GMAT Quant

91) How gmat tests divisiblity of certain integers in reverse, and 7

"91) np a) Can be tested in reverse i) A number has a ones digit equal to zero (1) Infer that number divisible by 10 ii) Sum of digits x=21 (1) Divisible by 3 but not 9 iii) Divisibility by 7 (1) Long division"

92) Factors and Multiples basic definitions

"92) np a) Factor i) Positive integer that divides evenly into an integer (1) 1,2,4,8 all factors of 8 (2) Also called divisors b) Multiple i) Integer formed by multiplying that integer by any other integer (1) 8,16,24,32 multiples of 8 ii) On the gmat multiples of an integer are equal or larger than that integer c) An integer is always both a factor and multiple of itself"

20) When can smart numbers be used

20) 4) Can be used only when problems contain unspecified variables a) Easiest examples contain variables, percents, fractions or ratios throughout or provide real numbers for the variables, even throughout

"Dividing decimals 12.42/3=4.14"

"4. 4)If there is a decimal in the dividend only 5. Bring straight up to answer and divide normally 6. 12.42/3=4.14 7. 8. = 4.14 9. If there is a decimal in the divisor a. Shift the decimal point in both the divisor and the dividend and make the divisor a whole number i. Then bring the decimal point up and divide A) 12.42/.3 B) 124.2/3=41.3 10. Simplifying division problems that involve decimals 11. Shift the decimal point in the same direction in both the divisor and dividend"

"40) Zero in the denominator X2+X-12/(x-2)"

"40) 2) Zero in the denominator: undefined a) X2+X-12/(x-2) i) (x-3)(x+4)/(x-2) ii) Solutions are x=3,x=-4 iii) Setting x=2 would make the denominator undefined and is illegal"

43) Positive/negative analysis-what is it, when can it be used and how do I use it?

"43) a) Positive/negative analysis i) Can be used when there is an inequality with 0 on one side of the inequality (1) Xy>0 (a) X and y are both positive or both negative (2) Xy<0 (a) X and y have different signs (one positive one negative) (3) X2-X<0 (a) X2<x, so 0<X<1"

"44) How do inequalities vary from equations Q: Given that 4-3x<10 what is the range of possible values for X?"

"44) a) When you multiply or divide an inequality by a negative number, the inequality sign flips i) Problem 128 (1) Given that 4-3x<10 what is the range of possible values for X? (a) 4-3x<10 (b) -3x<6 (c) X>2 (2) Do not multiply or divide an inequality by a variable unless you know the sign"

62) Rules for expressing rates

"62) wp a) Problem 42 b i) Elevator .25 floors/second (1) ALWAYS EXPRESS DISTANCE/TIME NOT TIME/DISTANCE ii) 2 minutes (1) .25 floors per second (2) Distance is unkown iii) Change time into seconds (1) 2 minutes by 60 sc/min iv) RT=D (1) .25(120) v) Answer=30 floors in 2 mins"

"73) median of sets containing unknown values X,2,5,11,11,12,33 X,2,5,11,12,12,33"

"73) 1) Median of sets containing unknown values a) May be able to determine even if some numbers are unknown bc median depends only on the one or two middle values i) X,2,5,11,11,12,33 (1) Median will always be 11 ii) X,2,5,11,12,12,33 (1) depends on x (a) if x is 11 or less 1) median is 11 (b) x between 11 and 12 1) median is x (c) if x is 12 or more 1) median is 12"

85) When can you extend a double set matrix table

"85) wp 1) Two sets, three choices: still double-set matrix a) Rarely may need to consider more than 2 options for one or both dimensions of chart b) As long as set of distinct options is complete and has no overlaps, you can extend chart i) Can respond yes, no or maybe to survey ii) Respondents are male or female Yes No maybe Total Female Male Total"

98) Prime number definition, unique qualities and first ten primes

"98)np Primes b) Any positive integer with exactly two factors, 1 and itself. i) 1 not prime because it has only 1 factor ii) 2 is first prime and only even prime c) First ten primes i) 2,3,5,7,11,13,17,19,23,29"

"99) Prime Factorization 72 Why is it important"

"99) np 4) Prime factorization a) Split number into two factors b) Repeat process on factors until the tree ends at a prime c) Fig 5.1 i) 72 (1) 6 (a) 2 (b) 3 (2) 12 (a) 2 (b) 2 (c) 3 d) Extremely important i) Once you know prime factors you can determine all the factors of that number, even large numbers. ii) Can be found by building all the possible products of the prime factors"

41) steps to solving combos and what to avoid (general)

"41) 1) Solving for a combination of variables a) Step 1: notice that the question asks for a combo i) Some are directly asked for and some are disguised b) Step 2: manipulate any given information to try to match the combo: i) X+Y/y=3 (1) since you're not solving for a variable, but the combo, you don't want to get rid of the fraction. Instead try getting rid of the denominator (2) x/y+y/y=3 (3) x/y+1=3 (4) x/y=2 (5) sufficient c) statement 2 i) y=4 (1) no information about x so not sufficient d) objective for DS is to find a single match for the desired combo e) avoid the C trap i) ds trap when both statements together appear to work together and in actuality one statement actually works"

"19) what is the formula for compound interest a. Q: A bank account with $200 earns 5% annual interest, compounded annually. If there are no deposits or withdrawals, how much money will the account have after 2 years. iii. How many times will interest compound b. If you have 8% annually compounded quarterly"

"19) a. Total amount=P(1+(r/n))^nt i. P=principal ii. R= rate (in decimal form) iii. N=number of times per year iv. T= years a. A bank account with $200 earns 5% annual interest, compounded annually. If there are no deposits or withdrawals, how much money will the account have after 2 years. iii. How many times will interest compound A) Once per year for 2 years=2 B) New value of account=105% of 200 a. 10%=20/2=10 C) At end of year one account will have 210 a. 10% of 210=21/2=10.5 b. 220.5 after 2 years b. If you have 8% annually compounded quarterly iv. 2% per quarter need 8 calcs c. CAN YOU DO (1/4)^8 AND MULTIPLY BY P"

23) When should I estimate?

