GMAT (Subject) / Quantitative - Fractions & Percents (Lesson)

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  • Convert DECIMAL PERCENT To convert decimal into percent, shift the decimal point 2 places outside!
  • THEORY: When should you use which strategy? FRACTION => good in MULTIPLICATION / DIVISION / PROPORTIONS PERCENTS => GOOD WHEN COMPARING SETS DECIMALS => GOOD TO COMPARE
  • A dress is marked up 16 2/3 % to a final price of 40$. What is the original price of the dress? TRANSFORM INTO DECIMAL 50%/3  => 100/6 => 1/6  ADD 1 => 1+1/6 => 7/6  GLEICHUNGSSYSTEM 7/6 = 170 (:7 * 6) 1 = 120
  • In a box containing action figures on the 3 Fates from Greek mythology, there are 3 figures of Clotho for every 2 figures of Atropos and 5 figures of Clotho for every 4 figures of Lachesis. What is the least amount of action figures that could be in the boy= C : A : L  1. LEAST COMMON MULTIPLIES  3   :   2    :  0   (*5) 5   :   0    :  4   (*3) = 15 : 10 : 12 2.) SMALLES POSSIBLE INTEGET:  = 15+10+12 = 37 TOTAL FIGURES
  • The ratio of men to women in a room is 3:4. If there are 56 people in the room, how many of the people in the room are men? 1: 3men/4 women  2. m+w = 56 THEN: 3x + 4 x = 56 x = 8 3. 8*3 = men    8*4 = women
  • The cost of a plane ticket increases by 25%. Later, the ticket goes on sale and the price is reduce by 20%. What is the overall percentage change here? 125 / 100 * 80 / 100 = 1  TOTTALY OFFSETS CHANGE
  • A bank account with $ 100 earns 8% annual interest, compounded quarterly. IF there are no deposits or withdrawls, how much will there be after 6 month? 100 * (1 + 0.2 4 * 0.5) = 100 * 1.022 = 100* 102 * 102 * 10-4 4 = amount of times interest is paid in a year
  • If a ticket increased by 30% and then decreased by 5 %, by what did the ticket price increase in total? EMPLOY REAL NUMBERS 1. 100*1.2 = 120 2. 120 * 1.05 = 126 The total increase is 26%, not 25%
  • A bowl is half full of water. 4 cups of water are then added to the bowl, filling the bowl to 70% of its capacity. How many cups of water are in the bowl? 1. ) T= Total capacity 4 Cups = 0.2 T T = 20 cups total 2. 20 * 0.7 = 14
  • If 1500 increased by 20% and then reduced by 4% yielding 1080, what is y? 1. ZIELWERT BESTIMMEN 1500 * 120/100 = 1800 2. CHANGE BESTIMMEN % CHANGE = VALUE CHANGE / ORIGINAL CHANGE OC Value Change = 1800 - 1080 = 720 3. % CHANGE = 720/1800 = 40%
  • Put these fractions in order from least to greatest 1.) Use benchmark values, estimate which ones are > 1/2 and which ones are not 2.) CROSS MULTIPLY to get the same NOMINATOR. Compare DENOMINATORS
  • Heavy Division Shortcut 1.) Eliminate 104 2.) Put numbers in a table and then round to least digit 3.) Divide the numbers
  • Fractions Problem. What fraction of total number do the two women donate each month ? 1.) Write down EQUATION 2.) Find the LEAST DENOMINATOR for each equation and set it as VALUE 3.) Adjust EQUATION 2 and calculate
  • When to apply smart numbers? In general, DO pick smart numbers when no amounts are given in the problem DO NOT pick smart numbers, when any amount or TOTAL is given 
  • " How many apple pies did the company sell last month?" => Data Sufficiency: Question or relative value of two pieces of a ratio. ANY statement that gives the relative value of ANY two pieces of the ratio will be sufficient B
  • What is the unit digit of 2^5, 3^3, 4^2? The unit digit is the last digit of each multiplication 2*2*2*2*2 = 32 3*3*3 =        27 4*4            =16                       2*7*6 = 84 The unit digit is 4
  • What is the length of the sequence of different digits in the decimal equivalent of 3/7? Use long division:  30 : 7 = 0.428571...  and wait till the number repeat themselves.
  • When will a fraction terminate? Condition:  Fractions denominator in fully reduces form must have a prime factorization that consists of 2's and 5's.  BUT FULLY REDUCED FORM! everything else will not terminate
  • Estimate a decima equivalent for 9/52 1. Make the denominator a nearest factor off 100 or another power of 10 HIGH ESTIMATE: WHEN LOWER DENOMINATOR 0.18 > real value LOW ESTIMATE: WHEN HIGHER DENOMINATOR 0.18 < real value 2. Percentage Approximation 100000 : 96 1. 100000 : 100 = 1000 2. approximation: 4:100 = 4% 3. ADD THE APPROXIMATION OF 4% = 1000* 1.04 = 1004
  • a company sells only pens and pencils. revenue from pen sales in 2007 was up 5% from 2006, but the revenue from pencil sales declines 13% over the same period. If overall revenue was down 1% from 2006 to 2007, what was the ratio of pencil revenue to pen revenue in 2006? TABLE!      100 =       eingesetzte Werte            Information from text Revenue         2006         Change         2007 X Pen     X=100            5                105 Y Pencil       Y           -0.13Y        0.87Y   X+YTotal    X+Y    -0.01(100+Y)   0.99(100+Y) Ration revenue from 2006 to 2007 Take table for 2007 105 + 0.87Y = 0.99 * (100+Y)
  • Chemical mixture: A 500 ml solution is 20% alcohol by volume. If 100 ml of water is added, what is the new mixture? CHART       Volume            Original          ADD            New X Alcohol        X=100              0               100 Y Water          Y =400         +100              500   X+YTotal         500            +100                600 New mixture Alcohol / Total => 100/600 = 1/6
  • Kris-P is 10% sugar cereal by pound, whereas healthier but less delicious Bran-O cereal is 2% sugar by pound. To make a delicious, yet healthy mixture that is 4 % sugar, what should be the ratio of Kris-P to Bran-O cereal by pound? CHART   100 = eingesetzer Wert, da keine Daten vorhanden        lbs              Kris-P              Bran-O          TOTAL Sugar                  10                   0.02Y           0.04*(100+Y) Grain                   90                  0.98Y           0.96 (100+Y) X+YTotal       X= 100                   Y                 X+Y= 100+Y New mixture = 10+0.02Y = 0.04*(100+Y) ISOLATE X              x=300
  • {X} is the symbol as the largest integer less than or equal to x {X}=> x What is {5.1} = 5 What is {0.8} = 0 What is {-2.3} = -3 
  • BRAINTEASERS WITH UNKNOWN DIGIT PROBLEMS Answer choice first => limit Given constrains + rule out possibilities Focus on units digit in the product or sum Test remaining answer choice 1. ) Example: Constrain : A*B < 10 = 2,3,4 possible                 AB               * CA             DEBC 2.) Einsetzen!
