Math Flashcards

Question

|a| < b as an inequality

Answer 1

then: -b < a < b

Answer 2

Always 1 (special)

Answer 3

4 cycles: 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 6 2^5 = 2 Exponent divisible by 4 --> 2^4 unit (6) Anything else --> remainder dictates which one to use e.g., R = 1 --> 2^1

Answer 4

4 cycles: 3^1 = 3 3^2 = 9 3^3 = 7 3^4 = 1 3^5 = 3 Exponent divisible by 4 --> 3^4 unit (1) Anything else --> remainder dictates which one to use e.g., R = 1 --> 3^1

Answer 5

2 cycles: (special) 4^1 = 4 4^2 = 6 4^3 = 4 4^4 = 6 Odd exponents = 4 Even exponents = 6

Answer 6

Always 5 (special) 5^1 = 5 5^2 = 25 5^3 = 125

Answer 7

Always 6 (special like 5) 6^1 = 6 6^2 = 6 6^3 = 6

Answer 8

4 cycles: 7^1 = 7 7^2 = 9 7^3 = 3 7^4 = 1 7^5 = 7 Exponent divisible by 4 --> 7^4 unit (1) Anything else --> remainder dictates which one to use e.g., R = 1 --> 7^1 = 7

Answer 9

4 cycles 8^1 = 8 8^2 = 4 8^3 = 2 8^4 = 6 8^5 = 8 Exponent divisible by 4 --> 8^4 unit (6) Anything else --> remainder dictates which one to use e.g., R = 1 --> 8^1 = 8

Answer 10

2 cycles (special like 4): 9^1 = 9 9^2 = 1 9^3 = 9 9^4 = 1 Odd exponent = 9 Even exponent = 1

Answer 11

FIRST SIMPLIFY before dropping all other digits!! (Same base, exponent rules!) = 6^36/6^30 = 6^6 Units = always 6

Answer 12

Long division is probably the fastest.

Answer 13

0** (0^2) **1 IS A PERFECT SQUARE (but not a prime number) 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 625

Answer 14

Because w could be equal to 0! (Cannot divide by 0) Instead, rearrange and solve as a quadratic equation: 3w^2 - w = 0 w(3w - 1) = 0

Answer 15

No need to expand! Recognize the perfect square Square root both sides: z + 3 = +/- 5 z = 5 - 3 = 2 OR z = -5 - 3 = -8

Answer 16

Difference of squares (x + y)(x - y) ---> SPECIAL PRODUCT (different signs) e.g., 9x^2 - 16 = (3x + 4)(3x - 4) e.g., a^2 - 1 = (a + 1)(a - 1) e.g., x^2 - 4 = (x + 2)(x - 2) Recognize special ones: (a + b + c + d)(a + b - c - d) = 16 ANY EVEN POWER can be converted to a difference of squares a^8 - b^8

Answer 17

M: 1 A: 2 (1, 1) (x + y)^2 = (x + y)(x + y) Tips: > Look for perfect squares > recognize x and sqrt(x), and reciprocals --> x - y = (sqrt(x) + sqrt(y))(sqrt(x) - sqrt(y))

Answer 18

M: 1 A: -2 (-1, -1) (x - y)^2 Tips: > Look for perfect squares > recognize x and sqrt(x)

Answer 19

a^2 + 2ab + b^2 e.g., x^2 + 6x + 9 = (x + 3)^2

Answer 20

Root in conjunction with a quadratic equation means SOLUTION A: k M: 8 x = -4 is a solution x + 4 = 0 (4, 2) K = 6 Or algebraically by subbing in x = -4: (-4)^2 + (k)(-4) + 8 = 0 k = 6

Answer 21

A collection of numbers in a set ORDER according to a specific RULE > ORDER MATTERS IN A SEQUENCE (don't treat it like a set where you can change the order from least to greatest) e.g., {1, 4, 9, 16, 25} --> each number is called a TERM Rule is An = n^2 (based on the term's POSITION) Special type of sequence (Recursive) e.g., An = An-1 + 2 (based on the PREVIOUS items Tips: > write out the first five terms of sequence to try to find out the PATTERN x 2(e.g., find the TERMS AND find the SUM, if asked about sums, or DIFFERENCE of terms if asked for difference) > use a number line for sum/subtraction questions (e.g., ranges of the sum of the first 10 terms)

Answer 22

Exponential Growth: Size = a1 * (3) ^ (time/10min) Helpful to create a table: (since "ago" makes time different to track) 20 mins ago ----> pop = 100 10 mins ago ---> pop = 100*3 = 300 NOW --> 900 10 mins --> 2700 20 mins --> 8100 30 mins --> 24,300 ***about 30 minutes In some cases, you might need to pick a SMART NUMBER as the starting point. OR formula: Population = A(rate of increase or decay)^(time/increment in time)

Answer 23

Inverse proportionality Q: yx = k (constant) --> products or y = k/x AR = AR 4R = A(1/3R) 4*3 = A A = 12

Answer 24

Direct proportionality Q: Think ratios y/x = k or y = kx y/x = y/x h/v^2 = h/v^2 4/16^2 = 9/v^2 CROSS MULTIPLY v = 24

Answer 25

Linear Growth problem = Arithmetic Sequence! Arithmetic Sequence Approach: An = a1 + (n - 1)*d ----> goal is to find d a1 = 4.5 a2 = 4.5 + d ---> height at 13 a4 = 4.5 + 3d = 1.2a2 --> height at 15 sub in a2 into a4 and solve d = 0.5 feet = 6 inches

Answer 26

Method 1) Sub in 1/x into the function to see if it simplifies to f(x) > f(x) changes for EACH answer! (don't compare to the incorrect equation!) Tip: use =? to compare LHS to RHS Method 2) Test cases --> see which function yields the same output for both x and 1/x (easier)?

Answer 27

CONCEPT - any multiple of 10 has a units digit equal to 0 e.g., 10, 20, 30, etc So if 6 is in the hundreds digit of 10n, then 6 is in the tens slot of n. Sufficient Also: 10n = 6 _ _ n = 6 _

Answer 28

Digit constraint! 1. Digits must be between 0 (or 1) and 9!! (Might not be mentioned explicitly in the question). 0 <= digit <= 9 e.g. 3P5 + 4QR = 8S4 and Q = 2P - We know that Q = 2P <10 or P < 5!! 2. For unknown addition or subtraction relating to VALUE, it is helpful to write out expressions with place values: e.g., 54 = 10*5 + 1*4 ab = 10*a + b 3. Sometimes equations can be helpful to show FACTORS of digits e.g., 11(B - 9A) = C ---> C is divisible by 11 (C = 0)

Answer 29

x^2 < x x is a fraction between 0 and 1 0 < x < 1 > number raised to an even exponent is SMALLER than original number only works if the fraction is between 0 and 1

Answer 30

NO - we don't know the sign of y so we don't know if we have to flip the sign x < 9/y or x > 9/y Applies to multiplication and division involving variables

Answer 31

Multiple inequalities combined into one inequality (two or more inequality signs) e.g., -4 < x < 4 TIP: Rearrange the inequalities so that the symbols point in the same directions > not exactly the same thing as adding two inequalities (still get one inequality with one inequality sign)

Answer 32

Perform operations on EVERY TERM (+ - * /) e.g., multiply every term by y (assume y > 0) 2y < xy < y^2

Answer 33

Yes, but only ADDITION (you CANNOT SUBTRACT or divide inequalities) > applies to compound inequalities too (e.g., 8 < x < 10) 1) Rearrange the inequalities so that the symbols face the same direction e.g., a < c b < d 2) Then add each side a + b < c + d

Answer 34

0) UNDERSTAND what it means to max or min the value of something > how high can xy go? (infinity) > how low can | a - b| go? (0) 1) Simplify inequalities, wherever you can 2) Look at the EXTREME ends of each range - pay attention to whether values can equal 0 or be negative. 3) Consider different scenarios that can lead to max or min values In this example, max xy = 30

Answer 35

Two cases: x >= 3 AND x <= -3

Answer 36

A > B OR A < - B

Answer 37

Linear inequality word problem 42 = r + c r= ? (1) r > 26 NS - r can be anything from 26 to 42 (2) r < 2c NS OR algebra: sub c = 42 - r (equality) into r < 2c (inequality) r < 2(42 - r) r < 84 - 2r 3r < 84 r < 28 (not sufficient) TOGETHER: 26 < r < 28 r = 27 (sufficient - C)

Answer 38

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 -+) > a / b > c / d AND b / a > d / c ONLY WHEN the two sides have different signs. e.g., -6 < 2 (different signs, don't flip the inequality) 1/-6 < 1/2 SPECIAL CASE: dealing with absolute values and positive powers --> always positive

Answer 39

Depends on if you KNOW the signs of both sides of the inequality (just like with reciprocals). Why? - because if there's a variable, squaring it means multiplying that side with a variable Rule of Thumb: If the signs are UNCLEAR or One side is Positive and One side is Negative, then you CANNOT SQUARE Otherwise: A) If both sides are known to be negative (e.g., x < -3), then squaring both sides needs to come with a FLIP of the inequality sign x < - 3 x^2 > 9 B) If both sides are known to be positive (e.g., x > 3), then squaring both sides does not need to flip the inequality sign x > 3 x^2 > 9 SPECIAL CASE: dealing with absolute values and positive powers --> always positive

Answer 40

One variable inequality: Sewing approach > cannot multiply both sides by x because we don't know what the sign of x is! Move 1/3 to the left side and do sewing approach: (x - 12)(3x) > 0 x < 0 , x > 12

Answer 41

x >= 3 OR x <= - 3 ----> x is negative, so flip the sign and add neg sign

Answer 42

This question has a HIDDEN INTEGER CONSTRAINT (people cannot be split into fractional parts) Max # accepted students = 50/3 = 16.667 --> 16 (not 17) Max # of rejected students = 50 - 16 = 34 (NOT 33!!)

Answer 43

Linear Inequality Word problem: Q: Is Nx > Ny? (1) We are given only info about Saturday and nothing else nx > 1000 ny < 1000 (n represents saturday) NS (2) nx < 0.2Nx ny > 0.2Ny We cannot figure out the actual number of books sold, Nx and Ny. NS (3) TOGETHER - COMBINE the inequalities 1000 < nx < 0.2Nx ---> 1000 < 0.2Nx Nx > 5000 0.2Ny < ny < 1000 ---> 0.2Ny < 1000 Ny < 5000 So Nx > 5000 (C) >>> Ny < 5000 < Nx

Answer 44

Divide both sides by ad/bc (positive number) ad/bc < 1 a/b < c/d

Answer 45

Concept: Rate (unit over time) * time = distance --> or set up ratios: Rate = Distance/time Distance Travelled East = Total distance travelled divided by 2. Total distance travelled = 2.75 + distance in 45 mins = 2.75 + (45/12) = 2.75 + 3.75 = 6.50 Distance Travelled East = 6.5/2 = 3.25km

Answer 46

Find the COMBINED rate at which the distance between the bodies changes A: Distance decreases at a rate of 5 + 6 = 11 mph. Extension - time to cover distance between --> use combined rate. - if the answer isn't as pretty --> use draw out technique to narrow choices down (see when two people have passed) B. Distance increases at a rate of 30 + 45 = 75 mph. C. Distance decreases (faster one is chasing slower one) at a rate of 3 mph Gap = (combined rate)*time

Answer 47

ADD (or subtract) the RATES to get combined rates e.g., Rate A + Rate B = Total Rate 1/3 + 1/2 = 5/6 of the task gets completed every hour. **to find total time to complete the task --> (5/6)*time = 1 time = 1/(5/6).

Answer 48

Ans is either D (each work) or E (neither work). - ans cannot be A, B, or C *Sometimes you have to simplify further to see if the equations are identical or different!

Answer 49

Rate Question: Both statements individually are insufficient to calculate distance! > while Al drove for a LONGER time, he could have had a faster speed and covered more distance than Barb > while Al drove at a slower speed than Barb, he could have driven longer and covered more distance Together sufficient --> 3/8 (because it is a RATIO)

Answer 50

Work Problem: M < N is asking whether Rate of M is faster than Rate of N (so Mary is spending LESS time than Nadir). We also know: combined time (t) < M combined time (t) < N Is m/2 < t < n/2 ? (1) m/2 < t Algebraically: Combined Rate = 1/m + 1/n Combined time = 1/(1/m + 1/n) Statement: 1/(1/m + 1/n) > m/2 SOLVE: (Hidden positive number constraint). n > m (sufficient) (2) t < n/2 Statement 1/(1/m + 1/n) < n/2 SOLVE: m < n (sufficient) SENSE CHECK: - if the combined time is greater than m/2, it means that Nadir is SLOWING Mary down (or spending more time). M < N - If the combined time is less than n/2, it means that Mary is FASTER (or spending less time). M < N

Answer 51

MUST BE TRUE PS with no measurements - you can choose any test case that's valid Or Solve Algebraically: * a, b, c must be positive numbers. Manipulate the inequality in question to get to the statements OR use test cases I) test case fails ii) b < c --> SUBTRACT a from both sides b - a < c - a (statement is true) III) b < c --> divide both sides by a b/a < c/a (statement is false)

Answer 52

Distance between -5 and 3 is 8. So middle value is -1. If you add 1 to every term: -5 + 1 <= x + 1 <= 3 + 1 -4 <= x + 1 <= 4 | x + 1 | <= 4

Answer 53

If given #s --> use the given #s If given percents (and asked to find a relative figure) --> Use 100 as the total If given fraction s--> Use a common denominator as the total (e.g., 1/3, 1/4 --> choose 12) Also PAY ATTENTION TO WORDING --> "10% of all cars are red and have heated seats" (Intersection of red/heated seats - 10% of TOTAL) vs. "10% of cars with heated seats are red" (intersection of red/heated seats - 10% of HEATED SEATS)

Answer 54

Eq'n 1: Old Avg = 800 = Sum/n Eq'n 2: New Avg = 900 = (Sum + 2000)/(n + 1) # of sales made = n + 1 Solve for n: 800n = Sum --> sub into eq'n 2 900(n + 1) = 800n + 2000 900n - 800n = 2000 - 900 n = 11 # of sales made = 12

Answer 55

*Always arrange the data in increasing order ODD Number of Values: > Median is positioned at the middle > Position: # of values/2 = fractional value ROUND UP > Median is a fraction with DENOM = 2 e.g., 13/2 = 6.5 ROUND UP = position 7 **Median is a number in the set. EVEN Number of Values: > Median is the average of the two middle values > Starting Position: # of values/2 > Ending Position = Starting Position + 1 e.g., 10/2 = median is average of # at 5 and # at 6. **Median is does not have to be a number in the set (unless two middle values are equal).

Answer 56

Usually No - Multiple possible combinations of numbers can result in the same average and same SD Exception: SD = 0 --> means all the numbers in the set are EQUAL - Can figure out what numbers are in the set (e.g., avg is 10, meaning all the numbers is equal to 10) - However, you still cannot figure out how many instances of 10 are in the set.

Answer 57

Likely won't be asked to calculate specific SD = sqrt( sum of squared deviations / N) 1) Changes to SD resulting from transformations to the set --> how does the data move relative to the mean? (Farther, closer, or stay the same?) > constant difference to all terms = no change in SD > factor applied to all terms = change in SD by the factor 2) Comparisons of SDs in multiple sets --> which set has numbers that are furthest away from the mean? > dot method Helpful to draw a picture (number line!) When adding a new term to a list, how do you INCREASE standard deviation? DECREASE? > Increase by adding a new term that is GREATER THAN +/1 standard deviation from the mean > Decrease by adding a new term that is LESS THAN +/- 1 standard deviation from the mean e.g., Original mean is 500, st dev = 50. Add new set with average 500, st dev 25 > will decrease standard deviation --> values are closer to the mean 500.

Answer 58

- Ans choices are close to one another --> need to do a more precise division = 190/21 21*7 = 147 21*8 = 168 21*9 = 189 (closer to 190) --> 9 21*10 = 210

Answer 59

Possibly, check three cases: 1) Unknown is less than middle value of the set 2) Unknown is the middle value of the set 3) Unknown is greater than the middle value of the set.

Answer 60

No --> compare relative to the simple average > which end pulls the average more? WA = (Data1*# of 1s + Data2*# of 2s)/(# of 1s + # of 2s) > if there are more 1s than 2s, then the WA will be pulled toward Data1 > To estimate percentages (PS) --> Ask yourself whether the average is closer to the extreme end or simple average (50/50) e.g., Brand A has 40% millet and Brand B has 65% millet. Customer purchases a mix of the two types of birdseed that has 50% millet, what percent of the mix is Brand A? > Simple average of millet is 52.5% (greater than 50% in mixture) > Mixture has more of Brand A, and difference from simple average is not that high --> Ans: 60% (not 85%). Alternatively do the visual approach: Wa > Wb Wa = 15/25 = 3/5 = 60%

Answer 61

Weighted Average Problem Simple Average of ticket prices = (10 + 25)/2 = 17.5 (1) 18.25 > 17.5 --> more A than C Sufficient (2) Simple Average of Revenue (assuming all 100 tickets were either A or C) = (100*10 + 100*25)/2 = 1750 1800 > 1750 --> More A than C Sufficient

Answer 62

Y represents the ADDITIONAL weight added to the truck that causes it to go from 1/4 to 7/8 In other words, 2/8*M + Y = 7/8*M Y = 5/8*M (Change in Weight!) M = 8Y/5

Answer 63

5% * (Sales - 1000)

Answer 64

33.333% (33 + 1/3)/100 (100/3)/100 = 1/3

Answer 65

Meaning of the question: A total of x bikes are sold each year from domestic and foreign producers! % of domestic production changed from 42% to 33% = 9% drop. ans: 9% of X WE DON'T CARE about the other years in between 1990 and 1993. So the DECREASE in the annual number of bikes produced and sold in Western Europe is just equal to 9%x

Answer 66

No Better to write it out: 15% more in the past -> Dec = 1.15Jan 85% of the past --> Jan = 0.85Dec ---> 1/0.85 = 1.18 (not 1.15)

Answer 67

COUNTING Consecutive Integers --> ints that go up by 1 (Last - First)/increment + 1 ---> because the lower extreme is subtracted out and not included in the count. = 765 - 14 + 1 = 752

Answer 68

COUNTING Consecutive Multiples (Last - First)/increment + 1 ------> adjust the Last and First bounds to be inclusive = (24-12)/2 + 1 = 7 For short ranges, it may be easier to list the terms of the pattern and count the number of integers. More generally: Consecutive ints (increments of 1) SUM = n + (n + 1) + (n + 2) + ...

Answer 69

COUNTING Consecutive Multiples (Last - First)/Increment + 1 ---> **using least and greatest multiples of 7 as Last and First Numbers =(77 - 14)/7 + 1 = 10

Answer 70

CONCEPT: EVENLY SPACED SET (multiples of 3) Mean = Median = (First + last)/2 = 7.5

Answer 71

CONCEPT: EVENLY SPACED SET (increments of 5) Mean = Median = (First + Last)/2 = (22)/2 = 11

Answer 72

CONCEPT: EVENLY SPACED SET (increments of 1) Mean = median = (first + last)/2 = (70)/2 = 35 > if not inclusive, adjust first and last so that they include it

Answer 73

CONCEPT: Evenly Spaced Set (increment of 1) --> SUM = Average * # of terms Average = Median = (First + Last)/2 = (300 + 200)/2 = 250 # of terms = (Last - First)/1 + 1 = (300 - 200) + 1 = 101 SUM = 250 * 101 = 25250 > if not inclusive end points, adjust first and last so that they include it (for both avg and # of term calculations)

Answer 74

Recognize the overlap. Real problem is: SUM of 100 to 124 (not 125!) = 2800 (using Evenly spaced set tips)

Answer 75

3 sets --> Use Venn Diagram and work from the inside out! Don't forget to subtract overlap for each of the overlapping parts! Ans: 10 + 23 + 29 + 1 + 5 + 2 + 4 = 74

Answer 76

5! = 5 * 4 * 3 * 2 * 1 = 120 CONCEPT - For any number of consecutive integers, the product of k consecutive integers is always divisible by k factorial. > Any series of 5 consecutive integers will contain at least one integer that is a multiple of 5, 4, 3, and 2.

Answer 77

If k number of consecutive integers is ODD --> yes - e.g., the sum of 5 consecutive integers is a MULTIPLE of 5, so the sum is DIVISIBLE by 5 (k) If k number of consecutive integers is EVEN --> No - e.g., the sum of 4 consecutive integers is NOT a multiple of 4, so the sum is NOT divisible by 4 (k)

Answer 78

Scheduling problem - consider extreme possibilities Gist of the problem --> don't know the exact date the product was purchased! (1) Shortest Warranty period --> Bought Jan 31, warranty, Feb 1 to March 1 (29 days later = 28 days in Feb + 1 day in March) Longest Warranty period --> bought is Jan 1, warranty from Jan 2 to March 31 (30 days in Jan + 28 days in Feb + 31 days in March = 89 days) NS (2) Shortest Warranty period --> Bought it May 1, warranty due May 2 (1 day) Longest Warranty period --> Bought it May 1, warranty from May 2 to May 31 (30 days) NS Together - NS

Answer 79

*Reason it out using previous concepts with multiples, divisibility, and consecutive multiples rst = 27*(m)*(m + 1)*(m + 2) - rst contains 4 3's - at least one of the numbers is EVEN 27 can be broken into its factors with the three 3's 54 can be broken into its factors with three 3's and one 2

Answer 80

Scheduling Question - consider smallest and largest times > however, the second statement restricts the largest time difference, so there is no need to consider the largest time difference imposed by statement 1 (saves time!) Statement 1 --> Eim is at least 10 hours ahead of N (min diff) Statement 2 --> Eim is at most 13 hours ahead. (max diff) So if it is 12pm Friday in N, Eim's time is between 10pm Friday and 1am Saturday.

Answer 81

DON'T DO LONG DIVISION Just simplify by cross multiplying and evaluating the inequality: 11*100 > 3 * 301 ---> so true

Answer 82

Consecutive integers and statistics: (1) if n is even, then the mean = median is NEVER an integer value (won't be a term in the set). > Therefore, avg cannot equal 1 > Sufficient (2) Test cases show that always false --> number of terms will be EVEN to make this statement true > Therefore, avg cannot equal 1 (will always equal 0.5) > Sufficient

Answer 83

NOT SUFFICIENT if n = 1 -- statement 1 says k > 2. If k = 3, then k > 3 is false. Otherwise, if k = 4, both inequalities are true. CONCEPT --> inequality DS questions need more simplification!

Answer 84

If the integer is divisible by 2 twice OR For larger numbers, the LAST TWO digits are divisible by 4

Answer 85

If the integer is divisible by 2 three times OR For larger numbers, if the last three digits are divisible by 8

Answer 86

If the sum of the digits is divisible by 9 (kind of like divisibility rule for 3). **powers of 9 are all divisible by 3

Answer 87

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Tip for identifying higher level primes: > see if it is divisible by 3 and 7 (and obviously must be odd)

Answer 88

If a is a factor of b, and b is a factor of c, then a is a factor of c. > An integer is divisible by its factors and the factors of ITS factors. For a^k to be a factor of b^m, a must be a factor of b AND k <= m

Answer 89

FACTOR BOX: SOMETIMES YES, SOMETIMES NO Partial prime box for r: 2 2 2 2 5 ...? > We DON'T KNOW if there are other prime factors in the box No example: r = 80 Yes example: r = 80 * 3 = 240

Answer 90

To get a negative number, you need an ODD NUMBER OF NEGATIVE SIGNS (watch out for 0!) > regardless of the # of positive numbers To get a positive number, you need an even number of negative signs

Answer 91

Properties of a product of numbers: is there an ODD # of negative numbers? (1) NS --> don't know how many negatives (2) NS --> while there are an odd number of negative numbers in set S, there could be 0 in the set. This would make the product = 0. TOGETHER --> If all the elements are negative and there are only 5 negative numbers --> Yes, the product is negative. Sufficient (c) Recall: 0 is NEITHER POSITIVE NOR NEGATIVE

Answer 92

Not necessarily! Concept: There is NO GUARANTEES for division (unlike Multiplication). e.g., 60/20 = 3 21/7 = 3

Answer 93

Odd/Even properties x is an integer (1) 2y + 2x = odd for this to be true, y must be a fraction with 2 in the dnom, and 2y must be odd. NS (2) Nothing is given about x Also same info as statement 1 NS TOGETHER: 2y + 2x = Odd Odd + 2x = Odd ---> x can be odd or even to make 2x even. NS

Answer 94

p is NOT a prime number (composite number) --> p has more than two factors (itself and 1)

Answer 95

At least one is POSITIVE!

Answer 96

At least one is NEGATIVE

Answer 97

x and y have the same sign

Answer 98

Yes = 2*3*x ---> 2 is a factor!! Same with any even integer * x

Answer 99

Even/Odd properties (1) even --> p and r are either both even or both odd NS --> need to know q o*e + o = o o*o + o = e e.g., 2*3 + 4 = Even 3*2 + 1 = Odd (2) odd --> q and r, one is odd one is even NS --> need to know p o*e + o = o o*o + e = o e*o + e = e e.g., 2*3 + 4 = even 3*3 + 4 = odd Together: > p and r have the SAME TYPE (either both even or both odd) > q and r have the OPPOSITE TYPE pq + r o*e + o = o e*o + e = e 3*2 + 5 = odd 2*3 + 4 = even E - not sufficient even together

Answer 100

72 --> this is NOT a factor question. This is a MULTIPLE question. 2(L + W) = P Lay out the multiples of 18 and 12. 18, 36, 54, 72, 90, 108 12, 24, 36, 48, 60, 72, 84, 96, 108 Divide each of the potential perimeters by 2 --> then see if the combo can be reached.

Answer 101

Combination Q or Permutation w Repetition: Anagram Grid Top row --> number of people Bottom row --> number of categories = n! / duplicate! * duplicate! ---> see if order matters! e.g., 7 people that can get 1 platinum medal, 1 gold medal, 2 silver medals, or 3 bronze medals: Permutation with repetition. 7! / (1!*1!*2!*3!) = 420

Answer 102

Arranging group --> Combination Q --> Anagram Grid Top: 1 2 3 4 5 6 7 8 Bot: Y Y Y N N N N M Cm,n = C8,3 (from 8 choose 3) = 8! / (3! * 5!) = 56 OR Slot method / # of duplicates = (8 * 7 * 6) / (6) ----> 6 duplicates because the same 3 people comprise of 3*2*1 = 6 arrangements = 3!

Answer 103

Combinatorics --> Multiple Decision Q (Combination Q) --> Anagram Grid with a twist ("at most" 2 colours) 1 colour OR 2 colours: 1 Colour Options: 7! / (1! * 6!) = 7 2 Colour Options: 7! / (2! * 5!) = 21 -----> C7,2 (from 7 choose 2) 7 + 21 = 28 options.

Answer 104

Permutation Order MATTERS --> There are NO duplicates (same type question) Two groups (Male AND Females) Males = 3! Females = 3! Together: 3! * 3! = 36 options

Answer 105

1/6 * 1/6 = 1/36 --> out of 36 possible combinations, (1,1) occurs ONLY ONCE!! Whereas a combination of 2 and 5 can happen twice, such as (2,5) and (5,2)

Answer 106

P(two doves) or P(two rabbits) P(two doves) = P(dove on first) and P(dove on second | dove on first) = (3/5) * (2/4) = 3/10 P(two rabbits) = P(rabbit on first) and P(rabbit on second | rabbit on first) = (2/5) * (1/4) = 1/10 P(two doves) or P(two rabbits) = 4/10

Answer 107

NO - the result is a NON-Multiple of N e.g., 18 - 10 = 8 (non-multiple of 3) 18 and 10 are both multiples of 2, so the result, 8, is also a multiple of 2 Recall: adding or subtracting a MULTIPLE of N and a MULTIPLE of N = Multiple of N

Answer 108

Not sure - could either be a multiple of N or a non-multiple of N e.g., 12 + 13 = 25 --> adding two non-multiples of 5 result in a multiple of 5

Answer 109

Factors: In other words: Is 6 a factor of z? (1) GCF = 3 12 factors: 1, 12, 2, 6, 3, 4 z factors = 1, 3... --> z does not have factors with 2 Therefore, z is NOT divisible by 6. Sufficient OR Prime Factor columns: GCF (lowest powers) If GCF = 3^1... 12 = 2*2*3 = 2^2 * 3^1 Therefore, z = 2^0 * 3^1 (2) GCF = 15 Prime Factor columns: GCF (lowest powers) If GCF = 3^1 * 5^1... 15 = 3^1 * 5^1 Therefore, z = at least 3^1 * 5^1 (could have a 2) A

Answer 110

Use Prime Column Tables 12 prime factors as powers: 2*2*3 = 2^2 * 3^1 * 5^0 40 prime factors as powers: 2*2*2*5 = 2^3 * 3^0 * 5^1 GCF = lowest powers = 2^2 * 3^0 * 5^0 = 4 LCM = largest powers = 2^3 * 3^1 *5^1 = 120

Answer 111

(1) Unique prime factors --> prime factorization, count unique primes 2000 = 2^4*5^3 ---> 2 unique primes (2) Length: Add exponents = 4 + 3 = 7 (3) total factors --> PRIME factorization, product of exponents plus 1 (includes 1 as a factor) (4+1)*(3+1) = 5*4 = 20 > kind of like finding the area of a table

Answer 112

> the PRIME factorization of perfect squares contains ONLY EVEN POWERS > the total number of factors is ODD (due to adding 1 to each power before multiplying)

Answer 113

Look at the number in prime factorization form - look at the EXPONENTS > all even (multiples of 2) --> perfect square > all multiples of 3 --> perfect cube > all multiples of 4 --> fourths

Answer 114

Factors > k^3 = three identical factor sets of k Prime Factorization of 240: 2^4 * 3^1 *5^1 > we have 3 2's --> k must be divisible by 2 > one left over 2 --> in order for k^3 to be divisible by 240, k must also two factors of 2 (plus, the k's are all identical) > we have incomplete factors of 3 and 5, but along the same logic as above (k*k*k each with identical factors), 3 and 5 must be present too So, least value of k is a product of its prime factors = 2*2*3*5 = 60

Answer 115

CONCEPT: Multiples of N added or subtracted together produces a result that is also a Multiple of N 10! -->a multiple of integers from 1 to 10 11! --> a multiple of integers from 1 to 11 So common factors include -- 1 to 10 Therefore, N! is a multiple of all the integers from 1 to N ALSO all numbers are a multiple of 1

Answer 116

Integer Form: 20/3 = 6 R 3 Fractional From: 20/3 = INTEGER + FRACTION = 6 + 2/3 Decimal form = 2/3 = 0.6667 = Remainder/Divisor 3/20 = 0 integer + 3/20 = 0 R 3

Answer 117

TEST CASES for REMAINDER questions Remember: Dividend = Divisor*quotient + remainder Dividend/divisor = quotient + remainder x 2 7 12, 17 (goes up by 5 = pattern) y 1 5 9 13 (goes up by 4 = pattern) find different combos 12 = 7 + 5 13 = 12 + 1 14 = ? 16 = 7 + 9 21 = 12 + 9

Answer 118

b/4 is an integer => 4 is a factor of b; b is a multiple of 4. a = b + 4 => a is also a multiple of 4 CONCEPT: For any two POSITIVE CONSECUTIVE multiples of an integer n, n is the greatest common factor of those multiples, so the greatest common factor of a and b is 4.

Answer 119

Combinatorics Question > Can you think of it as a Y/N arrangement? (Combination) > are you asked to ARRANGE different items WITHOUT DUPLICATES (Permutation) ---> > Are you asked to ARRANGE different items with some duplicates? (Permutations w Repetition) ---> "identical" or otherwise says there are duplicates > Are you asked to create a "box" of different items where order doesn't matter? (Not an arrangement) A) How many ways can you ARRANGE SSSEEE? > Arrange LETTERS (permutation with repetition) > n! / duplicates! * duplicates ! 6! / (3!)(3!) = 20 B) For SSSETG? = 6! / 3! (order six positions, divided by 3! ways to order S's) C) Combination This question is different than ARRANGING different items. You are asked to choose items to put into something where order doesn't matter. The "arranging" part comes in when you are picking which items to put in the box - you don't arrange the resulting items in the box. Approach --> do item by item = 3 S AND 1 E = C5,3 AND C3,1 = [5!/(3!2!)] * [3!/(1!2!)]