"23) 1) When to estimate a) When a question asks for an approximation b) When the answers are far apart c) when the answer covers "divided" characteristics i) if there are both pos or neg answers or fractions over and less than 1, you can eliminate answers and boost your chances at guessing right"

24) shortcuts for counting decimals

"24) 1) Number of decimal places in ac cubed decimal is 3x the number in the original 2) Number of decimal in a cubed root is 1/3 the number in the original a) (exponent is answers decimal places*original decimal places)"

"25) decimals and exponents- how to do higher powers .5^4=? .000027^1/3=?"

"25) a) To square or cube a decimal you can do it normally but to raise it to a higher power you can rewrite the decimal as a product of an integer and power of 10 then apply the exponent i) .5^4=? ii) .5=5*10^-1 iii) (5*10^-1)^4=5^4*10^-4 iv) 5^4=25^2=625 v) 625*10^-4=.0625 b) Solve for roots of decimals the same way i) .000027^1/3=? ii) .000027=27*10^-6 iii) .000027^1/3=(27*10^-6)^1/3 iv) 27^1/3*10^-6^1/3=27^1/3*10^-2 v) 27^1/3=3 vi) 3*10^-2=.03"

"26) repeating decimal rules 1/9="

"26) a) Dividing an integer by another integer yields a decimal that either terminates or repeats i) Use patters ie 1/9=.111 (1) 2/9=.2222 b) If the number is close to a power of 10-1 (9,99,999) the denominator tells you the repeating digits i) 23/99=.23232323 ii) 3/11=27/99=.2727"

42) Domain and range of functions

"42) (a) Domain of a function represents possible inputs (b) Range of a function possible outputs 1) F(x)=x2. Can take any input but never produce a negative number so the domain is all numbers, but the range is f(x)=>0"

28) When asked for the units digit or last digit

"28) 3) The last digit shortcut a) Sometimes asked to find a units number or a remainder after division by 10 i) What is the untis digit of 7^2*9^2*3^3 (1) I think its 3 ii) Last digit shortcut (1) To find the units digit of a product or a sum of integers only pay attention to the units digit of the numbers, drop all else (a) 7*7=49 (b) 9*9=81 (c) 3*3*3=27 (d) 9*1*7=63"

30) 4 rules to simplify expressions

"30) 1) Simplifying expressions a) Combine like terms i) 6z+5z=11z b) Find a common denominator i) 1/12+(3x^3)/4*3/3=1/12+(9x^3)/12=9x^3+1/12 c) Pull out a common factor i) 2ab+4ab=2b(a+2) d) Cancel common factors i) 5y^3/25y=y^2/5"

31) 6 operations that can be done to both sides of an equation

"31) 2) 6 operations that can be performed on both sides of an equation a) Must be performed on entire equation i) Add the same thing to both sides ii) Subtract the same thing from both sides iii) Multiply both sides by the same thing iv) Divide both sides by the same thing v) Raise both sides to the same power (1) √y=y+2 (2) (√y)2=(y+2)2 vi) Take the same root of both sides (1) X3=125 (2) 3√x3=3√125 (3) X=5"

32) How do I solve system of equations by combination

"32) 1) Either coefficient in front of one of the variables ( say x) is the same in both equations a) Subtract one equation from the other 2) The coefficient in front of one of the variables is the same but with opposite sign a) Add the two equations b) Accomplished by multiplying one of the equations by some number"

33) When does a negative sign when working with exponents

"33) i) Pay particular attention to PEMDAS ii) Unless the negative sign is inside the parentheses the exponent does not distribute (1) -24 (2) =-1*24=-16"

34) How many roots should I use with square roots and any even root

"34) 1) A square root has only one value a) Use only positive value for square roots if the square root symbol is provided b) If an equation contains a squared variable and I take the square root, use positive and negative Applies to any even root"

35) How to read fractional exponents

"35) 2) Roots and fractional exponents a) Link between roots and exponents b) Numerator i) What power to raise the base to c) Denominator i) Which root to take d) What is (1/8)(4/3) i) (1/8)(4/3)=8(4/3)=(3√8)4=24=16"

"5) Test Cases Q: steps and proving validity"

"1) 5)Step 1: what possible cases are allowed a) What restrictions have been placed on the basic problem 2) Step 2: choose numbers that work for the statement a) Only choose numbers that make the statement true 3) Step 3: try to prove the statement insufficient (ie multiple answers) a) Value i) Sufficient: single numerical answer ii) Not sufficient: two or more possible answers b) Yes/no i) Sufficient: always yes or always no ii) Not sufficient: maybe or sometimes yes, sometimes no"

"multiplying decimals q:.02*1.4"

"1. 2)Multiplication and division a. Multiplication i. Ignore the decimal point until the end just multiply normally and count the number of digits to the right of the decimal point in the starting numbers. The product should have the same number of digits the right of the decimal place A) .02*1.4 a. Count digits to the right of the decimal i. 3 (.02=2+1.4=1) b. Multiply normally i. 14*2=28 c. Move decimal 3 places left i. .028"

"100) Prime Boxes Q: Given that the integer n is divisible by 8 and 15 is n divisible by 12"

"100) np-problem 5) The Prime box a) Easiest way to work with factor foundation rule b) Can make any factor of a number with the larger numbers prime factors i) 72 prime factors are 2,2,2,3,3 (1) Is 27 a factor of 72 (a) 3x3x3 (b) 72 only has 2 3's so it is not c) Problem 21 i) Given that the integer n is divisible by 8 and 15 is n divisible by 12 (1) Factor both numbers (a) 8=2 x 2 x2 (b) 15= 3 x 5 (c) Because 12=2x2x3 then yes"

