Math Special Flashcards

Question

x^2 < 3/2

Answer 1

Squares are treated similarly with absolute value sign Think of positive case --> keep direction of the sign and square root it Think of negative case --> switch direction of the sign and add negative and square root x^2 < a means: -sqrt(a) < x < sqrt(a) so x^2 < 3/2 -sqrt(3/2) < x < sqrt(3/2)

Answer 2

Squares are treated similarly with absolute value sign Think of positive case --> keep direction of the sign and square root it Think of negative case --> switch direction of the sign and add negative and square root x^2 > a means: x > sqrt(a) or x < - sqrt(a) so x^2 > 3/2 means: x > sqrt(3/2) or x < - sqrt(3/2)

Answer 3

Multi-variable inequality Often will be testing PROPERTIES (e.g., >0, <0) So try to FACTOR and get one side equal to 0 (just like one variable inequality problems) > sewing approach isn't as effective here Or try to use the positions on the number line, especially for questions that have compound inequalities

Answer 4

One variable Absolute Value --> Find the range of VALUES of the unknown THINK OF CASES for the sign of the inside of the absolute value sign: | x | ---> x >= 0 --> | x | = x | x | ---> x < 0 ---> | x | = -x Keep only VALID cases > draw out a NUMBER LINE with markings representing the "roots" (what values of x makes the argument = 0) > Determine the cases to test > Don't forget to add = to one of the cases for each root WORKS FOR EMBEDDED ABSOLUTE VALUE SIGNS TOO --> for EACH absolute value sign, you have two cases Example: | 2x + 1 | > x + 1 Root --> x = -1/2 1) x >= -1/2 --> 2x + 1 >= 0 2x + 1 > x + 1 x > 0 ---> satisfies the test case and overrides it ("large choose large") 2) x < -1/2 --> 2x + 1 < 0 -(2x + 1) > x + 1 x < -2/3 ---> satisfies the test case and overrides it ("small choose small") so x > 0 or x < -2/3

Answer 5

Multi variable Absolute Value --> Distances using Number Line Characteristics: > typically SUBTRACTION between the terms in the absolute value signs | x | = distance that x is from 0 | x - 1 | = distance between x and 1 | x + 1 | = | x - (-1) | = distance between x and -1 Also | x - y | = | y - x | Focus on the COMMON POINT

Answer 6

Multi variable Absolute Value with Split --> SIGNS of the UNKNOWNS = only happens when unknowns have the same sign (xy > 0) | x + y | <= | x | + | y | ----> = only happens when x and y have the SAME SIGN (otherwise, x + y will be smaller) | x - y | >= | x | - | y | ----> = only happens when x and y have the SAME SIGN AND when | x | > | y | (otherwise, x - y will be larger) Three or more variables: > need to test cases (NOT the simple rule that if all the variables are the same sign, then =) > x - y must be >= z e.g., | 4 - 3 - 2 | =/ 4 - 3 - 2 e.g., | 10 - 2 - 1 | = 10 - 2 - 1 ---> x - y - z >= 0 e.g., | 11 - 6 - 5 | = 0 = 11 - 6 - 5 ---> x - y - z = 0

Answer 7

xy > 0 or x/y > 0 (if x and y =/ 0)

Answer 8

Keep only VALID cases > draw out a NUMBER LINE with markings representing the "roots" (what values of x makes the argument = 0) > Determine the cases to test > Don't forget to add = to one of the cases for each root Roots: x = 3, x = 4 1) x >= 4 --> (x - 3) + (x - 4) > 0 (x - 3) + (x - 4) < 2 2x - 7 < 2 2x < 9 x < 9/2 Therefore, 4 <= x < 9/2 2) 3 <= x < 4 --> (x - 3) > 0 and (x - 4) < 0 x - 3 + 4 - x < 2 1 < 2 (true) Therefore, 3 <= x < 4 3) x < 3 --> (x - 3) < 0 and (x - 4) < 0 3 - x + 4 - x < 2 7 - 2x < 2 5 < 2x 5/2 < x x > 5/2 Therefore 5/2 < x < 3 So in total (you can combine here): 5/2 < x < 9/2

Answer 9

n sides Sum of interior angles = (n - 2)*180

Answer 10

difference of the other two sides < side < sum of the other two sides

Answer 11

Angles correspond to sides > Largest angle is opposite from the largest side > Smallest angle is opposite from the smallest side a is opposite from 59 b is opposite from 60 c is opposite from 61

Answer 12

If a, b, c are the sides and a < b < c: a^2 + b^2 < c^2

Answer 13

If a, b, c are the sides and a < b < c: a^2 + b^2 > c^2

Answer 14

Arcs are related to a circle's CIRCUMFERENCE Arcs are defined by: radius and central angle The interior angle of one arc is EQUAL regardless of where the vertex is located > can be used to determine the value of the central angle Special example: Inscribed angle's vertex is on the same line as the center of the circle

Answer 15

Sectors are related to a circle's AREA Sectors are defined by: radius and central angle A sector can be broken into an isosceles triangle plus a curved part

Answer 16

Radius is key > isosceles or equilateral triangles (60 degrees!!) Diameter's inscribed angle = 90 degrees Angle rules (line = 180 degrees, parallel lines) Central angle = 2*inscribed angle sharing the same arc *don't be afraid to draw helping lines (based on connecting VERTICES on the circle)

Answer 17

Diagonal of the square = Diameter of the circle Diameter of the circle = side of the square > you can create a smaller square by drawing a line from the center of the circle to square's vertex = diagonal of the smaller square > circle's radius becomes the sides

Answer 18

Focus on the VALUE of angles, depending on which sides are equal (e.g., radius) Use symbols for angles, such as alpha, beta

Answer 19

Rely on EXTERIOR angles Also might need to calculate the "perfect regular shape" using sum of interior angles = (n - 2)*180 Also FYI parallel lines --> draw aiding parallel lines

Answer 20

Don't bother doing calculations. Start off by mapping out what info you need! e.g., radius, angle

Answer 21

Inscribed angle are all equal as long as they share the same arc

Answer 22

SSS SAS = side-angle-side are known and equal ASA = angle-side=angle are known and equal AAS = angle-angle-side are known and equal (NOT ASS or SSA) *with the above, the shape of the triangle is FIXED FOR RIGHT TRIANGLES: a^2 + b^2 = c^2 > Hypotenuse + 1 legs > 2 legs

Answer 23

6 faces 12 edges 8 vertices

Answer 24

Distance = sqrt[ (x2 - x1)^2 + (y2 - y1)^2 ]

Answer 25

Integer is the SUM of two perfect squares It may be possible to test values if you know the value of c Also in general if you are given a^2 + b^2 and asked to find ab --> likely a^2 - 2ab + b^2

Answer 26

Get into the habit of: > Grouping variables (highest to lowest exponent) > factoring out negative (positive coefficients on variables) > removing roots from the denom of a fraction (by multiplying top and bottom by the conjugate or just itself) ** do this BEFORE combining into one fraction > factoring out fractions out of EVERY TERM first > in equations: moving terms to one side and factoring e.g., ab = ac --> ab - ac = 0 ---> a(b-c) = 0 (a = 0 OR b=c)

Answer 27

Depends on sign of a: a > 0 (positive) --> U shaped a < 0 ---> downward shaped Max or Min depends on the sign of a > U shaped --> min > Downward shape --> max *Tip: Quadratic functions are symmetrical about the axis of symmetry > find the distance between the roots (if any) and divide by two to get the x coordinate of the max or min

Answer 28

(a, b) is a reflection of (b, a) Both points have the SAME distance from y = x (which acts as a perpendicular bisector of the line segment from (a,b) to (b,a)) REFLECTIONS deal with lines that are perpendicular bisectors of a line segment

Answer 29

(a, b) is a reflection of (a, -b) Both points have the SAME distance from the x axis (which acts as a perpendicular bisector of the line segment from (a,b) to (a, -b)) REFLECTIONS deal with lines that are perpendicular bisectors of a line segment

Answer 30

(a, b) is a reflection of (-a, b) Both points have the SAME distance from the y axis (which acts as a perpendicular bisector of the line segment from (a,b) to (-a, b)) REFLECTIONS deal with lines that are perpendicular bisectors of a line segment

Answer 31

MUST cross II and IV

Answer 32

MUST cross I and III

Answer 33

Radius means EQUAL DISTANCE from every point on the circle to the center If the center is the ORIGIN, you can create a right triangle to calculate x, y point on the circle and/or the radius > just need two info

Answer 34

Digit question If you are given any info about the VALUE of the integer, you can write it in the algebraic form **don't forget decimals can also be written in algebraic form e.g., 3.81 = 3 + 8/10 + 1/100 e.g. 1, if a and b were reversed, the resulting integer is 27 more than n. 10a + b + 27 = 10b + a (COMBINE THE REMAINING TWO VARIABLES) 10b - b + a - 10a = 27 9b - 9a = 27 b - a = 3 ----> we set a < b e.g., 2. abc - cba = a multiple of 7 ---> clearly this is a digit/multiple hybrid question 100a + 10b + c - 100c - 10b - a = 7*int 99a - 99c = 7*int 99(a - c) = 7*int ---> a - c must be a multiple of 7 (e.g., 9 - 2, 8 - 1) e.g. 3, you can determine the value of the UNITS digit 196 = 100(a - b) + 10(c + d) + (z - c) > know that z - c = 6 because the other terms are just multiples of 10.

