Quick Review. Free Book.. Not for sale!

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1 Quick Review 2013 Free Book.. Not for sale!

2 GRE Math Quick Review Percentages /100 or 1% /10 or 10% /5 or 20% /5 or 40% /2 or 50% /5 or 80% /4 or 25% /4 or 75% /3 or 33.3% /3 or 66.6% /1 or 100% /1 or 200% Percentages from fractions 1. 1/3 33.3% or.3(r) 2. 1/4 25% or /5 20% or /6 16.6% or 1.65 (r) 5. 1/8 12.5% or /3 66.6% or.6(r) 7. 2/5 40% or /5 60% or /8 37.5% /5 80% or /6 83.3% or.83(r) 12. 5/8 62.5% or /8 87.5% or.875

3 Percentages from fractions /4.75 3/10.3 3/ / / / /

4 Algebraic Concepts 1. (x/y)² = x²/y² 2. (x²)(x³) = x²+³ 3. (x²)(y²) = (xy)² 4. (x²)³ = x²*³ or x 6th 5. a² - b² = (a + b) (a - b) 6. Coefficient A number used in an algebraic term (3x where 3 is the C) 7. Equations Two expressions shown to be = (3x -3= 2y -2) 8. Expression One or more terms with coefficients and variables (3x² + 2y) 9. FOIL a way to multiply two expressions in Parenthesis. (First, Outer, Inner, Last) 10. Functions the variable f(x)= identifies a function and defines x 11. If both sides of an inequality are multiplied or divided by the same constant (#) If x does not = 0, then x to the 0 power = 1 the direction of the inequality is preserved if the constant is positive, but reversed if the constant in negative 13. PEMDAS (order of operations) Please Excuse My Dear Aunt Sally (Parenthesis, Exponents, Multiply, Dividie, Add, Subract) 14. quadratic equation = a² + bx +c = 0 (solve for x with x= -b +- b² - 4ac/2a) or factor out and solve for both expressions 15. Term a combination of a term and variables(s) (3x³) 16. When the same constant (#) is added or subracted from both sides of an inequality... the direction of the inequality is preserved 17. x-² = 1/x² 18. x²/x³ = x (²-³)

5 Number Theory and Algebra < xⁿ < 0 if -1 < x < 0, and n is a positive odd integer. The greater the exponent, the greater the value of xⁿ j Given the (j+2)(j-3), compute the "O" of FOIL method 3. (2) (-3) Given the (j+2)(j-3), compute the "L" of FOIL method 4. (a + b)ⁿ = aⁿ + bⁿ 5. (a b)ⁿ = aⁿ bⁿ False True 6. (a²)ⁿ = a²ⁿ True 7. (4 2 a³) 2ax (2a) 8. + y = (x+y) False 9. x equals x raised to the power of ½ < xⁿ < 1 if 0 < x < 1, and n is positive integer. The greater the value of n,the lesser the value of xⁿ < xⁿ < 1 if -1 < x < 0, and n is a positive even integer. The greater the exponent, the lesser the value of xⁿ j Given the (j+2)(j-3), compute the "I" of FOIL method 13. a + c > b + d If you have a>b and c>d, what will be true? 14. a > c If you have a>b and b>c, then you can infer a+c > b+c If you have a>b and you add a positive value to both sides, you get a b Given x² -x -6 and their factors (x+a) (x+b), - 6 is equal to a t < b t Given a > b and t < 0. what would happen if you multiply each side of the inequality by t 18. a t > b t Given a > b and t > 0. what would happen if you multiply each side of the inequality by t 19. a ⁿ equals 1 / aⁿ 28. j j Given the (j+2)(j-3), compute the "F" of FOIL method 29. j² -3j +2j -6 Given the (j+2)(j-3), compute the FOIL value. 30. negative number Multiplication-Division involving an odd number of negative terms results in a Number of factors of zero Infinite 32. Odd integer Even integer ± odd integer = 33. Odd integer Odd integer odd integer = 34. positive number Multiplication-Division involving an even number of negative terms results in a Stealth quadratic equations - type Stealth quadratic equations - type The greatest factor of a positive integer (other than itself) 38. unresolvable linear equation - type unresolvable linear equation - type 2 The same variable appears inside and outside the radical, x = 5x The same variable appears in the denominator and also appears elsewhere. 2 / x = 3 - x (the number) 2 If a linear equation (only one variable) can be reduced to 0=0 You have two equations with two variables, but they are actually the same equation. 40. xⁿ < 1 if x < -1, and n is a positive odd integer 41. xⁿ > 1if x < -1, and n is a positive even integer 42. xⁿ >1 if x >1, and n is positive integer 20. a⁰ equals a² aⁿ = a² ⁿ True 22. a² aⁿ = a²+ⁿ True 23. b and c Given ax² + bx + c, which terms can be zero. 24. Even integer Even integer ± even integer = 25. Even integer Odd integer ± odd integer = 26. Even integer Even Integer non-zero integer = 27. Inequality is reversed, a < b a > b, then you multiply both sides by -1, what would happen?