"14) How do I compare fractions 7/9,4/5"

"14) 1. Cross multiply a. The numerator of one fraction by the denominator of another fraction and vice versa b. 7/9,4/5 i. 7*5=35 ii. 4*9=36 iii. Since 35 is less than 36, 7/9<4/5"

"15) What are the rules for splitting complex fractions 1.53 15/5+10/5"

"15) a. Rules iv. You can split the numerator A) 15/5+10/5 B) Never split the denominator a. Simplify i. 15+10/7=25/7"

"17) How to translate percent questions; percent, of, is, what What is 70 percent of 120"

"17) 1. Translating percent questions a. Percent=/100 b. Of=multiply c. Is=equals d. What=uknown value 2. What is 70 percent of 120 a. What=x b. Is=equals c. 70=70 d. Percent=/100 e. Of=x f. 120=120 x = 70 /100 x 120 What Is 70 Percent Of 120"

"37) How to simplify roots Q: √25*16,ii) √144/16"

"37) a) When multiplying roots you can split up a larger product into its separate factors. b) 2 separate radicals and solve for each individually before multiplying can prevent having to compute large numbers i) √25*16=√25*√16=5*4=20 ii) √144/16=√144/√16=12/4=3"

93) Factor Pairs

"93) a) Factor pairs i) Easy way to find all factors of small integers ii) Pairs of factors that when multiplied together yield that integer (1) 8 (a) 1,8 (b) 2,4"

"18) What is the formula for percent change and decrease in percent Q: The price of a cup of coffee increased from 80 cents to 84 cents by what percent did the price change Q: If the price of a $30 shirt is decreased by 20% what is the final price of the shirt?"

"18) a. The price of a cup of coffee increased from 80 cents to 84 cents by what percent did the price change i. Percent change=change in value/original value A) x/100=4/80=1/20 B) 20x=100 C) X=5 ii. If the price of a $30 shirt is decreased by 20% what is the final price of the shirt? A) New percent=new value/original value a. Decreased by 20%=80% b. 80/100=x/30 c. 4/5=x/30 d. Take out 5's from denominator e. 4/1=x/6 f. 24=x g. Alt: find 20% of 30 and subtract"

"multiplying decimals-large and small numbers .02*1.4"

"2. 3)-large and small numbers adjust the numbers 3. Multiplication and division a. Multiplication i. Ignore the decimal point until the end just multiply normally and count the number of digits to the right of the decimal point in the starting numbers. The product should have the same number of digits the right of the decimal place A) .02*1.4 a. Count digits to the right of the decimal i. 3 (.02=2+1.4=1) b. Multiply normally i. 14*2=28 c. Move decimal 3 places left i. .028"

"38) When can you not combine or separate terms of roots √(16+9)"

"38) a) you can never separate or combine the sum or difference of roots. Only the products and quotients i) √16+9 does not equal √16+√9 ii) =√25"

"39) How do I factor quadratics? 1) X2+3x+8=12"

"39) 1) Factoring quadratics a) X2+3x+8=12 i) X2+3x-4=0 (1) Product is -4 (2) Sum is (a) X+4=0 1) X=-4 (b) X-1=0 X=1"

"45) How do I combine inequalities - Q: If x>8,<17 and x+5<19, what is the range of possible values for x?"

"45) 2) Combining inequalities: line em up a) May need to convert to compound inequality when there are multiple inequalities i) Series of inequalities strung together such as 2<3<4 ii) First line up the variables then combine b) Problem 129 i) If x>8,<17 and x+5<19, what is the range of possible values for x? (1) First solve any inequalities that need to be solved (only the last in this inequality (a) X+5<19 1) X<14 (2) Simplify the inequalities so the inequality symbols all point the same direction then line up the common variables in the inequalities (a) 8<X (b) X<17 (c) X<14 (3) Finally put the information together (a) Notice that x<14 is more limiting than x<17 1) (whenever X<14 x will always be less than 17 but not vice versa) (4) The range is 8<x<14 (a) Discard the less limiting inequality"

46) Rules for manipulating inequalities

"46) a) Adding inequalities is powerful i) Never subtract ii) Never divide iii) Only multiply if all values are positive"

"47) Solving factored expressions involving roots What is value of (√8-√3)( √8+√3)"

"47) 1) Foil with square roots a) Solving factored expressions involving roots i) Problem 160 (1) What is value of (√8-√3)( √8+√3) (a) Can be solved like a quadratic 1) First 1. √8 x √8=8 2) Outer 1. √8 x √3=√24 3) Inner 1. √8 x (-√3)= -√24 4) Last 1. (√3)(- √3)=-3 (b) 4 terms are 8+√24-√24-3 1) Simplified this=5"

48) Quadratic Formula

"48) 2) Quadratic formula a) Ax2+bx+c=0 i) b) X2+8x+13 i) ii) iii) iv)"

"49) solve with quadratic : Q) X2+8x+13"

"49) X2+8x+13 v) vi) vii) viii)"

"50) Information conveyed by the discriminant Which of the following has no solutions for x (1) X2-8x-11=0 (2) X2+8x+11 (3) X2+7x+11 (4) X2-6x+11 (5) X2-6x-11"

"50)Info conveyed by the discriminant (b2-4ac) i) If greater than 0 (1) 2 solutions ii) equal to 0 (1) one solution iii) less than 0 c) no solutions Question 161 i) Which of the following has no solutions for x (1) X2-8x-11=0 (2) X2+8x+11 (3) X2+7x+11 (4) X2-6x+11 (5) X2-6x-11 (a) None factorable (b) Evaluate answers 1) -4(1)(-11)=64+44=108 1. Positive plus positive so can stop calc early 2) 64-44=20 3) 49-44=5 4) 36-44=-8 5) 36+44=80 ii) Correct answer is D iii) Vast majority of quadratics can be done normally"

"51) Using conjugates to rationalize denominators Simplify 4/√2"