Answer 35

Questions related to powers, multiplication and addition (but you can have computations relating to subtraction, as long as there is tens column to borrow from) e.g., 9^19 - 7^15 e.g., 15^9 - 16^5 = u 5 - u 6 = 9 (not -1 --> units digit must be positive. And there is a tens column to borrow from). 1) Large exponents - BOTH MULTIPLICATION AND ADDITION e.g., units digit of 13^53 or product of large integers > focus on the pattern of the units digit raised to different powers e.g., 1^4 + 6^4 + x^4 + 4^4 = 16x4 e.g., (24^17 * 99^18) 2) Linear problems with two variables representing integers e.g., 15A + 3B = 100 3) Digit questions given the value of the units digit e.g., 2016 = 300A + 30B + a + b e.g., 2016 = A(B + 1) ** make sure the units digit is POSITIVE ** carry over is allowed (e.g., 6a = 6 units digit, a = 1 or 6) e.g., a + b > 0 e.g., z - q > 0 4) Remainder of an integer divided by 10 or 5

Answer 36

A RANGE of values > need to find the MAX and MIN > combine into one inequality (compound inequality) CONSIDER whether the end points are INCLUSIVE or not > typically if you solve for the max or min it will be inclusive

Answer 37

Write the fraction's numerator and denom as prime factors e.g., (5^2)/(2^4*3^2) Simplify as much as possible e.g., (5^2)/(2^4*3^2) If the denominator has only powers of 2 or 5, then the decimal is terminating e.g., (5^2)/(2^4*3^2) ---> Not a terminating decimal *integers are terminating decimals too **DS problems: > if you KNOW the denominator contains only 2's and/or 5's, then it is sufficient e.g., t/s, s = 4 However if the denominator contains anything other than 2's or 5's, then it is not sufficient e.g., t/s, s = 15 (we don't know if 3 is cancelled out)

Answer 38

P(at least __) = 1 - P(NOT_ and NOT_)

Answer 39

Total = A + B + C - (AandB + BandC + AandC) + center + other ** (could write them as %: 100% = A% + B% etc.) > then multiply the final percent by total A = all of A B = all of B C = all of C AndB = only AandB AND center ... Use the formulaic approach when the usual numbers approach (i.e., starting inside out) doesn't work To get # of people ONLY 2 (without center) --> AandB + BandC + AandC - 3*center To get # of people ONLY 1 criterion: Total - ONLY 2 - center

Answer 40

Write out the formula for mean x bar = sum/# of terms > often times there will be valuable info from the equation! For questions about consecutive integers and mean vs median? > mean = median ONLY for evenly spaced sets > If Sum1 = Sum2, then mean1 = mean2 only equals if the sums = 0 or if the two sets have equal number of terms. (BUT the medians DON'T have to be equal to each other!! e.g., {-5, 2, 3} vs {-1, 0, 1} Generally: > Info about the relationship between AVERAGES is NOT sufficient to know about the relationship between medians > Info about the relationship between MEDIANS is NOT sufficient to know about the relationship between averages > info about the relationship between MEDIANS is NOT sufficient to know about the relationship between # of TERMS

Answer 41

Statistics - set transformations If EVERY term of the set increases/decreases by delta (e.g., +3, -4) Mean changes by delta Median changes by delta Range DOES NOT CHANGE Standard deviation DOES NOT CHANGE

Answer 42

Statistics - set transformations If EVERY term of the set increases/decreases by a factor of k (e.g., *3, /4, *-2) Mean changes by *k Median changes by *k Range CHANGES by *k (keep the sign) Standard Deviation CHANGES by *k ** standard deviation is ALWAYS POSITIVE (so multiply std by | k |)

Answer 43

Determine whether the unknowns can equal 0 0 is a multiple of ANY number

Answer 44

If p is a prime number, then p has no factors n such that 1 < n < p In other words, the only factors (2) of p are 1 and p

Answer 45

Prime Factor Properties # of factors = (exponent on prime + 1)(exponent on prime + 1) etc. We NEED to know the value of x, otherwise, inconclusive > x could already be a prime factor (e.g., 2 and 3 for 6) > if x is NOT already accounted for, then the # of factors would DOUBLE

Answer 46

Rubber Band Geometry > once you FIXATE the shape, there must be one solution (it is possible to solve --> sufficient) > You don't have to know how to solve the shape, as long as you know the shape is FIXED e.g., SSS, SAS, AAS, ASA for triangles e.g., radius and angle for sectors and arcs e.g., length of a side in regular polygons

Answer 47

Total Simple Interest = Initial Inv * (rate per period) * # of periods **simple interest does not build on accumulated interest Total Inv Amount = Initial Inv + Total Simple Interest

Answer 48

Total Compound Interest = Initial inv * (1 + r per period)^n periods - Initial Inv = P(1 + r/m)^mt - P (where m = # of periods in a year, t = # of years) Total Inv Amount = Initial inv * (1 + r per period)^n periods

Answer 49

a = 2n b = 2n + 1 Application: > PS questions about odd/even properties > DS questions about remainders and divisibility e.g., p = odd = a^2 + b^2 > a and b have opposite types What is the remainder of p/4? (2n)^2 + (2n + 1)^2 = 4n^2 + 4n^2 + 4n + 1 ---> remainder always = 1

Answer 50

1 = P(m) + P(not m)

Answer 51

Writing it out in terms of DISTANCE (rather than a fraction): Distance = rate * time

Answer 52

Even or Odd E/E = O ---> E = O * E E/E = E ---> E = E * E e.g., 4/2 = 2 e.g., 14/2 = 7

Answer 53

Even number E/O = E --> E = E * O E/O = O --> E = O * O (Doesn't work) e.g., 10/5 = 2 Memorize: EOE

Answer 54

Odd number O/O = O --> O = O * O O/O = E --> O = O * E (Doesn't work) e.g., 21/3 = 7 Memorize: OOO

Answer 55

NOT POSSIBLE

Answer 56

Write out the place values and decimal e.g., d = _ . _ _ _ = _ . a b c is a >= 5?

Answer 57

Don't add brackets that aren't there - assume default > must factor out a -1 10^(-2a+2) = 10^-(2a - 2) = 1/[10^(2a - 2)]

Answer 58

Check to see if the non-10 integers can be multiplied into an integer ending with 0 > if yes --> reduce the power on 10 > if no --> keep power on 10 the same e.g., (0.0025)(0.002)*10^k = integer (25 * 10^-4)*(2*10^-3)*10^k = integer 50*(10^-7)*(10^k) = integer ---> 50 ends with 0 5*10*(10^-7)*(10^k) = integer 5*(10^-6)*(10^k) = integer k = 6 (not 7)

Answer 59

Since it is PS, there must be a solution. You can assume one valid case is a REGULAR polygon --> with equal sides and solve OR test any valid case **question must not have given you ANY measurements

Answer 60

Quantity of Ingredient Before the Mixture = Quantity of Ingredient After the Mixture e.g., % alcohol * volume = amount of alcohol after mixture > don't overcomplicate variables (e.g., removing P from original V) > removed liquid shares same characteristics as the entire mixture (e.g., if 2/3 of the mixture is alcohol, then 2/3 of the removed volume is alcohol) also Total Q of mixture + Total Q of mixture = Final Q of mixture e.g., If you mix 1 ton of A with 2 tons of B, you get 3 tons of a combined mixture **also keep in mind ratios can be converted to PERCENTS e.g., Juice to Water -> 3:2 --> 3/5 = Wj, 2/5 = Ww