6 Geometry 1. Area of a Circle Area=πr² 2. Area of a Sector Sector=πr² * x/360º 3. Circumference of a Circle Circumference=2πr or (πd) 4. Length of an Arc Arc=2πr * x/360º 5. Surface Area of a Cube Surface Area=6s² 6. Surface Area of a Cylinder Surface Area=2πrh + 2(πr²) 7. Surface Area of a rectangle Surface Area= Sum of the area of its Faces 8. The Diagonal of a Cube Diagonal=s 3 9. The Diagonal of a Rectangular Solid Diagonal= L² + W² + H² 10. Volume of a Cube Volume=s³ 11. Volume of a Cylinder Volume=πr²h 12. Volume of a Rectangle Volume=Length Width Height Geometry Formulas Triangle: x: x squared-root 3: 2x Triangle: x: x: x squared-root 2 3. Area of a circle: A=(Pi)r(squared) 4. Area of a parallelogram: A=bh 5. Area of a rectangle: A=lw 6. Area of a Sector: AS=(n/360)(Pi)r(squared) 7. Area of a square: A= (side) squared 8. Area of a trapezoid: A=1/2(b1+b2)h 9. Area of a triangle: A=1/2bh 10. Circumference of a circle: C=2(Pi)r or (Pi)d 11. Length of an Arc: LA=(n/360)2(Pi)r 12. Perimeter of a rectangle: P=2(l+w) 13. Pythagorean Theorem: a(squared) + b(squared) = c(squared) 14. Right Triangle Leg-to-leg Ratios: 3:4:5 and 5:12: Surface area of a Cylinder: SA=2(Pi)r(squared)+2(Pi)r(h) 16. Surface area of a Rectangular Solid/prism: SA=2(lw+lh+wh) 17. Volume of a Cube: V=e (cubed) 18. Volume of a Cylinder: V=(Pi)r(squared)h 19. Volume of a Rectangular Solid: V=lwh

7 Exponents 1. (x/y)^n = x^n/y^n 2. (x^m)^n = x^nm 3. 2^ ^ ^ ^ ^ ^ ^ ^ ^ ⁴ ^ ^ ³ ^ If -1<x<0, then 1/x is < If 0<x<1, then 1/x is >1 19. If x<-1, then 1/x -1<1/x<0 20. If x>1, then 1/x is 0<1/x<1 21. prime factorization of 12 (2)(2)(3)=2²(3) 22. prime factorization of 14 (2)(7) 23. prime factorization of 81 (3)(3)(3)(3)=3⁴ 24. prime factorization of 338 (2)(13)(13)=2(13)² 25. prime factorization of 800 (2)(2)(2)(2)(2)(5)(5)=(2)⁵(5)² 26. prime factorization of 1155 (3)(5)(7) 27. x^-n = 1/x^n 28. x^0 = x^m * x^n = x^m+n 30. x^m/x^n = x^m-n or 1/x^n-m

8 Hard Math Concepts 1. Factor this polynomial: a^2-2ab+b^2=: (a-b)^2 2. Factor this polynomial: a^2-b^2=: (a-b)(a+b) 3. Factor this polynomial: a^2+2ab+b^2=: (a+b)^2 4. Factor this polynomial: ab+ac=: a(b+c) 5. Ho do you count the number of possibilities?: Usually just systematically write down each possibility and count them. 6. How do you count consecutive integers? ex: How many integers are there from 73 through 419, inclusive?: The number of integers from A to B inclusive is B-A How do you determine a combined ratio? ex: Te ratio of a to be is 7:3. The ratio of b to c is 2:5. What is the ratio of a to c?: Multiply one or both ratios by whatever you need in order to get the terms they have in common to match. 8. How do you find a common factor?. ex: What factors greater than 1 do 130 and 225 have in common?: Break both numbers down to their prime factors to see what they have in common. Then multiply the shared prime factors to find all common factors. 9. How do you find a common multiple? ex: What is the least common multiple of 28 and 42?: Find all of the factors of each number. Then multiply one number by the non-common factors in the other number. 10. How do you find one angle or the sum of all the angles of a regular polygon?: if n= number of sides, sum of interior angles=(n-2)x180. divide by n to find the measure of one of the interior angles. 11. How do you find the area of a triangle?: 1/2 b*h even if it's not a right triangle 12. How do you find the average of consecutive integers?: It's the average of the smallest number and the largest number. 13. How do you find the sum of consecutive numbers? ex: What is the sum of integers from 10 through 50, inclusive?: Sum=(Average)x(Number of terms) 14. How do you handle a fractional power? ex: 4^(1/2): They relate to roots. e.g. sqrt How do you handle a linear equation? What is the formula?: y=mx+b where: m= the slope of the line=rise/run and b=the y-intercept 16. How do you handle the graph of a function, like a parablola? If you're asked to pick which formula describes the graph.: Pick out obvious points on the graph, plug these values into the answer choices, and eliminate answer choices that don't jibe with those values until only one answer choice is left. 17. How do you multiply and divide powers?: Add/subtract the exponents 18. How do you multiply and divide roots? ex: (2sqrt3)(7sqrt5): Deal with what's inside and outside separately. eg (27)(sqrt[35]) 19. How do you raise a power to a power to an exponent? ex: (x^a)^b=: Multiply the exponents. x^(a*b) 20. How do you solve a combination problem? eg the arrangement does not matter. ex: How many different ways are there to choose 3 delegates from 8 possible candidates?: C=n!/k!(n-k)! where n=# in larger group and k=number you're choosing 21. How do you solve a function problem? ex: What is the minimum value of the function f(x)=x^2-1: Plug numbers in for x to get various values/outputs. 22. How do you solve a group problem with either/or categories?: Organize the information into a grid. 23. How do you solve a permutation problem? e.g. the arrangement does matter. ex: There are five runners in a race. How many possible outcomes for gold, silver, and bronze medal winners are there?: P=n!/(n-k)! where n=# in larger group and k=number you're arranging 24. How do you solve quadratic equations?: Forget the formula. Solve until it =0 then break it into TWO simple formulas. 25. In a diagram with transversal across parallel lines, are the acute angles equal? Are the obtuse angles equal?: Yes and yes 26. What are the properties of an isosceles triangle?: Isosceles triangles have two equal sides and two equal angles. 27. What are the properties of similar triangles?: Corresponding angles are equal and corresponding sides are proportional. 28. What are the proportions of the sides of a triangle?: 1-2-sqrt(3) 29. What are the proportions of the sides of a triangle?: 1-1-sqrt(2) 30. What are two common triangle side proportions?: and What do you when you're solving for an inequality and you multiply or divide by a negative quantity?: Reverse the inequality sign. 32. What does a number raised to a negative exponent equal? ex: 5^-3: The reciprocal of that number raised to the exponent. eg. 1/(5^3) 33. What does zero raised to any nonzero exponent equal?: zero 34. What is a factorial? n!: The PRODUCT of all the integers from 1 to n. 4!=4x3x2x1=24. 0!=1!= What is the formula for the volume of a sphere?: (4/3)pi*r^3 36. What is the formula of a simple interest problem? ex: If $12,000 is invested at 6 percent simple annual interest, how much interest is earned after 9 months?: interest=(principal)x(interest rate as a decimal)x(time in years)