"51) 3) Using conjugates to rationalize denominators a) When problems contain sq roots in the denominator you can simplify by multiplying the numerator and denominator by sq root i) Problem 162 (1) Simplify 4/√2 (a) Remove root by multiplying numerator and denominator by square root 1) 4/√2(√2/√2) 2) 4√2/2=2/√2 b) Simplifying when the denominator contains sum or difference of a square root is more complicated i) Problem 162 b (1) Simplify 4/(3-√2) (a) Find conjugate 1) Conjugate expression defined 1. A+√b a. Conjugate=a-√b 2. A-√b a. Conjugate= a+√b 3. Change the sign of the second term (b) Multiply the numerator and denominator by conjugate 1) 4/(3-√2) x ((3+√2)/(3+√2)) 2) 4(3+√2)/(3-√2)(3+√2) 3) 12+4√2/9+3√2-3√2-2 4) 12+4√2/7 52) a) Some sequences easier to look at as patterns rather than rules i) If Sn=3n, what is the units digit of S65 (1) Cant multiply to 65 so must be a pattern (a) 3^1=3 (b) 3^2=9 (c) 3^3=27 (d) 3^4=1 (e) 3^5=243 (f) 3^6=729 (g) 3^7=2,187 (h) 3^8=6,561 (2) Units digit repeats (a) Multiple of 4 always 1 so that can anchor (b) 64 is a multiple of 4 1) Answer will be 3 since that always follows"

55) inverse proportionality

"55) 1) inverse proportionality a) change by reciprocal factors. i) Cutting the input in half doubles the output (1) Y=k/x 1. X=input value y is output and k is proportionality 2. Yx=k ii) Typically given before and after scenarios (1) Set up products not ratios to solve 1. Y1x1=y2x2"

56) relating AV to number lines

"56) 1) Positives and negatives a) Numbers can be either negative or positive except 0 (neither) i) Negatives to the left of 0 on number line ii) Positives to right of zero on number line b) A variable can have either a positive or negative value unless there is evidence otherwise 2) Absolute value: absolutely positive a) Absolute value answers the question: how far away is the number from 0 on the number line? i) I 5 I =5 ii) I -5 I =5 iii) Units from zero of both = 5 b) If x = -y i) X 0 y ii) Y 0 x (1) Cant tell which because you cannot tell which variable is positive or negative without more info"

"57) Rules for squaring inequalities Is x2>y2"

"57) 1) Need to know both sides' sign a) If both are negative: flip the sign i) X<-3 (1) Since both sides must be negative, square both sides (a) X2>9 (2) With x>-3 it is unclear (a) If x is negative x2<9 but if x is positive then x2 could be either greater than or less than 9 b) If both sides are positive do not flip inequality sign when you square c) If one side negative and the other positive, you cannot square i) X<y and x is negative y is positive you cannot square d) If signs unclear you cannot square 2) Problem 190 a) Is x2>y2 i) X>y ii) x>0 (1) both insufficient because y could be either positive or negative (2) correct"

58) story problem steps

"58) wp 1) Steps a) Step 1: glance read jot i) Is it problem solving or ds ii) Do the answers or statements give you any clues (1) Variables in the answers may lead you to choose smart numbers b) Step 2: reflect, organize, translate i) Turn the story into math c) Step 3: work, solve"

59) common unit relationships

"59) wp 2) Common relationships a) The following will assume to be mastered i) Total cost $=unit price ($/unit) x quantity (units) ii) Profit($)=revenue-cost iii) Total earnings ($)=wage rate ($/hour) X hours worked (hours) iv) Miles=miles per hour x hours v) Miles= miles per gallon x gallon"

"60) conversion factors Q: A certain med requires 4 doses/day if each dose is 150 g how many mg will a person have taken at the end of the third day"

"60) wp i) Conversion factors are fractions whose number and denominator have different units but same values (1) How many seconds are in 7 minutes? (a) 60 seconds/1 minute is conversion factor (b) 60 seconds/1 minute*7 minutes=420 seconds (c) Cancel minutes leaving you with seconds ii) Question 21 (1) A certain med requires 4 doses/day if each dose is 150 g how many mg will a person have taken at the end of the third day (a) Givens 1) 4 doses/day 2) 1 dose =150 mg (b) Combine 1) 3 days * (4 doses/1day)*(150 mg/1dose)=1800 mg"

61) primary components of rates and work

"61) wp 1) Primary components a) Rate time and distance (work) i) Rate x time =distance ii) Rate x time = work (1) RT=D (2) RT=W"

"63) Relative Rate problems 1) 2 ppl 14 miles apart a) Walk closer 2) A walks 3 mph 3) B walks 4 mph 4) When do they meet"

"63) wp 2) Relative rates a) 3 scenarios i) The bodies move towards each other (1) Two people decrease the distance between themselves at a rate of 5+6=11 mph ii) The bodies move away from each other (1) Two cars increase the distance between themselves at a rate of 30+45=75 mph iii) The bodies move in the same direction on the same path (1) Persons x and y decrease the distance between themselves at a rate of 8-5=3mph b) Can be dangerous because they can take a long time to solve w conventional strategies i) Crate a third RT*D equation for which the distance between the bodies changes c) Problem 45 i) 2 ppl 14 miles apart (1) Walk closer ii) A walks 3 mph iii) B walks 4 mph iv) When do they meet R x T = D A 3 X T = D B 4 x t = 14-d Alt A+B 7 x t = 14 (1) 7t =14 (2) T=2 v) Alternatively draw it out and keep track of distance at each hr"

"64) DS with average rates (relative to inputs) Average rate: find the total time i) Lucy walks 4 mph there ii) 6 mph back"