Answer 61

Work problems: x < y means that Machine x is more efficient (takes less time, has a faster rate) 1/x > 1/y The TOTAL combined time (t) is always LESS than each machine's individual time to complete t < x < y The RANGE of total time (t): > consider what t would be if you had TWO machine x's or TWO machine y's > t is smaller than the time it would take two machine y's and greater than the time it would take two machine x's x/2 < t < y/2 Proof: (1/x + 1/x)*t = 1 t = x/2 (1/y + 1/y)*t = 1 t = y/2 x/2 < y/2

Answer 62

No solution Parallel lines

Answer 63

Not solvable because of infinite number of solutions > SAME LINE

Answer 64

n = a^6 n has an exponent that is both divisible by 2 and 3. Derived from: n = (a^3)^2 = a^6 n = (a^2)^3 = a^6

Answer 65

Multiple and Factors: x must be a multiple of 5 y must be a multiple of 3 **if you know either x or y, you FIXATE the value of the other!! e.g., x = 5, then y must = 3 x = 15, then y must = 9 In the second example, we are NOT SURE if x is a multiple of y or m, or if x-1 is a multiple of y or m. *x and x-1 have zero common factors other than 1 *also applies to x and x+1 (have zero common factors other than 1) ANOTHER APPLICATION: h(100) and h(100) + 1 have zero common factors other than 1 > so the LEAST common prime factor must be LARGER than the current possible primes by one of the two numbers e.g., h(100) = 50!, so the smallest prime factor of h(100) + 1 must be greater than 50 (such as 53) > multiple + non-multiple = non-multiple

Answer 66

1) Possibly testing product of consecutive even or odd integers (n - 1)(n)(n + 1) 2) Possibly testing difference of squares 3) possibly testing FACTORS (if given # of the other side of an equation)

Answer 67

Share common time Dist travelled by A + Dist travelled by B = Total distance between them

Answer 68

Share common time AFTER the second person starts moving Dist Travelled by First mover at beginning + Dist travelled by first mover when other moves + Dist travelled by Second mover = Total Distance

Answer 69

Share common time after the faster person starts moving. Share common distance once faster person catches up Initial Gap + Dist travelled by slower person = Dist travelled by faster person

Answer 70

Share common distance once slower person catches up. Share common time when both are moving Dist travelled by faster person = Dist travelled by slower person when faster person is still moving + Dist travelled by slower person while faster person is waiting

Answer 71

Share common time Gap between two people = 2*pi*r Dis travelled by faster - Dist travelled by slower = 2*pi*r

Answer 72

Exponential function, has negative and positive values for x If a > 1 --> upward sloping --> can drop the base in algebra If 0 < a < 1 --> downward sloping

Answer 73

1) Odd/Even addition, subtraction, multiplication and division O +/- O = even E +/- E = even O +/- E = odd O*O = odd O*e = even e*e = even E/E = Odd or even E/O = Even O/O = odd O/E = not possible (not an integer) (2) Unknown +/- even or odd integer x + 1 --> opposite type of x x + 2 --> same type as x (3) Identifying if 2 is a factor of unknown (4) Consecutive integers (n)(n + 1) --> one is even

Answer 74

(a + b)^3 = a^3 + 3a^2*b + 3a*b^2 + b^3 (Exponent on "a" decreases from 3 to 0; exponent on "b" increases from 0 to 3) Middle terms have 3 as coefficient

Answer 75

Permutation Pm,n = From m choose n = m! / (m - n)! **assumes NO duplicates! > if there ARE duplicates, then you have to think about TIERS e.g., PEPPER --> _ _ _ , how many unique arrangements? P P P = 1 way P P other = 2*(3!/2!) E E other = 2*(3!/2!) P E R = 3! Total = 1 + 6 + 6 + 6 = 19 LONG WAY to think about it: > First determine the # of ways to choose 5 symbols from 10 > Then, multiply the # of ways by the # of ways to arrange 5 chosen symbols = (m!)/[(m-n)!*(n!)] * n!

Answer 76

Combination Cm,n = From m choose n = m! / (n! * (m - n)!) ** often part of complex probability questions e.g., 100 students, 20 girls, 80 boys. A group of three is formed. What is the probability that the group comprises only of girls? _ _ _ = G G G Approach 1: C20,3 / C100,3 Approach 2: P(G)*P(G)*P(G) = 20/100 * 19/99 * 18/98

Answer 77

Test two cases: Yes and No case > if both work, NS See if it is possible for the sum of the remaining numbers to meet the remaining sum Helpful to find the average of the remaining numbers and adjust each number so that the sum of the gaps = 0

Answer 78

Factor PAIRS (doesn't have to be a prime factor!!) Start with 1 * n You can also evaluate the INSIDE of factors too: (common for # of positive factor questions) e.g., (x + 1)*(y + 1) = 9 1 * 9 = 9 -----> x + 1 = 1 and y + 1 = 9 ----> x = 0, y = 8 3 * 3 = 9 -----> x + 1 = 3 and y + 1 = 3 -----> x = 2, y = 2 ** might have integer constraints on the variables

Answer 79

Yes, remainder is 0, regardless of the value of sqrt(C)

Answer 80

ODD # of Terms: # of terms other than x = n - 1 # of terms smaller than x = (n - 1)/2 # of terms larger than x = (n - 1)/2 Max VALUE (consecutive integers) = x + (n-1)/2 MIN value (consecutive integers) = x - (n - 1)/2 Median = 50th percentile (an equal number of terms are smaller and larger than the median, not including the median itself) > if it were 90th percentile, the value is at position 90% (position/n = 90%) e.g., 1 2 3 4 5 --> median is 3, 5 terms There are 5 - 1 = 4 terms other than the median. And there are 2 terms BELOW 3 and 2 terms ABOVE 3

Answer 81

EVEN # of terms: Same as odd # of terms: # of terms smaller than x: (n - 1)/2 ---> will be a fraction with denom 2 (ROUND UP) # of terms larger than x: (n - 1)/2 Max VALUE (consecutive integers) = x + (n-1)/2 MIN value (consecutive integers) = x - (n - 1)/2 Means the position of y is 60% --> Position of y / n = 60% e.g., 60%*10 = 6 (y is located at position 6) > There are 5 terms below y > There are 4 terms above y 1 2 3 4 5 6 7 8 9 10 y = position 6 = 6 6/10 = 60th percentile

Answer 82

No e.g., Py - Px = ?, is not sufficient to solve given Py = 2Px ---> 2Px - Px = Px = ? This is different from ratios --> sometimes the unknowns cancel out

Answer 83

(Price - Cost)/Cost = positive or negative percentage

Answer 84

(n!)/(n) = (n - 1)! ABCDE = EABCD ... > you don't need to divide by n factorial because the factorial accounts for DIFFERENT orders of the seats (we just want to account for the SAME ORDER, but moving around the circle)

Answer 85

If you KNOW the signs of both sides > flip sign if both sides are negative > keep sign if both sides are positive > Cannot square if sides have opposite signs HAVE NO FEAR If both sides are POSITIVE e.g., If x > y > 0, then x^2 > y^2 > 0 e.g., sqrt(x) > sqrt(y) x > y e.g., | x | > | y | x^2 > y^2 e.g., x > 3 x^2 > 9 e.g., sqrt( 63 + sqrt(3)) > x + y*sqrt(3), where x and y > 0

Answer 86

If both sides are POSITIVE

Answer 87

x and q both equal 0 | x | + | q | = 0 Any other number would make the statement false x - 3 = 0 to make the second statement hold true | x - 3 | + y <= 0

Answer 88

Always reword the question with probability stem: P( ) = ? If P( different items) AND not fixed already, you need to multiply by cases. e.g., P(A and B and C) = P(ABC) * 3! P(A and A and C) = P(AAC) * (3! / 2!) P( G and G and B and B) = P(GGBB) * (4!/(2!*2!)) If P( identical items), you don't need to multiply by cases e.g., P(AA) e.g., P(MM) If P(all different pairs method), you don't need to multiply by cases P(4 different numbers) = P(diff and diff and diff and diff)

Answer 89

"Identical" except colour --> duplicates (e.g., 3 red balls and 4 green balls. All the red balls are duplicates of one another). Think about if order matters among the same "type" e.g., order of performances where there are 5 songs and 3 dances --> yes, order matters (d1 then d2 is different from d2 then d1) e.g., order of MMM and FFF So differentiate between SAME TYPE and IDENTICAL > "same type" e.g., 6 junior partners and 3 senior partners ---> need a group with two J and 1 S e.g., 5 perennial flowers and 2 annual flowers ---> need a group with 4 P and 1 A