Answer 120

Combinatorics with Constraint --> Glue Method Permutation > Arrange LETTERS = Total # of Ways to Arrange - # of Ways to Arrange Sit Together = (6!) - (2*5!) = 480 5! because we treat Jean and Mark as one group (stuck together). 2*5! because each 5! ways could be in the order JM or MJ

Answer 121

Combinatorics and Probability (Domino-effect Q) Probability (B and G and R) --> no replacement each time! = (7/16) * (5/15) * (4/14) --> FOR ONE CASE, BGR = 1/24 # of ways to arrange BGR = 3! Total probability = 1/24 * (3! cases) = 1/4

Answer 122

Combinatorics and Probability (Domino-effect Q) P(JH and other)? = [P(JHM) + P(JHS) + P(JHN) ] * 3! ways to arrange each = (1/5*1/4*1/3) + (1/5*1/4*1/3) + (1/5*1/4*1/3) * 3! = 1/20 * 3! = 3/10 or P(JH and other) = (1/5*1/4*3/3)*6 = 3/10

Answer 123

Triangle is a RIGHT TRANGLE (90 degree angle is opposite to the diameter)

Answer 124

Base * height > Height --> shortest distance between base and opposite side, segment is perpendicular to base. Recall that the two parallel sides are EQUAL in length **max area of a parallelogram is a rectangle (perpendicular sides)

Answer 125

1/2 * (base 1 + base 2) * height In other words, take the AVERAGE of the two bases and multiply it by the height

Answer 126

RATIOS (remember they can be scaled) W/o special angles: 3-4-5 -----> key multiples (6-8-10, 9-12-15, 12-16-20) 5-12-13 ----> key multiples (10-24-26) 8-15-17 W special angles: 1-1-sqrt(2) >>>isosceles right triangle has two 45 degree angles (45-45-90 degrees) 1-sqrt(3)-2 >>> half of an equilateral triangle (30-60-90 degrees)

Answer 127

Line segment MN is PARALLEL to line segment OP

Answer 128

Sum of area of each side (6 sides!) Cube --> area of one side * 6 Rectangle --> area1*2 + area2*2 + area3*2

Answer 129

NOT SURE --> when dealing with fitting 3D objects into other 3D objects, knowing the respective volumes is NOT ENOUGH > need to know shapes e.g., 100 in^3 = 20 * 5 * 1 = 25 * 4 * 1 = 5 * 5 * 4

Answer 130

CONCEPT: Angles correspond to their opposite sides If two angles are equal, then the sides opposite to them are also equal in length Other related concepts: > Shortest side is opposite from the smallest angle > Longest side is opposite from the largest angle

Answer 131

Isosceles Right Triangle 1x - 1x - x*sqrt(2) Half of a square is an isosceles right triangle

Answer 132

1x - x*sqrt(3) - 2x Half of the equilateral triangle is a 30-60-90 degree triangle (so the height of an equilateral triangle has length x*sqrt(3))

Answer 133

Sketch the line. Make sure you find the x and y intercepts! y = -2x + 5 Points: (0, 5) (-1, 7) --> Quadrant II (1, 3) --> Quadrant I (10, -15) --> Quadrant IV

Answer 134

As long as you have at LEAST ONE of the following, you can solve for the rest! > Area of Circle (pi*r^2) > Circumference of a Circle (2*pi*r = pi*d) > radius, r > diameter, d = 2r

Answer 135

Area of a Sector (pie-shaped wedge) = % * Area of the Circle = (Central Angle/360) * Area of the Circle Length of an arc = % * Circumference = (Central Angle/360) * Circumference

Answer 136

Recall: Central Angle is formed when one of the vertexes is at the CENTER of the circle (and each segment is the radius) Inscribed Angle = 1/2 * central angle Both share a common arc (same two vertices) ***explains why an inscribed triangle with one side equal to the diameter must be a RIGHT TRIANGLE > two arcs, two different central and inscribed angles

Answer 137

V = pi*r^2 * h = area of circle * height SA = 2(pi*r^2) + 2*pi*r * (h) = area of two circles + area of rectangle

Answer 138

Permutation with same types (no duplicates because we treat each gnome and elf as unique) e.g., GEGEGE EGEGEG = 3! ways to arrange the Gnomes AND 3! ways to arrange the Elves * 2 starting positions = 2 * (3! * 3!) = 72

Answer 139

YES Half of the equilateral triangle is a 30-60-90 degree right triangle. Let's call one side y and the height h. 18*sqrt(3) = 1/2*(1/2*y)*(h) 36*sqrt(3) = (1/2*y)*(h) -----> Recall ratio of sides! (x-x*sqrt(3)-2) REWRITE HEIGHT AS A FUNCTION OF Y 36*sqrt(3) = (1/2*y) * (1/2*y)*sqrt(3) 36 = 1/4*y^2 144 = y^2 ----->y must be positive y = 12 (one of the equilateral triangle's sides) Height = (1/2*y)*sqrt(3) = 1/2*12 * sqrt(3) = 6*sqrt(3) or directly: Area of an equilateral triangle = s^2*sqrt(3) / 4

Answer 140

Basically a square that's been squished (or a diamond) > equal sides > Diagonals BISECT one another at 90 degrees (perpendicular bisectors) but are not equal in length Area = (diagonal1 * diagonal2)/2

Answer 141

SQUARE e.g., P = 36 MAX area --> square with each side equal to 36/4 = 9 Other types of dimensions: 2x14, 6x12 e.g., A = 100 MIN Perimeter --> square with each side equal to sqrt(100) = 10 (P = 40) Other types of dimensions: 50x2 (P = 104), 25x4 (P = 58)

Answer 142

Two sides must be PERPENDICULAR to each other *be wary of questions where perimeter is incomplete (e.g., one side of a fence is a wall) --> cannot use this approach.

Answer 143

[s^2 *sqrt(3)]/4 Just need to know the length of each side OR height to calculate area! (don't need to know height AND side!) This is because an equilateral triangle is comprised of two special 30-60-90 degree right triangles.

Answer 144

SHORT CUT: If the ratio of sides/heights/perimeters is a/b, then the ratio of areas will be (a/b)^2 or a^2 : b^2 CONCEPT: Similar polygons have PROPORTIONAL corresponding side lengths and EQUAL angles e.g., Triangle A is a right triangle with two sides equal to 9 and 12. Triangle B is a similar right triangle with two corresponding sides equal to 3 and 4. Ratio of sides a to b = 9:3 = 3:1 Ratio of areas a to b = 9:1 Check: Area of Triangle A = 9*12*0.5 = 54 Area of Triangle B = 3*4*0.5 = 6 54:6 = 9:1

Answer 145

side*sqrt(3) Derived from Pythagorean Theorem where the hypotenuse equals the diagonal, one leg equals the edge of the cube (s) and the other leg equals the diagonal of the square (s*sqrt(2)) Half of the diagonal is side/2 * sqrt(3) > from the CENTER of the cube

Answer 146

Slope of the perpendicular bisector = Negative Reciprocal Slope (Rule applies to ALL PERPENDICULAR LINES) Slope of Line Segment = -4/-2 = 2 Therefore, slope of the perpendicular bisector = -1/2 To find the y intercept, first find the point at which the lines intersect (or the MIDPOINT of the line segment) => midpoint coordinates are the AVERAGE of the x and y coordinates = (x1 + x2)/2 and (y1 + y2)/2 = (1, 0) y = -0.5x + 0.5

Answer 147

UNDERSTAND: > testing factors > two factors = prime > Rephrase Q: How many prime factors does x have? PLAN --> need to break down x into its prime factors = prime factorization. (don't just rely on Odd - Odd = even concepts to get to 2 and multiples of 3 method to get to 3. There could be more primes that are missing!) x = (3^2)^10 - 3^17 = 3^20 - 3^17 = 3^17(3^3 - 1) = 3^17(26) = 3^17(2)(13) Three primes

Answer 148

Digits Question: (1) z can be many values 9 - 6 = 3 ---> 9 + 6 = 15 --> z = 5 8 - 5 = 3 --> 8 + 5 = 13 --> z = 3 NS (2) y must be 1 (otherwise, f > 10) y = 1 a = 2 f = 6 2 b c d e 6 -------- x 1 z WRITE OUT in algebraic form: PAY attention to UNITS digit 200 + 10b + c + 100d + 10e + 6 = 100x + 10 + z 196 = 100x - 100d - 10b - 10e + z - c 196 = 100(x - d) - 10(b + e) + 1(z - c) Focus on the units column: (because others are powers of 10 and have units digit equal to 0) 6 = z - c (z, c) = (9, 3), (8, 2), (7, 1) However, the last two pairs contain integers that have already been used! z = 9 (sufficient).

Answer 149

Q: # of weekdays no rain / total number of weekdays = ? (1) We know % of Mondays, Tuesdays, and Wednesdays that have rain. We are missing Thursdays and Fridays => Not Sufficient (2) Like 1, we are missing, Wednesdays. (3) Together: Even if we know the % of days that do not have rain, we do not know how many weeks had no rain (i.e., days in which there are no rain could fall on different weeks!) => Not sufficient E

Answer 150

In other words, is y outside of the ranges? 7/11 > 0.5 --> we can only focus on the upper bounds. (1) 11/12 versus 7/11 We know that 7/11 < 8/12 and 8/12 < 11/12 So, 7/11 < 11/12 => NS (y is in the range) (2) 8/13 versus 7/11 ==> ugly comparison, CROSS MULTIPLY Is 8/13 < 7/11? Is 88 < 91? => Yes So 8/13 < 7/11 => S (y is not greater than 7/11)

Answer 151

Digits Question: Values --> algebraic form G: _x__ _y__ ----> G contains even digits H: _x/2_ _y/2_ G + H =? MAX value -> Since G and H are each two digit numbers, G + H < 88 + 44 = 132 (Eliminates 153, 150, and 137). MIN value --> G + H > 22 + 11 = 33 Write out algebraic form: 10x + y + 10(x/2) + y/2 = 10x + y + 5x + y/2 = 15x + 3/2*y = 3(5x + 0.5*y) ---> sum is a multiple of 3 (eliminates 137 and 89) *we know that 0.5*y is an integer, so 5x + 0.5*y is an int. Ans: 129 Check: 129/3 = 43 = 5x + 0.5y 40 = 5*x x = 8

Answer 152

p is an even integer, such that p cubed - p squared = 0 > since the units digit must be positive, it cannot be 0. > Units digit of p cubed and p squared must be equal! > Therefore p must be a multiple of 6 (always ends with 6) 6 + 3 has a units digit = 9

Answer 153

Strategy --> look at each bed separately AND each flower separately Bed A: S A A P P # of ways to pick 1 shrub: 7!/(1!*6!) = 7 # of ways to pick 2 Annual Flowers: 6!/(2!*4!) = 15 # of ways to pick 2 perennial flowers: 4!/(2!*2!) = 6 Total number of possible beds (order doesn't matter in the bed!) = 7*15*6 = 630 Bed B: SSAPP or SSAAP or SSAAA = 7!/(2!*5!) * [6!/(1!*5!)*4!/(2!*2!) + 6!/(2!*4!)*4!/(1!*3!) + 6!/(3!*3!)] ---> use prev calculations from Bed A = 2436

Answer 154

Solve by using BRANCHES (positive, negative, and ZERO case for what's inside the absolute value signs) > Same approach as solving square root of a square IDENTIFY "root" --> x = 4 Positive and Equal case: x - 4 >= 0 or x >= 4 x^2 - 8x + 21 = x - 4 + 5 x^2 - 9x + 20 = 0 (x - 4)(x - 5) = 0 x = 4 or x = 5 ---> x = 4, or 5 Negative case: x - 4 < 0 or -(x-4) or x < 4 x^2 - 8x + 21 = -(x - 4) + 5 x^2 - 7x + 12 = 0 (x - 3)(x - 4) = 0 x = 3 or x = 4 ---> x = 3

Answer 155

Algebraic way: M F T 3x + 5x = 8x (actual of students) M new / Total = 1/3 (3x - 5)/(8x - 5) = 1/3 x = 10 Total original = 80 OR WORK BACKWARDS > integer constraint --> total (8x) must be a multiple of 8 (eliminate 30, 90) > Test 8x = 24 --> x = 3 M = 9 F = 15 Ratio M/F = 9/15 = 3/5 M - 5 = 4 M - 5 / F = 4/15 (WRONG) > Test 8x = 48 ---> x = 6 M = 18 F = 30 M - 5 = 13 M - 5 / F = 13/30 (Wrong) Ans 80

Answer 156

SIMPLIFY and REWRITE n^3 - n = n(n^2 - 1) = n(n + 1)(n - 1) ----> Recognize this as the product of three consecutive integers!! = (n - 1)(n)(n + 1) Rule for product of 3 consecutive integers: Divisible by 3! = 6

Answer 157

test FRACTIONAL cases 3*fraction = even (e.g., 2) 3*z = 2 z = 2/3 3*(2/3) = 2 ---> z is not an even integer NS

Answer 158

Method 1) Create sets that conform to the facts and eliminate answers > Problem with this method - Takes TOO LONG Method 2) Pay attention to fractions and whether it makes sense. Also pay attention to Median rules > Median is either (1) int in the set or (2) fraction with a DENOMINATOR = 2 e.g., y must be 1, 2, 3, 4, 5 or 6 2/7 * y will yield a fraction without a 2 in the denominator > 2/7 y is wrong

Answer 159

Work problem Fraction of the whole job done by Peter = [(P's rate)*(Total Time worked)/1 Hourly Rates for each person: T = 1/6 jobs per hour P = 1/3 jobs per hour J = 1/2 jobs per hour First hour - T completes 1/6 of the job. 5/6 remains. Second hour - T and P complete 1/6 + 1/3 = 3/6 of the job. Total completed: 4/6. 2/6 or 1/3 remains. Last Time Segment (WE DON'T KNOW YET): > Hourly Rate of 1/6 + 1/3 + 1/2 = 1 job per hour Time to Finish 1/3 of a job: 1 * t = 1/3 t = 1/3. P's fraction = 1/3 + 1/3*1/3 = 4/9

Answer 160

THIS IS AN OVERLAPPING SET / DOUBLE MATRIX QUESTION Both statements are insufficient

Answer 161

Recall: Median splits the set into half. If Median > Mean, the numbers in the bottom half of the set must be farther away from the Median than the top half. e.g., {3, 50, 70} Mean = 123/3 = 41 (lower than 50) {2, 6, 7} Mean = 15/3 = 5 (lower than 6) No - Median = Mean just means that the elements in both the bottom and upper half of the set are equally close/far from the median. But the elements can be spread however far apart. e.g., {100, 200, 300} e.g., {1, 2, 3}

Answer 162

MEAN: When two sets are combined to form a composite set, the mean of the composite set must EITHER be BETWEEN the means of the individual sets or be EQUAL to the mean of BOTH of the individual sets. e.g., A = [3], B = [5] A + B = [3, 5] --> mean is 4 (greater than A and less than B) A = [5], B = [3] A + B = [3, 5] --> Mean is still 4 (this time, it is greater than B and less than A) MEDIAN: when two sets are combined to form a composite set, the median of the composite set must EITHER be BETWEEN the medians of the individual sets or be EQUAL to the median of one or both of the individual sets. > Median can equal or be greater than or less than its mean e.g., A = [3], B = [5] A + B = [3, 5] --> median is 4, which equals mean

Answer 163

- One inequality can IMPLY another inequality e.g., x > 5 ---> x > 0 (positive) - Watch out for positive and negative unknowns and how it affects the direction of the inequality - Inequalities can combine to yield a single answer e.g. 0 < x < 2 AND x is an integer ---> x = 1 - Many inequalities are actually disguised as positive/negative questions

Answer 164

Sewing approach: First, solve as if it were an EQUATION to get the "roots": a(a - 2)(a + 1) = 0 a = 0, or a = 2, or a = -1 Second, draw a number line using the roots and do sewing approach, switching signs at the roots (unless the exponent on the factor is even) Therefore a(a - 2)(a + 1) < 0 when a < -1 and 0 < a < 2.

Answer 165

No - due to weighted average principles > cannot use average of an average to get a number e.g., Average Rate =/ average of rates (1) NOT SUFFICIENT We want to know AUM = AUM/branch * 10 = average AUM per branch * 10 We are given: Customers/10 = 400 --> Total Customers = 4000 customers Each branch's Average AUM per customer = Sum AUM / N customers When you take Sum of Each Branch's Average Aum per customer / 10 = 400k per customer You cannot do 400k per customer * 400 --> this assumes that each branch has 400 customers (equal weight), which is doesn't!! 400 = customers/10 ---> some branches have more than others

Answer 166

Manipulate statement such that z/(x - y) versus 2(x - y)/y => formula for time!!! Statement 1 is sufficient

Answer 167

Key #1 --> t is a PREDICTION, not actual year the animal became extinct. Key #2 --> "incorrect by 2 years" means + or minus 2 E

Answer 168

Regular figures are those for which you only need ONE MEASUREMENT to KNOW EVERY MEASUREMENT > also have equal sides and angles Include: - Circles (radius = diameter = area = circumference = Sufficient) - Square (side = area = perimeter = diagonal) - 45-45-90 degree triangle (any side = all sides = area) - 30-60-90 degree triangle (any side = all sides = area) - Equilateral triangle (any side = area)

Answer 169

Arithmetic Sequence: Difference between terms = 7 (so each term is some multiple of 7 plus or minus something) An = a1 + (n - 1)*d = 10 + (n - 1)*7 = 7n + 3 n >= 1

Answer 170

The integer's UNITS DIGIT e.g., 25/10 = 2 remainder 5 = 2.5

Answer 171

300 integers in the set / 3 = 100 In other words, count the # of multiples of 3 between 1 and 300.

Answer 172

RECOGNIZE Special Factoring = Difference of Squares (any EVEN POWERS that are the SAME in both terms) a - b = (sqrt(a) + sqrt(b))*(sqrt(a) - sqrt(b)) (sqrt(a) + sqrt(b))*(sqrt(a) - sqrt(b)) = sqrt(a) - sqrt(b) sqrt(a) + sqrt(b) = 1 sqrt(a) = 1 - sqrt(b) a = 1 - 2sqrt(b) + b

Answer 173

Need to know signs Statement 1: Means that x + y and x - y have the SAME SIGN (++ or --) ****IMMEDIATLEY recognize this is INSUFFICIENT to determine if it is one of the three cases above (the two factors just have to be the same sign) ! Case 1) x + y > 0 and x - y > 0 ---> Yes Case 2) x + y < 0 and x - y < 0 ---> No

Answer 174

pqp - p = 0 p(pq - 1) = 0 ----> p = 0 or pq = 1 OR pq = 1 IF p does not equal 0

Answer 175

r^2 and | r | are both positive r^2 < | r | ----> only possible if -1 < r < 1 (and r does not equal 0) OR r^2 = | r | * | r | , so is | r | < 1 ----> is -1 < r < 1?

Answer 176

THIS IS NOT the same as a DIFFERENCE OF SQUARES Recall cool trick when you ADD Square of a Sum and Square of a Difference: (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) Also a^2 + b^2 = (a + b)^2 - 2ab

Answer 177

Recall cool trick when you SUBTRACT Square of a Difference FROM Square of a Sum: (a + b)^2 - (a - b)^2 = 4ab

Answer 178

Square of a Sum: a^2 + 2ab + b^2 = (a + b)^2 Square of a Difference: a^2 - 2ab + b^2 = (a - b)^2 Difference of Squares: a^2 - b^2 = (a + b)(a - b) (any EVEN POWERS that are the SAME in both terms, or 1) e.g., a^4 - b^4; a^8 - b^8; a - b **if given square of a sum or a difference equals a perfect square, you can calculate value of the inside! e.g., (x - 1)^2 = 16 x - 1 = 4 (if x > 1) 1 - x = 4 (if x < 1) Special Manipulations: Addition of Square of a Sum and Square of a Difference: (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) Subtraction: (a + b)^2 - (a - b)^2 = 4ab Disguised Quadratic Templates: Multiplication (e.g., 198*202 = (200 - 2)(200 + 2) = 200^2 - 2^2) a - b = (sqrt(a) + sqrt(b))(sqrt(a) - sqrt(b)) 1/a^2 + a^2 = (1/a + a^2) - 2 Right Triangles and Area --> just need to know the hypotenuse and SUM of the two shorter sides or DIFFERENCE of the two shorter sides to calculate AREA!! (a + b)^2 = c^2 + 4*area (a - b)^2 = c^2 - 4*area

Answer 179

Since the terms in the answer choices are quite large, we have to find a PATTERN in the sequence. A1 = 1 A2 = 100 A3 = between 1 and 100 (e.g., 50) A4 = between 100 and 50 (e.g., 75) A5 = between 75 and 50 (e.g., 60) etc. Strategy: DRAW A NUMBER LINE if the terms alternate from increasing to decreasing (+/-) Pattern Observed: Even terms are larger than Odd terms (All the answer choices are EVEN) > Largest Term = Smallest Even term (A100) > Smallest Term = Largest Even Term (A400) A400 < A300 < A200 < A100 A100

Answer 180

Parallelograms: Rectangles, Rhombuses, Squares

Answer 181

Square and Rhombus --> due to equal sides > every square is a rhombus (perpendicular bisectors, four equal sides)

Answer 182

Square, Rectangle ---> due to 90 degree angles

Answer 183

MUST BE 45-45-90 degree triangle with sides in ratio x-x-xsqrt(2)

Answer 184

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 185

Reciprocals Recognize this is related to square of a sum! (a + b)^2 = a^2 + 2ab + b^2 (1/a + a) = 3 --> square both sides (1/a + a)^2 = 9 1/a^2 + 2 + a^2 = 9 1/a^2 + a^2 = 7 = diameter Circumference = 2*pi*r = pi*d = pi*7 = 3.14*7 = 22

Answer 186

Disguised quadratic template - you can solve for area if you know (1) hypotenuse and (2) Sum or Difference of the sides (a + b)^2 = a^2 + 4(ab/2) + b^2 (a + b)^2 = c^2 + 4*area (12)^2 = 8^2 + 4*area 144 - 64 = 4*area area = 20

Answer 187

xy > 0 means xy have the SAME SIGN (++, --) (1) x > 2y ---> alarm bells should be ringing (inequality division of an unknown sign) If x and y are positive: x/y > 2 e.g., 3 > 2(1) 3 > 2 If x and y are negative: x/y < 2 e.g., -3 > 2(-4) 3/4 < 2 NOT SUFFICIENT

Answer 188

Goal: # painted cube faces / total number of cube faces DRAW OUT dimensions to help you figure out # of cubes 1) Determine the total number of cubes --> layer approach >Length of 4 in can have 8 cubes > Width of 1 in can have 2 cubes ----> base can have l*w = 16 cubes > Height of 1 in can have 2 cubes (2 rows) > Total # of Cubes = 8*2*2 = 32 2) Determine total number of faces = 6*32 = 192 3) Determine total number of painted cube faces --> Do it FACE BY FACE (surface area of the rectangular wooden dowel) = 2*(16 cubes) + 2*(4 cubes) + 2(16 cubes) = 2*(16 + 4 + 16) = 2*(36) = 72 4) Find the fraction = 72/ 192 = 3/8

Answer 189

Baseline approach --> focuses on differences from a chosen "baseline" (smallest number, e.g., 1942, median term, largest term, a round number near the range of values etc.) > keep signs 1942 - 1942 = 0 1954 - 1942 = +12 1980 - 1942 = +38 1999 - 1942 = +57 Then, COMPUTE THE AVERAGE of the differences = (0 + 12 + 38 + 57) / 4 = 107/4 = 26.7 Finally, ADD the average difference to the Baseline = Average = Baseline + Avg Difference = 1942 + 26.7 = 1968.7 Sometimes you want to find the TERM of a list, given the mean. Set mean as the baseline. The number above and below the mean should yield an average distance = 0.

Answer 190

Focus on the term you want to maximize or minimize Maximize Term --> minimize other terms --> draw out a number line detailing what each value can equal Minimize Term --> maximize other terms --> draw out a number line detailing what each value can equal _ _ 60 _ _ (median is 60, since it is an odd number of balls, 60 is a part of the list). a one more than a --> b = 1 + a Also mean < median, which indicates that the numbers above the median are CLOSER to 60 than the numbers below the median. > Minimum distance from median of the values greater than 60 = 61, 62 56 = (a + 1 + a + 60 + 61 + 62)/5 2a = 96 a = 48

Answer 191

1. In factored form --> ALL even exponents (divisible by 2) sqrt = 1/2 2. Odd number of factors (remember factors are always distinct) > One of the factor pairs is a repeat e.g. 4 --> factors: 1, 4, 2 e.g., 9 --> factors: 1, 9, 3

Answer 192

Slot method - multiply the # of possible numbers for each position e.g., Number of odd numbered four digit palindromes = (5)*(10)*(1)*(1) = 50 > last two spots is 1 because once you have chosen the first two spots, the last two must be fixed.

Answer 193

a^2 = | a |^2 | a | = sqrt(a^2) All are POSITIVE numbers > can freely divide both sides of an inequality without changing the direction of the sign > can freely square both sides of an inequality without changing the direction of the sign Use these interchangeably!

Answer 194

1) Largest area is 1/2 of the larger square (proven by using 45-45-90 degree triangle ratios) 2) Each of the four triangles are CONGRUENT triangles

Answer 195

1) Patterns in the numbers (X) that share the SAME REMAINDER (R) and DIVISOR (a) > X goes up by a e.g., x/5 = z + 1/3 ---> x = 5z + 1 x = 1, 11, 16, 21 (go up by 5 or a) 2) Possible remainders for a divisor, a = [0, 1, ... up to a - 1] > a (divisor) must always be LARGER than the remainder > R/a is always LESS THAN 1 > start off every remainder question by writing out a > R 3) X = aZ + R (X - R) = aZ ---> a and Z are factors of X - R ---> list them out! > combined with a > R, you should be able to figure out properties of a 4) If X/a has a remainder of X --> means that X < a, z = 0, X/a is the remainder e.g., 40/80 = 0 + 40/80 5) Just need one equation to come up with possible values for X. BUT if you know the remainders when X is divided by TWO DIFFERENT DIVISORS, you can come up with a COMBINED EQUATION FOR X. X = Least Common Multiple of a and b * Z + Smallest Value of X > LCM of DIVISORS 6) If you get X/a = int - R/a ---> you must convert the remainder into a positive one e.g., (48^2 - 1)/8 = int - 1/8 = 8*int - 1 ---> one less than a multiple of 8 Therefore, R = 7 7) special remainder formulas: (ab)/c --> Remainder = (Ra*Rb)/c (a + b)/c --> Remainder = (Ra + Rb)/c (a - b)/c --> Remainder = (Ra - Rb)/c > applies to real numbers too 8) Finding the remainder of an unknown integer with a known divisor (data sufficiency) e.g., X/8 has a remainder = ? > Set up the equation of the integer in the form X = a*Z + R > Divide both sides by the divisor > Recall the remainder of (a + b)/c = (Ra + Rb)/c > If a is a MULTIPLE of the divisor => Certain, fixed remainder > Otherwise if a is NOT a multiple of the divisor => UNCERTAIN remainder 9) Harder remainder problems are combined with even/odd property questions e.g., Even = even + even ---> remainder must be even 10) Divisor * decimal = Integer Remainder

Answer 196

SET UP: a > 0 int 81/a = z INTEGER + 1/a 81 = a*z + 1 80 = a*z ---> factors of 80, list possible values of a z a ------- 1 80 2 40 4 20 5 16 ... etc. (1) a/40 = int + 0 a is a MULTIPLE OF 40 --> a = 40, 80 (NS) (2) 40/a = int + 40/a 40/a = 0 + 40/a Meaning a > 40----> a = 80 (S)

Answer 197

Also 2 CONCEPT: A/m --> R A/n --> R Then A/least common multiple --> same R

Answer 198

sqrt(n) e.g., If n is a perfect square (odd # of factors), then half of the factors are above sqrt(n), and half of the factors are below sqrt(n) e.g., if n is not a perfect square (even # of factors), then half of the factors are above sqrt(n), and half of the factors are below sqrt(n) if n = 20, then half of the factors are above sqrt(20) = 2sqrt(5), and half of the factors are below sqrt(20). n = 2^2 * 5 ==> 6 factors (1, 2, 4, 5, 10, 20)

Answer 199

(R1 + R2)/c LONG WAY: a/c + b/c => remainder equals (R1 + R2)/c e.g., 10/3 = 3 + 1/3 --> R1 = 1 13/3 = 4 + 1/3 --> R2 = 1 (10 + 13)/3 = 23/3 ---> 7 R = 2 Formula - Remainder = (1 + 1)/3 = 2/3 --> R = 2

Answer 200

(R1 - R2)/c e.g., 10/7 = 1 + 3/7 --> R1 = 3 8/7 = 1 + 1/7 --> R2 = 1 (10 - 8)/7 = (2)/7 = 0 ---> remainder 2 Formula Remainder = (3 - 1)/7 = 2/7 --> remainder 2 (Must be positive differences)

Answer 201

(R1 * R2)/c e.g., (47*49)/8 e.g., (7^50)/8 ---> manipulate exponents so that you can get an integer in the numerator (part of it has remainder = 0) LONG WAY: (ab)/c = (a/c)*b ---> remainder equal to (R1/C)*b = R1*(b/C) = R1*(R2/c) = (R1*R2)/c

Answer 202

REMAINDER PROBLEM 1) Come up with GENERAL EQUATION for n We know that: n = 5z + 3 n = 6y + 2 n = LCM of divisors * m + smallest value of n > LCM --> list out multiples of 5 and 6, find the least common multiple. > From eq'n 1, n = 3, 8, 13, 15 > From eq'n 2, n = 2, 8, 14 > Therefore the smallest value of n = 8 n = 30m + 8 2) List out some values of n that meet the constraint n = 8, 38, 68 ---> n = 38 3) Find the remainder 38/4 = 9 + 2/4 ---> R = 2 ** For remainder questions, DO NOT SIMPLIFY the division (e.g., 38/4 keep it as is, do not simplify further)

Answer 203

One variable inequality - sewing approach Move terms to one side so that one side equals 0 y - 1/y < 0 ---> combine and factor (y + 1)(y - 1)/y < 0 In other words, (y)(y + 1)(y - 1) < 0 Roots = 0, -1, 1 y < - 1 0 < y < 1

Answer 204

x > sqrt(1) or x < - sqrt(1)

Answer 205

CONCEPT - consecutive integers and EXTREME values Compute the range of what K can be. n terms = 60 - 41 + 1 = 20 MAX value of K --> all 20 terms are 1/41 = 1/41 * 20 = 20/41 (less than 50%) MIN value of K --> all 20 terms are 1/60 = 1/60 * 20 = 1/3 Therefore, 1/3 < K < 20/41

Answer 206

___ ____ ____ 9 competitors = 3 are ABC + 6 others P(at least two win) = P(2 win) + P(3 win) CONCEPT --> When you see P(ABC) --> think DEPENDENT EVENTS = P(A) * P(B|A) * P(C|AB) 1) P(2 win) = (P(ABx) + P(ACx) + P(BCx))*6 ways to arrange each one = (1/9)*(1/8)*(6/7) * 3 * 6 = 3/14 2) P(3 win) = P(ABC) * 6 ways to arrange = (1/9*1/8*1/7)*6 = 1/84 3) SUM = 19/84 OR (1) P(2 win) = T T O = (3/9 * 2/8 * 6/7) * (3!/2! ways to arrange) = 3/14 (2) P(3 win) = T T T = (3/9 * 2/8 * 1/7) = 1/84

Answer 207

First get all terms into base 2 x = 2^b - (2^24 + 2^18) We want x to be closest to 0: 0 = 2^b - (2^24 + 2^18) 2^24 + 2^18 = 2^b Factor and evaluate so that 2^value = 2^b: 2^18(2^6 + 1) = 2^b 2^18(65) = 2^b Approx 2^18(2^6) = 2^b so 2^24 = 2^b b = 24

Answer 208

1) An isosceles right triangle is always a 45-45-90 degree triangle. 2) The altitude of an isosceles triangle is a PERPENDICULAR BISECTOR and splits the triangle into two CONGRUENT triangles (and bisects the ANGLE) > not necessarily special angle triangles

Answer 209

Properties of polygons: > Triangle: The sum of two sides must be GREATER than the third side

Answer 210

Sequences: Try to find the pattern (NOT just in the individual terms, but also the SUM) Ans: (10! - 1)/10!