"64wp-)Average rate: find the total time d) Problem 46 i) Lucy walks 4 mph there ii) 6 mph back iii) What is avg wlk rate (1) D= same iv) Cannot take 6+4/2 (1) If an object moves the same distance twice but at dif rates the avg rate will never be the avg of the two rates for the given journey bc the object spends more time traveling slower it will always be closer to the slower number. On ds that may be enough to answer v) use total combined times (1) avg speed=total distance/total time (2) pick a number for d (a) 12 works here as it's the lcm of the two rates rate x time = Distance Going 4 X 3 = 12 Return 6 X 4 = 12 total ? X 5 = 24 (3) R(5)=24=4.8"

"65) expressing basic work problems 3/7 of a room in 4.5 hrs How long to paint entire room"

"65)wp 3) basic work problems a) concerned with work instead of distances i) RT=W ii) R=w/t b) Express work per time not time per work c) Problem 48 i) M (1) 3/7 of a room in 4.5 hrs (2) How long to paint entire room ii) R (1) R iii) T (1) 9/2 iv) W (rooms) (1) (3/7) v) R=9/2=3/7 (1) 3/7*2/9=2/21 vi) 2/21/hr vii) R (1) 2/21 viii) T (1) T ix) W(rooms) (1) 1 x) 21/2t=1 xi) 21/2=10.5"

"66) What are overlapping sets and what are the categories? Q:Of 30 integers, 15 are in set A 22 in set B and 8 are in both sets A and B. How many of the integers are in neither a or B?"

"66) wp 1) Translation problems that involve two or more given sets of data that partially intersect with each other a) Problem 69. Nice i) Of 30 integers, 15 are in set A 22 in set B and 8 are in both sets A and B. How many of the integers are in neither a or B? (1) 2 sets (a) A (b) B (2) 4 categories (a) Numbers in A (b) Numbers in B (c) Numbers in A and B (d) Numbers in neither A nor B Find values for 4 categories"

"67) Overlapping sets and percents 70% of guests at co x are employees of co x 10% of the guests are women who are not employees of co. x If half the guests at the party are men, what percent of the guests are female employees of co. x"

"67) wp 2) Overlapping sets and percents a) Effective especially when choosing a smart number for the total i) Problems involving percents choose 100 ii) For fractions choose a common denominator b) Problem 70 i) 70% of guests at co x are employees of co x ii) 10% of the guests are women who are not employees of co. x iii) If half the guests at the party are men, what percent of the guests are female employees of co. x Men women Totoal Empty 70 Not empty 10 Total 50 50 100 (1) Calculate only what you need to answer the question (a) Female employees (b) Calculate women+ employee 1) 50-10=40 2) Percent of female employees =40/100=40% Men women Totoal Empl 30 40 70 Not emply 20 10 30 Total 50 50 100 (c) The last box filled in must work both horizontally and vertically (d) Can only assign fractions/smart numbers in no actual amount of ppl are given, only all fractions/percents"

"68) Overlapping sets and algebraic expression 10% of children est bet 8 and 18 and dislike soccer 50% of children who like soccer are between 8 and 18 If 40% of children in the world are between 8 and 18 what percentage of children are under age 8 and dislike soccer"

"68) 4) Overlapping sets and algebraic representation a) Pay careful attention to wording b) Problem 71 i) 10% of children est bet 8 and 18 and dislike soccer ii) 50% of children who like soccer are between 8 and 18 iii) If 40% of children in the world are between 8 and 18 what percentage of children are under age 8 and dislike soccer (1) Tempting to fill in the number 50 to represent percent of children who like soccer (a) Incorrect (2) 50=percent who like soccer who are between 8 and 18 (a) Fraction of unknown (b) Let x= number of children who like soccer 1) .5x 8 to 18 <8 Total Like .5x X Dislike 10 Total 40 100 1) .5x+10=40 2) X=60 8 to 18 <8 Total Like .5x=30 30 X=60 Dislike 10 30 40 Total 40 60 100 3) 30% of children under 8 dislike soccer"

"70) Using 2 average formulas Q: Sam earned 2000 commission on a big sale, raising his average commission by $100. If Sam's new avg commission is $900 how many sales has he made"

"70) wp-Using two average formulas i) Problem 80 c (1) Sam earned 2000 commission on a big sale, raising his average commission by $100. If Sam's new avg commission is $900 how many sales has he made (a) Set up a table 1) New avg is $900 1. Up by $100 2. Old avg was $800 3. Number and sum columns add up to give new cumulative values but the values in the average column do not add up Average X number =sum Old total 800 N 800n This sale 2000 1 2,000 New total 900 N+1 900 (n+1) 4. Right handed column gives correct equation a. 800 n+2000=900(n+1) b. 800n+2,000=900n+900 c. 1,100=100n d. N=11 e. Since you are looking for the new sales, which is n+1, you have made 12 sales"

"71) If average is known, if average is unknown The sum of 6 numbers is 90, what is the average term If the average of the set [2,5,5,7,8,9,x is 6.1 what is the value for x"

"71) wp 2) Using the average formula a) In general i) If avg unknown (1) A=s/n ii) If avg known (1) A x n= s b) When you see avg problem write formula i) Problem 80 (1) The sum of 6 numbers is 90, what is the average term (a) A=s/n 1) 90/6=15 ii) Problem 80 b (1) If the average of the set [2,5,5,7,8,9,x is 6.1 what is the value for x (2) A x n=s (a) 6.1 (7 terms)=2+5+5+7+8+9+x 1) 42.7=36+x 2) 6.7=x"

72) median definition and calculation

"72) 3) Median: the middle number a) Calculated one of two ways depending on number of data poinds i) Odd number of data pts (1) Unique middle value when data sets are arranged incr or decr order ii) Even number of values (1) Arithmetic avg of the two middle values when data are arranged in incr or decr order"

"74) Standard Deviation 5,5,5,5 2,4,6,8 0,0,10,10"