9 Hard Math Concepts 37. What is the formula to solve a group problem involving both/neither? ex: of the 120 students at school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither. How many are studying both?: Group 1 + Group 2 + Neither - Both = Total 38. What is the formula to solve a work problem? ex: If it takes Joe 4 hours to paoint a room and Pete twice as long to paint the same room, how long would it take the two of them, working together to paint hte same room, if each of them works at his respective individual rate?: 1/(Joe's hours to complete the room)+1/(pete's hours) = 1/t 39. When can you add and subtract, roots?: You can add/subtract only when the parts inside are the same Square Roots : 1: : : : : : : : : : : : : : If a>1, then 1/a? than 1: smaller 15. If 0<a<1, a? than a: bigger 16. If a>0 and a<1, then 1/ a? than 1: bigger 17. If a>1, a? than a: bigger

10 Math Formulas 1. Arc length 2. Area of a circle A=pi*(r^2) 3. Area of a rectangle A=l*w 4. Area of a sector X = Measure of interior angle Circumference of circle 360 X = Measure of interior angle Area of Circle Area of a trapezoid A= (1/2)h*(a+b) where a is the length of the bottom base and b is the length of the top base. 6. Area of a triangle A= (1/2)b*h 7. Area of parallelogram A=b*h 8. Circumference of a Circle 9. Convert 12.5% to a fraction 10. Convert 16.66% to a fraction 11. Convert 20% to a fraction 12. Convert 25% to a fraction 13. Convert 33.33% to a fraction 14. Convert 40% to a fraction 15. Convert 60% to a fraction 16. Convert 66.66% to a fraction 17. Convert 75% to a fraction 18. Convert 80% to a fraction 19. Convert 83.33% to a fraction 20. Find distance when given time and rate 21. Find hypotenuse of a right triangle given 2 side lengths c=2pir OR d*pi 1/8 1/6 1/5 1/4 1/3 2/5 3/5 2/3 3/4 4/5 5/6 d=rt so r= d/t and t=d/r Pythagorean Theorem: h^2= (S1)^2 + (S2)^2 22. First 10 prime #s 2, 3, 5, 7, 11, 13, 17, 19, 23, How to find the average of consecutive #s 24. How to find the sum of consecutive #s 25. How to recognize a # as a multiple of How to recognize a # as a multiple of How to recognize a # as a multiple of How to recognize a multiple of How to recognize if a # is a multiple of Perfect Squares Perimeter of a rectangle 32. Quadratic Formula 33. Side lengths of a right triangle 34. Side lengths of a right triangle Take the average of the largest and smallest # in the set. (i.e. for the set of consecutive #s would be (172+5)/(2)= 88.5) Sum= (Average of Consecutive #s) * (# of terms in set) The sum of the digits is a multiple of 3 The last 2 digits are a multiple of 4. (i.e is a multiple of 4, so 144 must also be a multiple of 4.) The sum of the digits is a multiple of 9. Sum of digits is a multiple of 3 and the last digit is even. 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.) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 P= 2L + 2w X= -b (+/-) Sqrroot [(b^2) -4ac)] 1-sqrroot of sqrroot of 2 2a

11 35. Slope given 2 points m= (Y1-Y2)/(X1-X2) 36. Surface area of a rectangular solid SA= 2( Lw + Lh + w*h) 37. Surface area of a right circular cylinder 2(pi(r^2))+ 2pirh 38. Surface area of a sphere SA= 4pi(r^3) 39. Volume of a rectangular box V=Lwh 40. Volume of a right circular cylinder pi(r^2)h 41. Volume of a sphere V=(4/3)pi(r^3) 42. When asked to find the distance between 2 points on a graph use this formula When dividing exponential #s with the same base, you do this to the exponents When multiplying exponential #s with the same base, you do this to the exponents... Distance formula. Distance= Sqrareroot[((Xa-Xb)^2) + ((Ya-Yb)^2)] Subtract them. i.e (5^7)/(5^3)= 5^4 Add them. i.e. (5^7) * (5^3) = 5^ When solving an inequality, flip the sign when you... divide or multiply both sides by a NEGATIVE number

