals

Reciprocal of 3/5 = 5/3 > fraction reciprocals get larger Reciprocal of 5 = 1/5 --> numbers > 1 get smaller Reciprocal x Itself = 1

Comparing Fractions Same denominator w/ bigger numerator = Bigger denominator w/same numerator = Smaller denominator w/ bigger numerator = Bigger denominator w/ smaller numerator = Increase num + denom by same amount

Same denominator w/ bigger numerator = bigger # Bigger denominator w/same numerator = smaller # Smaller denominator w/ bigger numerator = bigger # Bigger denominator w/ smaller numerator = smaller # Increase num + denom by same amount = same fraction

Simple v. Compound Interest

Simple: Same payment each time for X% - doesn't count amount earned to determine payment - i.e 5% on 1,000 = 50 --> 5% added is 50 every time Compound: Different payment each time for X% - payment changes each time - 5% on 1,000 = 50 --> 5% on 1050 =52.5 - multiplier for compound = 1.05

| x + 37 | = 25 For two solutions x1 and x2 D = | x2 - x1 | What is D?

X is a distance of 25 in both directions from 37 25 + 25 = 50 = D or 37+ 25= 62 37-25 = 12 62-12 = 50 = D

Cross Multiplication a/b = c/d

a/b = c/d ---> ad = bc

Fraction Rules ab/cd a+b/c a+b/c+d

ab/cd = a/c *b/d a+b/c = a/c + b/c *Does not work for add/sub in denominator* a+b/c+d = (a/c+d) + (b/c+d)

Proportion Rules -as fractions a/b = c/d

cross multiplication: a/b = c/x --> ax = bc cancelling: in num/denom of same fraction -both nums of each fraction -both denoms of each fraction ***CANNOT Diagonallly cancel*** ***NO CROSS CANCELING in proportions***

Squaring other numbers 64^2

n^2 - n - (n-1) 65^2 (4225) - 65- 64 = 4096

QC Strategy: Matching Operations i.e: x +10 vs. 3x+3 (x > 3)

x + 10 //-3 3x+ 3 //- 3 x + 7 // -x 3x//-x 7// /2 2x// /2 3.5 vs. x ---> answer is unknown

Factoring Rational Expressions x² + 4x- 21 / x² - 6x + 9

x² + 4x- 21 / x² - 6x + 9 --> (x + 7) (x-3) / (x - 3) (x-3)---> (x+7)/(x-3)

Compound Interest: Example $4,000 with 6% interest compounding annually for 8 years. Total after 8 years =?

6% = 1.06 4000 (1.06 ^8) May not need to solve - just get formula

Even Powers x^10

Any even power of x is the square of another power x^10 = x^(5+5) = x^5*x^5 = (x^5)²

Rules of Remainders

D/S = Q + R/S D = dividend (what is being divided) S = divisor (The number dividing the dividend) Q = Quotient (Integer answer) R = Remainder (Fraction form

Decimals to a Power (0.03) ^ 3

0.03^3 = .03 x .03 x .03 -6 decimal places 3^3 = 27 0.000027

Percent of Increase/Decrease Item increased from 60 to 102. %? Item decreased from 250 to 200. %?

% = new price/old price 1) 102/60 = 1.7 = 70% increase 2) 200/250 = .8 = 20% decrease

Basic Even/Odd Number Facts -General -Even -Odd

-0 is even -positive and negative numbers can be even/odd -only INTEGERS can be even/odd EVENS: 2k -always divisible by 2 - prime factorization always includes 2 ODDS: (2k+1) or (2k-1) -never divisible by 2 -no factor of 2 in prime factorization

Prime Factorization of Perfect Squares i.e. 12^2=144

-All exponents must be even --> can help in finding unknown number with factorization shown - Any factor once in N will appear twice in N^2 144 = 12x12 --> (3x2x2)x(3x2x2) ---> 2^4 x 3^2

Dividing Decimals: look for other patterns .00013/.025

.00013/.025 --> 0.0013/0.25 (0.0013 x 4) / (0.25/4) = .0052/1 =0.0052

Advanced Decimal Factoring 1) 0.9991

0.9991 = 1-.0009 = 1² - (0.03)² --> (1-.003)(1+.003) = 0.97 * 1.03 * Expresses the decimal as the product of 2 decimals

Difference of 2 Squares a²-b² (x²-49)

1) (a+b) *(a-b) a² +ab -b² -ab --> a²-b² 2) (x+7)(x-7)

Non-Integer Divisors i.e: 12/8

1) 1 R 4 = 8 goes into 12 once with 4 remaining 2) Fractions: 12/8 --> 3/2 = 1 1/2

2 Important Prime # rules

1) 1 is not a prime number 2) 2 is the only even prime number

Finding number of factors in a perfect square i.e. 144

1) 144 --> 2^4 x 3^2 2) 4, 2 3) 5, 3 4) 15 factors of 144 *Perfect square always has an odd number of factors - only integer with odd number of factors

Advanced Numerical Factoring 1) Prime factorization of 1599 2) Prime factorization of 2491