Answer 90

P(not prime and not prime) NOT 1 - P(prime and prime) ---> misses case where one of the numbers is prime Sets variation: Neither A or B = Total - (A + B - AandB)

Answer 91

Factorials Recall that you can expand the factorial: (n+1)! = (n+1)*(n)! = (n+1)*(n)*(n-1)! Therefore: (n+1)*(n)*(n-1)(n-2)! / (3!*(n-2)!) - (n)*(n-1)*(n-2)*(n-3)! / (3!*(n-3)!) (n+1)*(n)*(n-1)/3! - (n)*(n-1)*(n-2)/3! = 150 (n)*(n - 1)*(3) / 3! = 150 If you are given a NUMBER, it is generally SUFFICIENT TO SOLVE (simplifies into a quadratic function) > one variable, one equation (and negative in front of bx and c) **

Answer 92

Yes if there is one overlap e.g. 1) x = 5 or -2 2) x = 5 or -4 Together, x must be 5 to satisfy BOTH

Answer 93

TYPE 1: Solve for the range of values (max and min) > if you are asked what the VALUE of something falls in > keep track of end points Type 2: Solve for the max (or min) and then see which answer choice reflects the correct range > if you are asked for EITHER the MAX or the MIN (so you cannot solve for the "range") e.g., x max = sqrt(10) = 3.xx So 3.xx falls between 3 and 6.

Answer 94

No factors in common (other than 1)

Answer 95

Distance on number line > c is in the MIDDLE between a and b (if a =/ b) > or a = b (a and b are the same point) Also | c - a | = | a - c | When drawing scenarios, focus on the common point, c: c +/- #

Answer 96

M + F = total ---> might be sufficient to solve algebraically

Answer 97

Concept: a PRIME of n^3 MUST BE a PRIME of n Prime factorization: n^3 = 2*3^2*5^2 * y= n*n*n > three identical factor sets of n n must contain the MINIMUM number of integers such that n^3 = 450*y To guarantee 450 is a factor of n, n contains at least: > one 2 > one 3 > one 5 n = 2*3*5 (least possible value of n) n^3 = 2^3 * 3^3 * 5^3 = 450 * (2^2 * 3 * 5)*m where m is another possible factor that is a cube y = (2^2 * 3 * 5)*m > y is a multiple of 60

Answer 98

Maximize any value by MINIMIZING the other values > think of a number line and what the the max and min value a number can take a < b < c Maximize C: Minimize a and b (a = 1, b = 2) 1 < 2 < c Minimize C: Maximize a and b (b is 1 less c, a is 2 less c) c - 2 < c - 1 < c Maximize A: Minimize b and c (b is 1 more a, c is 2 more a) a < a + 1 < a + 2 a <= b <= c Maximize C: Minimize a and b (a = b = 1) 1 <= 1 <= c Minimize C: Maximize a and b (a = b = c) c <= c <= c Maximize A: Minimize b and c (b = c = a) a <= a <= a *often a constraint on c too (c is dependent on a --> then you just minimize b) Watch out for wording e.g., each integer can appear in the list at most twice (one OR two times!!)

Answer 99

WA = weight% * value + weight% * value WA = [(value*# of items) + (value*# of items)] / (total # of items) Application: mixture question, multiple averages e.g., Avg total = [(avgA * # of A) + (avgB * # of B)]/(A + B) > if you know the individual averages and total avg, you can solve for the ratio of # of A to # of B > avgA * # of A = Sum of A **Can also work for MORE THAN THREE ITEMS (you can still use visual approach or mathematical approach). WA*(QA + QB + QC) = PriceA*QA + PriceB*QB + PriceC*QC 22*QA + 7*QB = 5QC AND: 5QA + 5QB < 22QA + 7QB = 5QC QA + QB < QC ---> Max is C

Answer 100

SPOT THE PATTERN: y - x = - (x - y)

Answer 101

In each statement, test each symbol to see if the statement remains True. Keep only VALID cases and then evaluate the question. e.g., (1) 3#2 > 3: 3 + 2 = 5 > 3 (valid) 3 - 2 = 1 > 3 (invalid) 3 * 2 = 6 > 3 (valid) 3/2 = 1.5 > 3 (invalid) so # can either be + or *: Does (6+2) + 4 = 6 + (2+4)? ---> Yes Does (6*2)*4 = 6*2*4 ? ----> yes Always Yes --> sufficient (2) 3#1 = 3: 3+1 = 4 = 3 (invalid) 3-1 = 2 = 3 (invalid) 3*1 = 3 (valid) 3/1 = 3 (valid) So # can either be * or /: Does (6+2) + 4 = 6 + (2+4)? ---> Yes Does (6/2)/4 = 6/(2/4)? ---> No NS

Answer 102

All perfect squares are in the form: 4a or 4a + 1 > (even int)^2 ---> divisible by 4 > (odd int)^2 --> remainder 1 when divided by 4 3a or 3a + 1 > (multiple of 3)^2 --> divisible by 3 > (non-multiple of 3)^2 --> remainder 1 when divided by 3

Answer 103

Max # of bouquets = GCF = B 15 = # white roses per bouquet * B = 3 * 5 85 = # of red roses per bouquet * B = 17 * 5 B = 5

Answer 104

If you DON'T know the sign of x and y, you CANNOT take reciprocals Otherwise, the sign of x and y determine whether to flip the inequality or not. > FLIP the sign UNLESS x and y have DIFFERENT signs (+- or -+) e.g., -6 < 2 (different signs, don't flip the inequality) 1/-6 < 1/2 RULE: Same sign: Flip Opposite signs: Do not flip

Answer 105

Mean = Median = (First + Last)/2 Sum: Avg * # of terms # of terms = (Last - First)/increment + 1 ALL INCLUSIVE end points > e.g., if trying to find multiples of 7 and the end points are [10,80] --> change to [14, 77] APPLIES TO: - consecutive integers - any evenly spaced set (arithmetic sequence)

Answer 106

Integer = + whole number or - whole number or 0 Positive Integer: > 0 int Negative Integer: < 0 int 0 is NEITHER positive NOR negative

Answer 107

Yes -- we still have two different equations > eventually will get a quadratic function where ax^2 - bx - c (one positive answer) (negative in front of bx term and c, or just in front of c term - especially important for positive integer constraint questions!!) Long way proof: a = -1 - b (-1 - b)*b = -6 -b - b^2 = -6 b^2 + b - 6 = 0 (b + 3)(b - 2) = 0 Therefore, b = -3 or b = 2 If b = -3, a = 2 If b = 2, a = -3 We can tell that a =/ b. And we can see there are only TWO unique roots

Answer 108

PAY ATTENTION TO WORDING: A is 1/4 taller than B: A > B by a factor of 1/4 A = (1.25)*B CHANGE = A - B = (1/4)*B IN VARIABLE FORM: A = (1 + fraction)*B (if fraction < 1) or A = (fraction)*B (if fraction > 1) A is 1/4 *meters* taller than B: A > B by a difference of 1/4 A = B + 0.25 TIMES = MULTIPLICATION ALWAYS (ignore other stuff): A is 5/6 times B: A = (5/6)*B A is three times greater than B: > same as A is 3 times B A = 3*B There is twice as much A as B: A = 2B A is 10% greater than B A = (1 + 10%)*B A = 1.1B

Answer 109

GCF = product of the LEAST POWERS in each "column" > represents the DIVISORS of unknown > circle the number in each column that dictates the GCF > use descriptive exponents e.g., 1+ to indicate that there could be more > typically might have powers of 0 > COULD EXCLUDE FACTORS (but provides info about common divisors = multiples) LCM = product of the GREATEST POWERS in each "column" > the unknown is the divisor of the LCM > INCLUDES ALL FACTORS (but hides which factor belongs to which number) > if you are given LCM of two unknowns and asked to find LCM of the two unknowns plus one more integer, IT IS SUFFICIENT

Answer 110

A) Use "x" as the multiplier (could have an integer constraint) M: 2x F: 1x L/W = 3/2 --> L = 3x and W = 2x *you can perform operations on the "actual" amounts e.g., 2x - 10 --> subtract 10 from the number of men B) Max amount simply constrains the ACTUAL value: e.g., Blue, Yellow, and Red paint are in the ratio 2:3:1 Actual Yellow Paint = 3x > If max amount available is 10, then Actual <= 10, or 3x <= 10 C) Doubling ratio = 2*ratio Halving ratio = 0.5*ratio D) Yes, you can combine ratios --> make sure the individual ratios still work in the combined ratio E) Part vs whole --> you can FIGURE OUT THE RATIO of part to whole (but not know the actual amount) > recall that ratios and percentages are interchangeable e.g., # of men = 2/3 # of women M = 2/3*W M/W = 2/3 THEREFORE M/Total = 2/5