Answer 211

1. Angle sizes must match side lengths > hypotenuse has the LARGEST length 2. Sum of any two sides is larger than the third side. 3. Difference of any two sides is smaller than the third side.

Answer 212

Pattern in the remainder: 2^1/7 --> R = 2 2^2/7 --> R = 4 2^3/7 --> R = 1 2^4/7 --> R = 2 Remainder cycle [2,4,1] repeats every three terms. # of cycles = 50/3 = 16 R2 so Remainder of 2^50/7 = 4 Alternatively, Concept: Remainder of (ab)/c = (r1*r2)/c SIMPLIFY FIRST, get multiples of 7 in the numerator as a SUM of something, preferably with 1(which have a remainder = 0): = (2^3)^16 * 2^2 / 7 = (8)^16 * 2^2 / 7 = (7 + 1)^16 * 2^2 / 7 The remainder of (7 + 1)^16/7 = 1 > exponent is just a long multiplication! > (7 + 1) / 7 has a remainder of 1 > So (7 + 1)^16 / 7 has a remainder of (1*1*1*1....1)/7 The remainder of 2^2/7 = 4 So the overall remainder = (1*4)/7 = 4/7 --> 4

Answer 213

if n items are put into m containers, with n>m, then at least one container must contain more than one item. CONCEPT to solve: think of WORST CASE SCENARIOS E.g., Worst case scenario is that you draw 4 cards with different suits in a row, three times (a total of 12 cards drawn). Therefore, the 13th card must complete the set of one of the suits (4 of a suit).

Answer 214

When you see an inequality with the same terms on both sides --> rearrange first! Is 2xy < xy? Is 2xy - xy <0? Is xy < 0? (1) y < 0, however, we don't know the sign of x (2) Can divide both sides by (xy)^2 (since it is a positive value). xy < 1 --> NS either (3) together --> still don't know the sign or value of x E

Answer 215

Strategy: Set up a system of equations such that In = Out > inflow rate, r > outflow rate, x > SAME time > # of terminals = n Solve for an expression via elimination! Cow Grass Problem: Starting Grass + r*t = n*x*t --> asked for solve for t when there are 5 cows. Tank Problem: Starting Tank + r*t = n*x*t --> asked to solve for how LONG it takes to reach STARTING TANK amount e.g., L0 = 40*r ---> t = 40

Answer 216

Long way - add up the numbers and divide by 9. Recognize that the numbers are three sets of EVENLY SPACED integers (average = median) Find the formula: average = [(a1 + b1 + c1)/3 + (a2 + b2 + c2)/3 + (a3 + b3 + c3)/3]*3 then divide by 9 = (average1 + average2 + average 3)/3 = 551

Answer 217

No - N has a smaller standard deviation > M has values that are farther away/more spread out from the mean than set N Z has the same standard deviation as M. Concept: Two sets have the same standard deviation IF: > They have the SAME # of terms AND > The terms have the SAME GAP from the mean in each set (e.g., -3, -2, -1, 0, +1, +2, +3)

Answer 218

A1 * [(1 - r^n)/(1 - r)] where r is the shared factor where n starts at 1 (A1 = position 1)

Answer 219

Equal when x and y have SAME SIGNS e.g., | - 5 - 3 | = |-5| + |-3| Not equal when x and y have DIFFERENT SIGNS e.g., |5 - 3| =/ |5| + |-3| | x | - | y | <= | x + y | <= | x | + | y |

Answer 220

SA = pi*r^2 + pi*r*length > derived from calculating the area of the sector = (2*pi*r/2*pi*L) * (pi*L^2) V = 1/3 * volume of cylinder = 1/3 * (pi*r^2 * h)

Answer 221

SA: 4*pi*r^2 V: 4/3 *pi*r^3

Answer 222

Probability Q: # of prime integers in set: 2,3,5,7,11,13,17,19 = 8 How to get odd sum? = odd + even > Recall Odd + Odd = Even > therefore P(three odds) = (7/8)(6/7)(5/6) = 5/8 ----> there is NO NEED to multiply by the number of arranges (OOO has no arrangements). **Strategy: when multiplying probabilities dealing with a characteristic (e.g., odds)--> usually no need to multiply by cases e.g., P(draw no pairs in a row) = (1/1 * different number/total # * different number/total #)

Answer 223

Prime Factor problem: NEED TO BE IN A PRODUCT, NOT sum or difference!! First, start by factoring out the greatest common factor, 2^10, from each term Then, COMBINE the remaining terms in the bracket and re-do the prime factorization You will realize that 3 and 7 appear in the remaining output --> 7 is the greatest prime factor

Answer 224

CONCEPT: Use the average gap method mean = mean + avg gap of 0 _ _ _ _ _ _ 50 50 50 50 50 50 0 0 50 50 100 100 47 48 49 51 52 53 etc. ** question did not specify that the terms have to be different!

Answer 225

Probability Q: P(at least one pair) = 1 - P(no pairs) = 1 - (1/1*10/11*8/10*6/9) ----> NO NEED to multiply by # of cases because we did GENERAL approach = 1 - 16/33 = 17/33

Answer 226

Probability Q: P(only 1 correct) --> C W W W (arrange 4!/3! ways) = (1/4 * 2/3 * 1/2 * 1) * 4 --> 1/4 chance to get it correct * 2/3 chance to get second one wrong * 1/2 chance to get the third one wrong * 100% chance to get the last one wrong * 4 ways to arrange C W W W = 1/3

Answer 227

Starting Value * (growth factor)^# of periods > amount grows by a factor of X each period OR decays by a factor X each period > Periods increase by Y > Periods relate to TIME --> # of periods = t/y e.g., Periods go up by 2, while population increases by 2x Formula: A0 * (2)^(t/y)

Answer 228

A) Two different numbers = (1/1 on first role) * (n-1)/(n) on second role OR (1/n on first role)*(1/n on second role) * # of unique orders (n * n-1) B) Two equal numbers = (1/1 on first role) * (1/n) on second role OR (1/n on first role)*(1/n on second role) * # of unique orders (n)

Answer 229

Combination Q > order doesn't matter (groups of children) > remove 1 from x because Sally is guarantee to sit at the table n! / (duplicates! * duplicates!) = n! / [# chosen! * (n - # chosen)!] = (x - 1)! / [(y - 1)!(x - 1 - y + 1)!] = (x - 1)! / [(y-1)!(x-y)!]

Answer 230

Key info: > y is a prime number between 10 and 50: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 > x must be a multiple of 3 # of multiples of 3 between 10 and 21 ---> [12, 21] = (21 - 12)/3 + 1 = 4 # of integers between 10 and 21 --> (21 - 10 + 1) = 12 # of primes between 10 and 50 = 11 # of total products = 12 * 11 = 132 # of multiples of 3 = 4 * 11 = 44 P(xy is divisible by 3) = 44/132 = 11/33 = 1/3

Answer 231

Combination questions --> "Choose" r from a total of n > order doesn't matter Permutations --> arranging all the elements of n, taking into account duplicates ASK YOURSELF: 1) Does order matter? Y - permutation N - combination 2) Choose your method (slot or anagram) A - permutation - slot method or formula (m!/(m-n)!) B - permutation w repetition - anagram method C - combination - anagram method (Y/N)

Answer 232

Probability P(at least 3 draws) = 1 - P(1 draw or 2 draws) = 1 - (1/4) - (3/4)(1/4) ----> 2 draws mean prob not heart on the first draw and prob heart on the second draw. = 9/16 ** 13/52 = 1/4

Answer 233

1. The center of an inscribed polygon is also the center of the circumscribed circle. 2. The radius of the circle is simply any line connecting the center to a vertex 3. A line draw from the center to the MIDPOINT of any SIDE is PERPENDICULAR (perpendicular bisector, and the angle is bisected) Together, we can use the pythagorean theorem or for DS (SOH-CAH-TOA) 4. Regular polygons (equal sides and angles) --> sum of interior angles = (n - 2)*180 5. Center angles add up to 360

Answer 234

Carry over happens when b + d > 10 This also means b and d are GREATER THAN f If b and d are SMALLER than f, then there is NO carry over

Answer 235

Then either D (each sufficient) or E (neither sufficient) > cannot be C (sufficient together) > Cannot be A or B (if one is sufficient, the other will be sufficient too)

Answer 236

"The cannibal" Then the answer is either the: > NARROWER statement > D > or E

Answer 237

Think: X + Y and X - Y are either BOTH EVEN or BOTH ODD > X +/- Y produces the same type of number Product is either EVEN (e*e) or ODD (o*o)

Answer 238

1. Factorials are basically a PRODUCT OF consecutive Integers 2. Can rewrite factorials as a PRODUCT OF ITS FACTORS TOO - in this question, we care about the # of 5's among the factors of 200! > 5 alone, 25 has two pairs of 5s (+1 extra per multiple of 25), 125 has three pairs of 5s (+ 1 extra per multiple of 125)

Answer 239

Average Q DEFAULTS: > Equation for Sum: 85 = (A + B + C + D + E + F)/6 510 = A + B + C + D + E > Equation for Sum: 76 = (0.8A + 0.8B + C + D + E + F)/6 456 = 0.8A + 0.8B + C + D + E + F Two equations, subtract to get a value for the SUM of the original prices of the two products that were discounted: 54 = 0.2A + 0.2B 270 = A + B C + D + E + F = 240 *These sums are IMPORTANT *In this question, the coats could be equally priced > IF THEY ARE NOT EQUAL --> THERE IS SOME ORDER THAT CAN BE HELPFUL e.g., A Is A the cheapest? AVERAGE PRICE of the 4 non-discounted coats: 240/4 = 60 Case 1: A = 1, B = 269, four other ones are 60 each (can be equal to one another). > Yes, A is the cheapest one Case 2: A = 40, B = 230, four other ones are 60 each > No, A is not the cheapest one NS (2) Max price was 180 --> assign this to the most expensive one Is A the cheapest? Case 1) 270 = A + 180 A = 90 (minimum value if 180 is the most expensive one). Four others are 60 each > A is not the cheapest What else can A be? C + D + E + F = 240 = 180 + 20 + 20 + 20 270 = A + B Again, to make A as small as possible, make B as big as possible = 180. Regardless, A > average --> NOT THE SMALLEST SUFFICIENT

Answer 240

Average with inequalities To find if C > 1000, find the MIN value of C by maximizing a and b. 2700 = a + (a + 1) + 1.25(a + 1) a = 828 b = 829 c > 1000

Answer 241

distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2) = pythagorean theorem!

Answer 242

1) same slope (need two points to know) 2) same distance between all points on line 1 versus line 2 > distance is the length of the line that is perpendicular to one of the lines

Answer 243

Concept: Positive integer constraint --> use SUBTRACTION to limit options for the value of r and s 2^r * 2^2s = 2^4 2^(r + 2s) = 2^4 r + 2s = 4 (drop the bases) r = 4 - 2s ----> r > 0, so S can only = 1, R = 2 2r + s = 5

Answer 244

Volume INSIDE another Volume > start by DRAWING the picture of the rectangular paper Go through EACH OPTION to see if it can fit perfectly --> start with BASE # of shapes and build UP Ans: B

Answer 245

Concept: Factors > Candice and Sabrina are RELATED by their TOTAL TIME on the track Time spent by Candice on the track, if C represents her # of laps = 42*C Time spent by Sabrina on the track, if S represents her # of laps = 46*S 42C = 46S 21C = 23S ----> Since 23 is a prime number, C must be a multiple of 23. So the next time they are together at the starting point must be when C = 23

Answer 246

Prime Properties (factors: Perfect square of a prime: Odd number of factors --> must be a perfect square (HOWEVER MUST ONLY HAVE 3!!) Eligible perfect squares: 4, 9, 16 However only 4 and 9 have three factors

Answer 247

Sketch it out first For quadrilaterals: Calculate slopes to see which lines are PARALLEL and which ones are NOT, and which lines are perpendicular (negative reciprocal slopes) Distance formula = sqrt[ (x2 - x1)^2 + (y2 - y1)^2) = length of a line segment For triangles: Similar triangles Other tips: > Area of Parallelogram = easier to do 2 * area of triangles

Answer 248

CONCEPT: Set up EQUATION > IN MOST CASES: Variables represent the VALUE or PRICE Coefficients represent QUANTITY > sometimes if you are given value or price, the variables represent quantity e.g., 0.01*# of dimes = 1*# of dollars e.g., Let c, u and b equal the VALUE of each type of currency 12c = 8u + 5b > as long as you have two different equations, you can calculate an expression for u and b Ans D

Answer 249

1 + growth rate 1+0.5 = 1.5 (50% increase) 1+1 = 2 (100% increase) 1+1.3 = 2.3 (130% increase) 1+2 = 3 (200% increase) 1+5 = 6 (500% increase) 1-0.45 = 0.55 x original 1-0.25 = 0.75 x original % can also be found using PERCENT CHANGES! 50% increase => (P2 - P1)/P1 = 0.5 100% increase => (P2 - P1)/P1 = 1 45% decrease => (P2 - P1)/P1 = -0.45

Answer 250

Set up Ratios First convert all statements into the same units is area >= r is pi*r^2 >= r (square yards) is pi*r >= 1 is r >= 1/pi? (yards) (1) d > 2 ft or d > 2/3 yards 2r > 2/3 yards r > 1/(3) 1/3 is larger than 1/pi so SUFFICIENT (2) area > 2f (sqre feet) pi*f^2 > 2f (square feet) pi*f > 2 f > 2/pi (feet) ---> f feet = r yards and convert RHS to yards r > 2/pi * 1/3 r > 2/(pi*3) ---> 2/9.xx (smaller than 1/pi) NS

Answer 251

Sequence - find pattern in the SUM 1 = 1 (N) 1 + 2 = 3 (N) 3 + 3 = 6 (Y) 6 + 4 = 10 (Y) 10 + 5 = 15 (N) 15 + 6 = 21 (N) 21 + 7 = 28 (Y) 28 + 8 = 36 (Y) 36 + 9 = 45 (N) PATTERN of the SUM: NNYYNNYY > Every four numbers, there are two Y's > so between 1 to 200, there are 200/4 = 50 groups of four > so in total there are 50 * 2 = 100 even sums So from 1 to 200, one half are even = 100 Concept: > determine pattern > then determine # of groupings that exhibit that pattern (e.g., YYNN is one group) > Finally, multiply the # of groupings by the # of Ys (e.g., 50 * 2)

Answer 252

Think about drawing the DIAMETER and relying on the angle properties of isosceles triangles

Answer 253

Concept: Sequences, find the pattern in the SUM Recognize pattern in sum as the powers of 2: n = 1 --> 1 = 2^0 = 2^(n-1) n = 2 --> 2 = 2^1 = 2^(n-1) ... n = 3 --> 4 = 2^2 n = 4 --> 8 = 2^3 n = 5 --> 16 = 2^4 so v(g16)/v(g15) = 2^15 / 2^14 = 2^1 = g3

Answer 254

Focus on the term you want to maximize or minimize Maximize Term --> minimize other terms --> draw out a number line detailing what each value can equal Minimize Term --> maximize other terms --> draw out a number line detailing what each value can equal _ _ 50 _ _ a b 50 c d a <= b <= 50 <= c <= d ----> didn't say they have to all be different! To maximize d, minimize b and c and maximize a (since d = 5 + 3a) Minimum value of c = 50 Minimum value of b = a 50 = (a + a + 50 + 50 + 5 + 3a)/5 250 = 5a + 105 145 = 5a a = 29 d = 5 + 29(3) = 92

Answer 255

Concept: Remainders n/7 = z + R/7 ---> R =? Recall the remainder of (a + b)/n = (Ra + Rb)/n (1) n = 21m + 2 n/7 = (21m + 2)/7 -----> CERTAIN remainder 2/7 because 21 is a multiple of 7, so (0 + 2)/7 = 2/7 Sufficient (2) n = 5y + 3 n/7 = (5y + 3)/7 ---> UNCERTAIN REMAINDER If y was a multiple of 7, the remainder would be 3/7. Otherwise, the remainder varies. NS

Answer 256

1) Total after-tax amount needs: > Pre-tax prices/revenue and the tax rate; > Pre-tax prices/revenue and the tax dollar amount > Tax amount and tax rate 2) Tax dollar amount needs: > Total After-tax amount and Pre-tax prices; > Total After-tax amount and the tax rate; > The tax rate and Pre-tax prices 3) Pre-tax price needs: > Total After-tax amount and the tax rate; > Tax dollar amount and the tax rate > Total After-tax amount and the tax dollar amount

Answer 257

Weighted Average Mixture Problem Current Salt Amount in Solution = x% * 100 Current Salt Concentration as % = (x% * 100)/100 = x% Target Salt Concentration: (x + 10)% = (x% * 100)/(100 - y*time) -----> numerator is FIXED (fixed amount of salt in the solution) Solve: Rewrite "%" as "/100" 1000/(xy + 10y) OR alternatively Q of Salt BEFORE = Q of Salt AFTER x%*(100) = (x + 10)% * (100 - yt)

Answer 258

Define a variable x for the smallest integer e.g., 5 consecutive integers _ _ _ _ _ x x+1 x+2 x+3 x+4

Answer 259

For a*b to be even, at least one integer (a or b) must be even a^4 + b^4 = odd e + o = odd o + e = odd Recall that a^n has the same type as a, and b^n has the same type as b So yes, a*b is even

Answer 260

Consecutive Integers starting at n: n +/- odd number = Opposite type as n n +/- even number = Same type as n Factor first: n(n - 2) > n and n - 2 have the same type (both even or both odd) If n is even, then n(n - 2) = even*even = even Specifically, n*(n - 2) are two consecutive even integers => product is a multiple of 8 If n is odd, then n(n - 2) = odd*odd = odd

Answer 261

Concept: Even and Odd properties First: Collect like terms and factor y^2 + 2y = x^2 + x y(y + 2) = x(x + 1) y*(y + 2) have the SAME TYPE (o*o or e*e) x*(x + 1) have OPPOSITE types (o*e or e*o). Therefore, one of the factors must be even, so x*(x + 1) = even y*(y + 2) then = even and therefore are BOTH even Yes, y is even (sufficient)

Answer 262

Think: 1) The product of two consecutive integers 2) product is EVEN

Answer 263

Factors Question: PRODUCT of Point Values: e.g., 3 red beads have a product of 7*7*7 = 7^3 2 yellow beads have a product of 5*5 = 5^2 147,000 = 7^R * 5^Y * 3^G * 2^B Break up 147k into its prime factors (1000 = 2^3*5^3) 2^3 * 3^1 * 5^3 * 7^2 = 7^R * 5^Y * 3^G * 2^B Therefore, R = 2

Answer 264

Factors Question: Is n = multiple of 3? (1) n^2 = 36*int ---> int is a perfect square n = 6*int n = 2*3*int --> Yes, n is always a multiple of 3 Sufficient (2) 144 = n^2 * int ---> int is a perfect square 12 = n * int 2^2 * 3 = n * int ---> Unsure whether n is a multiple of 3 Not sufficient

Answer 265

Concept: Even and Odd properties a - b = even --> O - O or E - E (same type) a/b = even --> a = b*even ---> a and b must both be even REWRITE a as a MULTIPLE of b: a = 2*b*m = 4n (multiple of 4) Sub in a = 2bm (1) a/2 = (2bm)/2 = bm --> even (2) b/2 --> could be even OR odd e.g., 6/2 = 3, 4/2 = 2 (3) (a + b)/2 = (2bm + b)/2 = [b(2m + 1)/2] > 2m + 1 is odd > b/2 could be even or odd > not sure (4) (a + 2)/2 = (2bm + 2)/2 = bm + 1 ---> bm is even, bm + 1 must be ODD *** (5) (b + 2)/2 = b/2 + 1 --> b/2 could be even or odd

Answer 266

Concept: Even and Odd properties IF x is an integer, then x has the same type as x^2 = x^3 = x^n If x is NOT an integer, the result would NOT be even (1) x is even > x is an even integer > S = even + even + even .... = always even Sufficient (2) n is even > x could be an integer OR fraction > if x is an integer, S is even (e.g., e + e or o + o) > Otherwise, S is not even (fraction) Not sufficient A

Answer 267

Concept: Even and Odd properties (1) c(d + 1) = even means at least one of the terms is even NS (2) (c + 2)(d + 4) = even means at least one of the terms is even NS Note that c and c+2 are the SAME TYPE (int + even # is the same type) Note that d + 1 and d + 4 have DIFFERENT TYPES (int + odd # is the opposite type) (3) Together c and c+2 have the same type d + 1 and d + 4 have opposite types Test cases: Show that c must be even for both statements to be true. C

Answer 268

Concept: Even and Odd properties Pre-thinking: x could be odd, even, or a fraction (1) x = int^2 ---> x could be even or odd (we know that x is an integer) NS (2) x = int^3 ---> x could be even or odd (we know that x is an integer) NS (3) Together x is a square of an integer and also a cube of a different integer. x = (a^3)^2 = a^6 x = (a^2)^3 = a^6 Any value of a works. So insufficient

Answer 269

Concept: Factors (expressed as division) Rephrased, is x a multiple of y, or is y a factor of x? > if y is a factor of x, then y can be eliminated from the denominator (1) (2x)/y = integer NS --> y could be equal to 2 in which case integer = x, or y could not be a factor of x or 2. 2x = y * m x/y = m/2 ---> not sure if m is even or odd (2) y is odd --> NS (need to know about x) (3) Together (2x)/odd = integer y must be a factor of x (2/y is not an integer, unless y = 1. In that case, x/1 is also an integer) (C)

Answer 270

Concept: Even/Odd > is at least one term (x or y + 1) even? > is x even or y odd? (1) x and y are both primes > could be both odd = even > x could be even, y could be odd --> even > **x could be odd and y could be even --> odd*(odd) = odd NS (2) y > 7 NS --> y could be even or odd, unknown x (3) Together y > 7 and a prime --> y is odd Therefore, x(y + 1) is always even - Sufficient

Answer 271

Concept: Difference of Squares (factors) n = (3^4)^2 - (2^4)^2 = (3^4 - 2^4)(3^4 + 2^4) = (81 - 16)(81 + 16) = (65)(97) = (5)(13)(97)

Answer 272

Concept: Factors (expressed as division) Rephrase: is 7 a factor of n? Or is n a multiple of 7? (1) (3n)/7 = integer 3 is not a multiple of 7. Therefore n must be a multiple of 7 Sufficient Alternatively, 3n = 7*m ---> 3n must be a multiple of 7. So n must be a multiple of 7. (2) (5n)/7 = integer 5 is not a multiple of 7. Therefore, n must be a multiple of 7. Sufficient Alternatively, 5n = 7m --> 5n is a multiple of 7. So n must be a multiple of 7. D

Answer 273

Concept: Factors 8! = a^n * m (integer) Find what the value of a is. But first, rewrite 8! in prime factorization form: 8! = 2^7 * 3^2 * 5 * 7 (1) a^n = 64 2^6 = 64 4^3 = 64 8^2 = 64 All three cases work --> so not sufficient (2) n = 6 In the prime factorization form, only 2 shows up at least 6 times Sufficient B

Answer 274

Volume INSIDE another Volume To figure out if the statement is sufficient, there must be ONE SINGLE VALUE for the max # of rectangular blocks --> go through each option of which FACE the box is resting on. Case 1) Base dimensions are 60 x 30 Can fit 5 * 5 = 25 blocks in the bottom layer (12 by 6) and 5 height Case 2) Base dimensions are 60 x 20 Can fit 5 * 5 = 25 blocks in the bottom layer (12 by 4) and 5 height Case 3) Base dimensions are 20 x 30 Can fit 5 * 5 = 25 blocks in the bottom layer (6 by 4) and height 5 In all cases, the max # of blocks is 125.

Answer 275

Concept: Consecutive integers and Multiples n*(n + 1)*(n + 2) --> the product of ANY 3 consecutive integers is divisible by 3! and therefore a multiple of 3 (because one of the numbers MUST be a multiple of 3) > cycle repeats every 3 numbers Due to the cyclicality of multiples: If one of the numbers were a multiple of three, then the number +/- 3 would ALSO be a multiple of three Strategy: Determine whether there is a COMPLETE set of three CONSECUTIVE INTEGERS, keeping in mind the cyclicality of multiples Goal is to see if the numbers have the same properties as: (n - 1)*n*(n + 1) A) n + 2 = n - 1 so n*(n + 2)*(n - 1) has the same properties as n*(n - 1)*(n - 1) --> NO B) n + 3 = n ---> NO n - 5 = n - 2 = n + 1 so n*(n + 3)*(n - 5) has the same properties as n*n*(n + 1) C) n + 4 = n + 1 n - 2 = n + 1 so n*(n + 4)*(n - 2) has the same properties as n*(n + 1)*(n + 1) --> NO D) n + 5 = n + 2 = n - 1 n - 6 = n - 3 = n so n*(n + 5)*(n - 6) has the same properties as n*(n - 1)*(n) --> NO E) n - 4 = n - 1 so n*(n + 1)*(n - 4) has the same properties as n*(n + 1)*(n - 1) --> YES this is a product of three consecutive integers

Answer 276

Concept: Factors and consecutive integers To find the remainder, we have to see how much of 24 is cancelled out by factors in the numerator (1) n is ODD, so n - 1 and n + 1 are both even There are two consecutive even integers => product is a multiple of 8 > leaves 3 in the denominator and some other integer in the numerator --> NS (2) n is not a multiple of 3, so either n - 1 or n + 1 is a multiple of 3 (recall: product of three consecutive integers is a multiple of 3) > leaves 8 in the denominator and some other integer in the numerator --> NS (3) Together The product is both a multiple of 8 and multiple of 3 --> so 24 can get cancelled out completely, leaving a remainder = 0 S

Answer 277

CONCEPT: Remainders Decimal * divisor = remainder s/t = 64 + 0.12 s = 64t + t(0.12) Focus on t(0.12) --> REMAINDER (integer) --> rewrite as a fraction t(12/100) = t(3/25) Go through each answer option and see if t can be an INTEGER: E works t(3/25) = 45

Answer 278

CONCEPT: Remainders and Exponents > Find the PATTERN in the remainders when different POWERS are divided by the divisor 2^1 / 7 --> R = 2 2^2 / 7 --> R = 4 2^3 / 7 --> R = 1 2^4 / 7 --> R = 2 ---> Pattern repeats every 3 numbers 2^5 / 7 --> R = 4 5550/3 --> R = 0 ---> remainder of 2^5550 / 3 is 1

Answer 279

Look at the last TWO digits and see if they are divisible by 25

Answer 280

CONCEPT: Divisibility Rules and Factors See if n is a multiple of 12 = 2^2 * 3 * m (1) n = 4n = 2^2 * n --> NS (unsure if there is a 3) (2) _ _ _ Unsure if n itself is a multiple of 12 - to be a multiple of 12, the last two digits must be a multiple of 4 and the sum of the digits must add to a multiple of 3 e.g., 636 , NOT 616 (3) Together NS because of examples in statement 2 (E)

Answer 281

CONCEPT: Remainders 24/n = z + 4/n 24 = nz + 4 20 = nz ---> n is a factor of 20 *ALSO n > 4 FACTOR PAIRS OF 20: 1 20 2 10 4 5 Since n > 4, n = 5, 10, 20 I) n is not necessarily even II) n is a multiple of 5 III) n is a factor of 20 II and III

Answer 282

Remainder question (t + 1)^2 / 4 = z + R/4 (t + 1)^2 = 4z + R Goal is to see if there is a FIXED remainder (coefficients on the variables are multiples of the divisor) (1) t is a multiple of 2 --> SUB IN t = 2n (2n + 1)^2 / 4 = (4n^2 + 4n + 1)/ 4 = n^2 + n + 1/4 ---> REMAINDER is 1 (fraction!!) Also recall that any perfect square is either a multiple of 4 or multiple of 4 + 1 Sufficient (2) t is a multiple of 4 --> SUB IN t = 4n (4n + 1)^2 / 4 = (16n^2 + 8n + 1)/4 = 4n^2 + 2n + 1/4 --> REMAINDER is 1 (fraction!!) Sufficient

Answer 283

REMAINDER QUESTION n < 50 n/5 = a + 3/5 n = 5a + 3 n/6 = b + 5/6 n = 6b + 5 COMBINE into one general equation: n = LCM of divisors*m + smallest n n = 30m + 23 < 50 30m < 27 m = 0 so n = 23 23/7 = 3 R 2

Answer 284

ALGEBRA using rules a, b > 0 int Set a > b a + b = 24 a^2 - b^2 = 48 ab? a^2 - b^2 = (a + b)(a - b) = 48 sub in 24 for a + b: 24(a - b) = 48 a - b = 2 Now you have TWO EQUATIONS --> elimination a + b = 24 a - b = 2 ------------- 2a = 26 a = 13 b = 11 ab = 13*11 = 143

Answer 285

ALGEBRA using rules a, b > 0 int Set a > b a - b = 12 1/b - 1/a = 4/5 ab = ? --> combo question COMBINE the fractions (want to get ab denom): (a - b)/ab = 4/5 ---> sub in 12 12/ab = 4/5 ab = 15

Answer 286

Word Problem Application: Linear Equations Let A be the number of pencils and B be the number of pens (integer constraint) (1) 4.40 = 0.15A + 0.29B 440 = 15A + 29B ALTHOUGH there are two unknowns and one equation, this is SUFFICIENT (exception to the rule) > coefficients are close relative to one another > Two coefficients are multiples of 10 and 5 > to get a 0 units digit, must be either 0 + 0 or 5 + 5 ** write out the multiples of each term and see if more than one valid combo works ! > Multiples of 29 include: 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435 Possible values --> 145, 290 > Multiples of 15 include: 15, 30, 45, 60, 75... 440 - 145 = 15A 295 = 15A --> A is not an integer 440 - 290 = 15A 150 = 15A --> A = 10, B = 10 (Sufficient) (2) A = B --> not sufficient because we need to know the total value of pencils bought A

Answer 287

Quadratic equations and discriminants (looking for the value of the constant or coefficient) Discriminant: 1 solution: b^2 - 4ac = 0 x^2 + 2x - b = 0 (2)^2 - 4(1)(-b) = 0 4 + 4b = 0 4b = -4 b = -1

Answer 288

Quadratic equations and discriminants (looking for the value of the constant or coefficient) x^2 + px + 9 = 0 --> solutions x1 and x2 (1) x1 = x2 means there is ONLY 1 solution b^2 - 4ac = 0 p^2 - 4(1)(9) = 0 p^2 = 36 (sufficient) (2) x1 + x2 = -6 RULE: x1 + x2 = -(b/a) -(b/a) = -6 b/a = 6 p/1 = 6 p = 6 SUFFICIENT D

Answer 289

Quadratic equations and discriminants (looking for the value of the constant or coefficient) 1 solution means b^2 - 4ac = 0 ax^2 + bx + c = 0 b = -(m - 2) c = -(n - 4)^2 [-(m - 2)]^2 - 4(1)(-(n - 4)^2) = 0 (m - 2)^2 + 4(n - 4)^2 = 0 ---> sum of squares, must be equal to 0 --> each term must be equal to 0 m - 2 = 0 --> m = 2 n - 4 = 0 --> n = 4 mn = 8

Answer 290

Exponents Apply exponent rules: (3^3x)^2 = 8100 3^3x = 90 SUB into other equation: 3^(3x - 3) = (3^3x)/3^3 = 90/27 = 10/3

Answer 291

Exponents > solve for x and y (1) x^2y = 16 The prompt can be rewritten in this form: x^6y = (x^2y)^3 = (16)^3 Sufficient (2) xy = 4 xy are factors of 4 --> multiple values 1, 2, 4 --> unsure which is x and which is y NS