12 Math Concepts Review # cubed to the 4th to the 5th to the 6th cubed to the 4th cubed cubed squared squared squared squared squared squared squared acute angle angle is greater than 0 but less than 90 degrees 17. Area of a circle A=pi(r^2) 18. Area of a rectangle 19. Associative Property A=lw (a+b)+c=a+(b+c) 20. Average sum of all the numbers in the set divided by the number of elements in the set. 21. Circumference C=pi(d) or 2(pi)r 31. Length of an arc Arc=(x/360)x2(pi)r 32. Median the middle value in a set containing odd numbers; a set with even numbers it is the average of the two middle terms. 33. Mode the value that appears the most often in the set 34. odd + odd even 35. odd x odd odd 36. Perimeter of a rectangle P=2l+2w 37. pi 3.14 or 22/7 38. prime numbers 39. square root of square root of square root of supplementary angles 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39, 41, 43, two angles that form a straight line 43. Surface area S=6a^2: sum of the areas of all faces 44. vertical angles Happens when two lines intersect, vertical angles are equal 45. Volume V=lwh 22. complementary angles 23. Differences of Squares 24. distributive property two angles form a 90 degree angle a^2 - b^2 = (a+b) (a-b) ab+ac=a(b+c) OR ab-ac=a(b-c) 25. even + even even 26. even + odd odd 27. even x even even 28. even x odd even 29. Fractions or decimals less than 1 raised to a power 30. LCD - lowest common denominator will result in a number smaller than the original number Cross out any prime factor in the second denominator that already appears in the first denominator. then multiply the rest.

13 Math Concepts Review # /3 33 1/2% 2. 1/6 16 2/3% 3. 1/8 12 1/2% 4. 2/3 66 2/3% 5. 3/8 37 1/2% 6. 5/6 83 1/3% 28. Triangle 3 : Triangle 5 : 12 : Triangle x : x*sqrt(3) : 2x 31. Triangle x : x : x*sqrt(2) 32. x : x : x*sqrt(2) Triangle x : x*sqrt(3) : 2x Triangle /8 62 1/2% 8. 7/8 87 1/2% 9. Divisible by Divisible by Divisible by 4 if last digit is even if the sum of digits is a multiple of 3 if last 2 digits are divisible by Divisible by Divisible by Divisible by Divisible by Even * Even 17. Even * Odd 18. Even ^ Positive 19. Even +- Even 20. Negative # ^ Even 21. Negative # ^ odd if last digit is even & sum of digits is a multiple of 3 (rules of 2 & 3) if last 3 digits are dividible by 8 if sum of digits are divisible by 9 if sum of digits are a multiple of 3 & last 2 digits are divisible by 4 (rules of (3 & 4) Even Even Even Even 22. Odd * Odd Odd Positive # Negative # 23. Odd ^ Positive 24. Odd +- Even 25. Odd +- Odd Odd Odd Even 26. Triangle 5 : 12 : Triangle 3 : 4 : 5

14 MATH REVIEW x 10³: (move decimal 3 places to the right) = (¹/₁₆)² =: ¹/₂₅₆, A decimal or fraction less than 1, raised to a power will yield a # smaller than the original 3. (x/y) (y/x²) =: **Square the entire expression to get rid of the radical! = (x²/y²) (y/x²) = 1/y² Turn it back into a radical, (1/y²) = 1/ y 4. # is divisible by 2 if...: the number is even 5. # is divisible by 3 if...: the sum of its digits are divisible by 3 6. # is divisible by 4 if...: the last two digits are divisible by 4 7. # is divisible by 5 if...: it ends in 5 or 0 8. # is divisible by 6 if...: it is divisible by 2 and 3 9. # is divisible by 9 if...: the sum of its digits are divisible by : : : ²: ³: ⁴: ¹⁰: ²: ³: ⁴: ⁴: ⁵: ⁶: steps to solve an ALGEBRAIC INEQUALITY:: 1) multiply to get rid of fractions 2) isolate "x" by adding/subtracting 3) multiply by -1 to get positive "x" which will change direction of inequality ( < to >) steps to solve QUADRATIC EQUATION:: always has TWO answers 1) set it equal to "0" 2) factor it out 3) find values that make each factor = "0" 25. 3²: ³: ⁴: ²: ³: ⁴: ²: ³: ⁴: steps to solve for VARIABLE IN TERMS OF ANOTHER:: goal to isolate variable alone on one side 1) Cross-multiply to clear denominators 2) Remove () by distributing 3) Put variables on one side 4) Factor out the variable 5) Divide to get the variable alone 6) If necessary multiply by /2.5 =: (move decimals) 62.5/25 = ²: ³: ²: ³: ²: ³: ²: ³: ²: ³: ²: ³: ²: ³: is 33¹/₃% of what number?: 13 = 1/3X X = ²: ²: ²: ²: ²: ²: ²: ²: ²: ²: ²: ²: ²: ²: x 0.3 =: x 10 ⁴: (move decimal 4 places to the left) = A building has 2/5 of its floors below ground. What is the ratio of the # of floors above ground to the # below ground?: Pick Numbers! Pick a value for the total # of floors, one that is divisible by 2 and 5. Let's pick 10. 2/5 x 10 = 4 floors below ground & 6 above. Therefore, the ratio is 6:4 or 3:2