Answer 111

Ratios are flexible - you can invert them (because of cross multiplication)

Answer 112

A) Choose the answer that has the SMALLEST GAP from the desired number (i.e., 0.5) > calculate all the gaps, then choose the smallest one B) Set up the appropriate RANGE of values based on easy numbers. > for Square roots: try turning square roots into POWERS e.g., sqrt(569000) = x 569000 = x^2 700^2 < 569000 < 800^2 -----> 700 < x < 800 e.g., 4^1/3 > we know 8^1/3 = 2; 1^1/3 = 1 1 < 4^1/3 < 2 C) Remember DIFFERENCE of squares e.g., 0.99999999/1.0001 = (1 - 0.00000001)/(1 + 0.0001) = (1 - 10^-8)/(1 + 10^-4) = (1 - 10^-4)*(1 + 10^-4) / (1 + 10^-4) = 1 - 10^-4 **don't forget that you can WORK BACKWARDS from the answers (B, D) ** # of zeros (including before the decimal) = value of the exponent e.g., 10^3 = 1 with 3 zeros = 1000 10^-3 = 1 with 3 zeros = 0.001

Answer 113

RECALL: a/b = int + remainder/b a = b*int + remainder Multiply the decimal by the divisor to get an INTEGER REMAINDER e.g., s/4 = 3 + 0.75 s = 12 + 4(0.75) s = 12 + 3 ----> remainder is 3 (and s = 15) e.g., s/t = 96 + 0.12 s = 96t + 0.12t 0.12t must be an INTEGER --> (3/25)*t must be an integer, so t must be a multiple of 25. If you are given the REMAINDER, then you can SOLVE FOR the divisor.

Answer 114

Tangents - the rubber band leaves the circles at the SAME symmetrical spot

Answer 115

Write out: (P2 - P1)/P1 or P2/P1 - 1

Answer 116

1) Given RATIO (part to part relationship), you can calculate the PERCENT (or weights) or FRACION (part to whole) e.g., M/F = 2/1 or M = 2F M/(M + F) = (2F)/(2F + F) ---> F cancels out (M + F)/M = (2F + F)/2F ---> F cancels out 2) Given PERCENT (or weights), you can calculate the RATIO (via cross multiplication) e.g., M/(M + F) = 2/3 3M = 2M + 2F M = 2F MUST BE FOR TWO VARIABLES IN EITHER CASE ABOVE, you DON'T KNOW the QUANTITIES

Answer 117

3 triangles (two smaller, one larger) are SIMILAR Match up sides based on the angle > Larger triangle: 90 degrees, angle a, angle b > each mini triangle has 90 degrees and either angle a or angle b. So the third angle must be either angle b or angle a Review the proportions If h = height: h/x = y/h

Answer 118

1) Value question > Indicators: given quantities > often needs at least two info in each column or row to solve 2) Ratio question > Indicators: given percents or ratios > often can solve using just one equation that gives you the necessary ratios (read the question carefully!)

Answer 119

When two lines have slopes that are the SAME sign, the line with the LARGER ANGLE with respect to the positive x axis has the larger slope.

Answer 120

*you could see if the triangle is a right triangle by comparing the slopes and seeing if there are negative reciprocals Safer approach that works with ANY triangle: Trapezoid or Square or rectangle > Create a trapezoid or square comprised of the triangle in question and other smaller triangles or rectangles > draw helping lines to the closest axis (x or y) - depends on where the vertex lies Area of trapezoid = 1/2(base1 + base2) * height If one of the vertices is on the axis --> can create right triangles Area of the triangle = Area of trapezoid - area of smaller shapes

Answer 121

Concept: Must take into account thickness*2 into each length and width (and height) Length = inner length + 2*thickness Width = inner width + 2*thickness Height = inner height + 2*thickness AREA of BORDER = Area of larger rectangle - Area of smaller rectangle ***

Answer 122

Recognize factor patterns Difference of Squares

Answer 123

1) ONE VARIABLE - Solve for the value of the unknown (using statements) or Solve for an Expression of the unknown e.g., r = 10% (Sufficient) ----> as long as you know it is solvable, you can declare S e.g., x < 0.6 (Sufficient) or 2) ONE VARIABLE - Test two cases around the boundary (see if the statements remain valid or not) e.g., test case r = 8% and r = 9% --> both work, so Not Sufficient e.g., test c + d < 140 --> invalid and never possible --> Sufficient to know that c + d > 140 e.g., test F = 11 and F = 12 --> both work, so Not Sufficient or 3) MULTI VARIABLE - move all the variables to one side --> evaluate properties (>0 < 0)

Answer 124

Combine into ONE equation Proportional: y = kx Inversely Proportional: y = k/x *k is a constant e.g., c = k*(L^2)*(T)

Answer 125

MAKE SURE you don't forget the inclusive range, even if the data set starts after the start of the range or stops before the end of the range. e.g., [8am, 10am]

Answer 126

AT LEAST ONE is positive (Recall: x PLUS y > < 0 have "at least one") + + > 0 + - > 0 - + > 0 Often shows up on positive/negative questions > test a few cases if you are unsure > don't forget that you can COMBINE inequalities (addition if the signs are in the same direction) --> evaluating two of these together

Answer 127

Difference must be negative but... - + < 0 + + < 0 (| y | > | x |) - - < 0 (| x | > | y |) Often shows up on positive/negative questions > test a few cases if you are unsure > don't forget that you can COMBINE inequalities (addition if the signs are in the same direction) --> evaluating two of these together

Answer 128

Difference must be positive but... + + > 0 + - > 0 - - > 0 (| y | > | x | such as -1 - (-2) = 1) Often shows up on positive/negative questions > test a few cases if you are unsure > don't forget that you can COMBINE inequalities (addition if the signs are in the same direction) --> evaluating two of these together

Answer 129

x is an integer OR an irrational number (sqrt(2) - cannot be expressed as a simple fraction) TYPICALLY: Only an integer * integer = integer

Answer 130

AT LEAST ONE is negative (Recall: x PLUS y > < 0 have "at least one") - + < 0 + - < 0 - - < 0 e.g., 2x + 3y < 0 Often shows up on positive/negative questions > test a few cases if you are unsure > don't forget that you can COMBINE inequalities (addition if the signs are in the same direction) --> evaluating two of these together

Answer 131

I) single variable, one inequality sign > Sewing approach --> see if the answer must be true in light of the fact II) Single variable, multiple inequality signs and exponents > Functions --> draw it out very accurately (compare y values for A GIVEN value of x) III) Multi variable, multiple inequality signs > Number properties (fractions, positive/negative) -- test cases

Answer 132

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Full list of primes from 1 to 50: 15 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, 37 41, 43, 47

Answer 133

# Diagonals = [(n - 3)*n]/2 Logic: > each vertex can connect with n - 3 vertices > divide by 2 to get rid of duplicates Heptagon = (7-3)*7*0.5 = 14 Octagon = (8-3)*8*0.5 = 20 Nonagon = (9-3)*9*0.5 = 27 Decagon = (10-3)*10*0.5 = 35 **also the same approach for shaking hand questions # of shakes = [n*(n - # of ppl cannot shake)] / 2 e.g., 24 ppl need to shake hands, but you cannot shake hands with the people on your team (4 ppl per team) # of handshakes = (24 * (24 - 4))/2 = (24*20)/2 = 240 **Could also be viewed as a COMBINATION question Cm,r - restriction

Answer 134

Often about factors (i.e., combining into powers of 2, 5, multiples) **any factorial greater than 1 is EVEN e.g., 200! = p*10^q What is the max value of q? e.g., Is k a composite number? 13! + 2 <= k <= 13! + 13 13! + 2, 13! + 3, ... 13! + 13 ALL are MULTIPLES of an integer => composite!