Answer 292

Exponents x^8 + 1/x^8 --> Reciprocals --> RECALL rule of sum of squares = (x^4 + 1/x^4)^2 - 2 = [(x^2 + 1/x^2)^2 - 2]^2 - 2 (1) sufficient (2) sufficient

Answer 293

Exponents and roots > Even roots have arguments >= 0 > Even exponents HIDE the sign of the argument Since x < 0, we know that x - 3 < 0 Also -x is positive and |x| is positive, together, they are x^2 -(x - 3) - x = 3 - 2x

Answer 294

Roots Approximation Fastest way is to convert sqrt(5) into a decimal and test values 2 < sqrt(5) < 3 --> use 2.1 5 - 2.1 < sqrt(x) < 5 + 2.1 2.9 < sqrt(x) < 7.1 (1) x is even e.g, 10 -> sqrt10 is just over 3 but less than 7.1 e.g., 16 --> sqrt16 = 4 NS (2) sqrt(x) is an integer sqrt(9) = 3 falls between 2.9 and 7.1 sqrt(16) = 4 falls between 2.9 and 7.1 NS (3) sqrt(16) works sqrt(36) = 6 also works NS

Answer 295

Arithmetic Sequence, start at a1 (set equal to the most expensive pot, d = negative) a1 = most expensive pot a6 = cheapest pot An = a1 + (n - 1)*d SUM = 8.25 = (average)*# terms 8.25 = (a1 + a6)/2 * 6 33/4 = (a1 + a6)*3 33/(12) = a1 + a6 11/4 = a1 + a6 a6 = 11/4 - a1 a6 = a1 + (6 - 1)*(- 0.25) a6 = a1 - 5/4 11/4 - a1 = a1 - 5/4 11/4 + 5/4 = 2a1 4 = 2a1 a1 = 2

Answer 296

Geometric Sequence Don't use the sum formula - it is used to calculate the sum from A1 to An Instead, find the value of each term and then add together. An = a1 * r^(n - 1) r = 2 a1 = 1 A16 = 1 * 2^(15) A17 = 1 * 2^(16) A18 = 1 * 2^(17) 2^15 + 2^16 + 2^17 = 2^15( 1 + 2 + 2^2) = 2^15(7)

Answer 297

Sequence Problem: > all about rewriting the sequence in terms of p an = a1 + a2 + ... + an-1 = p an + 1 = a1 + a2 + ... + an-1 + an = p + an = p + p = 2p an + 2 = a1 + a2 + ... + an-1 + an + an+1 = 2p + an+1 = 2p + 2p = 4p

Answer 298

Geometric sequence Find this out by writing out a few of the terms and finding a pattern in the terms and sum: a1 = (-1)^2 * (1/2) = 1/2 a2 = (-1)^3 * (1/4) = -1/4 a3 = 1/8 a4 = -1/16 a2/a1 = -1/2 a3/a2 = -1/2 ---> constant factor between consecutive terms SUM is geometric = [a1(1 - r^n)]/(1 - r) = [1/2(1 - (-1/2)^10)]/(1 + 1/2) = ~1/3 or between 0.25 and 0.5

Answer 299

Functions: Symmetry > start by eliminating answers so you can save time on which functions to actual calculate (sub in 1-x into f(x) and see if the output simplifies to f(x)) > Also DO NOT OVERSIMPLIFY --> just try to recreate f(x) Eliminate A, B and E Test C and D: D --> f(1 - x) = (1-x)^2 * (1 - (1 - x))^2 = (1 - x)^2 * (x)^2 ---> MATCHES f(x)

Answer 300

Inequality 1 variable --> sewing approach > already in factor form with positive coefficients on x > roots = -5, 1, 4 > even exponent (4) --> no change in sign at the root 1 -5 < x < 4, x =/ 1

Answer 301

Inequality with 1 variable --> sewing approach Reframe the question stem as is 0 < x < 1 (1) sufficient (2) MOVE variables to one side and express as a product x^3 - x > 0 x(x^2 - 1) > 0 x(x + 1)(x - 1) > 0 ---> for which values of x satisfy this inequality? Roots -> x = 0, -1, 1 -1 < x < 0 x > 1 We know that x is not between 0 and 1 for sure --> always no --> SUFFICIENT

Answer 302

Fractional inequality --> use sewing approach to determine the sign of the fraction Step 1) MOVE all terms to one side and combine into one fraction (x^2 + 4x - 5)/(x^2 - x - 2) >= 0 Step 2) Factor, ensuring positive coefficients on x [(x - 1)(x + 5)]/[(x - 2)(x + 1)] >= 0 Step 3) Rewrite as a product, determine APPLICABLE ROOTS (x - 1)(x + 5)(x - 2)(x + 1) >= 0 Applicable roots that can make the fraction = 0 include 1, -5 only But for sewing approach, consider changes at ALL roots Step 4) Sewing approach x <= - 5 -1 < x <= 1 x > 2

Answer 303

Inequality word problem P = # of paperback books H = # of hardcover books DON'T FORGET P > 10 or since they are ALL integers, P >= 11 (CONVERT to equality signed inequality) So 8P >= 88 (1) 25H >= 150 H >= 6 NS (2) 8P + 25H < 260 , given 8P >= 88 (MIN value) So max value of H --> 25H < 260 - 88 25H < 172 H < 6.xxx NS (3) Together 6 <= H < 6.xxx and H is an integer, H = 6 (sufficient) C

Answer 304

Inequality word problem Was profit > 150000 Was 500000 - L - M > 150000 Is 350000 > L + M? (1) L + M = 3M L = 2M --> Not sufficient (2) Total Profit > L Not sufficient without material costs (3) Together L = 2M 500000 - L - M > L 500000 > 2L + M (sub in 2m for L) 500000 > 4M + M 500000 > 5M 100000 > M and sub in L/2 for M: 100000 > L/2 200000 > L So 300000 > L + M (sufficient)

Answer 305

Roots --> best to SIMPLIFY roots where ever possible = [sqrt(2) + 2 + 2sqrt(2) + 4] / [sqrt(2) + 2] = [3sqrt(2) + 6] / [sqrt(2) + 2] = 3

Answer 306

Squares and Exponents --> try to match the right hand side, find common bases sqrt(2.5/10^5) = 5 * 10^k *MULTIPLY top and bottom of the fraction by 10 --> just shifting the decimal by adjusting the denom sqrt(25/10^6) = 5 * 10^k 5 / (10^6/2) = 5 * 10^k 5 * 10^-3 = 5 * 10^k k = -3

Answer 307

Fractional Inequality - sewing approach > first move all terms to one side so that the other side is equal to 0 > then factor, ensuring that the coefficient on x is positive > then, treat the fraction as a product in order to evaluate the sign, keeping in mind valid and invalid roots [(x - 1)(x + 5)] / [(x - 2)(x + 1)] <= 0 Valid roots: x = 1, - 5 Invalid roots: x = 2, -1 -5 =< x < -1 -----> integers [-5, -2] = 4 integers 1 =< x < 2 ----> integers [1, 1] = 1 integer So a total of 5 integers satisfy the condition

Answer 308

Multi variable inequality --> testing properties (>0, <0) Reframe the question by moving all terms to one side: is (x - y)/(x + y) - 1 > 0 is (2y)/(x + y) < 0 ? -----> asking if top and bottom have OPPOSITE signs Or equivalently, is (2y)(x + y) < 0 (1) x > 0 NS (we need to know sign of y) (2) y < 0 We don't know if x + y > 0 or < 0 (3) Together 2y < 0 Even though we know x > 0 and y <0, we STILL don't know the sign of x + y: 1 - 5 = -4 < 0 10 - 1 = 9 > 0 NS E

Answer 309

Multi variable inequality --> testing properties (>0, <0) Reframe the question by moving all the terms to one side: is (m+x)/(n+x) - m/n > 0 is [x(n - m)] / [n(n + x)] > 0 ---> asking if the sign is positive Or equivalently, is x(n - m)*n(n + x) > 0 DON'T FORGET ABOUT QUESTION STEM INFO: GIVEN THAT m > 0 and n > 0, you can SIMPLIFY question stem FURTHER: Is x(n - m)(n + x) > 0 (1) m < n Both are positive 0 < n - m --> positive Don't know anything about x --> NS (2) x > 0 --> n + x > 0 Don't know anything about value of n - m (> 0 or <0) NS (3) Together n - m > 0 So sufficient (C)

Answer 310

Multi variable absolute value signs --> Distances using number line > common point: x Asking if the distance between x and y is greater than the distance between x and z (1) y is farther from 0 than z We have no info about where x is NS (2) x < 0 We have no info about positions of z and y NS (3) Again, we don't know where x is relative to y and z, or whether y and z are greater than or less than 0 NS E

Answer 311

Multi variable Absolute value signs --> Distance using number line > common point: y Asking if the distance between y and a is greater than the distance between y and b Number line order: a y z b (1) Distance between z and a < distance between z and b Distance between y and a is smaller than the distance between y and b --> | y - b | = | z - y | + | z - b | | y - a | = | z - a | - | z - y | Sufficient (2) Distance between y and a is less than the distance between z and b Again, this means that the distance between y and a is less than the distance between y and b Sufficient D

Answer 312

Multi variable Absolute value signs with split --> SIGNS of the unknowns Reframe the question stem: Do x and y have different signs? Is xy < 0 ? (1) xy < 0 (sufficient) (2) x > y --> NS whether they have the same or opposite signs A

Answer 313

Multi variable Absolute value signs with split --> SIGNS of the unknowns Reframe the question stem: Is x = y ? (1) x^2 - y^2 = 0 (x - y)(x + y) = 0 x = y or x = -y NS (2) | x + y | = | x | + | y | means that x and y have the SAME SIGN But this applies to ANY value of x and y (so they are not necessarily equal) NS (3) Together x and y have the same sign, so x = y (Sufficient) C

Answer 314

Special sequences: > Try to find the pattern yourself in the SUM a1 = 1/1 - 1/2 = 1/2 a2 = 1/2 - 1/3 = 1/6 a3 = 1/3 - 1/4 = 1/12 SUM 1 = 1/2 ---> n/(n + 1) SUM 1 to 2 = 4/6 = 2/3 ---> n/(n + 1) SUM 1 to 3 = 9/12 = 3/4 ---> n/(n + 1) So SUM 1 to 100 = 100/101 OR alternatively, notice how the middle terms CANCEL OUT: 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 ... 1/99 - 1/100 + 1/100 - 1/101 = 1 - 1/101 = 100/101

Answer 315

Special sequences: > Try to find the pattern yourself > NOTICE the difference of squares (2.5 + 1.5)(2.5 - 1.5) + (4.5 + 3.5)(4.5 - 3.5) + ... (100.5 + 99.5)(100.5 - 99.5) = 4*1 + 8*1 + .... + 200*1 Sum of the multiples of 4 or equally spaced sequence = (first + last)/2 * # of terms = (200 + 4)/2 * [(200 - 4)/4 + 1] = 102 * 50 = 5100

Answer 316

Special sequences: > Try to find the pattern yourself > this is a type of arithmetic sequence (actually, there are two arithmetic sequences here, depending on whether n is odd or even) n is odd: Bn = 45 + (n - 1)*7 n is even: Cn = 49 + (n - 1)*7 a1 = 45 = B1 a2 = 49 = C1 --> mult of 7 a3 = 52 = B2 a4 = 56 = C2 --> mult of 7 We know that all the multiples of 7 after 49 are in the sequence, such as 623, 630, 637. 630 = 49 + (n - 1)*7 83 = n - 1 84 = n ---> C84 C84 = 630 B85 = 45 + (84)*7 = 633

Answer 317

Functions > Simplify first if possible, then substitute and simplify (1) f(4) = 0 16a + 4b + c = 0 ---> Not sufficient without knowing a and b (2) f(-b/a) = 0 a(-b/a)^2 + b(-b/a) + c = 0 b^2/a - b^2/a + c = 0 c = 0 --> Sufficient

Answer 318

Functions > SIMPLIFY FIRST by moving all the terms to one side is: [f(w+h) - f(w)]/h - [f(w-h) - f(w)]/h > 0 [f(w+h) - f(w-h)]/h > 0 Given: f(w+h) = -(w+h)^2 + 1 = -(w^2 + 2wh + h^2) + 1 f(w-h) = -(w-h)^2 + 1 = -(w^2 -2wh + h^2) + 1 Therefore question stem simplifies to: Is -4wh / h > 0 ---> you can cancel h in top and bottom since h =/ 0 (without having to change direction of the sign) is -4w > 0 or is w < 0? (1) NS (2) Sufficient B

Answer 319

Multi variable inequality --> properties Use number line to see if x < y < z (1) NS --> unclear whether x < y and whether y < z (2) Sufficient x < x + 1 < y and y < z - 1 < z so x < y < z *Note the C trap (to add both compound inequalities)

Answer 320

Multi variable inequality --> Properties Is x > y or is x - y > 0 ? (1) x - y^2 > 0 x > y^2 --> x > 0 Test a few cases, including fractions y = 1/2, x > 1/4 --> x = 1/3 x < y y = 2, x > 4 --> x = 5 x > y NS (2) xy < 0 (opposite signs) NS (unclear whether x > y or x < y) (3) we know x > 0, so y < 0 Sufficient x > y C

Answer 321

Multi variable inequality --> Properties is 3x - 7y > 0 or Evaluate the ratio: is x/y > 7/3 (1) x > y + 4 --> could manipulate into the question stem x - y > 4 3x - 3y > 12 3x - 3y - 4y > 12 - 4y 3x - 7y > 12 - 4y ---> NS (unsure about y) (2) -5x < -14y x/y > 14/5 14/5 > 7/3 --> sufficient B

Answer 322

Multi variable inequality --> Properties Compound inequality: DRAW A NUMBER LINE mv < pv < 0 (all negative) Simplify into factored form: mv - pv < 0 v(m - p) < 0 Is v > 0 --> In other words, is m < 0 or p < 0? or is m - p < 0 ? > we know that m and p have the SAME sign and opposite sign from v (1) m < p m - p < 0 (sufficient) (2) m < 0 Sufficient

Answer 323

Multi variable inequality --> Properties (with statements that have splits) is x(y + z) >= 0 --> is y + z = 0, or do x(y + z) have same sign? (1) y and z have the same sign --> + + or - - NS (unknown x) > y + z =/ 0 (2) x and y have the same sign NS (unknown z) (3) x y z + + + ---> >= 0 - - - ---> >= 0 ** Sufficient C

Answer 324

Geometry - Irregular polygons Sum of interior angles of a quadrilateral = 360 degrees (1) 2 are right angles = 180 degrees, leaving 180 degrees left. NS - unsure whether one angle is 60 degrees and the other is 120 degrees. (2) DO NOT assume the quadrilateral is a parallelogram - sketch it out If a, b, c, d are the angles, we know a = 2b. so 3b + c + d = 180 degrees NS - unsure about whether one angle is 60 degrees or not. (3) Unclear whether 90 degrees is two times another angle (45 degrees) or whether 120 degrees is two times another angle (60). NS E

Answer 325

Geometry with triangles: Shaded Region = Area of Triangle ACD - Area of Triangle ABE Triangle ACD is an equilateral triangle with area = (s^2*sqrt3)/4 = (9sqrt3)/4 Triangle ABE is a 30-60-90 degree triangle, with sides in ratio x-xsqrt3-2x. x = 1, so the height of the triangle equals sqrt(3). Shaded region = (9sqrt3)/4 - (1*sqrt3)/2 = 7sqrt3/4

Answer 326

Triangles > Use side constraints to determine which sides could be 40, 9 and x > Longest side cannot be 9 > x cannot be equal to 40 (that would create an isosceles triangle that wouldn't be obtuse, or the obtuse angle would be opposite from 9, the smallest side) Basic side constraints: difference of two sides < x < sum of two sides 31 < x < 49 Obtuse triangle constraints: a^2 + b^2 < c^2 if longest side equals x: x > 41 81 + 1600 < x^2 1681 < x^2 However, 41^2 = 1681 (so longest side cannot be 41 or more) if longest side equals 40: 81 + x^2 < 1600 x^2 < 1519 39*39 = 1521 > 1519 33*33 = 1089 < 1519 ---> Ans B

Answer 327

Triangles > when you see 90 degrees, think of Pythagorean theorem > when you see constraints, think of side constraints of a triangle (1) If BE > 1/3, then the hypothenuse of the sub-triangles created must be greater than 1/3 (AB and BC). If AE > 1/3, then the hypotenuse of the sub-triangles created must also be greater than 1/3 (AB and AC). Therefore, all sides of the triangle are > 1/3, so Perimeter > 1 (sufficient) (2) If BE > 1/2, then the hypotenuse of the sub-triangles created must be greater than 1/2 (AB and BC) Therefore, the perimeter already exceeds 1 with just two sides. (sufficient).

Answer 328

Circles question about radius and diameter MAP OUT WHAT YOU NEED: Region = Area larger circle - Area smaller circle = (need AC or CE) - (need AB or BD) (1) AB = 3 = BD --> can calculate area of smaller circle BC = 2 --> AB + BC = AC --> can calculate area of larger circle Sufficient (2) CD = 1 DE = 4 ----> CD + DE ---> can calculate area of larger circle CD + DE = CE = AC AC + CD = AD --> can calculate area of smaller circle Sufficient D

Answer 329

Angles in Triangles and Circles > rely on side lengths to tell you about which angles are equal (equilateral triangles or isosceles triangles) > start off by mapping out radius > use symbols to mark which angles are equal > Also don't forget about EXTERIOR angles REMEMBER: within a triangle, equal sides have equal opposite angles. This doesn't apply to OTHER triangles with the same angle (i.e., sides are just similar, not congruent) b = 2a (1) BAO = 20 > exterior angles 60 = a + b b = 2a Sufficient (2) BAO = 20 b = 2a = 40 a = 20 Sufficient D

Answer 330

Properties of Squares and Right Triangles GUESS: 3-4-5 triangle with perimeter 12 Larger square has each side = 20/4 = 5 Smaller square has each side = 4/4 = 1 First FIGURE OUT which sides of the triangle are congruent with each other > 5 is the hypotenuse Let z be the length of the longer leg of the triangle. The shorter leg has a length equal to z - 1. Pythagorean theorem: 5^2 = z^2 + (z - 1)^2 z = 4 z - 1 = 3 So P = 12

Answer 331

Complex Geometry with inscribed triangles and sectors Don't bother doing calculations. Start off by mapping out what info you need! Area of shaded region = Area of sector ACD - Area of Triangle ADC - (Area of Sector ABC - Area triangle ABC) Area of sector ACD: > Need central angle = 90 plus angle BCD Area of Triangle ADC: > Need height Area of Sector ABC: > have everything (radius 5, angle 90) Area of Triangle ABC: > have everything (45-45-90 degree triangle) (1) Angle BAD = 15 ---> arc is BD So any inscribed angle sharing the same arc has an angle of 15 Therefore, the central angle for arc BD = 2*15 = 30 degrees = angle BCD ---> we know Area of sector ADC Just need height: > Triangle is a 30-60-90 --> we know hypotenuse is 5, so we know the height Sufficient (2) Angle DAC = 30 ---> we know Angle BAD = 15 (because angle BAC is 45 degrees) > Sufficient D

Answer 332

Circles: arcs > given radius --> isosceles and equilateral triangles > arcs and sectors --> need angles External Perimeter = Circumference*4 - 4*overlapping parts Circumference = 2*pi*1 There are actually FOUR equilateral triangles created inside the overlap --> angle is 60 degrees So the central angle of the arc we need to find is 120 degrees So external perimeter = 10pi / 3

Answer 333

Coordinate Plane Geometry: Distance *Don't forget that m and n are integers! "How many" --> cases or combinatorics Distance = sqrt(10) = sqrt[ (1 - m)^2 + (2 - n)^2 ] 10 = (1 - m)^2 + (2 - n)^2 ----> recognize the squares as perfect squares List out possible sums of perfect squares that equal 10: 10 = 1 + 9 Case 1) (1 - m)^2 = 1 ---> m = 0 or 2 (2 - n)^2 = 9 ---> n = -1 or 5 Case 2) (1 - m)^2 = 9 --> m = -2 or 4 (2 - n)^2 = 1 --> n = 1 or 3 So total number of possible points A: 2*2 + 2*2 = 8

Answer 334

Coordinate Plane Geometry: Distance Rephrase the question stem: is sqrt(r^2 + s^2) = sqrt(u^2 + v^2)? Or is r^2 + s^2 = u^2 + v^2? (1) r + s = 1 NS (no info about u and v) (2) u^2 + v^2 = (1 - r)^2 + (1 - s)^2 ---> FACTOR to see if equal r^2 + s^2 = 1 - 2r + r^2 + 1 - 2s + s^2 ---> reorder from highest to lowest exponent = r^2 + s^2 - 2(r + s) + 2 ---> NS, unknown r + s (3) Sufficient (yes) C

Answer 335

Coordinate Plane Geometry: Slope Is slope < 0 ? (1) Two points, calculate the slope of Line 1: y2 - y1 / x2 - x1 = (-s - r)/(r + s) = -(r + s)/(r + s) = -1 < 0 --> sufficient (2) Two points, calculate the slope of Line 1: = (-s - t)/(r - u) u < r --> r - u > 0 t < -s --> -s - t > 0 Therefore slope > 0 --> sufficient D

Answer 336

Coordinate Plane Geometry: Probability = # of possible points / area of the rectangle = # of possible points / 24 *DON'T FORGET about non-integer values. Therefore = Area / 24 Write out the EQUATION: x + y < 4 y = -x + 4 The possible points fall in an isosceles triangle with sides equal to 4. The area of the triangle = (4*4)/2 = 8 So probability = 8/24 = 1/3

Answer 337

Coordinate Geometry: Symmetry Keyword: Perpendicular bisector Point A is a reflection of Point B about y = x: A (2, 3) => B (3, 2) Point B is a reflection of Point C about the x axis, y = 0: B (3, 2) => C (3, -2)

Answer 338

Coordinate Plane Geometry: Circles and Distance *Center is at the origin --> Given the radius 50 and either x or y coordinate of a point on the circle, we can calculate the exact point (and therefore distance between P and Q) (1) Given the x coordinate, there are two values of y that can be calculated (+/- 40) > However, there is actually ONLY ONE distance that can be calculated (symmetry) > Sufficient (2) Given the y coordinate, there are two values of x that can be calculated (+/- 30) > However, there are still TWO DIFFERENT distances that can be calculated > Not sufficient A

Answer 339

Coordinate Plane Geometry is m = n? (1) m + n = -1 --> Not sufficient, m and n can be different things (two variables, one equation) (2) m*n = 1/4 --> Not sufficient (two variables, one equation) (3) Together (sufficient, two variables, two different equations) C m = -1 - n (-1 - n)*n = 1/4 -n - n^2 = 1/4 4n^2 + 4n + 1 = 0 (2n + 1)^2 = 0 n = -1/2, m = -1/2 Sufficient

Answer 340

Geometry *This is NOT a question about maximizing area by creating a square fence (because one side is a wall and thus doesn't use the fence material) Set up the equation for area: Area = L* W = x*(80 - 2x) = 80x - 2x^2 ---> to maximize area, take the first derivative and set it to 0 0 = 80 - 4x 4x = 80 x = 20 So the length of the side that is opposite to the wall = 80 - 2*20 = 40 Alternatively, find the max of the quadratic function f(x) = -2x^2 + 80x = -2x(x - 40) ---> roots are x = 0 and x = 40 Halfway through is x = 20 --> due to symmetry, the max is at this point.

Answer 341

Quadratic functions > Asking if P(x) opens downward and is beneath the x axis (1) a < 0 > P(x) opens downward, but we don't know if the max is above or below the x axis NS (2) b^2 - 4ac < 0 > P(x) has 0 roots > P(x) either opens up and is above the x axis or opens down and is below the x axis > NS (3) Together P(x) must open down and is below the x axis Sufficient C

Answer 342

Digit Question - given Value A = _ _ = 3 a B = _ _ = b 5 Digit constraint: 0 <= a <= 9 1 <= b <= 9 5A + 2B? --> find the value of A and B A*(B + 1) = 2016 --> Units digit is 6 a * (5 + 1) = units digit of 6 a = 1 or 6 so A = 31 or 36 Figure out which one is a factor of 2016: 2016/31 = NOT divisible 2016/36 = 56 Therefore A = 36, B = 55 5A + 2B = 5*36 + 2*55 = 180 + 110 = 290 ALTERNATIVELY, write out algebraic approach: A*(B + 1) = 2016 (300 + a)(100b + 5 + 1) = 2016 300b + 180 + 10ab + 6a = 2016 ---> since all other terms are multiples of 10, 6a must be = 6 (units digit) a = 1 or 6

Answer 343

Decimals - Rounding Asked for POSSIBLE values of E - S --> determine a RANGE: Minimum Difference < E - S < Maximum Difference 1/3*30 = 10 decimals with even tenths 20 decimals with odd tenths Even tenth max difference = +1 (i.e., 1.01 to 2) Even tenth min difference = +0.2 (i.e., 1.8 to 2) Odd tenth max difference = -0.9 (i.e., 1.9 to 1) Odd tenth min difference = -0.1 (i.e., 1.1 to 1) Max cumulative difference for even tenths: 1*10 = +10 Min cumulative difference for even tenths: 0.2*10 = +2 Max cumulative difference for odd tenths: -0.9*20 = -18 Min cumulative difference for odd tenths: -0.1*20 = -2 Therefore the range of differences: +2 - 18 <= E - S <= +10 - 2 -16 <= E - S <= 8 (due to rounding we can include -16) I and II

Answer 344

Digits - Pattern recognition Test out some values to see what resulting integer looks like: 100 - 74 = 26 (# of digits = # of 0s in 10) 1000 - 74 = 926 (# of digits = # of 0s in 10) so 10^50 - 74 has 50 digits, of which two 2 and 6 and 48 are 9s. SUM of digits: 9*48 + 2 + 6 = 440

Answer 345

Terminating Decimals > Once simplified in prime factored form, the denom only has powers of 2 and/or 5 FIRST LOOK AT THE DENOMS and see if there are any that ONLY have 2s or 5s A - (2*5)/(multiple of 3) --> No B - (3*5)/(14^2) --> No (because 7 in denom) C - (2^4)/(multiple of 3) --> No D - (5^2)/(12^2) --> No (because 3 in denom) E - (3*13)/(2^7) --> Yes (simplified with only powers of 2 in the denom)

Answer 346

Percent Question Set up an equation: 1000*(1 + x/100)*(1 + y/100) = z z? (1) xy = 20 NS --> x and y can be many different values, so z can be many different values (2) x + y + (xy)/100 = 9.2 BE CAREFUL with YOUR ALGEBRA: simplify the question stem 1000(1 + y/100 + x/100 + (xy)/10000) = z 1000 + 10y + 10x + (xy)/10 = z 1000 + 10(x + y + (xy)/100) = z Sufficient B

Answer 347

Percent question > understand WORDING > Let a be the cost of car 1 > Let b be the cost of car 2 profit percentage = (P - C)/C Car 1: Profit = 0.25*a = 20000 - a 1.25a = 20000 a = 20000/(1.25) = 20000 *4/5 a = 16000 Car 2: Profit = -0.2*b = 20000 - b 0.8b = 20000 b = 20000/0.8 = 20000*(5/4) b = 25000 Total profit = 4000 - 5000 = -1000

Answer 348

Double Matrix problem with inequality Find the max and min number of students who major in biology and chemistry MIN number of students: > maximize # of students who major in only chemistry (y) y <= 20 --> 20 > so min # of students who major in chem and bio = 130 - 20 = 110 MAX number of students: > minimize # of students only in bio = NOT 0 (because there is a constraint - CHECK TO SEE IF IT MATCHES THE ENTIRE TABLE!!!!!!!!) ---> = 20 > so max # of students who major in chem and bio = 150 - 20 = 130 110 < students in chem and bio < 130

Answer 349

Triple venn diagram - Formula approach Total = A + B + C - AandB - BandC - AandC + AandBandC + other 300 = (0.4*300) + (0.3*300) + (0.75*300) - (onlyAB + center) - (onlyBC + center) - (only AC + center) + center + 0 300 = (120 + 90 + 225) - (onlyAB + onlyBC + onlyAC) - 2*center 300 = 435 - (105) - 2center 2center = 30 center = 15 people So # of people experienced only one effect = Total - center - only two = 300 - 15 - 105 = 180

Answer 350

Statistics - set transformations WRITE OUT the mean formula: Mean Original = SUM/10 Mean New = (SUM + 5x + 5y)/10 = SUM/10 + (x + y)/2 Mean New - Mean Original = (x + y)/2 (1) sufficient (2) not sufficient (cannot get x + y) A

Answer 351

Statistics - set transformations Original Set: X's --> st dev = 4.6 New Set: Y's --> every term is multiplied by 5 and added to 30 > St Dev increases by *5 (but is unaffected by addition) Therefore, st dev of y coordinates = 4.6*5 = 23

Answer 352

Statistics - ST DEV range 8.1 +/- 1.5*stdev 8.1 +/- 1.5*0.3 8.1 +/- 0.45 7.65 < Data point < 8.55 So 11 out of the 12 cups fall within this range

Answer 353

Statistics - min/max MAX z by MINIMIZING other terms Keep in mind that the set contains different positive integers > minimum positive integer is 1 (set = x) 1 < 2 < y < z ---> minimum y is 3 1 < 2 < 3 < z 10 = (1 + 2 + 3 + z)/4 40 = 6 + z z = 34

Answer 354

Statistics with consecutive integers > write the terms in each set with starting variables > keep in mind that the terms could be positive, negative, or zero > consecutive integers --> mean = median = (first + last)/2 Set S: a a+1 a+2 a+3 a+4 Set T: t t+1 t+2 t+3 t+4 t+5 t+6 Rephrase question is a + 2 = t + 3? Or is avg of set S = avg of set T? (1) Not sufficient without set T info (2) Sum of consecutive integers = avg * # of terms Set S sum = (a + 2)*5 Set T sum = (t + 3)*7 Sums equal: (a + 2)*5 = (t + 3)*7 a + 2 =/ t + 3 =/0 BUT if a + 2 = 0 and t + 3 = 0, then they are equal ALTERNATIVELY: Sum S = Sum T So comparing Sum S/5 to Sum T/7 NS (3) Together sufficient C

Answer 355

Combinatorics question -- permutation with duplicates *PAY ATTENTION to the permitted directions (R and U) 1) Total number of paths from A to B > Determine the possible steps needed > Right four times and up three times R R R R U U U Total possible ways to array this: 7!/(4! * 3!) = 35 2) Total number of paths from A to C to B > "and" > Total number of paths from A to C AND Total number of paths from C to B = (R R U U) * (R R U) = (4!/(2!*2!)) * (3!/(2!)) = 6 * 3 = 18 OR do the pyramid approach (sum paths at each vertex) **if the paths connect at a COMMON point --> might need to do MULTIPLICATION at each node > count the first few to see if you are getting the right number

Answer 356

Combinatorics question - pyramid questions (# of paths) 1) Draw out the branches 2) For each decision point, determine how many paths there are to get to that point (summation) 3) Keep going until you reach the bottom There are a total of 20 ways to spell PASCAL

Answer 357

Inscribed Shapes in Circles > don't forget about ANGLES Total Sum of interior angles = (5 - 2)*180 = 540 Each interior angle = 540/5 = 108 >There are also five congruent triangles formed by connecting the center to each vertex > Each angle = 360/5 = 72 We need to be able to find out the value of each side. (1) radius = 4 > possible to calculate each side (due to sin-cos-tan) and FIXES shape (3 angles, one side) > altitude drawn from the center to each side is a perpendicular bisector Sufficient (2) Each diameter of the pentagon equals 8 > forms several isosceles triangles > AAS (with altitude drawn, perpendicular bisector) Sufficient

Answer 358

Odd Even Properties Rewrite X = 2n --> n could be even or odd Sub X = 2n into each option to see which one must be odd (2n + 1) E is ans 3(2n)^2 / 2 + 1 = (3(4n^2))/2 + 1 = 3*2*n^2 + 1 --> odd