15 MATH REVIEW 68. A line drawn tangent to a circle is: to the radius at the point of tangency 69. A negative # raised to an even power: a positive number Ex: (-3)² = A negative # raised to an odd power: a negative number Ex: (-3)³ = A triangle inscribed in a semicircle such that one side of the triangle coincides with the diameter of the semicircle is a?: Right triangle 72. Acute angles =: < Add & Subtract Roots: You can add/subtract roots only when the parts inside the radical are identical = = cannot be combined! = Al plants a straight row of trees with 5 meters b/w each tree. If 15 trees are planted, what is the # of meters b/w the 1st tree and the last?: Fence-Post Problem Distance-(# of posts-1) x (distance b/w posts) (15-1)(5) = 70 meters* 75. An inscribed angle will always make an arc that is...: twice the size of the degree of the inscribed angle 76. Anything raised to the zero power =?: 1 Ex: 4⁰ = Arithmetic Sequence =?: An = A₁+(n-1)d (the difference between each term and the next is constant) 78. CHORD: a straight line connecting two points on a circle... can never be longer than the diameter 79. Circumscribed shapes are...: outside another shape and all the vertices touch the edge of the inside shape 80. Compare 5/7 and 9/11: Quick Tip: Cross-multiply! 5x11>9x7 81. Complementary angles =: Compound Interest Rate Formula: A=P (1+ (r/n))^nt A-Final Amount P-Principle r-rate of interest expressed as a decimal n-# times compounded annually 83. Consider the sequence 1,4,2,8,5,7,1,4,2,8,5,7...what is the 500th term of the sequence?: When a group of k numbers repeats itself, to find the nth number, divide n by k and take the remainder r. The rth term and the nth term are always the same. So n=500, & k=6 (6 #s repeat) *500/6 = 83.3 *Ignore the decimal, the quotient is 83, since 83x6=498, the remainder is =2 *The 500th term = 2nd term (rth term) = Customers can choose among 5 different flavors & either a sugar or waffle cone. How many distinct triple scoop cones w/ 3 different flavors are available? (scoop arrangement doesn't matter): C=n!/r!(n-r)!, n=# in the larger group r=number you're choosing 5!/3!(5-3)! = 10 x 2 (types of cones) = Diagonal Length of a Cube is...?: s Diagonal Length of a Rectangular Solid: d= ( l² + w² + h²) 87. Distance Problems: Distance = rate x time 88. Dividing Decimals: move decimals point in the divisor (outside #) until the # is a whole number, move the decimals point the same # of places in the dividend (inside #) 89. Dividing Fractions: Invert the 2nd fraction and multiply 90. even# + even# =: even# 91. even# + odd# =: odd# 92. even# x even# =: even# 93. even# x odd# =: even# 94. Factoring Trinomials ex: a²-5a+4 =: (a-4)(a-1) *The product of the 2 missing terms will be the last term in the polynomial & the sum of the 2 terms will be the middle term 95. Find the dimensions or area of an Inscribed or Circumscribed Figure: *remember a circle's diameter is also the squares diagonal which is n 2 (diagonal creates two triangles) thus the circumference = n(π) Find the fraction: 12.5%: 1/8 97. Find the fraction: 16²/₃%: 1/6 98. Find the fraction: 30%: 3/ Find the fraction: 33¹/₃%: 1/ Find the fraction: 37.5%: 3/ Find the fraction: 40%: 2/ Find the fraction: 60%: 3/ Find the fraction: 62.5%: 5/ Find the fraction: 66²/₃%: 2/ Find the fraction: 80%: 4/ Find the fraction: 83¹/₃%: 5/ Find the fraction: 87.5%: 7/ Find the median of a set of numbers: FIRST put the numbers in increasing order! Then find the middle number (for even sets of #s, average the middle 2 numbers) 109. Find the percent: 1/3: 33¹/₃% 110. Find the percent: 1/5: 20% 111. Find the percent: 1/6: 16²/₃% 112. Find the percent: 1/8: 12.5% 113. Find the percent: 1/20: 5% 114. For 0<x<1, Which is larger; ³ x OR x?: ³ x, X will always be smaller than it's square root, and the cube root will be less than the square root ex: ¼ = ½ 115. For a given area, the rectangle with the smallest perimeter is a...: Square 116. For any numbers a and b, with a b, a b=(a+b)/(a-b); What is 25 15?: = 25+15/25-15 = 40/10 = 4