Answer 135

Immediately write out the "other" percent e.g., Mrs. Lee's May income was 60% of her family's total income THEREFORE, her other family member's income was 40%

Answer 136

Units digit of 2 has a cycle of 4: [2, 4, 8, 6] Every EXPONENT that is a multiple of 4 has a units digit of 6 Exponent in this question: 8r + 6 > exponent is always 6 more than a multiple of 8 > or always two more than a multiple of 4 > so the remainder is FIXED regardless of the value of r (8r + 6)/4 = 2r + 6/4 Exponent is remainder 2, so the units digit is 4

Answer 137

Either a question about: 1) Slopes 2) SIGN of x and y coordinates > to be in the same quadrant, the x coordinates must have the same sign AND the y coordinates must have the same sign

Answer 138

Some overlap = A + B - overlap Complete overlap means that every member of B is ALWAYS a member of A > however, not every member of A is a member of B Total # of ppl = A + B - B = A (outside circle) **DON'T FORGET THE OUTSIDE PPL > always draw a box If you are trying to minimize the # of ppl --> maximize overlap # of ppl = A + B - overlap + neither

Answer 139

Time (therefore rate = output/time) or Rate (so rate*time = output) CONNECT RATE AND TIME in DS: rate = job/time

Answer 140

x% = x/100, where x is an value greater than 1 (e.g., 2%) 1-x% = (100 - x)/100 = 1 - (x/100) E.g., 1998 = 1997_amount * (1 + x/100) 1999 = 1997_amount * (1 + x/100) * (100 - x)/100

Answer 141

A) 3600 seconds in an hour B) 1440 minutes in a day C) VOLUME 1m = 100cm (start with the basic unit conversion, then transform) 1m^3 = 100^3 cm^3

Answer 142

TOTAL absolute value is IRRELEVANT Type 1) Percent represents VALUE > sum of individual percents adds to the total percent e.g., _ _ _ (n = 3) 25% , 25%, 50% (percents are individual TERMS) Type 2) Percent represents POSITION (e.g., percentile) > Focus in on PLACEMENT or ORDER of these values e.g., _ _ _ (less than 20), (20), (over 20)

Answer 143

#1) Draw out a few possible combos or perms first > if the answer choices are quite small, you can probably just list them out > other times, do in terms of GROUPS of options #2) Usually need to combine options with "+" #3) Combinations: Sometimes you may be GIVEN the # of arrangements and you have to WORK BACKWARDS to solve for the # of terms (m) e.g., Set A has 6 subsets with exactly 2 terms in them each. 6 = Cm,2 e.g., What is the least number of letters that can be used to create enough codes for 12 employees? 12 <= Cm,1 + Cm,2 #4) Total # of Possible Groups (or Subsets) = 2^n where n = # of terms > 2^n - 1 is for at least one term chosen > often needed when you are given the total number of SUBSETS #5) For perms, when in doubt - SLOT METHOD (# of options for each position * each other) > e.g., digit questions (sometimes it is helpful to organize by digit -- 7 in 000s, 8 in 000s, then 9 in 000s) ---> "paths" #6) For coordinate plane geometry questions, focus on determining the # of possible options for EACH POINT (often relative to the first one) e.g., Once you fixate point P (based on # of possible x * # of possible y), the other two points are fixed either vertically or horizontally #7) For combination AND permutation hybrid questions, you might need to divide by duplicates if order doesn't matter (The number of ways in which n different items can be divided equally into x groups, each containing y objects and the order of the groups is NOT important) C8,2 * C6,2 * C4,2 * C2,2 = # of possible teams of four with 2 people each, and ORDER of TEAMS matters (e.g., [1,2]-[3,4]-[5,6]-[7,8] is currently treated differently from [3,4]-[5,6]-[7,8]-[1,2]) > need to DIVIDE by # of groups! (e.g.., 4!)

Answer 144

0 - multiplication (pos/neg), exponent (a^0 = 1), multiple (0 is a multiple of every number) 1 - factor (1 is a factor of every number!!), exponents (1^n = 1) > e.g., 81/x 2 - even/odd, only even prime number

Answer 145

Perfect squares of DECIMALS 1.44 = (1.2*1.2) 14.4 = (1.2*1.2)*10 1.69 = (1.3*1.3) 1.21 = (1.1*1.1) 2.25 = (1.5*1.5)

Answer 146

Using compound inequalities e.g., When distance is rounded to the nearest integer,, Sally travelled 12 m 11.5 <= Distance < 12.5 e.g., when rounded to the nearest multiple of 100, 500 is closest to x 450 <= x <= 549

Answer 147

Integer * decimal (between 0 and 1) = Integer > rewrite decimal as a FRACTION > DENOM of the fraction must be entirely cancelled out Ex: 9x = integer x = (1/9), (2/9), (3/9), (4/9), (5/9)....(8/9) Ex: 10(x + 2y)/(x + y) = k x + y = 10 to cancel out the 10 in the numerator x < y: 1, 9 ---> k = 19 2, 8 ---> k = 18 3, 7 ---> k = 17 4, 6 ---> k = 16

Answer 148

Max height of the tunnel = radius Max Height of a Truck < Radius (because the truck has a WIDTH) Max Height of a Truck (assuming 0 space between height of the truck and the tunnel) = calculate using Pythagorean Theorem > hypotenuse = Radius > smaller leg = half the width of the truck > longer leg = max height

Answer 149

Cylinders: both need height V = (pi*r^2) * h -------> area of base * height SA: 2(pi*r^2) + 2*pi*r*h ----> 2 circles + rectangle Cone: V: 1/3*volume of a cylinder = (1/3)*(pi*r^2) * h SA: circle + sector = pi*r^2 + pi*r*L (symmetry) **Length is the hypotenuse in a right triangle with Height and Radius Sphere: V: (4/3)*(pi*r^3) SA: 4*pi*r^2 (no height for sphere) > both have "4" Hemisphere: 1/2 of a sphere V = (2/3)*(pi*r^3) *******SA: 2*pi*r^2 + pi*r^2 = 3*pi*r^2 = area of a sphere/2 + base Radius = Height **Differentiate between "curved part" and flat parts

Answer 150

RATIOSSSS (Ratio of sides) = constant factor = k > typically do large side/small side 2D: Ratio of Areas = (Ratio of Sides)^2 = (k)^2 3D: Ratio of Surface Areas = (Ratio of Sides)^2 = (k)^2 3D: Ratio of Volumes = (Ratio of Sides)^3 = (k)^3

Answer 151

FACTOR First one: a^2 - a + 6 ---> Quadratic factoring > Same method is used to factor square of a sum and square of a difference Second one: Difference of Squares Third one: factor out w Fourth one: Difference of squares pattern x - 11 = [ sqrt(x) + sqrt(11) ] * [sqrt(x) - sqrt(11)] = sqrt(x) - sqrt(11)

Answer 152

ax^2 +/- bx MINUS C ---> will have ONE positive and one negative solution > sufficient for positive constraint questions ax^2 +/- bx PLUS C --> will have to factor out to see if you get: > One solution (e.g., (a - b)^2) > Some other constraint in the question (e.g., n > 5)

Answer 153

Integer/x = integer -----> x is a divisor or FACTOR (list out factor pairs, INCLUDING 1) > x could also be a fraction x/integer = integer -----> x is a MULTIPLE of something x/y = integer ----> x is a multiple of y (y can be cancelled out)

Answer 154

#1) Try to SOLVE as much as possible for you to confidently declare Sufficient or Not Sufficient > e.g., 3 variables, 2 equations --> might be sufficient if THINGS CANCEL OUT #2) Make sure the equations are DIFFERENT and not ratios #3) Beware of traps (e.g., only one possible solution when there are integer variables, combo is sufficient)

Answer 155

ENOUGH = Inequality !!! Budget (up to $x to spend) = inequality Example: Spending <= Budget # of codes >= 12

Answer 156

1. What types of terms? > integers, positive integers, decimals 2. Can the terms be the same or must they all be different? a < b < c or a <= b <= c

Answer 157

= P(A) + P(B) - P(AandB) *** don't forget to subtract the SHARED COMPONENTS IF non-mutually exclusive (BUT you have to know WHAT is the RELATIONSHIP between A and B > 0 overlap > independent events: P(A)*P(B) > complete overlap: one circle inside the other (smaller probability) = 1 - P(not A and not B) ----> 1 - neither A nor B = 1 - (1 - P(A)) * (1 - P(B)) Therefore: 1 = P(A) + P(B) - P(AandB) + P(notA and notB) Think of two circles with possible overlap (intersection) MIN intersection is 0 MAX intersection is if one circle is INSIDE the other (choose the smaller probability)