Answer 359

Inequality Word Problem with Max/Min > integer values > all diff values Did F > 11? ----> see if the MINIMUM value of F exceeds 11. Sum = 90 TEST values for F around the inequality (1) Thurs = 8 F > 8 Case 1) F = 10 3 + 4 + 5 + 6 + 7 + 10 + 55 (works) Case 2) F = 12 3 + 4 + 5 + 6 + 7 + 12 + 53 (works) NS (2) Saturday = 38 90 - 38 = 52 left Case 1) F = 10 Sun + Mon + Tues + Wed + Thurs = 42 Avg = 42/5 = 8.4 5 + 6 + 7 + 8 + 9 =/ 42 INVALID Case 2) F = 12 Sun + Mon + Tues + Wed + Thurs = 40 Avg = 8 6 + 7 + 8 + 9 + 10 = 40 (works) SUFFICIENT B

Answer 360

Prime Properties "defined" - special function --> focus on UNDERSTANDING what it means if p is a prime number, then the integers less than p ALL have no common factors with p aside from 1 e.g., p = 5, the positive integers would be 4, 3, 2 AND 1 p - 1

Answer 361

Work problem Set up the equation and SOLVE *RHS of the equation isn't 1 anymore y: t days for w wedges x: t + 2 days for w wedges Ry = w/t wedges per day Rx = w/(t + 2) wedges per day [w/(t + 2) + w/t] * 3 = (5/4)w ---> immediately cancel out w *don't be afraid of ugly factoring ---> just list out factor pairs and test > don't forget about a = 5 (not 1) t = 4 t + 2 = 6 Time to complete 2w = 12

Answer 362

Rate Question > you need at least TWO info together (rate, distance, or time) to be sufficient > e.g., if calculating distance, just relative rates alone or relative time alone is insufficient (1) TWO DIFF EQUATIONS, two unknowns, SUFFICIENT Total T = 10 = t1 + t2 Total D = 530 = 50(t1) + 60(t2) (2) t1 = t2 + 4 D1 = 50(t1) = 50(t2 + 4) = 50(t2) + 200 D2 = 60(t2) Not sufficient --> need relative distances A

Answer 363

Digit Question > Think about CARRY OVER > k = _ _ = a b (could be three or more digits) > digits between 0 and 9, inclusive 10a + b + 5 => tens digit is 4 What is a? (a = 4 or 3) Or is b >= 5? (1) k > 35 NS b can be >=5 or < 5 (2) b > 5 Sufficient b + 5 > 10 --> carry over a = 3 B

Answer 364

Exponents Rephrase question stem: What is k or what is n? (1) only one value of n fits this constraint k = 51,000 Sufficient (2) 2.601 * 10^9 is a positive INTEGER and a perfect square. k is also a positive integer. So we can take the square root to find k. Sufficient Algebraically: k^2 = 2.601 * 10^9 = (5.1 * 10^n)^2 n = 4 Sufficient D

Answer 365

Exponents > find the value of a > Constraint: a is an integer (+,-,0) Is a < 4? (1) Rewrite as fractions > don't assume the bracket around 2a and 2! You have to factor out -1. 1/[10^(2a-2)] < 1/10^3 ---> both are positive bases 2.5 < a a >= 3 (since a is an integer) NS OR alternatively, know that 10^x is an exponential function where the exponent x can be positive or negative. Drop the bases immediately. (2) a > 2 NS (3) a >=3 , statements 1 and 2 are the same E

Answer 366

Probability - overlap > "possible value" --> need a range P(AUB) = P(A) + P(B) - P(AandB) = 1/2 + 1/3 - P(AandB) Unknown relationship between A and B: Min P(AandB) = 0 (no overlap) ---> P(AUB) = 5/6 (max) Max P(AandB) = 1/3 (B inside A) ---> P(AUB) = 1/2 + 1/3 - 1/3 = 1/2 (min) So 1/2 <= P(AUB) <= 5/6 ** INCLUSIVE END POINTS II and III

Answer 367

Word Problem Application: Linear Equations > variable represents PRICE (in cents) (1) 5 = 8*(r/100) + 6*(d/100) 500 = 8r + 6d ----> SIMPLIFY FURTHER 250 = 4r + 3d ---> r and d are INTEGERS Find two valid cases to prove NS: 250 = 40 + 210 250 = 220 + 30 (2) 10 = 16*(r/100) + 12*(d/100) 1000 = 16r + 12d 500 = 8r + 6d 250 = 4r + 3d --> same as Statement 1 NS (3) NS together E

Answer 368

Rates Question: > Write two equations and solve > Common distance tu = 0.5 + td 90 = (v - 3)*tu = (v - 3)*(0.5 + td) 90 = (v + 3)*td td = (v - 3)/12 ---> SUB BACK INTO one of the original equations v = 33 td = 2.5

Answer 369

Geometry PS: > Since there is always a solution and NO measurements are given, you can solve as if the shapes are regular polygons and the rectangle is perfectly along the diagonal. (perfectly symmetrical) - isosceles right triangles Ratio = sqrt(2)

Answer 370

Triangle Angle Question > think about LINES, triangles, and exterior angles Exterior angle approach: x + y + z + w + v = 180 degress Alternatively, PS without measurements -- assume all angles are congruent 5x = ? Each interior angle of the pentagon = 540/5 = 108 Each angle equals = (180 - 108)/2 = 36 5(x) = 5*36 = 180

Answer 371

Word Problem Application: Linear Equations > simplify wherever possible > convert decimals to integers 6 = Pc*c + Pd*d d = ? (1) 2Pd= 3Pc - 0.1 Pc = (2Pd + 0.1)/3 Eq'n becomes 6 = (2Pd + 0.1)/3*c + Pd*d --> 3 variables still, NS (2) (Pc + Pd)/2 = 0.35 Pc + Pd = 0.7 Pc = 0.7 - Pd Eq'n becomes 6 = (0.7 - Pd)*c + Pd*d --> 3 variables still, NS (3) Together, two equations > can solve for Pd and Pc 60 = 3c + 4d ---> No unique solution NS E

Answer 372

Evenly Space Sets: Sum of integers [2, 100] = 2550 Sum of integers [102, 200] ---> every term in the previous set plus 100. = 2550 + 100*50 = 2550 + 5000 = 7550

Answer 373

Work problem: > since this is a PS question that gives us NO measurements and asks for "MUST BE TRUE", we can test cases Logically: c < h 1/c > 1/h t must be lower than c and h (combined time is smaller than individual time) and greater than 0 time. I) If there were two cold leaks, t = c/2 If there were two hot leaks, t = h/2 Since c < h, c/2 < h/2: c/2 < t < h/2 III)

Answer 374

GCD is the same as Greatest Common Factor: > Find the smallest powers OR get each integer into product form and see if there are fixed common factors > Remember multiple rules > Simplify wherever possible > Test cases or multiple rules (1) x = 12u ---> sub into the equation in the question 12u = 8y + 12 12u - 12 = 8y 3u - 3 = 2y ----> 2y is a multiple of 3. Therefore, y must be a multiple of 3 2y = 3(u - 1) x = 2^2 * 3 * u y = mult of 3 = (3u - 3)/2 ---> u must be odd x and y could have different common factors TEST two cases to prove NS: u = 3 x = 36 y = 3 GCF = 3 u = 5 x = 60 y = 6 GCF = 6 (2) y = 12z --> sub into the equation x = 8(12z) + 12 --> x is a multiple of 12 x = 12(8z + 1) --> factor out 12 y is a multiple of 12 Both depend on z y and x don't have any common factors other than 12 > 8z + 1 has no common factors with z (n and n+1 have no common factors) > 8z is a multiple of z. So 8z + 1 is not a multiple of z. Sufficient B

Answer 375

Mixture problem: Ingredient = Alcohol Original solution has 50% alcohol Let V be the original volume. Let P be the amount of liquid replaced. What is P/V = ? Q of ingredient before mixing = Q of ingredient after mixing 0.5(V - P) + 0.25P = 0.3(V) 0.5V - 0.5P + 0.25P = 0.3V 0.2V = 0.25P P/V = 0.2/0.25 = 0.8

Answer 376

Consecutive Integers and Statistics: Median = Mean for a set of consecutive integers = 140 M - 1 terms fall before or after the median. (M - 1)/2 terms fall after the median. Largest integer = 140 + [(M - 1)/2]*1

Answer 377

Statistics: *Real numbers INCLUDE DECIMALS "Could be" --> RANGE _ _ _ _ _ _ _ 2 _ _ 6 _ _ 20 Since 3 occurs the most often in the list, TWO 3's must exist: 2 3 3 6 _ _ 20 Let x and y be real numbers between 6 and 20. 6 < x, y < 20 12 < x + y < 40 Average = (2 + 3 + 3 + 6 + 20 + x + y)/7 = (34 + x + y)/7 I) Could average = 7? 49 = 34 + x + y 15 = x + y (possible in range) II) Could average = 8.5? 59.5 = 34 + x + y 25.5 = x + y (possible in range) III) Could average = 10.5? 73.5 = 34 + x + y 39.5 = x + y (possible in range) ALL possible

Answer 378

Statistics: Standard Deviation * no need to perform calculations * > Compare each list and see which one has numbers that are the furthest apart from the MEAN > Tip: Dot plot, one set at a time (what are the TRANSFORMATIONS?) > smaller range = smaller st dev 5588 and 5599 --> 5599 5668 and 5599 --> 5599 5669 and 5599 --> 5599 5789 and 5599 --> 5599 5599

Answer 379

Statistics: Standard Deviation Zero overlap between Set A and Set B: StDev > 7 + 13 StDev > 20 Some overlap between Set A and Set B: 13 < StDev < 20 Complete overlap between Set A and Set B (A inside B): StDev = 13

Answer 380

Statistics: Range Max Height - Min Height (1) Min Boy = Max Girl - 3 > Min Boy = Max Boy - 14 = Max Girl - 3 > The max height = max boy's height > Min height = min girl's height Max Height = Min B + 14 Min Height = Max Girl - 12 Min B + 14 - Max Girl + 12 = Max Girl - 3 + 14 - Max Girl + 12 = 23 Sufficient (2) Tallest boy is 75 inches NS (don't know relative heights between boys and girls).

Answer 381

Min/Max Problem: > Minimize the score of a team by MAXIMIZING the score of the two other teams (6) Possible points based on place: 1st = 6 - 1 = 5 2nd = 4 3rd = 3 4th = 2 5th = 1 6th, 7th, 8th, 9th = 0 6 = 5 + 1 + 0 6 = 4 + 2 + 0 LAST TEAM = 3 + 0 + 0 = 3

Answer 382

Remainder Question > Determine what the fractional part is n/k = z + r/k (1) FACTOR out n = k^3 + 3k^2*1 + 3k*1^2 + 1^3 n = k^3 + 3k^2 + 3k + 1 n/k = k^2 + 3k + 3 + 1/k Remainder is 1/k ---> since k > 1, this remainder is FIXED (sufficient) (2) k = 5 NS (unknown n) A

Answer 383

Sets and subsets - NOT a double matrix problem (no overlap) Cast Ballot = For + Against Did not cast Ballot Cast Ballot + Did not cast Ballot = Registered Voters For = ? (1) 25000 = Did not cast Ballot = 1/4*registered voters Registered Voters = 100,000 NS --> unknown split between For and Against (2) 55% * Cast Ballot = Win 55% * (3/4 * Registered Voters) = win NS --> unknown # of registered voters (2) Registered voters = 100,000 55% * 3/4 * 100,000 = win (sufficient) C

Answer 384

Combinatorics - Permutation > first have to choose the letters, then arrange them > # of groups of 5 * # of ways to arrange group of 5 Use formula: P10,5 / P10,4 = [10!/(10-5)!] / [10!/(10-4)] = 6!/5! = 6 to 1

Answer 385

Probability Question --> fixed order (only 1 case) P(WRRW)? = (2/4) * (2/3) * (1/2) * (1) = 1/6

Answer 386

Inequality Word Problem with Max/Min > integer values 75 = A + B + C + D + G + F A = 10? (1) If G sent 41 reps, then F > 41 --> exceeds 75 Therefore the most number of reps sent by a country is 41, leaving 34 reps left to account for. A + B + C + D + G = 34 TEST VALUES OF G Case 1) G = 10 A + B + C + D = 24 (and A, B, C, D < 10) Avg of 4 variables is 6 (just ADJUST gaps so gap = 0) 4 + 5 + 7 + 8 = 24 Works Case 2) G = 9 A + B + C + D = 25 (and A, B, C, D < 9) Avg of 4 variables is ~`6.2 4 + 6 + 7 + 8 = 25 Works NS (2) G < 12 Both test cases above work NS (3) NS - both test cases above work E

Answer 387

Statistics: Average Average = Sum/n = ? (1) 6ft 2.5in = (Sum of the first third largest heights)/(n/3) 5ft 10in = (Sum of the two third heights)/(2n/3) Can find average = (Sum of first third + Sum of two third)/n > n cancels out Sufficient (2) Don't know n NS A

Answer 388

Coordinate Geometry SLOPE = (y2 - y1)/(x2 - x1) or graphical approach (1) NS --> either line n or p could have the greater slope (2) y int of n > y int of p NS (depending on where they intersect, or if they are parallel) (3) Algebraic approach: Each line has two points: (5, 1) and y int slope n = (1 - ny)/(5 - 0) slope p = (1 - py)/(5 - 0) ny > py ny = 1 - 5(n) py = 1 - 5(p) So: 1 - 5(n) > 1 - 5(p) 5p > 5n p > n (Sufficient)

Answer 389

Digit question what is the value of x, where z = _ . _ x (1) 100z = _ . 2 _ z (smaller than 100z) = _ . _ _ 2 ---> NS to know hundreths digit (2) 1000z = 2 . _ _ z (smaller than 1000z) = _ . _ _ 2 NS to know hundrethds digit (3) Same info from statement 1 and 2 NS E

Answer 390

Word Problem: > make sure you have TWO DIFFERENT equations to solve for two variables (otherwise, NS) 360 = b + v + g v = ? (1) b = 40% * 360 = 144 216 = v + g (NS) (2) b = 2/3 * (v + g) 360 = 2/3*(v) + 2/3*(g) + v + g 360 = 5/3*(v) + 5/3*(g) 216 = v + g NS (3) NS (same equation)

Answer 391

Linear Equation word problems > Normally you need n different equations to solve, but watch out for exceptions What is 4b + 2g = ? > what is 2(2b + g) = ? b and g are positive integers. (1) 6b + 3g = 42 3(2b + g) = 42 ---> sufficient (combo exception) (2) 5b + 7g = 53 CHECK to see if there is only one solution: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 7, 14, 21, 28, 35, 42, 49 25 + 28 = 53 ---> one solution for b and g Sufficient D

Answer 392

Double Matrix > Use variables and rewrite total as M + F 2/5*(M + F) = business majors 2/5*(M) + 2/5*(F) = business majors F=? (1) 2/5*(M) = Male business majors 2/5*(F) = female business majors NS (cannot solve for F without values) (2) NS without knowing proportion of males who are business majors > cannot assume 2/5 males are business students (3) Sufficient C

Answer 393

Statistics: Percentiles 90th percentile means A is at the POSITION representing 90% of the terms > 90% * 80 = 72 (A is at position 72) > 71 grades below A and 8 grades above A In the second class, A is NOT one of the grades. So there are 19 grades above A and 81 grades lower than A In the combined class: > # of grades lower than A: 71 + 81 = 152 > A in the middle > # of grades above A: 8 + 19 = 27 A's percentile = A's Position/Total grades = (152 + 1)/(180) = 85th percentile

Answer 394

Work Problem: > Quantity is not 1 job Rate of J: 60 candies/2 minutes = 30 candies/min Rate of K * time = 120 candies time = 120 / rate of k = ? (1) NS (given an unbounded inequality) time > 10 minutes for 120 candies (2) (30 + rate of k)*(1.5) = 60 rate of k = 10 candies/min (sufficient) B

Answer 395

Inequalities (max/min) with exponents and primes *x and y CAN BE decimals!!! -9 < x < 9 -5 < y < 5 Max possible value of x - 2y? > max x, min y x < 9 y > -5 x - 2y < 9 - 2(-5) x - 2y < 19 ---> largest prime is 17

Answer 396

Profit word problem: Same selling price, $96. DIFFERENT COSTS. **Selling price is kind of like a markup or markdown First share: Profit of 20% Profit % = (Price - Cost)/Cost 0.2 = (96 - C)/C 0.2C = 96 - C 1.2C = 96 C = 96/1.2 = 80 Profit = 96 - 80 = 16 Alternatively (markup approach): Price > Cost: 96 = 1.2C Second share: Loss of 20% Profit % = (Price - Cost)/Cost -0.2 = (96 - C)/C -0.2C = 96 - C 0.8C = 96 C = 96/0.8 = 120 Loss = 96 - 120 = -24 Alternatively (markdown approach): Price < Cost 96 = 0.8C Profit = 16 - 24 = -8 (loss)

Answer 397

Statistics with inequalities (max/min) a 70 is d + e > 140 ----> find what the MINIMUM sum equals (min d and e) (1) c = 70 min d and e = 71 + 72 > 140 Sufficient (2) (a + b + c + d + e)/5 = 70 a + b + c + d + e = 350 min d and e ---> MAX a + b + C (see if d + e can be < 140) Test case: a + 67 + 68 + 69 + 70 = 350 a = 74 --> not possible Therefore, d + e > 140 Sufficient

Answer 398

Combination question _ _ _ 10 people, 5 couples = Total # of ways - # of ways to arrange with a married couple = (C10,3) - (5 * 1 * 8 ) = 80 5 possible couples * 1 fixed partner * 8 ppl who are not a couple with the chosen pair > no need to multiply by the number of arrangements (this is a combination question and is not a probability question)

Answer 399

Permutation _ _ _ _ _ = 5! * 2^5 ----> AB and BA within each couple = 3840

Answer 400

Multi variable Absolute value signs - Distance OR split zy < xy < 0 -----> z and x have the same sign, opposite from y Either z and x < 0 and y > 0, or z and x > 0 and y < 0 Rearrange: is | x - z | = | z | - | x | ? **Question stem ITSELF is sufficient | x - z | = | z - x | = | z | - | x | when z and x are the same sign AND | z | > | x | (1) z < x zy < xy ---> sign doesn't change direction Therefore y > 0 and z and x < 0 Yes, sufficient (2) y > 0 (same info as 1) Sufficient D

Answer 401

Roots Algebra Compare inequalities and ask yourself if it must be true. x + y > 0 I) Is sqrt(x + y)/2x > 1/[sqrt(x + y)]? Is x + y > 2x Is y > x? ----> not always! False II) Is [sqrt(x) + sqrt(y)]/(x + y) > 1/[sqrt(x + y)]? *We need to get rid of the root in the DENOM > multiply top and bottom of the RHS by sqrt(x + y) > becomes x + y ---> can cancel both denoms Is sqrt(x) + sqrt(y) > sqrt(x + y)? *Square both sides (since they are both positive, there is no need to change the inequality sign) Is x + y + 2sqrt(xy) > x + y Is 2sqrt(xy) > 0? -----> always yes! True III) Is [sqrt(x) - sqrt(y)]/(x + y) > 1/[sqrt(x + y)]? Is sqrt(x) - sqrt(y) > sqrt(x + y)? Opposite from statement II --> FALSE

Answer 402

Even/Odd properties (1) x(y + 5) = even Either term can be even or odd NS (2) 6y^2 is even 25 is odd E + O + O = even ---> 41y must be Odd, y must be odd NS - nothing about x (3) x(odd + 5) = even x(even) = even x can be even or odd NS E

Answer 403

Factors Problem with Exponents k > 1 int or k >= 2 int r > 0 Basically asking if k is composed of only 2's as factors. (1) k = 2^6 * m NS - m could be 2 or another integer (2) k has no odd factors so k is composed of only 2's as factors Sufficient B

Answer 404

Multi-variable Inequalities - properties SIMPLIFY the question: Is -y > y Is 0 > 2y ? or is y < 0 ? GIVEN: x - y > 10 (1) x = 8 8 - y > 10 -2 > y (always negative) SUFFICIENT (2) y = -20 (always negative) Sufficient D

Answer 405

Odd/Even Properties SIMPLIFY the question: Is w/x + y/z = odd? Is (wz + xy)/xz = odd ? (1) Use Even/Odd properties wx + yz = odd wx could be odd --> both w and x are odd, so w/x = Odd yz could be even --> y/z = even OR odd odd + even = odd odd + odd = even NS (2) SUB in wz + xy odd/xz = INTEGER (we know that w/x and y/z are both integers) xz must be odd, and o/o = odd (sufficient)

Answer 406

Factors and exponents a, b, k, and m > 0 int is b^m = a^k * int (1) a is a factor of b NS --> unknown exponents e.g., 4^2/2^5 is not an integer (2) k <= m NS --> unknown a and b (3) e.g., 4^2/2^1 4^2/2^2 a^k is always equal to less than b^m e.g., b^3/a^2 = bbb/aa = b (all of a will cancel out) Sufficient C

Answer 407

Probability _ _ [13, 27] > 13, 17, 19, 23 (4 primes) > 27 - 13 + 1 = 15 numbers > 11 composite numbers P(neither prime) = P(not prime AND not prime) = (11/15)*(10/14) = 11/21

Answer 408

Combinatorics - Permutation (order matters) > two types - songs and dances > NOT identical objects though _ _ _ _ _ _ _ _ (8 objects in total) 1) 8! = 40,320 2) 6! * 3! = 4320 3) Total - # arrangements dances performed one after the other = 40320 - 4320 = 36000

Answer 409

Probability DS Replace question with simplified version: 10 = w + m P(ww) = (w/10)*[(w-1)/9] is (w/10)*[(w-1)/9] > 1/2 Is w^2 - w - 45 > 0 ? (cannot factor further) (1) 5 < w < 10 (integer) Test case: w = 6 --> false w = 8 --> true NS (2) P(mm) = [(10 - w)/10]*[(9 - w)/9] < 1/10 (w^2 - 19w + 81 < 0 (cannot factor) DON'T BE THROWN off if you cannot factor > test cases > also recognize that m >= 2, so w <= 8 ***** w = 7 is valid --> false w = 8 is valid --> true NS (3) NS (for same reasons as above) E

Answer 410

Least Prime Factor = 3*5*....45*47 + 2 = O + 2 = Odd The sum 3*5*...45*47 + 2 is NOT divisible by any of the prime factors between 3 and 47. > recall: mult of n + non-mult of n = non-mult of n y (least prime factor) must be greater than 50

Answer 411

Rate Question Compare distances = rate * time (1) Ta = 1 + Tb Ra = Rb - 5 NS - nothing about M (2) Tm = 9 Rm = 50 Dm = 50*9 = 450 NS - nothing about A and P (however, we know that M did not drive the farthest distance --> 1050 m unaccounted for) (3) Da + Db Da = (1 + Tb)*(Rb - 5) = Rb - 5 + Tb*Rb - 5Tb Db = Tb*Rb 1050 = Rb - 5 + Tb*Rb - 5Tb + Tb*Rb 1050 = Rb - 5 + 2Tb*Rb - 5Tb NS to figure out which one is larger ---> 2 variables one equation E

Answer 412

Work problem X: Tx hours to complete 1 job ----> Rx = 1/Tx Y: Ty hours to complete 1 job ----> Ry = 1/Ty Ty - Tx = ? (1) (1/Tx + 1/Ty)*(2/3)*(Tx) = 1 (1/Tx + 1/Ty)*(Tx) = 3/2 1 + Tx/Ty = 3/2 Tx/Ty = 1/2 2Tx = Ty NS --> we need the values for both Tx and Ty Ty - Tx = Ty - Ty/2 = Ty/2 (2) (1/Ty)*(2Tx) = 1 2Tx = Ty (same as statement 1) NS (3) E

Answer 413

Algebra - Combo question wx = y (1) wx^2 = (wx)(x) = 16 xy = 16 Sufficient (2) wx = 4 x(4) = ? NS without x A

Answer 414

Least Prime Factor H(100) + 1 = (2*4*6*...*100) + 1 = Even + 1 = Odd Factor out the two to see clearly the other possible prime factors: = 2(1*2*3*4...*50) + 1 A multiple of N + non-multiple of N = non-multiple of N So the LEAST prime factor must be greater than 50 (53)

Answer 415

Set Problem (Not double matrix because we have multiple categories - Employed/Not employed, 65+ or younger than 65, Men/Women) Is (Employed and 65+)/65+ >= 10% ? > Ratio! (1) 11.3% * Pop = 65+ NS - nothing about employment (2) (65+)*(20%)*(Men%) + (65+)*(10%)*(1 - Men)% = Employed 65+ crosses out from top and bottom of the ratio SIMPLIFY FURTHER (don't worry about unknown M) Is (20%*M% + 10% - 10%*M%) >= 10%? Is (10%*M% + 10%) >= 10%? Always Yes --> Sufficient Alternatively: Look at JUST the 65 year old population = M who are 65+ plus W who are 65+ 20% * men 65+ > 10% * men 65+ -------> add 10% * women 65+ to both sides 20% * men 65+ + 10% * women 65+ > 10% (men 65+ + women 65+) (sufficient) B

Answer 416

Number properties and Inequality: 0 < x < 10 Is z > (x + 10)/2? You could expand and simplify this question stem into number properties (e.g., > 0) (1) GIVEN something about DISTANCE Therefore, z must be greater than the average of x and 10 (closer to 10). Also z > 5 (2) z = 5x Not sufficient A

Answer 417

Linear word problems > variables represent price (can be NON-INTEGER e.g., 2.5 dollars) p = ? N = # of loaves of bread bought Total charged = p + (N - 1)*q (1) Total charge per loaf = Total Charged/N [p + (2 - 1)*q]/2 = 0.9*[p] (p + q)/2 = 0.9p p + q = 1.8p q = 0.8p NS (2) 10 = p + (6 - 1)*q 10 = p + 5q ---> p and q can be NON-INTEGERS NS (3) 10 = p + 5q q = 0.8p Sufficient C

Answer 418

Prime properties > p is an odd prime number (1) 100 prime numbers between 1 and p + 1 > we can count 100 prime numbers starting from 1 and determine p > p is the 100th prime number > p + 1 is not a prime number (even) (2) Again we can count the number of prime numbers between 1 and 3,912. D

Answer 419

Work problem Statistics > Jane's credit card account has a balance of either $600 or $300 > We need to determine WHICH DAY her payment was credited from her account during the 30-day period. avg = ((30 - # of days)*600 + (# of days)*300)/30 = (Sum of daily balances)/30 (1) Sufficient > 20 days with 600 and 10 days with 300 (2) Sufficient avg through 25th day = (Sum of daily balances prior)/25 > we know the payment was credited already, so her balance for the remaining days is $300) avg through 25th day * 25 = sum of daily balances prior Therefore: avg = [(540 * 25) + 300*5]/30 Sufficient D

Answer 420

Counting Question with Digits - do it systematically --> start with largest column and keep track of numbers already accounted for = Total # of odd integers - # of odd int with 5 = 450 - (# of odd with 5 in hundreds) - (# of odd with 5 in tens) - (# of odd with 5 in ones) # of odd with 5 in hundreds: [500, 599] = 100/2 = 50 odd integers > we are done with the 500s # of odd with 5 in tens (excluding 500s): 51, 53, 55, 57, 59 for 100s, 200s, 300s, 400s, 600s, 700s, 800s, and 900s = 5*8 = 40 odd integers # of odd with 5 in ones column (excluding 500s and 50s): 5, 15, 25, 35, 45, 65, 75, 85, 95 for 100s, 200s, 300s, 400s, 600s, 700s, 800s, and 900s = 9*8 = 72 Total = 450 - 50 - 40 -72 = 288

Answer 421

Linear INEQUALITY word problem > Keyword: "enough to" --> means you don't have to spend ALL of the money Price per pen = a Price per pencil = b $10 >= 5a + 3b Is $10 >= 4a + 4b? Is 5 >= 2a + 2b? (1) a < 1 TWO INEQUALITIES ARE KNOWN: - you cannot assume a = 1 is the max and then solve for min b due to the direction of the sign - instead, set a value of "a" and then solve for the RANGE of values for b - then find the RANGE of values for the sum 2a + 2b Test case: a = 0.9 10 >= 5(0.9) + 3b 5.5 >= 3b 1.8 >= b Therefore: 2a + 2b <= 2(0.9) + 2(1.8) 2a + 2b <= 5.3 (NS to know whether 2a + 2b <= 5) (2) 10 >= 11a a <= 10/11 < 1 (same as statement 1) NS E

Answer 422

Absolute Value signs: Distance | A - C | = 5 | B - C | = 20 > since both of these info are relative to C, we can draw out SOME scenarios based on C: C +/-5 = A C +/- 20 = B B A C A B Is C between A and B? (1) | A - B | = 25 Sufficient C must be between A and B for this to work (2) NS > We don't know the distance between A and B A

Answer 423

Statistics: Mean and Ratios > write the equation in terms of sum = mean * n > weighted average = (averageA * A + averageB * B)/(A + B) D/(M + D) = ? (1) MeanM = MeanT - 5000 (SumM/M) = (SumM + SumD)/(M + D) - 5000 NS --> cannot get D alone (2) MeanD = MeantT + 15000 (SumD/D) = (SumM + SumD)/(M + D) + 15000 NS --> don't know what the values of SumD, SumM and SumD are **likely will have to CROSS Out stuff (3) MeanM = MeanT - 5000 MeanD = MeantT + 15000 Try: MeanT = MeanT MeanM + 5000 = MeanD - 15000 --> won't work Try: MeanT = (SumM + SumD)/(M + D) MeanT = (MeanM*M + MeanD*D)/(M + D) MeanT(M + D) = MeanM*M + MeanD*D MeanT(M) + MeanT(D) = (MeanT - 5000)*M + (MeanT + 15000)*D THINGS CROSS OUT 5000M = 15000D C

Answer 424

Word Problem: Ranges "Could be" - range UNDERSTAND THE GAME: Integer on the card * (int on card + 1) = product = x*(x + 1) 15 < x*(x + 1) < 200 MAX and MIN value of x? x >= 4 (4*5 = 20, 5*6 = 30 etc.) x <= 13 (13*14 = 182, but 14*15 = 210) Therefore ans: C

Answer 425

Xy Plane geometry: Is m * n = -1? (1) Sufficient (2) WHILE this seems to show that they are NOT perpendicular, there is one special case: m = 1 and n = -1 Is (-n) * n = - 1 Is - n^2 = -1? is n^2 = 1? (only the case if n = +/- 1) So we have a YES NO situation NS A

Answer 426

Factors n > 1 (int) is n = 2? (1) n is a prime NS (2) E - O = O or O - E = O > n cannot have more than 1 even or odd factor --> only has one even and one odd factor > n = 2 Sufficient B

Answer 427

Multi-variable Inequality a - b =/ 0 NOTICE the pattern: b - a = -(a - b) > still move all the terms to one side to evaluate PROPERTIES Is 1/(a - b) < -(a - b) Is 1/(a - b) + a - b < 0 is (1 + (a - b)^2)/(a - b) < 0 ? is a - b < 0 ? Numerator is always positive (1) a unsure the sign of a - b

Answer 428

Multiples x and y > 1 (int) Is x = y*m, where m is an integer? (1) Can factor out y on the RHS --> x is a multiple of y Sufficient (2) x(x - 1) = y*m (x*(x - 1))/y = integer > either x is a multiple of y or x-1 is a multiple of y We don't know if x is a multiple of y or if x-1 is a multiple of y x and x-1 have NO common factors other than 1 NS A

Answer 429

Exponents z = ? > z can be positive OR negative Either n = 0 and z can be any integer Or z = 1 and n can be any integer Or z = -1 and n can be any even integer ** (1) n =/ 0 z is either 1 or -1 NS (2) z > 0 NS - z = 1 and n could be anything, or z = any integer and n = 0 (3) z > 0 and n=/0 z = 1 Sufficient C

Answer 430

Mixture problem Q of ingredient Before = Q of ingredient after 0.1x + 0.02y = 0.05z ----> z = x + y 0.1x + 0.02y = 0.05(x + y) 0.05x = 0.03y 5x = 3y x = (3y)/5 x = ? ---> solve for x or y (1) y = 10 sufficient (2) z = 16 = x + y (doesn't give you y directly, but you NOW have TWO different EQUATIONS) x + y = 16 x = (3y)/5 Can solve for x and y --> Sufficient Ans D