16 MATH REVIEW 117. Geometric Sequence =?: Gn= G₁r ⁿ ¹ (the ratio between one term and the next is constant, not the actual difference. Ex: 3, 9, 27, 81) 118. How do you change a decimal to a percent?: Multiply by 100 (move decimal 2 places to the right) ex: 0.17 = 17% 119. How do you change a percent to a decimal?: Remove the percent symbol and divide by 100 (move decimal 2 places to the left) ex: 0.8% = How do you compare fractions? Ex: Which fraction is bigger, A) 1/3 or B) 3/8?: Cross-Multiply! Multiply numerator of A with denominator of B & put this number in the A column, multiply numerator of B with denominator of A, & put this in B column, then COMPARE! 1x8 = 3x3 *8 < 9, so B is bigger 121. How do you convert a percent to a fraction?: Put the given percent over 100, remove the percent symbol (%) and simplify if needed How do you count consecutive integers? ex: How many integers are there from 73 through 419, inclusive?: The number of integers from A to B inclusive is B-A How do you find the area of a sector of a circle?: A sector is a wedge bound by 2 radii and an arc of the circle. Area=(x/360 )(πr²), where x=the measure of the central angle 124. How do you find the average (mean) of a group of numbers?: Average = sum of values/number of values 125. How do you find the length of an arc of a circle?: An arc is part of the circumference. Length of the arc = (x/360 )(2πr), where x=the measure of the central angle 126. How do you find the remainder when one integer is divided by another?: *Use long division *Ignore everything after the decimal portion *Multiply quotient by the divisor (outside #) *Subtract from dividend to get the remainder 127. How do you find the slope of a line?: Slope = y₂-y₁/x₂-x₁ If you are given an equation, y=mx+b, you can solve for the slope (m) but you must first set = y 128. How many 3-digit numbers can be written using only odd digits?: Counting Principle There are 3 jobs: selecting the hundreds, tens, & ones digit. Each job can be done in 5 ways (since you can only choose 1,3,5,7,9 for each digit) So, 5x5x5=125 ways 129. How to find the average of consecutive #s: Take the average of the largest and smallest # in the set. (i.e. for the set of consecutive #s would be (172+5)/(2)= 88.5) 130. How to find the sum of a group of numbers: Sum of values = average value x number of values 131. How to solve linear equations?: y=mx+b; where m= the slope of the line=rise/run and b= the y-intercept Solve by isolating a variable! 132. How to solve quadratic equations?: Set equation equal to zero, ax²+bx+c = 0 Solve, using one of two methods: 1) Factor and then solve 2) Use the quadratic formula: x=[-b±( (b²-4ac)]/2a You will end up with two answers 133. How to tell difference between a permutation problem and a combination problem: Permutation problems involve order, combination problems do not care about order If "a" is inversely proportional to "b" and b=5 when a=3, what is the value of "b" when a=10?: By definition, "a" is inversely proportional to "b" if the product "ab" is a constant. 1) 3x5 = 15, the constant is 15 2) 10 x b =10b 3) 10b=15 *b=³/₅* 135. If 2 students are chosen at random from a class with 5 girls and 5 boys, what's the probability that both students chosen will be girls?: *Probability that the 1st student chosen will be a girl = 5/10 *Probability that the 2nd student chosen will be a girl = 4/9 *Probability that both students chosen will be girls = 1/2 x 4/9 = 2/ If ³/₅ of the pets in a store are dogs, and ¼ of the other pets are cats, what fraction of the pets is neither cats nor dogs?: 1) Choose a total # of pets, lets say 20. 2) Then ³/₅ of 20 =12, so there are 12 dogs. 3) Of the remaining 8 pets, 2 (¼ of 8) are cats. 4) So 6 pets are neither dogs nor cats: 6/20=3/ If a class of 10 boys & 15 girls took a test & the average grade of the boys was 80, and for the girls was 90, what was the class average?: Calculate the weighted averages! (10x80) + (15x90) / (10+15) = 86 *b/c the class is 40% boys & 60% girls, calculate the average as: (.4x80)+(.6x90)= If a coin is flipped 4 times, what is the # of possible outcomes?: If one task has n possible outcomes, and another has m, then the joint occurrence of the 2 tasks has (nxm) possible outcomes. 2x2x2x2 = If a fair coin is tossed 4 times, what's the probability that at least 3 of the 4 tosses will come up heads?: *2 possible outcomes/toss x 4 tosses = 16 possible outcomes *5 possible desired outcomes Probability = 5/ If a polygon has n sides, what is the sum of the measures of its n exterior angles, taking one angle at each vertex?: = If a=bc, which of the following is equal to b/c? A) a/bc B) ab/b C) a/c D) a/c²: Pick three easy numbers to plug in: a=6, b=2, c=3 So, b/c=2/3. Then test the answers to see which one gives you 2/3. Only choice D works.

17 MATH REVIEW 142. If the measures of the two acute angles of a right triangle are in the ratio of 5:13, what is the measure of the larger angle?: 1) Let the measures of the 2 angles=5x & 13x 2) Remember the sum of all 3 angles = 180 3) 5x+13x+90=180 4) x=5, so the larger acute angle is 13x5= IF you have 12 shirts in a drawer and 9 of them are white, what is the probability of picking a white shirt at random?: Probability = 9 white shirts/12 shirts total = 3/ Important Binomial Products: (a+b)(a-b) = a²-b² (a+b)² = a²+2ab+b² (a-b)² = a²-2ab+b² 145. In a triangle: if you know the length of the longer leg...: To find shorter leg: divide by 3 To find the hypotenuse: multiply the length of the shorter leg by In a triangle: if you know the length of the shorter leg...: To find longer leg: multiply by 3 To find the hypotenuse: multiply by In a triangle: if you know the length of the hypotenuse...: To find shorter leg: divide by 2 To find longer leg: multiply the shorter leg by In a right triangle, solve for the legs by...: dividing the hypotenuse by In a right triangle: solve for the hypotenuse by...: multiplying the length of a leg by 2 Ex: In an isosceles right triangle, what is the hypotenuse length if the legs are 4? = 4( 2) 150. In a classroom of 30 students, the ratio of the boys in the class to students in the class is 2:5. How many boys?: Multiply the ratio by the total = 2 boys/5 students x 30 students = 12 boys 151. In any triangle...: *longest side is opposite largest angle *shortest side is opposite smallest angle *sides w/same length are opposite (=) angles *sum of lengths of any 2 sides is > 3rd length 152. in FRACTIONS with the SAME DENOMINATOR...: the larger numerator is the larger fraction 153. In FRACTIONS with the SAME NUMERATOR...: the smaller denominator is the larger fraction 154. Inscribed shapes are...: inside another shape and all the vertices touch the edge of the outside shape 155. Integer: Whole numbers and their opposits. {...-3,-2,- 1,0,1,2,3...} These are rational numbers. Negative, positive, include 0 (neither negative or positive) 156. Jay can finish the job alone in 5 hours, and Ray can finish in 4 hours. How long will it take them if they work together?: ¹/₅ + ¼ = ¹/t **t=2²/₉ hours 157. John travels 30 miles in 2 hrs and then 60 miles in 3 hrs. What is his average speed in miles/hr?: Average A per B = Total A/Total B = (30+60)miles/(2+3)hrs = 18mph 158. MULTI-STEP PROBABILITY problem=: (Prob of Event 1) x (Prob of Event 2) 159. Multiple Event Probability where each individual event must occur a certain way (2 independent events happening simultaneously): 1. Figure out the probability for each individual event 2. Multiply the individual probabilities together 160. Multiple Event Probability where individual events can have different types of outcomes: 1. Find total number of possible outcomes (find total for each individual event and multiply those together) 2. Find number of desired outcomes 161. Multiply & Divide Roots: To multiply/divide roots deal with what's inside the and outside the separately. (2 3) (7 5) = (2x7) ( 3x 5) = = (10/5) ( 21/ 3) = Multiplying Decimals: Multiply the #s and then move the decimal point the total # of places in the answer 163. Negative# * Positve Fraction=: a larger # 164. New Average (when a number is added or deleted) =?: Use sum of the terms of the old average to find the new sum New Average = New Sum / n* n* = # of terms being averaged 165. Obtuse angles =: 90 <x< odd# + odd# =: even# 167. odd# x odd# =: odd# 168. Original Average & New Average (to figure out what was deleted or added): number added = (new sum) - (original sum) number deleted = (original sum) - (new sum) 169. Percent Increase/Decrease Equations: % increase/decrease = (change/original amount) x 100% 170. Percent Problems Equation: Percent x Whole = Part 171. Permutation Equation **Figure out how many possible outcomes to arrange elements sequentially: npk =n! /(nk)! n= (# in the larger group) k= (# you're arranging) 172. Positive# * Positive Fraction =: a smaller # 173. Powers of 10: 10⁶ = 1,000, Powers of 10: Prime Numbers: Numbers that can only divide evenly by 1 or themselves 176. Probability Equation: # of desired outcomes/# of total possible outcomes 177. Probability of an event that cannot occur is?: Properties of 0: 1) only # that is neither +/- 2) a multiple of every integer 3) is an even # 4) a/0 = undefined 5. 0/a = 0