1) 1599 = 1600-1 = (40²)-(1²) ---> (40+1)(40-1) --> (41)(39) ---> (41)(3*13) 41, 13, 3 = 3 prime factors of 1599 2) 2491 = 2500 - 9 --> (50²)-(3²) --> (50-3)(50+3) --> 47 and 53 = prime factors

Find the Least Common Multiple i.e. LCM of 24 and 32

1) Find prime factorization of each - 24 --> 8x3--> 2^3 x 3 - 32 --> 4x8 ---> 2^5 2) Find GCF - 2^3 --> 8 3) Rewrite each number as GCF x a Factor - 24 = 3x8 - 32 = 4x8 4) Take the product of the 3 factors for the LCM - 3 x 4 x 8 = 96 = LCM

Percent Increases Y increased by 30% 800 increased by 20% =? A 30% increase is 78. What was original?

1) Multiplier = Y whole + increase --> 1+.3 = 1.3 2) 800 * 1.2 = 960 - 10% of 800 = 80 *2 = 160 3) X * 1.3 = 78 --> 78/1.3 = 780/13 = 60

QC Strategy: Estimation 3/7 + 2/5 vs. 13/27 + 41/97

13/27 > 13/28> 12/28 = 3/7 41/97 > 41/100 > 40/100 = 2/5 B is greater

Dividing by decimals 2.35 / .01

2.35/.01 = 2.35 x 100 = 235

QC Strategy: Estimation 32.8% of 5929 vs. 41.6% of 5041

32.8 < 33.33% = 1/3 5929 < 6000 1/3 of 6000 = 2000 (answer slightly smaller than 2000) 41.6 > 40% 5041> 5000 2/5 of 5000 = 2000 (slightly larger) B is larger

Ratios 3 boys to 4 girls --> 3:4 Class of 5:8 boys to girls. 40 girls then boys =?

3:4 = 3/4 5/8 = X/40 1) Cancel ---> 5/1 = X/5 2) Cross multiply --> X = 25

Ratios with Scale Factor Boys to girls is 3:7. If there are 32 more girls than boys how many boys are there

3:7 --> 3n and 7n 1) Find difference of n = 7n-3n = 4n = 32 2)Find n --> 32/4 = 8 = n 3) Plug in n --> 3 * 8 = 34

Factors

A*B = C A and B are factors of C - they can be multiplied by another integer to result in C -1 is a factor of every integer - every integer is a factor of itself

Factor Pairs I.e. How many factors are there of 42

Any number below 100 you can list all the factor pairs for and find the # of factors 1 - 42 2 - 21 3 - 14 6 - 7 8 Factors of 42

QC Strategy: Matching Operations i.e: 35/8 vs. 13/3

Can add/subtract/multiply/divide by same amount on both sides 32/8 = 4 & 12/3 = 4 35/8 - 4 --> 35/8 -32/8 = 3/8 13/3 -4 --> 13/3 - 12/3 = 1/3 3/8 --> 9/24 1/3 --> 8/24 A is bigger

Fraction Multiplication 5/14 * 7/15

Cancel numerators with denominators -i.e. vertical and diagonal cancellation only Cannot cancel num w/ num or denom w/ denom 5/14 * 7/15 --> 1/14 * 7/3 1/14* 7/3 --> 1/2 * 1/3 = 1/6

Properties of 0 -division -0 product

Cannot divide by 0 - 0/9 = 0 vs. 9/0 = not possible If A * B = 0 then A or B is 0

3+ Ratios Problems Make concrete with 1:2:3 ratio of cement, sand, gravel. 150 kgs of sand allow for how much cement

Concrete: 6 parts = whole Sand: concrete = 2:6 = 1:3 1/3 = 150/x = 450 kgs of concrete

FOIL Method (2x + y) (x + 2y)

First, Outside, Inside, Last 2x^2 + 4xy + yx + 2y^2 ---> 2x^2 + 5xy + 2y^2

Divisibility by 4 1) 126,739,096 2)123,693,055

If last 2 digits form a number divisible by 4 then entire number is divisible by 4 1) 96/4= 24 --> YES 2) 55/4 ---> NO

Divisibility by 6

If the number is divisible by both 2 and 3

Multiplying/Dividing by Powers of 10 350 x 1000 1235/ 100 .064 x 10^-2

Multiply - Move decimal to the right equal to the number of factors of 10 350 x 1000 = 350,000 Divide - move decimal left equal to factors of 10 1235/100 = 12.35 Multiply by decimal/neg power - move decimal left equal to factors of 10 .064 x 10^-2 = .064 x .001 = .00064

Dividing by 5 625/5

Multiply by 2 Divide by 10 i.e. 625 x 2 = 1250/10 = 125

Finding Greatest Common Factor i.e. GFC of 360 and 800

The largest factor that 2 numbers share 1) Find prime factorization of each number - 360 --> 2^3 x 3^2 x 5 -800 --> 2^5 x 5^2 2) Find the highest factor in common of each integer - 3 factors of 2/ 0 factors of 3/ 1 factor of 5 - 2^3 / 3^0 / 5^1 3) Multiply common factors together - 2^3 x 5 = 8x5 = 40 GFC = 40