Answer 431

Distance | A - B | = 18 | C - D | = 8 ----> 4 points | B - D | = ? (1) | C - A | = | C - B | ----> common point C ---> either in the middle of A and B or A = B > | A - B | = 18, so C is in the middle of A and B > D is in between A and B too > NS to find the distance between B and D (two points for D) (2) A < D Too many scenarios for where B and D are NS (3) NS - in statement 1, A is already to the left of D E

Answer 432

Factor k = 3^m * 7^n, where m and n >= 1 (int) 6 = (m + 1)*(n + 1) ----> write out the possible values of m and n (m, n) = (0, 5) --> not possible to have m = 0 (m, n) = (1, 2) (m, n) = (2, 1) (1) 3^2 is a factor, so m = 2. n must equal 1 Sufficient (2) 7^2 is not a factor of k, so n must equal 1 and m must equal 2 Sufficient D

Answer 433

Linear Equations > don't forget what you are solving for Solve for QUANTITY (int) P = C + 30 (1) 270 = P*Q ---> do not know what P is ---> three unknowns, two equations not enough NS (2) P = 1.5C P = C + 30 We can solve for P and C --> BUT STILL not enough to find Quantity NS (3) We know P, C and now we can calculate Q Sufficient C

Answer 434

Digits n: _ _ n > 20 ***** Is n not a prime factor? (1) n: a a*int e.g., 22, 24, 26 etc. > Primes greater than 20: 23 (invalid), 29 (invalid), 31 (invalid), 37 (invalid) Always yes --> sufficient (2) n: 2 _ NS --> 22 is a composite but 23 is a prime A

Answer 435

Word problem Company wants to produce a total quantity = 1000*n n = ? (1) 1000*n = 600*5 + rate*(n - 5) > two unknowns NS (2) 1000*n = 1500*(4) + rate*(n - 4) > two unknowns NS (3) Even though we have two different equations, I am unsure if the rates are the same in statement 1 and 2 Also: 1000*n = 600*5 + 1500*4 + rate*(n - 9) ---> still have the two variables

Answer 436

LCM --> largest powers in each column (1) 30 = 2*3*5 6 = 2 * 3 x must have 5, could have up to one 2 and one 3. LCM x, 6 and 9? 6 = 2 * 3 9 = 2^0 * 3^2 x = 5 > largest power on 3 is now 2 We can figure out the largest powers on every factor: LCM: 2*3^2*5 = 90 (sufficient) (2) 45 = 3^2 * 5 9 = 3^2 x = must have 5, could have up to two 3s (no 2s) LCM x, 6 and 9? 6 = 2 * 3 9 = 2^0 * 3^2 x = 5 > largest power on 2 is 1 We can figure out the largest powers on every factor: LCM: 2 * 5 * 3^2 (sufficient) D

Answer 437

Algebra properties Is m/r = x/y? Or is my = rx ? (1) mr = xy ---> impossible to get to my = rx Yes case: m = r = x = y = 1 Not sure if there is a no case --> cannot answer the question so the statement is INSUFFICIENT on its own NS (2) (m + x)*y = (r + y)*x ---> CROSS MULTIPLY my + xy = rx + xy my = rx (sufficient) B

Answer 438

Set Problem - Double-Matrix-like question with percents 1400 = M + F 42*1400 = 580 = Yes F = ? (1) 0.36*M + 0.5*F = Yes = 580 36M + 50F = 580 ALSO: M + F = 1400 (sufficient - two equations for two unknowns) Sufficient (2) 288 = Men%*580 Therefore Females who said Yes = 292 NS to find total Females surveyed

Answer 439

Even/Odd properties xy + z = odd = E + O or O + E Is x even? SYSTEMATICALLY CHECK WHETHER x can be even or odd: (1) xy + xz = Even = E + E or O + O If x = even: xy (even) + z (odd) = odd (valid) If x = odd, then y and z must be even: xy (even) + z (even) = even (invalid) If x = odd, then y and z are odd: xy (odd) + odd = even (invalid) Therefore x must be even Sufficient (2) y + xz = Odd = E + O or O + E If x is even, then y must be odd. Z even xy (even) + z (even) = even (invalid) If x is even, then y must be odd. Z odd xy (even) + z (odd) = odd (valid) If x is odd, then y must be odd and z is even: xy (odd) + z (even) = odd (valid) NS (x can be even or odd) A

Answer 440

Coordinate Plane Geometry: Linear functions y = mx + b b =? (1) m = 3b b = m/3 NS (2) (-1/3, 0) 0 = m(-1/3) + b b = (1/3)*m NS (3) b = m/3 and b = m/3 (same equation) NS E

Answer 441

Number properties: 0 Are all the numbers 0? _ _ _ ... (1) Find a "no" case: 1 0 0 1 * 0 = 0 1 * 0 = 0 0 * 0 = 0 If you have one nonzero number, it works. NS (2) Find a "no" case: 1 -1 0 1 - 1 = 0 -1 + 0 = -1 (invalid) ONLY two zeros creates a sum = 0 for ANY two numbers Always Yes S B

Answer 442

Distance Question > 0 doesn't have to be in the middle of r and s r < s < t Question: Are r and s opposite signs? Or Is rs < 0? (1) s > 0 NS --> r could be pos or neg (2) | t - r | = | t - -s| ---> common point: t (use as reference point!) -s could be = r or an equal distance from t as r is from t. > if r = -s ---> r and s have opposite signs > if r =/ -s ---> -s is positive, s is negative, r is negative (r and s have the same sign and both are negative) NS (3) s > 0, so -s < 0 Sufficient r = -s

Answer 443

Remainder Question with Even/Odd properties: **P is ODD p/4 => R = ? (1) p/8 = z + 5/8 p = 8z + 5 p/4 --> fixed remainder (sufficient) (2) p = a^2 + b^2 = odd a and b have opposite types Remainder = (Ra + Rb)/4 even perfect square/4 --> remainder always = 0 odd perfect square/4 --> remainder always = 1 Sufficient D

Answer 444

Rates Question: KEEP TRACK OF THE CORRECT SPEEDS AND PEOPLE: ORDER in the question changes to trick you! Common time: t (after Al starts driving) Distance travelled by Ben = (20mph)*(3 hours + t) Distance travelled by Al = (40mph)*(t) 240 = Dist by Ben + Dist by Al 240 = 20*3 + 20*t + 40*t 240 - 60 = 60t 180 = 60t 3 = t 11am + 3 hours = 2:00Pm

Answer 445

Word problem: ROUND UP *no need to overcomplicate the problem and track both start and end dates. *Try a few at the beginning to find the PATTERN Book 1 takes 6 days Book 2 takes 3 days (cumulative = 9 days) Book 3 takes 3 days Book 4 takes 4 days Book 5 takes 4 days Book 6 takes 1 day Book 7 takes 5 days Book 8 takes 2 days (cumulative = 28 days) She will have finished 8 books.

Answer 446

Inscribed Shapes in Circles > center of the shape = center of the circle > radius = center to each vertex > there are 3 congruent triangles formed when the radius is drawn from the center of the circle to each vertex: - each central angle = 360/3 = 120 degrees 24 = (central angle/360)*(2*pi*r) 24 = (240/360)*(2*pi*r) r = 18/pi d = 36/pi = 36/3.14 = 36/3 = 12 (actual answer is slightly smaller than 12) Alternatively: 24 represents 2/3 of the total circumference 24 = (2/3)*(2*pi*r)

Answer 447

Statistics 15 homes Mean = 150,000 = SUM/15 > SUM = 2250000 (get rid of the trailing zeros) = $2250 Median = 130 (at position 8) TESTING CASES TO SEE IF THE SUM is possible: One case: 8 homes were sold for 130 2250 - 130*8 = 1210 left Avg price of the remaining 7 homes: 1210/7 = 170ish I) Not true (the minimum selling price could be 130) II) Not true - we could have 8 homes sold for 130 and 7 homes sold for 170 III) True - see if it is possible to sell a home for less than 165 --> MINIMIZE max selling price but maximizing other prices Homes 1 to 8 sold for: 130 Remaining homes sold for an average of 170ish --> 165*6 = 990 Remaining price of the max house: 2250 - 130*8 - 990 = 220 > therefore, at least one house was sold for more than 165 Only III must be true

Answer 448

Perfect square properties and max/min: Max mp by maximizing both m and p ---> equal to each other Approach #1) m^2 + m^2 < 100 2m^2 < 100 m^2 < 50 (both positive) m < sqrt(50) ----> m < 8 AND m is an int So max m and p = 7 mp = 49 Approach #2) Since this is PS - start from the largest answer choice > 51 is not possible to achieve 51 = 3 * 17 17^2 already exceeds 100 Algebraically: RECOGNIZE THE PATTERN (m - p)^2 >= 0 m^2 - 2mp + p^2 >= 0 2mp <= m^2 + p^2 -----> m^2 + p^2 < 100 2mp < 100 mp < 50

Answer 449

Ratios - observe the PATTERN and the LINKAGE Pre-think: x/y = c/d = a/b I) y/x = b/a is the same as x/y = a/b (T) II) x/a = y/b is also T III) y/a = x/b is NOT true > xb = ya --> cannot get y/a = x/b I and II

Answer 450

Approximation - choose the fraction that has the SMALLEST gap from 0.5 > recall: Smallest fraction is when numerator is small and denom is large A) 4/7 - 1/2 = 1/14 B) 5/9 - 1/2 = 1/18 C) 6/11 - 1/2 = 1/22 D) 7/13 - 1/2 = 1/26 ****smallest fraction E) 9/16 - 1/2 = 1/16

Answer 451

GCF - lowest powers in each column (watch out for 0 exponents) > Pro of GCF: get common divisors > Con of GCF: might miss other factors that unknowns have (1) m is a prime - NS (nothing about n) (2) 2n = 7m (n and m both don't equal 0) > m is a multiple of 2 > n is a multiple of 7 Test case 1: m = 2, n = 7 GCF: 1 Test case 2: m = 4, n MUST BE 14 GCF = 2 NS (3) m is a prime and multiple of 2 --> m must be = 2 THEREFORE n MUST BE 7 Sufficient (GCF = 1)

Answer 452

Word problem: Pay attention to TIMES 365 days (no leap year) 12 months 52 weeks THP = monthly take home pay x = fraction saved per month 1 - x = fraction not saved per month Total Amount of Money Saved at the END of the year = x*THP*12 Amount of that portion of monthly take home pay that is not saved = (1 - x)*THP *** do not multiply by 12 x*THP*12 = 3*(1 - x)*THP 12x = 3 - 3x 15x = 3 x = 1/5

Answer 453

Set question > 5 circles... we care ONLY about the overlap between IBM and ATT Actual drawing: I + A - IandA + Neither = 200 # people who own stock in neither IBM nor ATT = Total - (# people who own IBM + # people who own ATT - # of people who own both) = 200 - (30 + 48 - 15) = 137

Answer 454

Remainder Question with Factors (GCF and LCM) and Even/odd properties Pro of GCF - know about common factors Con of GCF - hides info about other factors each unknown has Pro of LCM - know about combined factors Con of LCM - hides info about each specific unknown p/m = z + r/m PRE-THINK: to get r = 1, m and p must be opposite types differing by 1 e.g., Odd/Even (15/14) or Even/Odd (12/11) (1) GCF = 2 > both m and p are multiples of 2 > both are EVEN p = mz + r even = even + r ---> r must be even, r > 1 Sufficient (2) LCM = 30 = 2*3*5 > not sure which unknown has which factor Test case 1: m = 5 p = 2*3 6/5 --> R = 1 Test case 2: m = 3*2 p = 2*5 10/ 6--> R = 4 NS A

Answer 455

Sequence (order matters) How many terms are in this sequence? Let a = # of 7s Let b = # of 77s n = a + b 350 = 7a + 77b 350 = 7(a + 11b) 50 = a + 11b Test b = 1, a = 50 - 11 = 39 > Total number of terms = 1 + 39 = 40 (E) If you test b = 2, a = 28 > Total number of terms = 2 + 28 = 30

Answer 456

Work problem - READ and understand the question carefully Rate of 1 Machine R = 1/36 jobs per hour Rate of 1 Machine S = 1/18 jobs per hour Let R = # of machine Rs Let S = # of machine Ss R = S = N (1/36)*N*2 + (1/18)*N*2 = 1 (1/36)*N*2 + (1/18)*N*2 = 1 (1/36 + 1/18)*N*2 = 1 N = 6 ** Do not set X as the total number of machines and then divide X by 2. It is better to set up the equations for EACH MACHINE first.

Answer 457

Geometry Word Problem with Rates: "Edge" of a circular lake --> the minimum diameter of the jogging track is 2 miles. Pre-think: Find either the Range of t or the value of t (and see which range fits) diameter = 2 radius = 1 Rate = 3 m/h Circumference of the lake = 2*pi*r Circumference of the lake <= Distance she travels 2*pi*1 <= 3*t (2pi)/3 <= t 6.18/3 <= t ~2.0x <= t We cannot find the max distance --> so select the answer that represents the correct range in which t falls in C Upper range is 2.5 because 7.5/3 = 2.5 and 6.18 < 7.5

Answer 458

Approximation > use appropriate RANGES for each term (including 1 powers) sqrt(4) = 2 (4)^1/3 > we know that (8)^1/3 = 2 (4^1/3 is smaller than 2) > we know that (1)^1/3 = 1 (4^1/3 is greater than 1) (4)^1/4 = (2^2)^1/4 = (2)^1/2 = 1.4 Therefore, sqrt(4) + (4)^1/3 + (4)^1/4 MIN value: 2 + 1 + 1.4 = 4.4 Max value: 2 + 2 + 1.4 = 5.4 Correct range is E

Answer 459

Remainder question x/y = z + 9/y And x/y = 96 + 0.12 The remainder (int) is 9 = y*0.12 y = 9/0.12 = 9/(3/25) = 75

Answer 460

Weighted Average Problem A = 0.4 B = 0.6 10 = Qa + Qb 0.56 = (0.4*Qa + 0.6*Qb)/10 5.6 = 0.4*Qa + 0.6Qb New avg = 0.52 Same Qa, Different Qb Change in Qb? USE VISUAL APPROACH: Weight of A = 20%, so Qa = 10*20% = 2 and Qb = 8 New weight of A = 40%, so Total # of fruit = 2/0.4 = 5 5 = 2 apples + 3 oranges Change in Qb = 8 - 3 = 5

Answer 461

XY plane geometry: Trapezoid approach (you don't have to check if the triangle is a right triangle) Area of trapezoid - area of two right triangles = area of triangle > two 3-4-5 triangles > area of trapezoid = 1/2*(3 + 4)*7 = 24.5 > area of two 3-4-5 triangles: 2*(3*4*0.5) = 12 Area of the triangle = 24.5 - 12 = 12.5

Answer 462

Word Problem: Inequality (max/min) and sets Let a = # of teachers who teach 3 classes Let b = # of teachers who teach 2 classes Let c = # of teachers who teach 1 class Teachers: 37 = a + b + c Classes: 64 = 3a + 2b + c Solve for max and min value of a: Two equations, three variables: However c can be ELIMINATED 27 = 2a + b a = (27 - b)/2 Max a is if b is minimized and = 0: > however, a must be an integer > Try b = 1 a = 13 (int) (works) --> sub in a = 13 to solve for b and c (b CANNOT be 0 because then a is not an integer) 37 = 13 + b + c 64 = 3*13 + 2b + c b = 1 c = 23 Min a is if b is maximized and = 27: b = 27 c = 10 a = 0 (works) 0 + 27 + 10 = 37 teachers 3*0 + 2*27 + 10 = 64 classes

Answer 463

Combinatorics problem - permutation with same type FIXED ORDER: MFMFMF (we are NOT looking for the TOTAL possible lineups, just the number of possible lineups in this ORDER) > no duplicates among people so you don't divide by anything = 3! * 3! = 36

Answer 464

Multiples Question 1 year = 365 days (without a leap year) or 366 days (with a leap year). Weeks increase by a multiple of 7 365/7 --> Remainder of 1 366/7 --> Remainder of 2 Therefore, in regular years, Nov 16th moves by 1 day. In a leap year, Nov 16th moves by 2 days. Ans: Sunday

Answer 465

Geometry: solids > "thickness" --> subtract 2*thickness from every side Actual Length = 10 - 2*0.5 = 9 Actual Width = 8 - 2*0.5 = 7 Actual Height = 12 - 2*0.5 = 11 *We have to figure out the LARGEST volume of the cylinder first before we can determine the associated radius #1) Bottom: 9 by 7 --> diameter is constrained by 7 --> r = 3.5 --> height = 11 V = pi*(3.5)(3.5)*11 = pi*(12.25)*11 = pi*132 #2) Bottom: 9 by 11 --> diameter is constrained by 9 --> r = 4.5 --> height = 7 V = pi*(4.5*4.5)*7 = pi*(20.25)*7 = pi*140 *** largest volume r = 4.5

Answer 466

Interest Question with Inequality > Two approaches: either solve for r or the inequality, OR test cases around the boundary Is r > 8? (1) IT IS SUFFICIENT TO SOLVE FOR r 210 = 1000*(1 + r/100)^2 - 1000 1210 = 1000(1 + r/100)^2 1.21 = (1 + r/100)^2 1.1 = (1 + r/100) 0.1*100 = r r = 10 (sufficient) (2) (1 + r/100)^2 > 1.15 *hard to take the square root --> test values of r to see if the statement is still sufficient or not #1) r = 8 (1.08)*(1.08) = 1.1664 (valid) #2) r = 9 (1.09)*(1.09) = 1.1881 (valid) NS

Answer 467

Sequence Question: Arithmetic an = a1 + (n - 1)*k **MEDIAN = position 8 (a8) # of terms greater than 10? > need to know distribution of values and whether k +/- (1) a1 = 24 a2 = 24 + k (NS) > I don't know the sign of k or the value of k (2) a8 = 10 There are exactly 7 terms before and after a8. (AND no other term equals 10). REGARDLESS of the value of k, we know there are 7 terms greater than 10 e.g., k < 0, then 7 terms above 10 e.g., k > 0, then 7 terms above 10 Sufficient B

Answer 468

Combinatorics question - combination _ _ _ Two stems: A or B A) 3 notepads of the same colour and size = 2 sizes * C4,1 = 2 * 4 = 8 B) 3 notepads of the same size but different colours = 2 sizes * C4,3 = 2 * 4 = 8 8 + 8 = 16

Answer 469

Word Problem: Inequality Store Purchase Cost: p + 0.06*p = 1.06p Online Purchase Cost: q Is 1.06p > q? Pre-think: > either solve for values of p and q or inequality > or test cases (1) q - p < 50 q < 50 + p Case 1) p = 10, q < 60 --> q = 50 Is 10.6 > 50 (No) Case 2) p = 100, q < 150 --> q = 10 is 106 > 10 (yes) NS (2) q = 1150 NS (p ?) (3) q < 50 + p 1150 < 50 + p p > 1100 Therefore 1.06p > 1100*1.06 1.06p > 1166 > 1150 (sufficient) c

Answer 470

Combinatorics question _ or _ _ n letters Since ab and ba only has one valid order (ab) --> combination question (not permutation!) Cn,1 + Cn,2 >= 12 (n!)/(n - 1)! + (n!)/(2! *(n - 2)!) >= 12 (n + n^2)/2 >= 12 Test values of n n = 5 --> 15 >= 12 (works) n = 4 --> 10 >= 12 (invalid)

Answer 471

Word Problem (similar to arithmetic sequence) WE DO NOT KNOW THE STARTING BALANCE AMOUNT Range = Max bal - Min bal ---> we need to know WHEN Carl switched from +120 to -50 > max bal = month before -50 > min bal = we can also compute it May = 2600 (1) April < 2625 May - 120 = 1400 (April's balance) - possible May + 50 = 2650 (April's balance) - NOT possible > however, we do not know when Carl switches NS (2) June < 2675 May + 120 = 2720 (June's balance) - Not possible May - 50 = 2550 = (June's balance) - Possible > however, we do not know when Carl switches (could have been earlier) NS (3) Together, we know Carl switches after May Sufficient

Answer 472

Word Problem: Inequality > solve for an inequality containing x, using the equations (sub stuff in) 1 = x + y c = 6.5x + 8.5y (cost per kg) is x < 0.8? SIMPLIFY FIRST: sub 1 - x = y c = 6.5x + 8.5(1 - x) c = 6.5x + 8.5 - 8.5x c = 8.5 - 2x (1) y > 0.15 y = 1 - x (sub in) 1 - x > 0.15 x < 0.85 (NS) (2) c >= 7.3 (sub in cost equation) 8.5 - 2x >= 7.3 1.2 >= 2x 0.6 >= x (sufficient) B

Answer 473

Inequality PS - reciprocals and square root n < 0 n^2 < 1/100 Therefore n < 1/10 (if n positive) or 0 > n > - 1/10 n is negative, so is -1/10, therefore flip the sign when you find the reciprocal: 1/n < -10

Answer 474

Probability > combination because we care about the PRODUCT and ab is the same as ba = # of options that fit the criteria / total # of options = 1 / C4,2 = 1 / 6

Answer 475

Difference of squares! 0.99999999/1.0001 - 0.99999991/1.0003 = (1 - 10^-8)/(1 + 10^-4) - (1 - 9*10^-8)/(1 + 3*10^-4) = (1 - 10^-4) - (1 - 3*10^-4) = 10^-4 * ( 3 - 1) = 10^-4 *(2)

Answer 476

Digit question: N: _ _ > 0 a b > 0 Constraints: 1 <= a <= 9 and 0 <= b <= 9 is a < 4? (is a = 1, 2, or 3)? (1) b = 6 + a a = b - 6 > find the max value of a by maximizing b (9): a = 9 - 6 = 3 Sufficient (2) 10a + b = 4b - 4 10a + 4 = 3b Test cases: CHOOSE VALID cases a = 8, b = 28 (invalid) a = 5, b = 18 (invalid) a = 2, b = 8 (valid) Sufficient D

Answer 477

Statistics Question: Avg = (# students at event 1 + # of students at event 2 + ..)/8 = (Sum of the TOTAL number of tickets sold)/8 (because each student could go to more than 1 event) (1) NS to know the numerator (2) 4 = (# of events for student 1 + # of events for student 2 + ...)/200 800 = (Sum of the TOTAL number of tickets sold) Sufficient (A simple problem will show you that the numerators are the SAME) B

Answer 478

Absolute value signs question and positive/negatives: is y = 0? (sub in to figure out value of x) is 0 = | x | + x ? (Only possible if x < 0) Is x < 0? (1) Sufficient (2) y < 1 (DON'T RULE THIS ANSWER OUT IMMEDIATELY - could be a trap where there is ONLY one valid case) e.g. y = 0, y = -1 If y = 0, then x < 0 (valid) If y = -1, then - 1 = | x | + x -1 - x = | x | Case 1) x > 0 -1 - x = x -1 = 2x x = -1/2 (invalid) Case 2) x =< 0 -1 - x = -x -1 = 0 (invalid) y cannot equal -1, so y must equal 0. Sufficient D

Answer 479

Percent PS Two approaches: 1) Smart numbers Set other family member's income = 100 2) Algebraically, recalling that her other family member's income = 40%*total family income Ans A

Answer 480

Absolute Value Signs > EACH absolute value sign requires TWO cases > Since we have 2 signs, we need FOUR CASES > don't forget to include the equal case Case 1) |x-3| - 6 >= 0 |x-3| >= 6 A) x - 3 >= 0 x >= 3 Therefore: ((x-3) - 6) = 6 x = 15 (valid x2) B) x - 3 < 0 x < 3 Therefore: (-(x-3) - 6) = 6 (-x + 3 - 6) = 6 x = -9 (valid x2) Case 2) |x-3| - 6 < 0 A) x - 3 >= 0 x >= 3 Therefore: -((x-3) - 6) = 6 -(x - 9) = 6 x - 9 = -6 x = 3 (valid x2) B) x - 3 < 0 x < 3 Therefore: -(-(x-3) - 6) = 6 -x + 3 - 6 = -6 x = 3 (invalid) # of solutions = 3

Answer 481

Double Matrix > Pay attention to WORDING Goal is to find: (defectiveAndaccepted)/(total accepted) We are not given the total number of light switches produced, so we can set total = 1 or 100 etc. Ans 5%

Answer 482

xy plane geometry (quadrants) > Goal is to see if -x has the SAME sign as both -a and -b, and if y has the same sign as both a and b (-a, b) and (-b, a) are in the same quadrant: > -a and -b have the same sign > a and b have the same sign (ab > 0) (1) xy > 0 > means that x and y have the same sign > NS because we don't know whether x has the same sign as a and b or y has the same sign as a and b (2) ax > 0 > means a and x have the same sign --> x coordinates match > NS because we don't know anything about the y coordinate (3) xy have the same sign a and x have the same sign Therefore, y has the same sign as a and b as well Sufficient C

Answer 483

Venn Diagram Set question > Special twist: German is INSIDE of English > overlap is between English and Spanish only How many people speak only two languages? = # ppl who speak EnglishAndGerman + # ppl who speak EnglishAndSpanish Given: > 70 only speak Spanish > 200 ppl = 70 + Only English + GermanAndEnglish + EnglishAndSpanish + None We need to know 200 - 70 - Only English - None (1) Seems sufficient...BUT there are people who DON'T SPEAK ANY OF THE LANGUAGES NS (2) 20 = None NS (3) Sufficient C

Answer 484

Inequality Word problem with Digits Bill: a b > a is between 1 and 9 inclusive > b is between 0 and 9 inclusive Tip = 2a Was 2a > 0.15(bill)? Was 2a > 0.15(10a + b)? Was 2a > 1.5a + 0.15b Was 0.5a > 0.15b Was a > 0.3b? (1) Test case Bill = 20 > a = 2, b = 0, a > 0 (yes) Bill = 29 > a = 2, b = 9, a > 2.7 (No) NS (2) 2a = 8 a = 4 Test case: Bill = 40 > a = 4, b = 0, a > 0 (yes) Bill = 49 > a = 4, b = 9, a > 2.7 (yes) Sufficient (max value of a is 49.99 or 50, and 2a > 0.15(50)) B

Answer 485

Inequality - one variable, multiple signs, diff exponents > Must be true DRAW out the functions carefully x^2 is a quadratic function 2x is a linear function 1/x is an inverse function Only I and II are true

Answer 486

Ratio/Inequality word problem > multi variable, one inequality sign ---> MOVE TO ONE SIDE and evaluate properties > Denom will always be positive (sum of positives) Is Fx/Px > (Fx + Fy)/(Px + Py)? Is: Fx/Px - (Fx + Fy)/(Px + Py) > 0? (FxPx + FxPy - FxPx - FyPx)/(Px*(Px+Py)) > 0? (FxPy - FyPx)/(Px*(Px+Py)) > 0? (just focus on the numerator): Is FxPy - FyPx > 0? (1) Fy/Py < (Fx + Fy)/(Px + Py) Fy/Py - (Fx + Fy)/(Px + Py) < 0 (FyPx + FyPy - FxPy - FyPy)/(Py*(Px+Py)) < 0 (FyPx - FxPy)/(Py*(Px+Py)) < 0 Numerator: FyPx - FxPy < 0 Therefore: FxPy - FyPx > 0 (Sufficient) (2) 0.5*(Fx + Fy) < Fx 0.5*(Px + Py) < Py Fx + Fy < 2Fx ---> Fy < Fx Px + Py < 2Py -----> Px < Py All are > 0: Fx > Fy FxPy > FyPy And Py > Px: FxPy > FyPy > FyPx FxPy > FyPx (sufficient) ALTERNATIVELY LOGIC IT OUT: For Fx/Px > (Fx + Fy)/(Px + Py), Fx/Px must be > Fy/Py: Is Fx/Px > Fy/Py ? Or is Fy/Py < (Fx + Fy)/(Px + Py) ? (1) Fy/Py < (Fx + Fy)/(Px + Py) Sufficient (2) Fx > (1/2)*(Fx + Fy) ----> Fx > Fy Py > (1/2)*(Px + Py) ----> Py > Px Therefore: (based on fraction properties) Fx/Px > Fy/Py Sufficient

Answer 487

Double Matrix - Value Question > pay attention to what you are solving for - draw it out twice if you have to > use smart variables to represent quantities > fill out the remaining boxes will variables 48 with patio; 27 without patio How many houses have a swimming pool? (1) 38 = Pat and No Pool 10 = Pat and Pool However, still NS (we need at least two knowns in each column and row) (2) Set a = PatandPool = NoPatandNoPool THIS IS SUFFICIENT x = a + 27 - a = 27 B

Answer 488

Arithmetic Sequence Word Problem Jan = a1 March = a3 = ? an = an-1 + x = a1 + (n - 1)*x a3 = a1 + (2)*x or a3 = a2 + x (1) a12 = 310000 NS to find a2 or x (2) a9 = 30000 + a6 a1 + 8*x = 30000 + a1 + 5x 3x = 30000 x = 10000 NS to find a2 or a1 (3) x = 10000 310000 = a1 + 11*x Sufficient C

Answer 489

Double Matrix - Ratio question > READ carefully (for EACH colour, there is an equal number of red and green toys) > red small = x > green small = x > red large = y > green large = y > also looking for fraction of the TOTAL GREEN toys that are large y/(x + y) = ? ---> need to know ratio x to y (1) x = 400 NS to know y (2) we HAVE A RATIO Simplifies to x = 2y SUFFICIENT B

Answer 490

Ratios > read carefully so you don't MESS UP S: A: W 2: 50: 100 S/A = 2/50 2*(S/A) = 4/50 S/W = 2/100 0.5*(S/W) = 1/100 NEW RATIOS: S: A: W 4: 50: 400 50x = 100 cm^3 (actual value) ---> x = 2 Therefore 400*(2) = 800 E

Answer 491

Probability Inequality question b + w + r (all positive integers) Is r/(b + w + r) > w/(b + w + r)? Is r > w? (1) Multi-variable inequality (MOVE TO ONE SIDE) > denominator is always positive (so you can ignore when determining the sign) (rb + r^2 - wb - w^2)/[(b+2)*(b + r)] > 0 FACTOR NUMERATOR (r - w)*(b + r + w) > 0 -----> therefore r > w (sufficient) (2) b - w > r b > r - w (NS to know whether r - w > 0 or < 0) A

Answer 492

Statistics question: > total expenditures figure is not relevant - FOCUS ON THE PERCENTS > percents represent INDIVIDUAL TERMS 67.8% = SUM of individual percents that belong to the category Was X among the 6 highest contributing countries? (1) 67.8% = 56% + 5th and 6th highest contributions NS to know percentage contributed by X (2) X's contribution = 4.8% NS - could be or could not be one of the 6 highest contributing countries (3) Yes case: 67.8% = 56% + 7% + 4.8% No case: 67.8% = 56% + 6% + 5.8% NS E

Answer 493

Rates Question: t? d = 400 = rate*t > with rate, you can find time (1) NOTHING is known about whether the rate is CONSTANT NS (2) r2 = r1 + 20 t2 = t1 - 1 SAME DISTANCE: r2*t2 = r1*t1 (r1 + 20)*(t1 - 1) = r1*t1 ---> two variables PLUS we know 400 = r1*t1 Two variables, two different equations ---> SUFFICIENT

Answer 494

Combinatorics question - permutation _ _ _ _ _ _ a in EACH ROW - heights increase from left to right > second row must be taller than the person standing in front of him or her **notice how the answer choices seem pretty small (5, 6, 9 etc.) ---> you can probably test cases and count them Recognize that a AND f are both FIXED #1) d e f a b c #2) b e f a c d #3) c e f a b d #4) b d f a c e #5) c d f a b e Only 5

Answer 495

Geometry - 2D polygons (NOT solids!!) > utilize properties of squares - x - x - x*sqrt(2) > utilize properties of isosceles triangles SYMMETRY Area of the Triangle BCE = (Area of the triangle BED - Area of triangle BCD)/2 Ans = sqrt(2)/4