18 MATH REVIEW 179. Properties of 1: 1) ax1 = a 2) a/1 = a 3) 1ⁿ = 1 4) is a factor of every integer 5) is the smallest positive odd integer 6) is NOT a prime # 180. Properties of a parallelogram: *Opposite sides are congruent *Opposite angles are congruent *Sum of the measures of any 2 consecutive angles = 180 *The diagonals bisect each other 181. Properties of a rectangle: *The measure of each angle = 90 *The diagonals are congruent 182. Properties of a square: *The measure of each angle = 90 *The diagonals are congruent *All 4 sides have the same length *The diagonals are to each other *Each diagonals divides the square into 2 isosceles right triangles 183. Rules for Powers: a³x a⁴ = a³+⁴ a³/a⁴ = a³ ⁴ (a³)⁴= a³ ⁴ (a/b)³ = a³/b³ (ab)³ = a³b³ a ³ = 1/a³ 0ⁿ = Rules for Roots: (a/b) = ( a)/( b) (ab) = a x b (a)^(¹/n) = ⁿ a a^(m/n) = ⁿ a^m 185. Similar triangles...: 1) Corresponding angles are equal 2) Lengths of corresponding sides are in proportion 186. Simple Interest Rate Formula ex: If $12,000 is invested at 6 percent simple annual interest, how much interest is earned after 9 months?: Interest = Prt P-principle r-interest rate expressed as a decimal t-time in years 12,000(.06)(9/12) = $ Simplify a Radical: Look for perfect squares (4,9,16,25,36...) inside the. Factor them out and "unsquare" them. ex. 48= 16 x 3 = 4 3 ex2. 72= (36x2) = 36 x 2 = Simultaneous Equations: Questions involving 2+ equations, add them! Ex: If a+b=3, a+c=5, b+c=10, what is the average of a,b, and c? Add equations = 2a+2b+2c=18 Then average a, b, and c. = 3** 189. Solve this inequality: 3-(x/₄) 2: Solve the same as you would solve an equation EXCEPT: if the inequality is multiplied/divided by a negative #, the direction of the inequality changes! -(x/4) 2-3 -x (-1)(4) =-4 (then divide by (-1) & swap inequality**) = x Sum of all 3 angles in a triangle =?: Sum of the 3 exterior angles of a triangle =?: Sum of the angles around a point =?: Supplementary angles =: The answer is D (not enough information provided) if...: you have 2 variables and neither is proven to be positive 195. The greatest angle in a traingle...: lies opposite the longest side 196. the length of any side of a triangle is...: less than the sum of the other 2 sides AND greater than the difference of the other 2 sides 197. The measure of a straight line =?: The measure of an exterior angle of a triangle is equal to?: the sum of the measures of the two opposite interior angles 199. The price of an antique is reduced by 20% and then this price is reduced by 10%. If the antique originally cost $200, what is its final price?: 1) 20% = X/$200, X = $40 2) $200-$40=$160 3) 10% = X/$160, X=$16 4) $160-$16 = *$144* 200. The PROBABILITY of an event =: (# of desired outcomes)/ (# of possible outcomes) 201. The probability of an event that must occur is?: the ratio of the area of similar triangles...: is the square of the ratio of their sides 203. The Slope of a Line: 1) a horizontal line = 0 2) a vertical line = undefined 3) a line that goes up from left to right = + 4) a line that goes down from left to right = The sum of 3 consecutive integers is less than 75, what is the greatest possible value of the smallest one?: n + (n+1) + (n+2) < 75 3n + 3 < 75 3n < 72 n < 24 The most n can be is There are 36 marbles in a bag containing only blue and red marbles. If there are 3 red marbles for every blue marble, how many blue marbles are in the bag?: 1 : [1+3] or 1 : 4 Using b, for blue marbles, set up the proportion: ¼ = b/₃₆ 4b = 36 b = 9 blue marbles