"74) wp 4) standard deviation a) spread or variation of data b) how far from average the data points typically fall i) small sd indicates set closely clusters around avg ii) large sd indicates that data are spread out widely with some points appearing far from mean c) consider the following sets i) 5,5,5,5 (1) Avg spread = 0 (2) sd=0 ii) 2,4,6,8 (1) As=2 (2) Sd=moderate (√5=2.24) iii) 0,0,10,10 (1) As=5 (2) Sd= large (technically sd=5) (a) Every every absolute difference from the mean is equal than the sd equals that difference (3) Same mean (4) Very different sets iv) Do not have to calculate sd unless there is a shortcut (sd=0) v) Average spread variation (1) Changes in sd when a set is transformed (2) Comparison of sd of two or more sets (3) More spread out the larger the sd"

"75) When asked about sd, ask myself: e) Problems 83 i) Which set has greater sd (1) 1,2,3,4,6 (2) 441,442,443,444,445 ii) If each data point in a set is increased by 7, what does the set's sd do? iii) If each data point is increased by a factor of 7, doe sthe set's sd incr, decr or remain constant."

"75) wp-problem a) When problems ask about changes in the sd ask i) Do changes move data (1) closer to the mean (2) Farther from the mean (3) Neither b) Problems 83 i) Which set has greater sd (1) 1,2,3,4,6 (2) 441,442,443,444,445 ii) If each data point in a set is increased by 7, what does the set's sd do? iii) If each data point is increased by a factor of 7, doe sthe set's sd incr, decr or remain constant. iv) Answers (1) First has greater sd (a) Observe gaps between numbers are on avg sligtly bigger than second set (b) Th sd will not change 1) Increased by 7 means 7 is added to each data point 2) Will not affect the gaps between data points and thus not affect how far dat pts are from mean (c) the sd will increase 1) make all the gaps between points 7 x as big as they originaly were so each point falls 7 pts further from mean. Sd will incr by factor of 7. Specified diff numbers bc sd would be zero if they were the same"

"76) Weighted average algebraic method Final exam 60% or 3/5 weight (1) Score=100% Midterm=40% or 2/5 weight (2) Score=80%"

"76) wp 1) The algebraic method a) Multiply each score by its weight i) Final exam 60% or 3/5 weight (1) Score=100% ii) Midterm=40% or 2/5 weight (1) Score=80% iii) 100*3/5+80*(2/5) (1) 60+32=92 iv) Weighted avg=92 b) How it is calculated/works i) Think of the average formula as two equally weighted items (1) 100(1/2)+80*(1/2)=(100+80)/2=50+40=90 ii) Weighted average=component 1*weighting 1+component 2*weighting 2"

"77) weighted average teeter totter 100% on final 80% on midterm final worth 60%"

"77) wp 2) The teeter totter method a) 100% on final 80% on midterm final worth 60% i) If you did not have a weighted average the teeter totter would perfectly balance at 90, halfway between 100 and 80 ii) The weighted average "slips" toward the heavier side (1) Can infer it is closer to 100 than 80 (2) Enough for some DS b) To calculate exact i) The two ends are 100 and 80 ii) The difference is 20 iii) The final exam is responsible for 60% of that length of 20 (1) 60% of 20=12 (a) Start counting from higher (lighter side) of teeter totter (b) Imagine final exam weighs down his side by 12 (c) Therefore the average is 80+12=92 c) Don't have to draw full teeter totter but do draw sloped line d) Know which side is heavier and calculate "length" i) Calculate value of heavy line ii) Start counting from smaller side iii) If the situation were reversed and the midterm was worth 60%, you would subtract at last step iv) 12 units from the lighter side (1) 100-12=88"

78) evenly spaced sets, consecutive multiples, consecutive integers

"78) wp 1) Evenly spaced sets a) The values go up or down by the same amount (increment) i) 4,7,10,13,16 b) Consecutive multiples i) Special cases of evenly spaced sets (1) All values are multiples of the same increment (a) 12,16,20,24 (b) Must be composed of integers c) Consecutive integers i) All values increase by 1 and all values are multiples of 1 (1) 12,13,14"

16) Converting decimals and percents

16) 2. Convert percents to decimals by moving two decimal spaces left 3. Decimals can be converted to percent by moving the decimal point two spaces right (the percent is always 'bigger' than the decimal

79) counting integers: add one-how to count extremes

"79) wp-problem 2) Counting integers: add 1 before you are done a) How many integers are from 6-10 i) 1 right? Wrong ii) 5 (1) 6,7,8,9,10 iii) Both extremes must be counted iv) formula (1) Last-first+1 b) Counting extremes i) Subtract and add one ii) Integers 6-10 iii) 10-6=4+1=5 c) Problem 108 i) How many integers are there from 14 to 765 inclusive ii) Last-first+1 iii) 765-14+1=752"

"80) finding integers dealing with consecutive multiples (a) How many multiples of 7 are between 100 and 150"

"80) wp problem a) Dealing with consecutive multiples i) If you subtract largest from smallest and add one you overcount (1) All even integers between 12 and 24 inclusive=7 but this will yield 13 (2) ((last-first)/increment)+1 (3) The bigger the increment the smaller the result (4) Problem 108 b (a) How many multiples of 7 are between 100 and 150 1) 105 is first number 2) 147 is last number 3) Last-first /increment+1 1. (147-105)/7+1 a. 6+1=7"

"81) Properties of evenly spaced sets What is arith mean of 4,8,12,16,20 What is arith mean of 4,8,12,16,20,24 What is the arith mean of 4,8,12,16 and 20 What Is arithmetic mean of 4,8,12,16,20,24"

"81)wp problem 3) Properties of evenly spaced sets a) Arithmetic mean and median are equal to eachother i) Problem 109 (1) What is arith mean of 4,8,12,16,20 (a) Media is 12. (b) Evenly spaced set so mean is also 12 ii) Problem 109 b (1) What is arith mean of 4,8,12,16,20,24 (a) Mean of 2 middle numbers so 14 b) The mean and median of the set equal to the average of the first and last terms i) Problem 109 c (1) What is the arith mean of 4,8,12,16 and 20 (a) 20+4/2=12 ii) Problem 109 d (1) What Is arithmetic mean of 4,8,12,16,20,24 (a) 24+4/2=14 c) Formula=For all evenly spaced sets avg=(first+last)/2"