Answer 496

Remainder Question r = 5m + R s = 5q + R t = 5p + R R < 5 (R = 0, 1, 2, 3, 4) t = ? (1) r + s = t (sub in equations) 5m + R + 5q + R = 5p + R 5(m + q) + 2R = 5p + R 5(m + q) + R = 5P *a multiple of 5 + R = a multiple of 5 ---> R must be a multiple of 5 ----> remainder must be 0 (possible remainders when divisor is 5 is only 0, 1, 2, 3, 4) OR 5(m + q) = 5P - R (R = 0) r, s, t are all MULTIPLES Of 5 NS (2) 20 <= t <= 24 NS (3) 20 is the only multiple of 5 in the range Sufficient C

Answer 497

Number properties and inequalities (CONDITIONS ARE KEY in this question) x and y are BOTH negative AND x < y x-y = ? (1) (x+y)(x-y) = 5 x+y must be < 0 Case 1) -1 * -5 x+y = -1 x-y = -5 ------------ 2x = -6 x = -3 y = 2 (Not possible) Case 2) -5*-1 (only other case left) Sufficient x+y = -5 x-y = -1 ---------- 2x = -6 x = -3 y = -2 (works) (2) xy = 6 -6*-1 -3*-2 NS A

Answer 498

DOUBLE-MATRIX PROBLEM > set T = total number of geese > set x = total number of migrating geese > x and T will cancel out Ans 7/12

Answer 499

Combinatorics question - "subsets" = different possible groups What is m (# of terms in set A)? (1) 6 different subsets have 2 terms Cm,2 = 6 Sufficient (2) 16 total possible subsets = empty subset + complete subset + # of single terms + # of two terms + ... 16 = 2^n ----> each term has two options (Yes include me or No don't include me) Sufficient D

Answer 500

Quadratic equations "Same fraction" - set as y e.g., y = 1/2 (20% increase), 1/4 (25% increase) so: 10*(1 + y) = x x*(1 + y) = 14.4 Therefore: 10*(1 + y)^2 = 14.4 **** recognize this as a perfect square (1 + y)^2 = 1.44 1 + y = 1.2 y = 0.2 Therefore: x*(1.2) = 14.4 x = 14.4/1.2 = 144/12 = 12 A

Answer 501

Word Problem: Linear equations > Seems to be a mixture problem, BUT IT IS NOT THE USUAL TYPE Still the TOTAL VOLUME = sum of individual volumes Let volume = 1 + 3 + g = 4 + g Amt of Blue Paint: (x/100)*(4 + g) = 1 Amt of Red Paint: (z/100)*(4 + g) = 3 Amt of Green Paint: (y/100)*(4 + g) = g g = ? SIMPLIFY each equation g = (100 - 4x)/x g = (300 - 4z)/z g = (4y)/(100 - y) (1) x = y ---> we now have TWO different equations for g and two unknowns (x and g) Sufficient (2) z = 60 --> we can solve for g Sufficient D

Answer 502

Number Properties (pos or neg) ax + b = 0 ----> ax and b must have opposite signs ax = -b is x > 0? (1) a + b > 0 At least ONE term is positive ax + b = 0 + + ---> x < 0 + - ---> x > 0 - + ---> x > 0 NS (2) a - b > 0 Could be: + + ---> x < 0 - - ---> x < 0 + - ----> x > 0 NS (3) Shared possible signs: + + ----> x < 0 + - ---> x > 0 NS E

Answer 503

Perfect Squares and Factors m > 0 int Is sqrt(13m) an integer? Is 13m a perfect square? Does m = 13*int^even powers? (1) 117m = (int)^2 3^2 * 13 * m = (int)^2 = perfect square m must equal 13 * int^even powers Sufficient (2) m/117 = (int)^2 m/(3^2 * 13) = int^2 m = int^2 * 3^2 * 13 Also m must cancel out 3^2 and 13 and the remaining integers must be a perfect square Sufficient D

Answer 504

Factors and Multiples Is k a composite number? > k must be a MULTIPLE of some integer (1) k > 24 NS k = 25 (Yes) k = 29 = (No) (2) 13! + 2 <= k <= 13! + 13 ----> FACTOR the end points to SEE that both are MULTIPLES 2(13*12*11*...3*1 + 1) <= k <= 13(12! + 1) 13! + 2 to 13! + 13 will ALWAYS BE A MULTIPLE of some integer Sufficient B

Answer 505

One variable inequality with one inequality sign > Sewing approach > compare the solution of each inequality to 0 < x < 1 ans I, II, and III

Answer 506

Rounding (to the nearest multiple of 100) and Compound Inequalities 450 <= x <= 549 350 <= y <= 449 x + y rounded to the nearest 100? (1) x < 500 --> revise the inequality 450 <= x < 500 350 <= y <= 449 ------> ADDing two inequalities is OK ---------------------- 800 <= x + y < 949 x + y rounded to the nearest 100 is either 800 or 900 NS (2) y < 400 --> revise the inequality 450 <= x <= 549 350 <= y < 400 ------------------------ 800 <= x + y < 949 x + y rounded to the nearest 100 is either 800 or 900 NS (3) Two statements above are identical NS E

Answer 507

Absolute value signs y >= 0 (1) Case 1: x >= 3 x - 3 >= y x >= y + 3 (NS) Case 2: x < 3 -x + 3 >= y x <= 3 - y (NS) (2) BEFORE You do cases, look at the NEGATIVE -y <= 0 Absolute values are ALWAYS positive or 0. So -y = 0 and x - 3 = 0 x = 3 Sufficient

Answer 508

Statistics # terms of S = # terms of T = n All positive integers is med of S > avg of T? (1) Sum S > Sum T Sum S/n > Sum T/n Avg S > Avg T However, we do not know anything about MEDIAN e.g., S: 10 20 100 --> median 20 T: 10 20 30 ---> avg 20 No S: 10 20 100 --> median = 20 T: 10 11 12 --> avg = 11 Yes NS (2) S consecutive even numbers: 2n, 2n + 2, 2n + 4 T consecutive odd numbers: 2m + 1, 2m + 3, 2m + 5 Median = Avg Is 2n + 2 > 2m + 3? NS (3) Avg S > Avg T Median = Avg Median S > Avg T (sufficient) C

Answer 509

Exponents and Factors > turn into fractions > you can cross multiple to get rid of the fractions What is (1/a^2)*(1/b^3)? (1) a^3 * b^2 = 36 (after cross multiplication) NS --> a and b could be integers OR fractions Also b^2 means b could be + or - (2) a/b = 1/6 > this is a RATIO ---> a and b can be ANYTHING as long as the ratio is met Or: b/a = 6 NS (3) a/b = 1/6 ---> 6a = b a^3 * b^2 = 36 Sufficient (two variables, two different equations)

Answer 510

Remainder Question > do NOT SIMPLIFY the fractions in remainder questions n > 0 int n/12 = z int + R/12 R < 12 (1) n/6 = m int + 1 n = 6m + 1 n/12 = (6m+1)/12 NS (because variable m remains) (2) R > 5 5 < R < 12 NS (3) n/12 = (6m + 1)/12 Possible remainders are 1 and 7 Since R > 5, then R = 7 (sufficient) C

Answer 511

Geometry - area Total Area = 9*9 = 81 Area of Shaded = Area of Unshaded > cross out areas that you know are equal > you are left with the BOTTOM row 2 shaded = 1 unshaded Bottom row dimensions are: x, 6 - x (9 - 3 - x), and 3 Ans C = 1.5

Answer 512

Decimals and Digit Question x: 0. a _ is a =/ 0? (1) 16x = integer ---> rewrite x as a fraction, where the DENOM must be CANCELLED OUT 16(1/16) ----> 1/16 = 0.01xx (No) 16(2/16) = 16(1/8) ---> 1/8 = 0.125 (Yes) NS (2) 8x = integer ---> rewrite x as a fraction, where DENOM must be cancelled out 8(1/8) ---> 1/8 = 0.125 (smallest fraction) Always Yes B

Answer 513

Inequalities - range ("possible value") > combine into one fraction on RHS > Since we need NUMBERS in our range, we need to cancel out x + y in the denom > manipulate the expression to cancel out x + y in the denom (via factoring) 10x + 20y = 10x + 10y + 10y 10 + (10y)/(x + y) = k 10 + (10y + 10x)/(x + y) > 10 + (10y)/(x + y) 20 > 10 + (10y)/(x + y) = k (max value of k is 20) 10 + (10y)/(x + y) > 10 + (5x + 5y)/(x + y) (because x < y, so 5x + 5y < 10y) k = 10 + (10y)/(x + y) > 15 (min value of k is 15) 15< k < 20 k = 18 ALTERNATIVELY: Since K is an integer, x+y must be 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 Only 18 is in the list of answers given.

Answer 514

Geometry - Triangles is 0.5*base*height < 20? (1) Triangle is NOT a right triangle > we are not given any lengths (NS) (2) x + y < 13 > we are not given any lengths e.g., x = 2, y = 4 (max area is right triangle) 2*4*0.5 = 4 (yes) x = 6.5, y = 6.4 6.5*6.4*0.5 > 20 (no) NS (3) Triangle is NOT a right triangle x + y < 13 Max area is > 20, but we don't know if the triangle's area is STILL greater than 20 or less than 20. NS e.g., x = 6.5, y = 6.4, height = 6.4 6.4*6.4*0.5 > 20 (no)

Answer 515

Geometry: tunnel problem (semi-circle) Need to know: > Max width of the truck > Max height of the truck (1) NS without max height (2) NS without max width (3) Sufficient ** if this were PS, you would have to double check that the max height of the truck is SMALLER than the permissible height (calculated using Pythagorean Theorem).

Answer 516

Divisibility, Factors P, N both integers, neither are multiples of 8 P/N = int (is it odd)? (1) P = 4m P/N = int = 4m/4m = 1 (yes) P/N = int = 4m/2m = 2 (no) NS (2) N = 4m ---> P must also be a multiple of 4 P/N = int = 4m/4 = m (must be ODD, otherwise, 4m is a multiple of 8) Sufficient B

Answer 517

Digit question x: _ 7 _ y: _ 5 _ x < y x + y: _ _ _ _ III) Since x < y and x + y is a four digit integer, y must be at least 501 (T) I) Test: 75 + 55 = units digit 0 (Not True) II) Not true (what if there is carry over) Test: 475 + 555 = 1030 Only III (For these types of "must be true" questions, you just need to find ONE INVALID case)

Answer 518

Xy plane geometry: slopes Slope of line I = 2/3 y = (2/3)*x + b Ans A and C both have slope 2/3 HOWEVER A is actually the SAME LINE as Line I (so it is not parallel to Line I) Ans C Tip: always scan the other answers before moving on!

Answer 519

Xy plane geometry: Equation of the line: 4x + 3y = 60 (creates a triangle) To find which point lies inside region R, TURN IT INTO AN INEQUALITY: 4x + 3y < 60 ---> easier to plug in (turns into y < (-4x)/3 + 20) RHS = manufactured y (on the function) LHS = actual coordinate y

Answer 520

Divisibility Word Problem: Is N/M = integer? IMPORTANT CONSTRAINT: 3 but 16/6 not int (3*18)/6 = 9 --> and 18/6 = int (2) (13N)/M = Integer M < 13, so M cannot be cancelled out by 13. N must be a multiple of M Sufficient B

Answer 521

MUST BE TRUE - either find a disproving case or prove must always be true SIMPLIFY the equation: xy + z = xy + xz *** z = xz z - xz = 0 (because z could equal 0) z(1 - x) = 0 ---> Therefore z = 0 OR x = 1 Ans E **be careful with simplification

Answer 522

Remainders Question [1, 100] --> pick 7 Sum of remainders? > need to know ALL of the remainders (1) Range OF THE REMAINDERS = 6 > we already know the possible remainders are 0, 1, 2, 3, 4, 5, 6 > so we know there is at least one remainder 0 and one remainder 6 > however, we DO NOT KNOW ANYTHING ABOUT THE MIDDLE NS (2) Seven consecutive integers > will always have a multiple of 7 and the set of remainders is COMPLETE [0, 1, 2, 3, 4, 5, 6] Sufficient B

Answer 523

Word Problem - remainders and cost minimization > READ CAREFULLY and Understand "minimum possible total cost" 391 need to be shipped in boxes For the MINIMUM cost option, you CAN MIX-AND-MATCH!! Case 1) Ship all 391 bags in 5-capacity boxes # of boxes needed = 391/5 = 78 + 1/5 = ROUND UP 79 boxes Cost = 79*$1 = $79 Case 2) Ship 391 bags in THE MINIMUM TOTAL COST Cheapest box --> 20 handbags per box > 5-capacity is 5 bags per dollar > 12-capacity is 6 bags per dollar > 20-capacity is 6.6 bags per dollar # of 20-capacity boxes needed = 391/20 = 19 boxes For the remaining 11 boxes, find the cheapest option: > 5-capacity (we need 3 * 1 = $3) > 12-capacity (we need 1 * 2 = $2)** > 20-capacity (we need 1 * 3 = $3) Therefore cost = 19 * $3 + 1 * $2 = $59 Total extra cost = 79 - 59 = $20

Answer 524

Geometry - Similar Solids Cone SA: pi*r^2 + pi*r*L Cone V: (1/3)*(pi*r^2*h) Ratio of SA = (Ratio of Sides)^2 2 = (k)^2 sqrt(2) = k Therefore: radius2/radius1 = height2/height1 = sqrt(2) V1 = (1/3)*(pi*r^2*h) V2 = (1/3)*(pi)*(r*sqrt(2))^2*(sqrt(2)*h) V2 = 2*sqrt(2)*V1 Therefore V2 = 2*sqrt(2)*24 = 48*sqrt(2) OR: Ratio of V = (Ratio of Sides)^3 V2/V1 = (sqrt(2))^3 V2 = 24*(2*sqrt(2)) = 48*sqrt(2)

Answer 525

Permutation and Digits - slot method _ _ _ _ _ 5 5 5 Start with units digit: > 15 = 7 + 8 or 8 + 7 ---> 2 options (there is carry over) Tens Digit: > 5 - 1 from carry over = 4 > 14 = 5 + 9 or 9 + 5 ----> 2 options (there is carry over) Hundreds Digit: > 5 - 1 from carry over = 4 --> only one option Therefore: 1 * 2 * 2 = 4 options

Answer 526

One var Inequality - sewing approach Is: x^3 - 16x > 0 x(x^2 - 16) > 0 x(x + 4)(x - 4) > 0? is -4 < x < 0 OR x > 4? (1) x <= 4 NS - sign changes (2) x >= 0 NS - sign changes (3) 0 <= x <= 4 THIS is an example of ALWAYS NO -- Sufficient

Answer 527

Factoring - recognize the PATERN Let xy = A A^2 - A - 6 = 0 ----> QUADRATIC FACTORING (A + 2)*(A - 3) = 0 A = -2 xy = -2 y = -2/x or A = 3 xy = 3 y = 3/x II and III

Answer 528

Statistics Question: SumS = SumT > could be a set of positive or negative or 0 integers Is Ns > Nt? (1) MeanS < Mean T SumS/Ns < SumT/Nt HOWEVER we do not know whether the sums are >0 or <0 NS (2) MedianS > MedianT e.g., S: -2 -1 0 1 2 ---> sum = 0, median = 0, 5 terms T: -3 -2 5 --> sum = 0, median = -2, 3 terms e.g., S: -2 -1 0 ---> sum = -3, median = -1, 3 terms T: -6 -5 -2 5 5 --> sum = -3, median = -2, 5 terms 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 529

Divisibility and factors Is x/3 an integer? > Pre-think: x could be 0, +, -, int, or fract > x/3 is only an integer if x is a MULTIPLE of 3 (1) 72/x = int ---> x could be cancelled out by any factor of 72 (2, 3 etc.) NS (2) 81/x = int 81 = x*int ---> x is a FACTOR of 81 Factors of 81: 1, 81, 3, 27, 9 1 is NOT a multiple of 3, all other factors are multiples of 3. so NS ALSO x could be a fraction e.g., 81/0.5 = 162 (3) x could still be 1, a fraction, or a multiple of 3 NS E

Answer 530

DON'T BOTHER DOING CALCULATIONS Focus on the last two digits - perfect square? Only B --> 25 is a perfect square B

Answer 531

Probability, equally spaced sets: n(n + 1)(n + 2) is the product of three consecutive integers. The product is divisible by 8 IF: > n is even --> e*odd*e > or n+1 is a multiple of 8 **** [1, 96] Total # of terms = (last - first)/1 + 1 = (96 - 1) + 1 = 96 # of even terms --> [2, 96] = (96 - 2)/2 + 1 = 47 + 1 = 48 # of multiples of 8 (n + 1): [8, 96] ---> n could be 7 to 95 = (96 - 8)/8 + 1 = 11 + 1 = 12 Therefore probability = (48 + 12)/96 = 60/96 = 15/24 = 5/8

Answer 532

Statistics and Sets: Max possible range --> infinity Least possible range --> MAXIMIZE overlap ---> set A is inside of Set B Therefore least possible range = larger range = 320

Answer 533

Percent Linear Equations > 3 variables (J, K, p) --> however J and K CANCEL OUT (Always try to solve as much as possible for you to declare S or NS) > the variables do not have to be integers here ($) p = ? 1995: J, K 1998: J*(1 + p/100) K*(1 + p/100) (1) 1995: K = 2000 + J NS (no specific values given) (2) 1998: K(1 + p/100) = 2440 + J(1 + p/100) --->three unknowns NS (3) Together K = 2000 + J K(1 + p/100) = 2440 + J(1 + p/100) (2000 + J)*(1 + p/100) = 2440 + J(1 + p/100) --> two variables, one equation HOWEVER KEEP SOLVING 2000 + (2000p)/100 + J + (Jp)/100 = 2440 + J + (Jp)/100 ---> J and (Jp)/100 cancel out 2000 + (2000p)/100 = 2440 ----> one variable, one equation Also recognize the COMBO: K - J = 2000 (1 + p/100)*(K - J) = 2440 Sufficient C

Answer 534

Inequality and number properties R, P and Q can be anything: > +, -, 0 (except for Q), fract, int Is R < P? > test cases (1) P > 50 > NS - nothing is known about Q e.g., R = 100/1 = 100 --> R = P (No) e.g., R = 100/2 = 50 --> R 1 > NS - nothing is known about P e.g., R = 100/2 = 50 --> R R > P (no) (3) P>50 Q>1 e.g., R = 100/1.5 < P (always Yes) e.g., R = 51.5/2.5 < P (always Yes) C

Answer 535

Probability - sets 1 = P(A or B) + P(not A and not B) Also wording: P(A or B) = P(A) + P(B) - P(AandB) **we need to KNOW the relationship between events E and F to know if there is an INTERSECTION (1) NS without P(F) and relationship (2) NS without P(E) and relationship (3) We STILL DO NOT KNOW the relationship between event E and event F > could be 0 overlap --> P(E or F) = 1 > could be independent events --> P(E or F) = 0.4*0.6 > could be event F is inside event E --> P(E or F) = 0.6 NS E

Answer 536

Statistics a < b < c Could be int, fracts, + or - Is: b = (a + b + c)/3? 3b = a + b + c? 2b = a + c? (1) Range = c - a = 2*(c - b) c - a = 2c - 2b 2b = a + c (sufficient) (2) a + b + c = 3(one number) **since a < b < c, we know that: 3a < a + b + c < 3c Therefore, a + b + c CAN ONLY equal 3b Sufficient D

Answer 537

LCM question (make the answer an integer) Let N = total number of students enrolled (3/8)N = sophomores (1/50)N = biology majors We need a single value for N that will turn the above figures INTO INTEGERS > LCM (helps you find the least common denominator) LCM = 8*25 = 200 So any MULTIPLE OF 200 will work A (7000) is the only multiple of 200

Answer 538

Xy-plane geometry: Any coordinate point that satisfies the inequality works: 2x + 3y <= 6 Is: 2r + 3s <= 6? **You can visualize this by drawing it! (downward sloping line with shaded region below the line) (1) 3r + 2s = 6 ---> NS (not exactly what we are looking for) Also you can draw this out --> not parallel lines (some portion will be above and some portion will be below) (2) r <= 3 and s <= 2 > again, NS (some portion will be above and some portion will be below) e.g., r = 2, s = 2 (3) Again, some portion will be above and some portion will be below > you can find the intersection of the lines and prove that the intersection occurs above the line in question E

Answer 539

Linear equations "R-scale and S-scale are related linearly" --> y = mx + b Or: S = m(R) + b Two points: (6, 30), (24, 60) Question: (R, 100)? Step 1) Slope m = (30)/(18) = 5/3 Step 2) y int y = (5/3)x + b 30 = (5/3)(6) + b 20 = b Step 3) S = (5/3)R + 20 100 = (5/3)R + 20 (80*3)/5 = R R = 16*3 = 48 OR ESTIMATE: R scale goes up by 18, and S scale goes up by 30. Next points: (42, 90), (60, 120) R must be between 42 and 60 --> 48

Answer 540

Geometry: Right Triangles *Answer choices are RANGES --> try to find the range > START WITH MINIMUM x < y < z ---> y must be greater than x Therefore, y will be greater than the case where x = y: (y^2)/2 = 1 y = sqrt(2) (minimum value) There is no max Ans: y > sqrt(2)

Answer 541

Geometry - overlapping circles Strategy: Area of the Overlap = (Area of each sector * # of sectors) - Area of the Square *then multiply the area of the overlap by 95% = ((Pi/2)*4 - 4)*0.95 = (2.28)*0.95 approx 2.2

Answer 542

Digits Question: place values M > 0 int M^3 has how many digits? Test cases... (1) M = _ _ _ e.g., 100 --> 100^3 = 1000000 (7 digits) e.g., 500 --> 500^3 = 125000000 (9 digits) NS (2) M^2 = _ _ _ _ _ e.g., 100^2 = 10000, then 100^3 has 7 digits e.g., 300^2 = 90000, then 300^3 has 27000000 has 8 digits NS (3) NS because of test cases in 2 E

Answer 543

Estimation Word Problem: Was t est +/- 0.5 = t actual ? t = d/r (1) d est +/- 5 = d actual NS --> nothing is known about speed (2) r est +/- 10 = r actual NS --> nothing is known about distance (3) Nothing is known about the ACTUAL value of d and r > literally could be ANYTHING (and the range, 0.5, is very tiny) e.g., d est and r est = d act and r act --> Yes e.g., d est and r rest =/ d act and r act --> No d act = 20 r act = 20 t act = 1 d est = 25 r est = 10 t est = 25/10 = 2.5 (No) E

Answer 544

Overlapping sets > we need to INCLUDE n (sum [n, m]) Sum n to m = (Sum 1 to m) - (Sum 1 to n minus 1) = [m(m+1)]/2 - [(n-1)(n)]/2

Answer 545

Rates Question: > READ CAREFULLY and make sure you MATCH the numbers City R = 25m/g ----> will drive 10m Highway R = 40m/g -----> will drive 50m Total distance driven = 60m Gallons used in the city: Set up a Ratio (25m/g) = (10m/x) x = 10/25 Gallons used on the highway: (40m/g) = (50m/y) y = 5/4 Total R = (Total Distance/Total Gallons used) = 60/( 10/25 + 5/4) = 400/11 = approximately 36

Answer 546

Ratio Question > set up equations - don't panic! > then recognize that this is a FACTOR question N = # of staff members? Each person received x, y, and z items (1) Let w be the multiplier (integer) Each person received: x = 2w y = 3w z = 4w NS - still cannot determine N (2) 18 pens = N*x = 2*3^2 27 pencils = N*y = 3^3 36 pads = N*z = 2^2 * 3^2 FACTORS ---> N could be 3 or 3^2 NS (3) 18 pens = N*x = 2*3^2 = N*(2w) 27 pencils = N*y = 3^3 = N*(3w) 36 pads = N*z = 2^2 * 3^2 = N*(4w) Therefore: N*w = 3^2 N*w = 3^2 N*w = 3^2 N could be 3 or 9 ---> NS E

Answer 547

Percent word problem: 24 min 18 sec represents the TIME LEFT ---> 90% left (10% has been completed) 24 min 18 sec = 1458 seconds = 0.9*Total Time Total Time = 1620 seconds Therefore when the readout indicates 40%, he still has 60% LEFT: 1620*0.6 = 972 seconds or 16 min 12 seconds

Answer 548

Permutation / Digit - Slot Method Tip: Organize by hundreds digit > make sure you deduct 1 from 7 _ _ (because 700 is invalid) > make sure you DO NOT INCLUDE ALL identical digits (e.g., 777) 1) 7 7 _ = 9 7 _ 7 = 9 7 _ _ = 9 - 1 = 8 Total = 26 2) 8 8 _ = 9 8 _ 8 = 9 8 _ _ = 9 (second slot is FIXED based on the first one) Total = 27 3) 9 9 _ = 9 9 _ 9 = 9 9 _ _ = 9 Total = 27 Grand Total = 26 + 27 + 27 = 80

Answer 549

Ratio MIXTURE problem: We care about: M/T M1/T1 = 3/5 (Original Group) M2/T2 (Added people) Is (M1 + M2)/(T1 + T2) > 3/5? Recall: If M1/T1 < M2/T2: M1/T1 < (M1 + M2)/(T1 + T2) < M2/T2 Therefore, we need to see if M2/T2 > 3/5: (1) M2/T2 > 1/2 ----> NS (we care whether M2/T2 > 0.6, not 0.5) (2) (T1 + T2) = (6/5)*(T1) ---> NS (nothing is known about M's weighting) (3) We still don't know whether M2/T2 > 3/5 E

Answer 550

Xy-plane Geometry With Circles X intercepts of the circle: (1, 0) and (-1, 0) DRAW accurate pictures (1) X-intercept of line k > 1 NS - could intersect, could not (2) Slope of line k = -0.1 (pretty shallow and flat) NS - could intersect, could not (if y intercept is pretty high) (3) NS - could intersect, could not (if y intercept is pretty high) E

Answer 551

Rates Question Max Distance = 250 miles When the two trains pass each other, their combined distance = 250 Pass each other after 2 hours. Need to know DISTANCE of each train after 2 hours = individual average rate * 2 hours (one distance is ENOUGH) (1) P avg rate = 70 = Distance Travelled/2 Distance Travelled by P = 140 m Therefore, Distance Travelled by Q = 250 - 140 = 110 m Sufficient (2) Q avg rate for the ENTIRE trip = 55 = 250/total time NS to know avg speed of Q in the first 2 hours (Q could have been very fast or very slow) A

Answer 552

Double Matrix Problem --> beware of DENOM (1/5)A = AandB (1/4)B = AandB Q: B/Total Readers ---> total readers DOES NOT equal A + B (there are people who read both) Total readers = sum of the four inner squares (1/5)A = (1/4)B 4A = 5B We also know the "opposites" of the above fractions: (4/5)A = A,NotB (3/4)B = B,NotA Total Readers = (1/5)A + (4/5)A + (3/4)(4/5)A + 0 = A + (3/5)A = 8/5 A B/Total Readers = (4/5)A / (8/5)A = 1/2

Answer 553

Xy-plane Shaded Region Question > differentiate between "manufactured y" (on the function) and actual y (b) y = x^2 - 4x is a parabola Shaded Region is represented by area ABOVE the function (and below y = 0): y > x^2 - 4x --> RHS sub in x coordinate, RHS get "manufactured y" (on the function) ---> compare with LHS y (actual y coordinate) --> region is where actual y is greater than manufactured y Given (a, b) is the point: See if b > a^2 - 4a (we know that b < 0) (1) 0 < a < 4 --> NS without knowing b (could be in the region or outside of the region) (2) a^2 - 4a matches simplified question stem Since b < 0, region is the shaded one B

Answer 554

Set Question UNDERSTAND the table: > the columns are MUTUALLY EXCLUSIVE (i.e., for each candidate, you either choose "Fav", "Unfav" or "Not Sure" > There might be some overlap across rows Possible options: (M, N) - 9 options Fav, Fav Fav, Unfav Fav, NS Unfav, Fav Unfav, Unfav Unfav, NS NS, Fav NS, Unfav NS, NS Sum = 100 We are asked for Fav, Fav (1) Not Fav NEITHER M nor N = 40 = (Unfav, Unfav) + (Unfav, NS) + (NS, Unfav) + (NS, NS) Think about just the favorable set: Total = Fav M or Fav N or both + Unfav/NS neither M nor N 100 = (40 + 30 - BothMN) + 40 BothMN = 10 Sufficient (2) Unfav, Unfav = 10 NS to know the split for favorable All we know is: UnfavM + UnfavN - Bothunfav = 20 + 35 - 10 = 45 A

Answer 555

MUST be less than 1 > sounds like an inequality > calculate the MAXIMUM for each option - if the max is less than 1, True. Otherwise, False. (Test cases might not get you the right answer because you could be missing a test case! "could be true" =/ "must be true") I. r/s ---> max by max r/min s = 0.99/1.01 = < 1 True II. rs ---> max by max r * max s = 0.99 * 1.99 = > 1 False III. s - r ----> max by max s - min r = 1.99 - 0.1 = > 1 False Only I

Answer 556

Worst case scenario question > KEEP track of the digits!! (must use them ALL) [0, 9] = 10 DIGITS = 10 pieces of paper Possible Combos to get to 10: 1 + 9 2 + 8 3 + 7 4 + 6 Non-used numbers: 0 and 5 WORST case order: 0 5 1 2 3 4 Anything else will get you a sum of 10 on two slips = need 7 draws

Answer 557

Rates Question - inequality Is t1 > t2? r1*t1 = d1 r2*t2 = d2 Therefore: Is (d1/r1) > (d2/r2)? > all positive (1) d1 = 30 + d2 NS without speeds (2) r1 = 30 + r2 NS without distances (3) SUB IN IS: (30 + d2)/(30 + r2) > d2/r2? ---> CROSS MULTIPLY 30*r2 + r2*d2 > 30*d2 + r2*d2 ? 30*r2 > 30*d2? (DOES NOT CANCEL OUT!!) NS without exact speeds and distances E

Answer 558

Triple Venn Diagram > tip - keep percents until the last calculation! Max Bargain Prices ONLY by minimizing overlap and minimizing other group (0%) B = Bonly + UB + FB - Center Bonly = B - UB - FB + center = 42% - UB - FB + center 100% = U + F + B - UF - FB - UB + center + other 100% = 56% + 48% + 42% - 30% - UB - FB + center + 0% 100% = 116% - UB - FB + center -16% = - UB - FB + center Therefore: Bonly = 42% - 16% = 26% 26%*1200 = 312 Alternatively: > Overlap UB = 0% > Overlap FB = 0% > overlap other = 0% U + F - UF = 56% + 48% - 30% = 74% 100% = U + F - UF + B only 100% = 74% + B only 26% = B only

Answer 559

Number Properties (positive, negative) inequality Is xy > 0? (do x and y have the same sign: + + or - -) (1) x - y > -2 Could be: + + (yes) + - (no) NS (2) x - 2y < -6 Could be: + + (yes) e.g., 1 - 2(10) = -19 - + (no) e.g., (-1 - 2(10) = -1 - 20 = -21) NS (3) x - y > -2 x - 2y < -6 ----> two inequalities, can COMBINE if in the same direction via addition x - y > -2 -6 > x - 2y x - y - 6 > -2 + x -2y y > 4 ---> sub into equations to find range of x e.g., y = 4 x - 4 > -2 x > 2 Both x and y are positive Sufficient

Answer 560

Digit Question with Exponents > this is a UNITS DIGIT question 1^4 + 6^4 + k^4 + 4^4 = 16k4 RHS units digit = 4 LHS units digit: 1 + 6 + k's unit digit + 6 = 3 + k's unit digit Therefore 3 + k's unit digit = 4 k's unit digit = 1 Ans B (3^4 = 81)

Answer 561

Digit question with some permutation qualities Tip: Since the answer choices are very different, start with the thousands column. _ _ _ _ # of 1000s = fixed * 3 * 2 * 1 = 6 # of 2000s = fixed * 3 * 2 * 1 = 6 # of 3000s = fixed * 3 * 2 * 1 = 6 # of 4000s = fixed * 3 * 2 * 1 = 6 SUM = 6000 + 12000 + 18000 + 24000 = 60000 However, the sum must be GREATER than 60,000 because we have other digits Ans E