19 MATH REVIEW 206. to solve for larger fractions with unlike parts...: cross multiply keeping numerators of the same side OR find a common numerator or denominator 207. Weighted Average =?: (weight1 x score) + (weight2 x score) / (weight1 + weight2) Don't just average the averages 208. What are factors?: The factors (divisors) of a number are the positive integers that evenly divide into that number (ex: 36 has nine factors: 1,2,3,4,6,9,12,18,36) 209. What are multiples?: An integer that is divisible by another integer is a multiple of that integer (ex: 12 is a multiple of 3, since 12 is divisible by 3) 210. What are the formulas for the area of a parallelogram, rectangle, and square?: Parallelogram: A=base x height (bh) Rectangle: A=length x width (lw) Square: A=side² (s²) OR diagonal²/2 (d²/2) 211. What are the formulas for the diameter, circumference, and area of a circle of radius r?: D=2r C=πd or 2πr A=πr² 212. What are the formulas for the surface area of a rectangular solid and the surface area of a cube?: Rectangular solid: A=2(lω+lh+hω) Cube: A=6e² 213. What are the formulas for the volumes of a rectangular solid and the volume of a cube?: Rectangular solid: V=lwh Cube: V=e³ (e=edge) 214. What are the Pythagorean Triplets?: Right triangles with sides 3,4,5 (6,8,10, or 9,12,15) and 5,12,13 and 7,24,25 and 9,40,41 and 11,60, What does it mean if 7 is directly proportional to x?: By definition, y is directly proportional to x if the quotient y/x is a constant What is 25% of 36?: =.25 x 36 = What is a chord?: a line segment joining 2 points on a circle The diameter of the circle is also the longest chord 218. What is a factorial? n!: The PRODUCT of all the integers from 1 to n. ex: 4!=4x3x2x1=24 ex2: 0!=1!= What is a function?: y=f(x)=2x+3, to every number x, the number 2x+3 is assigned. Ex: F(10)=2(10)+3 = What is a Prime Number?: an integer greater than 1 that has no factors other than 1 and itself (0 and 1 are not prime #s) 221. What is important about a triangle?: 1) Right triangle 2) Hypotenuse = 2x length of the shorter leg 3) Ratio of lengths of the 3 sides = x:x 3:2x 222. What is important about a triangle?: 1) Right triangle 2) Isosceles 3) Ratio of lengths of the 3 sides = x:x:x What is Standard Deviation?: measure of how spread out a group of numbers is. The more spread out the numbers, the larger the SD What is the Absolute Value of a number?: It is the number of units a number is away from 0 on a number line What is the counting principle?: If two jobs need to be completed and there are m ways to do the first job and n ways to do the second job, then there are m x n ways to do one job followed by the other What is the formula for the area of a triangle?: A = ½ base x height base-any side of the triangle height-the altitude drawn to the base from the opposite vertex 227. What is the formula for the area of an equilateral triangle with side s?: A = (s² 3)/ What is the formula for the distance b/w two points?: ( (x₂-x₁)² + (y₂-y₁)²) 229. What is the formula for the volume of a cylinder?: V=πr²h, where r is the radius of the circular base and h is the height 230. What is the formula to solve a group problem involving both/neither? Ex: Out of the 120 students at school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither. How many are studying both?: Group 1 + Group 2 + Neither - Both = Total 1) x=120 students 2) 169-x=120 3) x=49 are studying both 231. What is the formula to solve a work problem? Ex: If it takes Joe 4 hours to paint a room & Pete twice as long to paint the same room, how long would it take the two of them, working together to paint the same room, if each of them works at his respective individual rate?: 1/(Joe's hrs to finish) + 1/(Pete's hours) = 1/t 1) ¼+¹/₈ = ¹/t 2) ³/₈ = ¹/t 3) 3t=8 **t=2²/₃ hours 232. What is the Greatest Common Factor (GCF) of a pair of numbers?: The largest number that will divide evenly into both (factor) 1,2,4,8 = factors of 8. 1,2,3,6 = factors of 6. GCF = What is the Pythagorean Theorem?: In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the hypotenuse: a²+b²=c² 234. What is the smallest prime #?: The number 2 is the smallest prime and the only even prime number 235. What is the sum of the measures of the four angles in a quadrilateral?: What is the sum of the measures of the n angles in a polygon with n sides?: =(n-2)(180 )

20 MATH REVIEW 237. What is true about the area of similar triangles?: *ratio of the lengths of their corresponding sides = k, then the ratio of their areas = k² Ex: Each side of triangle B is 2x the length of the corresponding side of triangle A. So triangle B must have 2², or 4x area of A 238. What is true about the measures of the 4 angles formed by two intersecting lines?: They form vertical angles and their measures are equal to their opposite What is true about the slop of perpendicular lines?: The slop of one line is the negative reciprocal of the slope of the other line What is true about the slope of parallel lines?: They are equal 241. When a pair of parallel lines is intersected by a third line (transversal), what is true about the 8 angles that are formed?: 1) 4 of the angles are acute, 4 are obtuse 2) All acute angles are equal 3) All obtuse angles are equal 4) Sum of measures of any acute angle and any obtuse angle is 180

21 Multiplication Tables 20x x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: 48

22 Multiplication Tables 20x x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: 180

23 Multiplication Tables 20x x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: 120

24 Multiplication Tables 20x x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: 342

25 Multiplication Tables 20x x 19: x 20: x 1: x 2: x 3: x 4: x 5: x 6: x 7: x 8: x 9: x 10: x 11: x 12: x 13: x 14: x 15: x 16: x 17: x 18: x 19: x 20: 400