Answer 158

LIST THEM OUT! > easier for you to visualize properties

Answer 159

a < b < c 3a < a + b + c < 3c Generalized: Smallest term*n < Range of sum for n terms < Largest term*n

Answer 160

Utilize your knowledge about the area of the SQUARE Area of the Overlap = (Area of each sector * # of sectors) - Area of the Square What about a large circle with four overlapping smaller circles inside? Area shaded = Area of Larger Circle - Area of Center Square - 4(area of half circles)

Answer 161

Hundreds digit must be 1 At most, you carry over 1 into the next column

Answer 162

Characteristics of Ratio Mixture Problems: > Group 1 and Group 2 have similar "features" that can be expressed as a RATIO or PERCENT (e.g., male/female, male/total, female/total, part-time/full-time) > Group 1 and Group 2 are COMBINED to form a larger group > The combined group has a related RATIO on the same features > Group 2 is often the NEW members added E.g., M/F Ratio of Group 1: M1/F1 Ratio of Group 2: M2/F2 If M1/F1 < M2/F2: M1/F1 < (M1 + M2)/(F1 + F2) < M2/F2 If M1/F1 = M2/F2: M1/F1 = (M1 + M2)/(F1 + F2) = M2/F2 In other words: If (M1 + M2)/(F1 + F2) > M1/F1, then M2/F2 MUST BE larger than M1/F1

Answer 163

Differentiate between "manufactured y" (output from the function given a value of x) and actual y (point you want to compare) > manufactured y is also ON THE FUNCTION It is helpful to express the equation of the line in the form: y = x > sub in a value of x on RHS > RHS gives you output (manufactured y, on the function) > LHS y is the ACTUAL y you want to compare Ex: 3x + 4y <= 60 Region 4y <= -3x + 60 y <= (-3x + 60)/4 ---> Means actual y must be equal to or less than manufactured y to fall in the acceptable region Therefore, whatever point (s, t), (r, s) etc. satisfies this inequality, the point falls in the region

Answer 164

If there is a Yes/No test case --> NOT ALWAYS TRUE Strategy - try to find a NO case to INVALIDATE the option - or solve algebraically to prove MUST BE TRUE

Answer 165

0.2r dollars for each additional hour OR fraction of an hour = round UP e.g., 7 hours --> Charge = 0.2r * (7 - 4) e.g., 7.5 hours ---> Charge = 0.2r * ( 7.5 - 4) = 0.2r * (8 - 4)

Answer 166

Complicated PS question with UNKNOWN -- plug in any value of x (e.g., 0 or 1) to evaluate > Use Smart Numbers when asked about UNKNOWNS and not given any information about the unknowns e.g., x = 0 then the sum simplifies to 30 A 1 B 2 C 3 D 4 ----> not a factor of 30 ** E 5

Answer 167

1) Even/Odd question x + y and x - y have the SAME TYPE --> E*E or O*O 2) Factor question > List out the factors of the integer > then solve the system of equations for x and y (might have an integer constraint)

Answer 168

1) No integer appears more than twice in the list => max # of duplicates is 2 (up to 2 repeats) 2) X is divisible by exactly 4 positive integers less than X => X has 4 factors OTHER THAN itself

Answer 169

Geometry circle questions > # of rev is often combined with RATES # rev * time per rev = Total Time # rev per time * Circumference per rev = Distance per time (speed)

Answer 170

For equations (=) ---> SQUARE BOTH SIDES For inequalities --> if both sides are POSITIVE, you can SQUARE BOTH SIDES In general, whenever you see a square root: > rationalize with conjugate > simplify and solve > or square both sides

Answer 171

Not A = Only B + NeitherANorB Not B = Only A + NeitherANorB 1 - P(A or Not B) = Only B 1 - P(B or Not A) = Only A

Answer 172

Divisibility Rule for 3 --> divisible if the sum of the digits is a MULTIPLE of 3 > otherwise, Remainder is FIXED = Sum of Digits - closest multiple of 3 e.g., 22 - 21 = R 1 e.g., 787/3 = 262 R1 Divisibility Rule for 9 --> divisible if the sum of the digits is a MULTIPLE of 9 > otherwise, Remainder is FIXED = Sum of Digits - closest multiple of 9 e.g., 22 - 18 = R 4 e.g., 787 / 9 = 87 R4

Answer 173

YES Regardless of carry over > look at sum of the digits a + 1 + b + 5 + c + 1 = 3*int a + b + c + 7 = 3*int a + b + c = 3*int - 7 Pattern in Remainder: 0 1 2 0 1 2 0 1 2 0 3int - 7 --> start from 0, move left 7 times. R = 2

Answer 174

Characteristics: > typically a RANGE PS problem (need to set up inequality, max and min) > typically relative to ONE PRICE > then either use range or percent change (+ or -) to calculate the range of values for the closing stock price READ CAREFULLY (relative to what price)

Answer 175

Try different groupings: e.g., 10z^2(z - 1) - 90(z - 1) = 0 (z - 1)(10z^2 - 90) = 0 10(z - 1)(z^2 - 9) = 0 10(z - 1)(z + 3)(z - 3) = 0 z = 1, -3, 3

Answer 176

Nonnegative integers 0! = 1 1! = 1 2! = 2*1 Any factorial greater than or equal to 2 is EVEN

Answer 177

Easier to view as a PRODUCT e.g., | j | * j = 1 ----> j must be > 0 j^2 = 1 j = + 1

Answer 178

NO > side lengths DO NOT have to be integers (6-8-10) Could be anything that satisfies the Pythagorean theorem: 6^2 + b^2 = c^2 e.g., 6^2 + 1^2 = (sqrt(37))^2

Answer 179

If you know: - n terms and either the A) First term, B) Last term, C) Average or Median NS if you just know: - range and n terms (gap between first and last term)

Answer 180

Line drawn from one vertex to the midpoint of the opposite side Each median divides the triangle into smaller triangles with the SAME AREA Median of a right triangle's length (drawn from the 90 degree to hypotenuse) = 1/2 * hypotenuse Proof: > Circle method --> diameter represents the hypotenuse, 90 degree angle is on the circle > each segment represents the radius Special application: Right Isosceles triangle's median = splits triangle into two congruent right isosceles triangles (follows isosceles altitude property) AND median's length = 1/2 hypotenuse > half of a square

Answer 181

Opposite angles are equal Sum of other two angles in Triangle 1 = Sum of other two angles in Triangle 2

Answer 182

Concave angle = sum of three angles

Answer 183

Choose one variable to be "x", then make all other variables the coefficient. Factor using methods: - criss cross - difference of squares - simple factoring/grouping e.g., let x be "x" x^2 - (6y)x - (5z)x + 9y^2 + 15yz + 6z^2 x^2 - x(6y + 5z) + (9y^2 + 15yz + 6z^2) ---> factor last term using criss cross method x^2 -x(6y + 5z) + 3(3y^2 + 5yz + 2z^2) x^2 - x(6y + 5z) + 3(y + z)(3y + 2z) ---> factor using criss cross method (x - 3y - 3z)*(x - 3y - 2z) = 0

Answer 184

Two angles at the corner = sum to 90 degrees Four congruent triangles (ASA)

Answer 185

Prove the certain angles are equal (angle rules) 1) Corresponding angles (F shape) are equal 2) Alternate angles (Z shape) are equal 3) Interior angles are (C shape) complements (add to 180 degrees) Likewise, if you KNOW two lines are parallel, angle rules apply

Answer 186

Altitude of an isosceles triangle with the incongruent side as the base = Median > splits the triangle into two congruent triangles > incongruent angle is split into half PROOF: > Median bisects the base (incongruent side) so that SAS = two triangles are congruent > To prove 90 degrees, use exterior angles Angle 1 + 2 + alpha = 180 degrees Angle 3 + 4 + beta = 180 degrees Angle 1 + 3 + 2 + 4 = 180 degrees Therefore: 180 - alpha + 180 - beta = 180 alpha + beta = 180 AND alpha = beta (because of congruency) Therefore alpha = beta = 90 degrees Special applications: > Equilateral triangle --> altitude splits into two congruent triangles > all 3 medians/altitudes are the SAME

Answer 187

Lines drawn from the same point on an angle bisector and are perpendicular to each ray are EQUIDISTANT Proof: Two congruent triangles --> AAS (90-angle-shared side)

Answer 188

Use the Angle Bisector Theorem - Lines drawn from the same point on an angle bisector and are perpendicular to each ray are EQUIDISTANT Prove that the third segment bisects the angle