Answer 562

Number Properties: > since a > 1, you can drop the base and keep the direction of the signs Is x > -2y? Is x + 2y > 0? (1) x + y > 1 ----> at least one is positive + + -----> Yes + - ------> No e.g., 4 - 2 > 1 ---> 4 + -4 = 0 - + NS (2) x < 2y x - 2y < 0 + + ---> 1 - 2(5) < 0 --> 1 + 5 > 0 (Yes) - + ----> -5 -2(1) < 0 ----> -5 + 2(1) (no) NS (3) COMBINE inequalities (add, signs in the same direction) x + y > 1 2y > x x + y + 2y > 1 + x 3y > 1 y > 1/3 Test: y = 1 x + 1 > 1 x > 0 ---> x + 2y > 0 (Yes) Test: y = 10 x + 10 > 1 x > -9 ----> x + 2y > 0 (Yes) Always Yes, Sufficient OR Use the Middlemen: Is 1 - y > -2y? 1 - y - (-2y) = 1 - y + 2y = 1 + y (since y > 1/3, 1 + y > 0) Yes, 1 - y > -2y x > 1 - y > -2y Sufficient

Answer 563

Permutation with duplicates and selection _ _ _ PEPPER --> 3 P's, 2 E's, 1 R Typically we would use: Pm, n = m! / (m - n)! However, the formula assumes NO duplicates. Since there are duplicates, it is best to do this GROUP BY GROUP (options): P P P = 1 way to arrange P P other = (3!/2!) * 2 cases for other = 6 E E other = (3!/2!) * 2 cases for other = 6 P E R = 3! = 6 In total = 1 + 6 + 6 + 6 = 19

Answer 564

Xy plane Geometry and Combinatorics > first SKETCH a picture for xy plane questions (especially with the boundaries > then calculate the length of the boundaries (watch out for INCLUSIVE or NOT INCLUSIVE) ---> typically will be integer points 10 integers [-4, 5] 11 integers [6, 16] Triangle PQR is a right triangle > does not have to be in the same ORIENTATION --> P and R could be flipped etc. Logic: once you fixate P's x and y, then R and Q also have either a fixated x coordinate or fixated y coordinate # of possible triangles = # of possible P * # of possible R (based on that P) * # of possible Q (based on that Q) # possible P = anywhere in the region and borders = 10*11 = 110 # possible R (same horizontal line as P, could be L or R of P) = 10 - 1 = 9 # of possible Q (same vertical line as P, could be above or below P) = 10 Therefore: 110*9*10 = 9900

Answer 565

Geometry: Acute Triangle (all angles < 90degrees?) Is the triangle an acute triangle? is a^2 + b^2 > c^2? is a^2 + b^2 - c^2 > 0 ? (1) With area, we can calculate radius and diameter = each side length --> we can see if a^2 + b^2 > c^2 Sufficient (2) c < a + b < c + 2 (compound inequality) 0 < a + b - c < 2 a + b < c + 2 ----> since both sides are positive, SQUARE both sides (to match question stem) (a + b)^2 < (c + 2)^2 a^2 + 2ab + b^2 < c^2 + 4c + 4 a^2 + b^2 - c^2 < 4c + 4 - 2ab RHS depends on value of c relative to a and b --> NS e.g., c = 1, ab = 2 4 + 4 - 2 = 6 (could be positive or negative) A

Answer 566

Geometry and Factor Question n^2 - k^2 = 48 (n - k)*(n + k) = 48 ----> FACTOR QUESTION (need valid values for n and k = integers) k <=n, so n - k >= 0 1 * 48 --> n - k = 1 and n + k = 48 2n = 49 --> n is not an integer, INVALID 2 * 24 ---> n - k = 2, n + k = 24 2n = 26 --> valid 3 * 16 ---> n - k = 3, n + k = 16 2n = 19 ---> invalid 4 * 12 ---> n - k = 4, n + k = 12 2n = 16 ---> valid 6 * 8 ---> n - k = 6, n + k = 8 2n = 14 ---> valid 3 pairs

Answer 567

Digit Question with Combinatorics > digits cannot equal 0 (only from 1 to 9, inclusive) abc - cba = 7*int ----> VALUES => algebraic method 100a + 10b + c - 100c - 10b - a = 7*int 99a - 99c = 7*int 99(a - c) = 7 * int Since 99 is not a multiple of 7, a - c must be a multiple of 7 e.g., a - c = 9 - 2 8 - 1 Therefore the possible values of abc: a = two options b = can be any digit from 1 to 9 = 9 c = ONE OPTION (fixed based on a) 2 * 9 * 1 = 18

Answer 568

Statistics > "no integer appears more than twice" = up to 2 times! _ _ _ _ _ _ _ _ _ _ MAX integer a by MINIMIZING all 9 others (min is 1) 1 1 2 2 3 3 4 4 5 a mean = 10.1 = SUM/10 101 = 25 + a 76 = a

Answer 569

Factors Question > Wording: looking for the # of factors of x MINUS 1 (itself) e.g., x = 4, factors are 1, 2, 4 --> 2 positive integers less than 4 (itself). (1) x^2 is divisible by exactly 4 positive integers less than x^2 x^2 has 4 factors + itself = 5 factors Let x = a^m * b^n (where a and b are prime numbers) x^2 = (a^m * b^n)^2 = a^2m * b^2n (2m + 1)*(2n + 1) = 5 1 * 5 ---> m = 0, n = 2 x = b^2 ---> 3 factors, or 2 factors other than itself (sufficient) (2) 2x is divisible by exactly 3 positive integers less than 2x > typically 2x is inferior to x^2 because we DON'T KNOW if x already contains 2 > if Yes: then 2 just increases the exponent by 1 > if No: then 2 actually doubles the number of factors NS A

Answer 570

Counting Question > do it systematically --> this time it makes sense to start with units digit and make your way up. > however, you CAN COUNT some integers MULTIPLE TIMES (because you are counting the 5's, not the integer itself!!) [5, 1500] --> increments of 5 1) # of Unit digit 5 5, 15, 25, 35, 45, 55, 65, 75, 85, 95 = 10 * 15 = 150 (or every odd multiple of 5) [1, 299] = (299 - 1)/2 + 1 = 150 2) # of Tens Digit 5 50, 55, 150, 155 etc. = 2 * 15 = 30 3) # of hundreds digit 5 500, 505 ... 595, 1500 [500, 595] = (595 - 500)/5 + 1 = 20 + 1 for 1500 21 Total: 150 + 30 + 21 = 201

Answer 571

Double Matrix > we need to know x and y to understand the breakdown of the totals (otherwise, the info is useless! we already know the split is either x/100 or (100 - x)/100) If z is an integer representing a percent, we want to know (z/100)*N = ? (1) we know the total = 1000 150 experienced neither SE nor RC > not helpful to figure out x and y (2) N is actually irrelevant (cancels out) z + 30 = y NS (3) we actually get the same info as (2) z + 30 = y E Therefore, the key is to STAY NEAT

Answer 572

Square Root PS --> either rationalize, simplify/solve or SQUARE both sides > there is nothing to rationalize and nothing really to simplify ---> SQUARE BOTH SIDES 63 - 36sqrt(3) = (x + y*sqrt(3)^2 63 - 36sqrt(3) = x^2 + 2xy*sqrt(3) + y^2*3 NOTICE the sqrt(3) term on both sides --> they must be equal (since x and y are integers) -36sqrt(3) = 2xy*sqrt(3) -18 = xy (FACTOR PAIRS) xy can be: -1 * 18 or 1 * -18 -2 * 9 or 2 * -9 -3 * 6 or 3 * -6 -----> x + y = -3 C

Answer 573

Probability and Sets question ~B represents everything OUTSIDE of B's circle --> only A + NeitherANorB P(B) + P(~B) = 1 P(A) = ? = Only A + Intersection (1) P(AUB) = 0.7 = P(A) + P(B) - P(AandB) ** if there is an intersection NS - no other values (2) P(AU~B) = 0.9 = P(A) + P(~B) - P(Aand~B) ** NS - no other values and info about intersection (3) P(AU~B) = 0.9 = P(A) + P(~B) - P(Aand~B) ** there is an intersection (because "only A" is not part of B) Only B = 0.1 Therefore: P(B) - P(AandB) = 0.1 0.7 = P(A) + P(B) - P(AandB) 0.7 = P(A) + 0.1 P(A) = 0.8 Sufficient C (in this case, regardless of the relationship between A and B, we can calculate the B only part) 1) Complete Overlap (B inside A) 0.1 = P(B) P(AandB) = P(B) ---> cancels out completely, able to find P(A) 2) Zero overlap 0.1 = P(B) P(AandB) = 0 ---> able to find P(A) 3) some overlap (above)

Answer 574

Remainder, Divisibility, and Digit Question a b c ? Recall divisibility rule for 9 --> divisible by 9 if the sum of the digits is a multiple of 9 (1) a + b + c = 15 ---> not a multiple of 9. ALSO remainder is FIXED (R = 6) Always No --> sufficient e.g., 348/9 = 38 + R 6 (2) abc + 111 = 9*int ----> VALUES = Algebraic format 100a + 10b + c + 100 + 10 + 1 = 9*int 100(a + 1) + 10(b + 1) + c + 1 = 9*int We can get the DIGITS of abc + 111: a + 1 + b + 1 + c + 1 = 9*int a + b + c + 3 = 9*int a + b + c = 9*int - 3 > Fixed remainder (R 6) (What if there is carry over? e.g., 999 = abc) > doesn't matter --> sum of digits gives you a FIXED REMAINDER Sufficient D

Answer 575

Weighted Average Problem - 3 items Which drink had the highest quantity? (Max) (1) Avg = 4 Logically: > Simple average = (1.8 + 4.5 + 3.3)/3 = 3.2 > Therefore, Drink B must have been purchased the most to bring the average up to $4. Mathematically: 4*(A + B + C) = 1.8*A + 4.5*B + 3.3*C ---> SIMPLIFY to see relationships 4A + 4B + 4C = 1.8A + 4.5B + 3.3C 2.2A + 0.7C = 0.5B ---> Get rid of decimals 22A + 7C = 5B Since: 5A + 5C < 22A + 7C = 5B 5A + 5C < 5B A + C B is the max Sufficient (2) 60 = 1.8A + 4.5B + 3.3C NS because we don't know weighted average A

Answer 576

Geometry - will it fit? (need DIMENSIONS, not area) > rubber band geometry --> fixate the shape = sufficient Only one possible outcome after the fold --> symmetrical fold Concept: REGULAR HEXAGON = comprised of 6 equilateral triangles (one vertex is the center) = two trapezoids (1) each side = 4 inches = side length of ALL the 6 equilateral triangles > we have FIXATED the shape --> we can tell if it will fit or not Proof: > diagonal = 2*4 = 8 (length of the folded card) > width = height of two equilateral triangles --> we can figure out (s/2 * sqrt3)*2 Sufficient (2) Area of 3 equilateral triangles or Area of 1 trapezoid < 36 square inches > area is not fixated, nothing known about side lengths, usually won't be sufficient TEST cases: #1) Longest length = 9, shorter length = 3, width = 4 (height of trapezoid) area = 0.5(9 + 3) * 4 = 24 < 36 --> fits #2) Longest length = 10, shorter length = 4, width = 4 area = 0.5(10 + 4) * 4 = 24 < 36 --> does not fit NS A

Answer 577

Statistics consecutive integers Since the median is NOT an integer --> there are an EVEN NUMBER (m is not a part of the list) k is even I. Not necessarily true (disproving case) _ _ = 1 + 2 = 3 (not even) II. True (same with odd number of consecutive integers) Min Value = median - (# of terms - 1)/2 III. Not true Max value should be = median + (# of terms - 1)/2 Only II

Answer 578

Closing Stock Price Word Problem - Range (inequality) Relative to the FIRST TRADING DAY = P1 Pc? --> set up a range P2 = $10 P2 was very close to the max: P1*(1.5) > P2 P2 was very close to the min: P1*(0.5) < P2 0.5*P1 < P2 < 1.5*P1 0.5*P1 < $10 < 1.5*P1 What can P1 = ? 0.5*P1 < $10 P1 < $20 1.5*P1 > $10 P1 > 20/3 = 6.67 Therefore: 20/3 < P1 < $20 Pc Range (not inclusive) Min Pc = Min P1*0.5 = 20/3*0.5 = 10/3 = 3.33 Max Pc = Max P1*1.5 = 20*1.5 = 30 Therefore: 10/3 < Pc < 30 A) $3.00 is outside this range

Answer 579

Closing Stock Price Word Problem - Range (inequality) Range (R) = Max P - Min P (don't worry about finding exactly the 52nd week --> R is the range for the ENTIRE YEAR) If P = Max P: Then R = P - MinPc Min Pc = P - R If P = Min P: Then R = Max Pc - P Max Pc = P + R Therefore: we have two inequalities to combine into a compound inequality P - R <= Pc Pc <= P + R P - R <= Pc <= P + R Ans C, P - (1/2)*R, is in the range

Answer 580

Digits question - all about trailing zeros and non-zero digits > first get expression into PRIME FACTORIZATION FORM 2^17 * 5^16 * 11^2 # of 10's --> # of 2*5 pairs --> 16 Non zero digits = 2*11^2 = 242 IN total we have: 3 non zero digits + 16 0s = 19 digits

Answer 581

Geometry Pythagorean Theorem --> need to solve system of equations > set good variables > a + 14 - a = 14 Eq'n 1: h^2 + a^2 = 15^2 Eq'n 2: (14 - a)^2 + h^2 = 13^2 a = 9 h = 12

Answer 582

Factors > remember the facts in the stimulus (p, q > 0 int)!! 1) p^2 = pq + p p^2 = p(q + 1) --> p =/ 0 p = q + 1 ---> p and q have zero factors in common other than 1 Sufficient 2) p^2 = q^2 + 2q + 1 p^2 = (q + 1)^2 ---> both are positive integers! p = (q + 1) Same as (1) Sufficient D

Answer 583

Inequality > remember: try to solve the inequality (see if a > or < 0) (1) a > b NS case 1: 5 > 2 25 > 12 (yes) case 2: -10 > -20 -50 > -120 (no) 2) 4b - 5a = 24 ---> rearrange for b so that you can sub b into the fact 4b = 24 + 5a b = (24 + 5a)/4 therefore: 5a > 6(24 + 5a)/4 10a > 72 + 15a -72 > 5a -72/5 > a (a is always negative) Sufficient B

Answer 584

Mixture and Ratio Question > solve a system of equations > ratios can be turned into percents (weights) Let x be the weight of the first bar, let y be the weight of the second bar 8 = x + y We also have info about gold: Q of Gold Before = Q of Gold After (2/5)*x + (3/10)*y = (5/16)*(x + y) Simplify: 0.4x + 0.3(8 - x) = 2.5 0.4x + 2.4 - 0.3x = 2.5 0.1x = 0.1 x = 1 Ans: 1 kg

Answer 585

Prime Numbers: > best to list them out so you can test cases First 15 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (1) 3x + 1 = prime ---> easiest way is to find valid cases of x (positive integers) x = 2, then 3x + 1 = 7 (valid) --> yes x = 6, then 3x + 1 = 19 (valid) --> no NS (2) 5x + 1 = prime x = 2, then 5x + 1 = 11 (valid) --> yes x = 6, then 5x + 1 = 31 (valid) --> no (3) NS because x can be 2 and 6 E

Answer 586

Exponent and integer question Integer IF 7^(7 - x) = integer x can be positive, 0, or negative as long as x is an integer 1) NS because x could be a fraction or an integer 2) | x | = x^2 (both sides are positive) --> SQUARE x^2 = x^4 0 = x^4 - x^2 0 = x^2*(x^2 - 1) 0 = x^2*(x + 1)*(x - 1) x = 0, -1, 1 Always an integer Sufficient

Answer 587

Permutation Step 1) Figure out how many groups of 5 digits sum to a multiple of 3 (remember, just add or subtract 3 to get to a multiple of 3 from an existing one!) 5 4 3 2 1 = 15 5 4 2 1 0 = 15 - 3 = 12 ** > however, you cannot have the first digit equal to 0 Step 2) Calculate the # of arrangements and Sum. 5 4 3 2 1 --> 5! = 120 5 4 2 1 0 ---> 5! - 4! = 96 Total = 120 + 96

Answer 588

Factors > given a large number (12!), FACTOR first > pay attention to wording ("product of THE LEAST five different prime numbers" =/ "product of AT LEAST five different prime numbers" 12!/m = 2^n * int Therefore: m = 2 * 3 * 5 * 7 * 11 12! / m = (12 * 10 * 9 * 8 * 4 * 3 * 2) = (2^9)*(3^4)*(5) Max value of n = 9

Answer 589

Double matrix/sets question > bucket segments of people into variables (e.g., Morning only, Afternoon only, Both) 128 = Morning only + Afternoon only + Both Morning only = ? (1) (3/4)*128 = Both Therefore, (1/4)*(128 = Morning only + Afternoon only NS (2) (7/8)*128 = Afternoon only + Both Therefore, (1/8)*128 = Morning only Sufficient B

Answer 590

None of them Recall: Standard deviation of a set does NOT change if the set is transformed using addition or subtraction

Answer 591

Distance question UNDERSTAND the scenario: Fly travels for 3 hours at 10km/h. Therefore, TOTAL DISTANCE travelled is just d = r*t = 10*3 = 30 km LONG WAY: 1) Fly meets hiker #2 after 2 hours, travelling 20 km 2) Fly returns to hiker #1 after 2/3 hours, travelling 20/3 km = 6.xx -- hikers have not yet met up (5*2 h + 5*2/3 = 10 + 3.33 = 13km, not 15km) 3) Fly returns to hiker #2 then goes back to hiker #1 when they intercept

Answer 592

Number properties: xy > 0 xy = p * int Is p = 2? (1) x^2 * y^2 = even ---> xy = even HOWEVER, we don't know whether p is even or whether int is even e.g., xy = 6, then p = 2 (Y) or 3 (N) NS (2) xp = 6 ---> p = 2 (Y) or 3 (N) NS (3) x is even, then p is odd (N) But if x is odd and y is even, then P is even (Y) NS E

Answer 593

Probability > think about the possible cases (don't be overwhelmed) > we want probability 10 < K < 15 Case 1) Kate gets +1 +1 +1 -1 -1 = net +1 ---> K = 11 Case 2) Kate gets +1 +1 +1 +1 -1 = net +3 ---> K = 13 *must multiply each case by # of permutations Case 1) P(T T T H H) = (1/2)^5 * [5! / (3! * 2!)] Plus Case 2) P( T T T T H) = (1/2)^5 * [5! / 4!] Ans B = 15/32

Answer 594

Number properties / factors / remainders p = 7*z + 2 is p = 8*int? ** First think about the possible values of p (pattern!): p = 2, 9, 16, 23, 30, 37, 44, 51, 58, 65, 72 (2) p < 100 --- NS p = 9 (N) p = 16 (Y) (1) p = 6*m= 7*z + 2 Test cases for m: m = 5 --> p = 30 (N) m = 12 --> p = 72 (Y) NS (3) NS because p = 30 or 72 E

Answer 595

Digits and inequality > first simplify > don't forget DIGIT constraint (digits are greater than 0 and less than or equal to 9) Is 3(2 + X/1000) > 2(3 + Y/1000)? Is 6 + 3X/1000 > 6 + 2Y/1000? Is 3X > 2Y? (1) Sufficient (always no) (2) Multiply both sides by 3 to get LHS to match 3X 3X < 3Y - 9 Now see if RHS equals or is LESS than 2Y (hint: see statement 1) Is 3Y - 9 =< 2Y? Is Y - 9 =< 0? Is Y =< 9? ----> always true So 3X < 3Y - 9 <= 2Y ---> 3X <= 2Y (always no) S Ans: D

Answer 596

Digits 0 <= A =/ B <= 9 3A * 2B = 864 A*B = unit digit 4 A = ? LIST POSSIBLE values for A and B: **make sure you test 3A*2B = 864 TOO*** 1 * 4 2 * 7 3 * 8 4 * 1 4 * 6 6 * 9 And vice versa (1) A + B = 10 3 + 8 or 8 + 3 4 + 6 or 6 + 4 HOWEVER only A = 6 and B = 4 works (multiply to 864) Sufficient (2) A*B = 24 3*8 or 8*3 4*6 or 6*4 HOWEVER only A = 6 and B = 4 works (multiply to 864) Sufficient Ans D *** sometimes only ONE possible option works > don't fall into the trap!!

Answer 597

Rates question --> set up capacity (distance) equation capacity (z) = rate * time Info #1) Pipe empties a pool in 4 hours outflow = (1/4) pools per hour Info #2) Takes 6 hours to empty a pool when there is rain inflow (units = liters) > (1/4) pools per hour * z liters per pool = (1/4) * z liters per hour Net outflow = outflow rate - inflow rate = (1/4 * z ) - 3 Together: z = rate * time z = ((1/4 * z) - 3 ) * 6 z = 36 hours

Answer 598

Number properties > need to know whether p > 0 AND p is a multiple of 11 > p can be +, - or 0 (1) p is a prime number N - P = 2 Y - P = 11 NS > by definition, p > 0 (2) 2p = 11*int p must be a multiple of 11 ---> however, p could be positive, neutral or negative *RECALL: 0 is a MULTIPLE of every integer and can be divisible by EVERY INTEGER except for itself

Answer 599

Number properties (integers) FIRST SIMPLIFY question stem: (a/b)/c = a/(bc) (1) a/c = 3 a = 3c ---> not sufficient (no info about b) also sub in: 3c/bc = 3/b --> only an integer if b = 1 or 3 (2) a = b + c ---> sub in (b + c)/bc = 1/c + 1/b ----> c and b are DIFFERENT integers, so the sum will never be an integer (c = b = 1 for the sum to be an integer) Always no, sufficient B

Answer 600

Absolute values: sqrt ( x^2 ) = | x | Ans: 63/20

Answer 601

Number properties - exponents, factors, integers (denominator must cross out) Rewrite 256 as a power of 2: 2^98 = 2^8L + N DIVIDE both sides by 2^8, because you know an important condition is that L and N are integers: 2^90 = L + N/2^8 N/2^8 must be an integer ---> N >= 2^8 Since 0 <= N <= 4, N must be = 0

Answer 602

System of equations > 3 equations, 3 variables ---> solvable > you reach a dead end if you try adding up the three equations --> NEED TO SUBTRACT TWO equations first to get a new expression > R =/ Q means that R - Q =/ 0 (can divide by R - Q or Q - R) Equation 2 - Equation 3: R^2 + PQ = 10 - (Q^2 + PR = 10) R^2 - Q^2 - P(R - Q) = 0 (R + Q)(R - Q) = P(R - Q) ------> can cross out R - Q R + Q = P Now add up all 3: P^2 + Q^2 + R^2 - QR + PR + PQ = 30 P^2 + Q^2 + R^2 - QR + P(R + Q) = 30 ----> Sub in P^2 + Q^2 + R^2 - QR + P(P) = 30 P^2 + Q^2 + R^2 = 30 - P^2 + QR ----> Equation 1 P^2 + Q^2 + R^2 = 30 - 10 = 20

Answer 603

Geometry / x-y plane: > IF one f the vertices was on the x or y axis, you can solve via finding the area of a trapezoid then subtract other right triangles > In this case, we can first find the area of a SQUARE, then subtract OTHER RIGHT TRIANGLES (Same approach, utilizing right triangles) Area of triangle = 6

Answer 604

Rates question: > draw a visual to help When Buster and Charlie pass each other, is the distance that Buster traveled greater than / equal to / less than the distance Charlie travelled? Let t be time in hours it takes for Buster and Charlie to pass each other, after Charlie started travelling Total distance = (B/3 + B*t) + C*t Is (B/3 + B*t) > C*t? (1) Total d = C*(55/60) > no info about how long it takes B, when they meet (t) etc. NS (2) Logic - since Charlie left after Buster left AND they both arrive at the destinations at the SAME TIME, Charlie has a FASTER SPEED (C > B) Now we have to see whether distance travelled by Buster in the first 20 mins + distance travelled by Buster while Charlie is travelling > distance travelled by Charlie Let T be the time it takes for both to reach their final destinations after passing each other: Charlie must travel C*T = B/3 + B*t and Buster must travel B*T = C*t Since C > B, C*T > B*T, or B/3 + B*t > C*t----> sufficient

Answer 605

Number properties > best strategy is to figure out the POSSIBLE values of X and if there are two values that work, not sufficient > TAKE YOUR TIME DON'T GIVE UP ON TESTING 0 instinct is to solve for x √(x+1)-1 = 2 ----> x = 8 √(x+1)-1 = 3 ----> x = 15 NS (3) √(x+1)-1 = 3 ----> x = 15 (divisible by 3 and 5) -- valid √(x+1)-1 = 4 ----> x = 26 (divisible by 13) -- invalid √(x+1)-1 = 5 ----> x = 35 (divisible by 5 and 7) -- valid OR sub in values of x = 15, 21, 35 NS E

Answer 606

Combinations / Geometry: > First find the total # of possible GROUPINGS of 3 from 9 using combinations > then subtract away non-right angled triangles and lines C9,3 = 84 Less 6 lines Less 2 diagonals Less 2 * 12 single line/one row triangles Less 2 * 4 single line/two rows triangles = 44

Answer 607

Work problem > break it down into steps and understand what is being asked > don't worry about decimals!! keep going and might have to round and interpret answers 1) Daily rate per crew member (rate)*110 days*10 = 1 rate = 1/1100 2) 5 days of rain before day 61 means that first 55 days did not have any rain Work done = 1/1100 * 10 * 55 = 1/2 Work left = 1/2 3) Since the job was done in 100 days in total (including rain days), 40 days are left to do 1/2 work with 16 ppl 40 - time it takes to finish 1/2 work = rain Time it takes to finish 1/2 work: (1/1100) * 16 * days = 1/2 t = 34.3 ---> NEED TO ROUND UP because will need to finish in 35 days Therefore, days of rain after day 60 = 40 = 35 = 5 days

Answer 608

Statistics > understand the problem and DO THE CALCULATION CORRECTLY > notice how the answer choices are VERY CLOSE to each other --> pay attention to what you need (easiest to set up inequality) Total points after 7 games = 61.5*6 + 47 = 369 + 47 = 416 Total number of points to get to ABOVE 500 total points: 416 + x > 500 x > 84 x = 85 (A)

Answer 609

Geometry (volumes) > UNDERSTAND THE PROBLEM FIRST Start with the conversion: 1 m^3 = 1000 L Goal is to make volume = 1,000,000 L So in cubic meters = 1000 m^3 Meaning each SIDE = 10m We have sheets of paper 4 by 2 meters. For each cube's side... > # of sheets needed along length = 2.5 > # of sheets on the along width = 5 TOTAL # of sheets needed PER cube's side = 2.5 * 5 (not addition) = 12.5 sheets Total # of sheets for cube: 12.5 sheets per side * 6 sides = 75 (E) Alternatively, surface area of the cube tank = 100*6 = 600m^2 Each metal sheets has surface area = 8m^2 # of metal sheets needed = 600/8 = 75

Answer 610

Conditional probability > think about decision trees (shows options) Two branches: Ben wins or Ben loses If Ben loses: > Mike wins and Rob loses > Rob wins and Mike loses > Neither Mike nor Rob win P(Mike wins) + P(R wins)? P(Mike wins) = Mike wins AND Rob loses AND Ben loses = (6/7 chance Ben loses) * (1/4 chance Mike wins, given Ben loses) = 3/14 P(R wins) = Rob wins AND Mike loses AND Ben loses = (6/7 chance Ben loses) * (1/3 chance Rob wins, given Ben loses) = 2/7 Therefore: P(Mike wins) + P(R wins) = 3/14 + 2/7 = 7/14 = 1/2

Answer 611

Ratios (Not linear equations) Concept: conversion from part to whole > you don't need to know EXACT number of shares, just the FRACTION F shares = 2/3*(L + A + W) F/(L + A + W) = 2/3 THEREFORE F fraction of TOTAL number of shares = 2/(2 + 3) = 2/5 L = 3/7*(F + A + W) L/Total = 3/10 A = 4/11*(F + L + W) A/Total = 4/15 F + L + A's fraction of total = 2/5 + 3/10 + 4/15 =12/30 + 9/30 + 8/30 = 29/30 So W/T = 1/30 Therefore, Werner received $3.6M * 1/30 = 120,000 C

Answer 612

Geometry > write out the formulas first (i.e., watch out that you need to divide by 2 for the half circle) ans 2/1

Answer 613

Geometry > draw diagram > we need to "fix" vertex B, however, there can be MULTIPLE values of B that yield the SAME AREA (as long as the HEIGHT is the same) A and C form the BASE of the triangle (1) | x | = y = 10 (10, 10) and (-10, 10) are two valid points > same base, same height (y coordinate) SUFFICIENT (2) x = | y | = 10 (10, 10) and (10, -10) > same base, and same height (mirrored y coordinate)

Answer 614

Geometry > can be a CIRCLE (doesn't have to be a square or rectangle > a circle has the minimum possible perimeter for a given area, then in order to minimize the length of a rope it should enclose a circle. RECALL: pi = ~22/7 154 = pi(r^2) 154 = (22/7)*r^2 2*7*11 = (22/7)*r^2 49 = r^2 r = ~7 Circumference = length of a rope = 2*pi*r = 2*(22/7)*(7) 44 ans E

Answer 615

Combinatorics and probability AAABBBCCC n = 9 First think what it logically means to "taste all of the samples" = At least one sample is tasted twice If you DON'T taste all of the samples, it means you MUST taste TWO samples (b/c you taste 4 cups and each sample has max 3 cups) P(does not taste all the samples) _ _ _ _ Option 1) = 1 - P(taste all the samples) = 1 - P(A and B and C and A)*cases = 1 - (3/9 * 3/8 * 3/7 * 2/6)*(3)*(4!) = 5/14 Option 2) Denom = C9,4 Numerator = C3,2 (from 3 samples, choose 2) * C6,4 (for each sample pairing, choose 4 cups out of 6 cups) = (C3,2 * C6,4)/(C9,4) = (3 * 15)/(9*7*2) = 5/14

Answer 616

Geometry (frame question) > Don't forget to subtract the inner square to get the area of the border!! > also notice the special numbers related to squares 25 = 5^2 > 25 + 39 = 64 = 8^2 Step 1) Draw diagram and variables Let x be the length of one side of the entire plaque Let w be the length of the width of the wooden strip Let y be the length of one side of the brass square Area of the brass square = y^2 Area of the entire plaque = x^2 Area of the wooden strip = x^2 - y^2 Step 2) With the squares in mind, calculate the ratio of the area of the brass inlay to the area of the entire plaque = 25/(25+39) = 25/64 = y^2/x^2 therefore y/x = 5/8 or y = 5/8 * x Step 3) calculate an equation for w We also know that x = y + 2w Therefore: x = (5/8)*x + 2w 3/8 * x = 2w w = (3/16)*x Step 4) Check whether the answer choices are valid Can w = 1? Yes --> x = 16/3 Can w = 3? Yes --> x = 16 Can w = 4? Yes --> x = 64/3

Answer 617

Work question > instead of calculating time, figure out # of units produced Machine N's rate: 1/2 per minute In 1 hour, Machine N can produce 1/2*60 = 30 widgets (1) x < 1.5 Machine K's rate < 1/1.5 or 1/3/2 or 2/3 per minute In 1 hour, Machine K produces 2/3*60 = 40 widgets Already Machine K and N produce 30 + 40 = 70 widgets > 50 widgets Sufficient (2) y < 1.2 Machine M's rate < 1/1.2 or 1/6/5 or 5/6 per minute In 1 hour, Machine M produces 5/6*60 = 50 widgets Already Machine M and N produce 30 + 50 = 80 widgets > 50 widgets Sufficient

Answer 618

Geometry > make sure you get the ANGLES correct > test cases if you are absolutely unsure of the rules Isosceles triangle --> two sides are equal and two angles are equal > since one of the angles is already 90 degrees, question becomes whether AB = BC or Yes Isosceles Case 2: No isosceles NS (2)