"82) summing consecutive integers What is the sum of 20 to 100 inclusive"

"82) wp-problem 4) The sum of consecutive integers a) Problem 109 i) What is the sum of 20 to 100 inclusive (1) Formula for sum of an evenly spaced set (a) Sum=average x number of terms (b) Average first and last term to get average 1) 100+20=120 and 120/2=60 (c) count number of terms 1) 100-20=80+1=81 (d) Multiply average by number of terms 60*81=4,860"

"84) What are hidden integer constraints and where should I look for them Q: Enough spaces on a school team to select at most 1/3 of 50 students What is greatest number of students could be rejected while still filling al available spaces"

"84) wp-problem 1) Maximizing and minimizing a) Problem 122 i) Enough spaces on a school team to select at most 1/3 of 50 students ii) What is greatest number of studnts could be rejected while still filling al available spaces (1) At most 50/3 or 16 2/3 can be accepted. (a) I should be stronger on why this is correct math 1/3*50 makes sense it just didn't make sense immediately to write it as this (2) Cant accept 2/3 of a student (3) 16 is most (4) 50-16=34 iii) Has a HIDDEN INTEGER CONSTRAINT (1) Many word problems have similar constraints"

"86) Venn Diagram-3 overlappigng sets: describe how to solve and things to watch for Workers grouped by area of expertise placed on at least 1 team 20 on marketing 30 on sales 40 on vision 5 on both marketing and sales 6 on sales and vison 9 on both the marketing and vision 4 on all 3 How many workers are there in total"

"86) wp-propblem 2) 3 set problems: venn diagrams a) 3 overlapping sets i) Prominent with teams/clubs etc b) Problem 134 i) Workers grouped by area of expertise placed on at least 1 team (1) 20 on marketing (2) 30 on sales (3) 40 on vision (4) 5 on both marketing and sales (5) 6 on sales and vison (6) 9 on both the marketing and vision (7) 4 on all 3 (8) How many workers are there in total ii) Draw venn diagram w 3 overlapping circles (1) 7 sections (a) A= all 3 overlap (b) B=sales and vision (c) C=vision and marketing (d) D=sales and marketing (e) E=vision (f) F=marketing (g) G=sales (h) App 3.10 (2) Begin by filling in workers on all teams (a) A=4 (3) Next fill in workers on two teams (a) Remember to subtract workers who are on all 3 teams (b) B=6-4=2 (c) C=9-4=5 (d) D=5-4=1 (4) Last workers on one team only (a) Must subtract workers who are on two or 3 teams 1) 20 on marketing includes 1. 1 worker who is on sales and marketing 2. 5 who are on marketing and vision 3. 4 who are on all 3 (b) E=40-2-5-4=29 (c) F=20-1-5-4=10 (d) G=30-1-2-4=23 (5) Add all 7 numbers together 19+5+4+2+1+23+10=74"

87) Factor Foundation rule

"87) wp a) Factor foundation rule i) Every number is divisible by all the factors of its factors ii) If there is always a multiple of 3 in a set of 3 consecutive integers, the product of 3 consecutive integers will always be divisible by 3 iii) There will also always be a multiple of 2 iv) Therefore, the product of 3 consecutive integers will always be divisible by 3! (1) 3x2x1=6 v) The same logic applies to sets of 4 consecutive integers, 5 etc"

"88) sums of consecutive integers and divisibility Find the sum of 5 consec integers Is k^2 odd-review (rush)"

"88)wp -problem 3) Sums of consecutive integers and divisibility a) Find the sum of 5 consec integers i) 4+5+6+7+8=30 ii) 13+14+14+16+17=75 (1) Both the sums are divisible by 5 b) For any set of consecutive integers with odd number of items, the sum of all the integers is ALWAYS a multiple of the number of items. i) The sum equals the average multiplied by number of items c) Find sum of 4 consec integers i) 1+2+3+4=10 ii) 8+9+10+11=38 (1) Neither is divisible by 4 iii) For any set of integers with even number of items, the sum of the items is never a multiple of the number of items (1) The sum never equals the average multiplied by number of items iv) Problem 136 (1) Is k^2 odd-review (rush) (a) K-1 is divisible by 2 (b) The sum of k integers is divisible by k 1) Statement 1 tells you k-1 is even k is odd k^2 will be odd 1. Sufficient 2) Sum of k integes divisible by k sum of divided by k is an integer. Sum of k variables/k=average of k 3) Sum of kintegers/k=avg of k integers=integer 1. Sufficient 4) Correct answer is d"

89) Divisibility rules: 2,3,4,5

"89) np a) Divisibility rules i) Divisible by 2 (1) If the integer is even ii) Divisible by 3 (1) If the sum of the integers digits is divisible by 3 (a) 72 1) 7+2=9 iii) Divisible by 4 (1) If the integer is divisible by 2 twice or if the last two digits are divisible by 4 (a) 28/2=14/2=7 (b) 23,456 1) 56 divisible by 4 iv) Divisible by 5 (1) If the integer ends in 0 or 5"

90) divisibility rules: 6, 8,9,10

"90) np i) 6 if the integer is divisible by both 2 and 3 (1) 48 (a) Divisible by 2 since it is even (b) Divisible by 3 since 4+8=12/3=4 ii) Divisible by 8 (1) Divisible by 2 three times of if the last three digits divisible by 8 (a) 32 bc divide by 2 3 x and get an integer 16 8 4 (b) 23,456 1) 456 divisible by 8 iii) Divisible by 9 (1) Sum of the integers digits is divisible by 9 (a) 4185 1) 4+1+8+5=18 iv) Divisible by 10 (1) If the integer ends in a zero"