Answer 189

INcenter = inscribed circle in a triangle Property: Angle Bisectors Rule: Angle Bisector Theorem > Lines drawn from the same point on an angle bisector and are perpendicular to each ray are EQUIDISTANT (AAS proof) > shared point = incenter = center of the circle > equidistant segments = radius

Answer 190

CIRCUMcenter = Circle is outside the triangle (inscribed triangle) Property: Perpendicular bisectors (from the circumcenter to each side of the triangle) Rule: Perpendicular Bisector Theorem > Each line drawn from a point on the perpendicular bisector to an endpoint of the segment IS EQUAL in length > creates isosceles triangles

Answer 191

Each line drawn from a point on the perpendicular bisector to an endpoint of the segment IS EQUAL in length

Answer 192

> Can be on the line connecting A and B (midpoint) > Can be on ANY POINT on the line that is a perpendicular bisector of the line (recall perpendicular bisector theorem)

Answer 193

No > Smallest prime number is 2

Answer 194

| x | | x - 3 |

Answer 195

Cross multiply: x^2 = y^2 or | x | = | y | or x = +/- y and y = +/- x

Answer 196

Power rule - sprinkle (multiply) outside exponent by inner exponents (x^a)^b = x^(ab) = (x^b)^a Product of a power rule - sprinkle (multiply) outside exponent by inner exponents (x^a * y^b)^c = x^ac * y^bc Power of a fraction rule - sprinkle (multiply) outside exponent by inner exponents (x / y)^a = x^a / y^a Long way - multiply the argument in the brackets b number of times E.g., (5^sqrt2)^2 becomes (5^sqrt2)*(5^sqrt2) > then you can PULL THE EXPONENT OUT = (5*5)^sqrt2 (because the exponent sprinkles onto the stuff in the brackets NOTE: > REVERSE RULES APPLY TOO > with multiplication and powers, you can rearrange order (as long as the outcome is the same) e.g., 5^(2sqrt2) = (5^2)^sqrt2 = 25^sqrt2

Answer 197

Product rule --> same base, add exponents x^a * x^b = x^(a + b) Quotient rule --> same base, subtract exponents in order x^a / x^b = x^(a - b) Therefore, REVERSE RULES APPLY TOO

Answer 198

Product of a power rule - sprinkle (multiply) outside exponent by inner exponents (x^a * y^b)^c = x^ac * y^bc

Answer 199

a and b can be integers or fractions > don't be afraid to test square roots given the square e.g., a = sqrt(24), b = 1 e.g., a = sqrt(27), b = 2

Answer 200

This is a COMBINATION QUESTION > any 3 points chosen can create a triangle C6,3 = 20 For more complicated XY plane questions (e.g., right angled triangles, dots can be on the same line), start with the overall number (combination), then subtract lines and other invalid shapes

Answer 201

No = cannot deduce that fraction squared = int means that the fraction is an int, UNLESS we know K and L themselves are integers K and L could be irrational numbers e.g., K = sqrt(2) and L = 1

Answer 202

Choose one equation as the main one (e.g., Equation #1) Then subtract remaining two equations from Equation #1 Then see what the results are and solve for each variable

Answer 203

Independent events: P(A AND B) = P(A) * P(B) Conditional (dependent) events: P(A AND B) = P(A) * P(B | A) Tip: it is helpful to think of DECISION TREES (or branches) to visualize the different outcomes > each branch represents a different outcome ("or") > Each branch has several events that must happen for the final outcome to occur ("and")

Answer 204

Logic question Ans is sufficient with option A (only statement 1) Think about the options if you pick ANY 2 snakes: {V,V} or {V,C} or {C,C} > since there is AT LEAST one viper, {C,C} is NOT a valid option > and we know C>=1 so {V,V} is NOT an option > therefore, C = 1 Lesson: If you think the answer is C, double check each statement individually especially as you learn more > or just always definitely prove the statement is insufficient

Answer 205

1) Rule for POSITIVE FRACTIONS (proper and improper): If you ADD the exact same number to both the numerator and denominator, the resulting fraction gets closer to 1 (could increase or decrease towards 1) Order from least to greatest: 1/2 < 2/3 < 11/12 < 111/112 2) Another Rule: when COMPARING fractions, CROSS MULTIPLY and compare the products > this works because we are essentially finding the common denominator and comparing numerators e.g., Is 2/3 < 11/12? Is 24 < 33 (Yes, so 2/3 < 11/12)

Answer 206

Max # people only 1 (only raspberry) --> minimize # of overlap Formula: 100% = A + B + C - (AB + BC + CA) + center + other 100% = 56% + 44% + 40% - 30% - AR - SR + center + other 100% = 110% - AR - SR + center + other BECOMES: 100% = 110% - ARonly - center - SRonly - center + center + other 100% = 110% - ARonly - SRonly - center + other To minimize the overlap with R, we want AR only, SRonly, and center to be as small as possible: ARonly + SRonly + center = 10% + other ----> make other = 0% Therefore we know R total = 40% So R only = R total - SRonly - ARonly - center = 40% - 10% = 30%

Answer 207

Both ADDITION and MULTIPLICATION have the commutative and associative property Commutative Property: Order does not matter a + b = b + a a * b = b * a Associative Property: Bracket placement does not matter (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Subtraction and division both LACK commutative and associative property

Answer 208

(1) long division (2) convert fraction to a denominator that is a power of 10 > allows you to move the decimal place e.g., 53/5000 = 106/10,000 = 0.0106

Answer 209

What could the MAX value of x and y be? > set other variable equal to 0 Max value = 10

Answer 210

Either convert to the SAME base or SAME exponent e.g., 2^300 = 8^100 3^200 = 9^100 So 3^200 > 2^300 OR scale down exponents (taking root where n is GCF) e.g., GCF of exponents is 100 so becomes 2^3 vs 3^2 ---> 3^2 and therefore 3^200 is bigger

Answer 211

Approach using TIERS > consider whether to count duplicates (i.e., if we are counting DIGITS) or exclude them (i.e., if we are counting INTEGERS) Example: how many times does the digit 7 appear from 1 to 1000? HUNDREDS digit: [700, 799] = 100 numbers with 7 in hundreds digit TENS digit: [70, 79] for all numbers between 1 to 1000 = 10 numbers with 7 in tens digit per each hundred range * 10 hundred ranges = 100 ONES digit [07, 17, 27, 37, 47, 57, 67, 77, 87, 97] = 10 * 10 = 100 TOTAL = 300 times 7 appears as a digit

Answer 212

Treat like algebra, works for rectangles and squares e.g., 1 cm = 4m --> square both sides to get area relationship 1 cm^2 = 16m^2 PROOF: > if you use that linear relationship to get two cm and multiply the two together to find area of rectangle, you will get same area relationship a cm = b m a * x = b * x a * m = b * m therefore: a^2 * x * m = b^2 * x * m a^2 = b^2 (same as if you squared a = b) Another example: 2 cm = 8m 3 cm = 12m area if 6 cm^2 = 96m^2

Answer 213

Extend the side (visually with dotted line) Exterior angle = 180 - interior angle Fun fact: sum of exterior angles of a quadrilateral also equals 360 The sum of exterior angles of a triangle is equal to 360°

Answer 214

No (at least for GMAT no)

Answer 215

Remember that you can MULTIPLY positive values to both side without changing sign Case 1) Assume Lf < Lg Multiply both sides by positive value Wf to get Lf*Wf < Lg*Wf Then separately we know Wf < Wg Multiply both sides by positive value Lg to get Lg*Wf < Lg*Wg COMBINE inequalities together: Lf*Wf < Lg*Wf < Lg*Wg Lf*Wf < Lg*Wg Area F < Area G (False) Case 2) Assume Lf > Lg Perform same steps to arrive at Area F > Area G (True) Sufficient

Answer 216

Express units as ratio and numbers as ratio too cm (C) to meters (M) or as a fraction C / M = 1 / 5 Cross multiply to get 5C = M If C = 1, M = 5 If C = 2, M = 10

Answer 217

x*sqrt(3) Proof: Each face is a square that follows special right angle x-x-x*sqrt(2) Diagonal is also a right triangle with x and x*sqrt(2) as legs More broadly: > if you know surface area (6*x^2), you know volume (tied by knowing side)

Answer 218

Concept: Similar triangles mean the triangles have PROPORTIONAL side lengths and heights, and EQUAL ANGLES Just prove that TWO angles are equal (AA rule) Ex: think of overlapping triangles with a parallel line through the triangle

Math Special Flashcards

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