94) step by step factor pairs: 72

"94)np i) Step by step for factor pairs of 72 (1) Make a table with two columns labeled small and large (2) Start with 1 in the small column and 72 in large column (3) Test the next possible factor of 72 which is 2, (is 2 a factor of 72) divide 72 by 2 to find the factor pair:36 write 36 in large column (4) Repeat until numbers in small and large column run into each other small Large 1 72 2 36 3 24 4 18 6 12 8 9"

95) Few factors more multiples

"95) np 2) Fewer Factors, more multiples a) Mnemonic can help you remember the difference i) Every integer has a limited number of factors (1) Factors divide the into the integer and therefore less than or equal to the integer (a) Only 4 factors of 8 ii) Every positive integer has infinite multiples (1) Multiply out and are there fore greater than or equal the integer (a) 8,16,24,32,40 iii) Closely related (1) 3 is a factor of 12 means 12 is a multiple of 3 and that 12 is divisible by 3"

96) gmat divisibility terminology

"96) np a) On the gmat the terminology is used interchangeably to make problem seem harder than it actually is i) Try to convert all divisibility statements to same terminology (1) Gmat terminology for division (a) 12 is divisible by 3 (b) 12 is a multiple of 3 (c) 12/3 is an integer (d) 12 is equal to3n when n is an integer (e) 12 items can be shared among people so that each person has the same number of items (f) 3 is a divisor of 12 or 3 is a factor of 12 (g) 3 divides 12 (h) 12/3 yields a remainder of 0 (i) 3 goes into 12 evenly"

97) Divisibility and addition/subtraction

"97) np 3) Divisibility and addition/subtraction a) if you add or subtract multiples of N you will get another multiple of N i) 35+21 (multiples of 7)=56 ii) 35-21=14"

"1) Multiplying and dividing negative exponents Q: 39742*10^3= 89.507*10= 4,192.2/10^2= 89.507/10="

"a) 1) Patterns with powers of 10 In words Thousands Hundreds Tens Ones Tenths Hundredths Thousandths In Numbers 1,000 100 10 1 .1 .01 .001 In powers of ten 10^3 10^2 10^1 10^0 10^-1 10^-2 10^-3 b) When you mult by a positive power of ten move dec right that number of places i) 39742*10^3=3,974.2 ii) 89.507*10=895.07 c) When you divide by a power of 10 move to the left the specified number of places i) 4,192.2/10^2=41.692 ii) 89.507/10=8.9507"

83) general facts about sums and averages of consecutive integers

"a) 83) wp b) General facts about sums and averages of consecutive integers i) The average of an odd number of consecutive integers (1,2,3,4,5) will always be an integer because middle number will always be a single integer ii) Average of an even number of consec integers will never be an integer because there is no true "middle number""

10) Why and how do you convert mixed numbers to fractions

10) Required to multiply or divide mixed numbers b. Multiply the whole number by the denominator and add the numerator

"How do I multiply fractions and mixed numbers 7/3*33/5"

11) Convert to improper fraction b. Simplify i. 7/3*33/5=7/1*11/5 Multiply numerator by numerator and denominator by denominator

12) How do I divide fractions

12) Change the divisor into the reciprocal (second number) b. Multiply the fractions

13) What are my options when I see double decker fractions

13) b. Take the reciprocal of the bottom fraction and multiply c. Or multiply both the top and the bottom by a common denominator

21) Are these reciprocals (definition)

21) do these equal 1 (x*y=1)

22) What are prime numbers to 100

22) 1) Know first ten primes but better to know all up to 100 a) 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

27) When will decimals terminate or not terminate

27) c) Positive powers of 10 have only 2's and 5's as prime factors and will be in the denominator i) Every terminating decimal shares this characteristic. If, after being fully reduced, the denominator has any prime factors besides 2 or 5 then the decimal will not terminate. If the denominator has only 2 and 5 it will terminate

29) biggest difference between equations and expressions

29) 2) Equations contain an equals sign and expressions do not

36) when can you simplify a root

36) 5) When can you simplify a root a) When connected via multiplication or division

53) solving direct and inverse proportionality

53) 2. The two quantities always change by same factor and same direction 1) Tripling the input triples the output 2) Y=kx 1. K=constant 2. y/x=k 3) typically given before and after scenarios 1. set up ratios to solve a. y1/x1 for before and y2/x2 for after b. y1/x1=y2/x2 since both are equal to constant k (2) inverse proportionality 1. change by reciprocal factors. 1) Cutting the input in half doubles the output 1. Y=k/x a. X=input value y is output and k is proportionality b. Yx=k 2) Typically given before and after scenarios 1. Set up products not ratios to solve a. Y1x1=y2x2

54) direct proportionality

54) 1. Direct proportionality 2. The two quantities always change by same factor and same direction 1) Tripling the input triples the output 2) Y=kx 1. K=constant 2. y/x=k 3) typically given before and after scenarios 1. set up ratios to solve a. y1/x1 for before and y2/x2 for after b. y1/x1=y2/x2 since both are equal to constant k

6) When to test cases

6) When Data sufficiency allows multiple starting points

69) arithmetic mean and variation

69) 1) Averages a) Arithmetic mean b) Average=sum/# of terms i) A=S/n (1) S refers to the sum of all terms in the set (2) N=number of terms (3) Average=arithmetic mean ii) Commonly used variation (1) Average x # of terms =sm

7) If you increase the numerator of a fraction while holding the denominator constant

7) the fraction increases in value

8) If you increase the denominator of a fraction while holding the numerator constant

8) the fraction decreases in value as it approaches zero

9) Adding the same number to both the numerator and the denominator

9) brings the fraction closer to 1 regardless of the fractions value i. If the fraction is less than 1 it increases ii. If the fraction is less than 1 it decreases in value