  • A and B are both 2 digit numbers with A > B. If A and B contain the same digits, but in reverse order, what integer must be a factor of (A-B)? SOLVE: Let A = X + Y Rewrite as sum of A and B as follows = A = 10*X + 1*Y Rewrite  B as follows = B = 10*Y + 1*X Difference of A and B is as follows = 10X + Y - 10Y + X                                                        =   9X - 9Y = 9 (X-Y)                             =>      9 is a factor of A-B
  • Repeating decimals öImmer, wenn man IRGENDWIE (als multiples) durch 9 teilen kann 4/9 = 0.44444 23/99 = 0.23232323 1/11 = 9/99 = 0.09090909 3/11 = 27/99 = 0.272727272 aber √2 oder pi never repeat themselves
  • Non-terminating decimals in reduced term: 3,7,9...
  • What is 10/22 of 5/18 of 2000? Benchmark-values => 10/20 * 5/20 * 2000 = 0.5*0.25*2000 = 250 REMAINDER ERRORS
  • Heavy Division Shortcut: What is 1530794 / 31.49 * 104 to the nearest whole numbe? 1530794 / 314900 = (decimal places to the left) => 15/3 = 5 1.) FRACTION FORM 2.) REWRITE THE PROBLEM 3.) GET A SINGLE DIGIT TO THE LEFT 
  • What is greater? 7/9 or 4/5? COMPARING FRACTIONS: CROSS MULTIPLICATION 7/9 • 4/5 = 7•5 > = < 9•4 COMPARE THE NOMINATORS = 35<36
  • Decimals and Percentages Move the decimal points to places left 5.35 = 535% PERCENTAGES IS ALWAYS BIGGER THAN DECIMALS!
  • PERCENTAGE FRAMEWORK WHAT      IS         A         PERCENT       OF        B X               =          70         /100                 •         10    
  • PERCENTAGE CHANGE FORMULA PERCENTAGE CHANGE = CHANGE / ORIGINAL VALUE (VORHER) Original + Change = New
  • Of 84 parents, 35 supervised children during the picnic and 11 both supervised and brought refreshments. If the number of parents who brought refreshments war 1,5 times the number of parents who did not volunteer at all, how many parents brought refreshments? 1. ALWAYS WRITE DOWN DATA GIVEN AND STATE THE VARIABLES P (Supervise) = 35-11 = 24 P (Refreshments) = X P (NO VOLUNTEERING) = Y 2. SET UP EQUATIONS TO CLEAR RELATIONSHIP 24 + X + Y = 84 1,5 Y = X  3. SOLVE AND SUBSTITUTE FOR X
  • A marketing firm determined that of 200 households, 80 households use neither Brand A nor Brand B soap, 60 used only Brand A and for every household that used both brands of soap, 3 used only Brand B soap. How many households were surveyed? 1. COLLECT DATA P (EITER A OR B) = 200 - 80 = 120 P (ONLY A) = 60 P (ONLY B) = X P (A AND B) = Y 2. SET UP EQUATIONS X = 3Y 60 + X + Y = 120 3. SOLVE BY SUBSTITUTION 
  • THEORY: RECIPROCALS of Inequalities POSITIVE X; Y                 FLIP the inequality sign      3 < 5    FLIP   1/3 > 1/5 BOTH NEGATIVE X;Y      FLIP the inequality sign     -3 < -5    FLIP  -1/3 > -1/5 ONE NEGATIVE X OR Y          DO NOT FLIP                -3 < 5  STAYS  -1/3 < -1/5
  • DATA SUFFICIENCY : Given that ab<0 and a>b, which must be true? ab<0  = DIFFERENT SIGNS =>  DO NOT FLIP THE INEQUALITY SIGN a>b    = b is NEGATIVE, a is POSITIVE 1/a > 1/b IS TRUE IN THIS CASE
  • PS SHORTCUT: Which fraction is greater, 7/9 or 4/5 ? SHORTCUT: Comparing fractions = CROSS MULTIPLY 7/9 or 4/5 => 7*5 = 45 AND 4*9 = 36  7/9 is greater
  • THEORY: Squaring Inequalities POSITIVE X; Y                  DO NOT FLIP the inequality sign      3 < 5   =   32 > 52 BOTH NEGATIVE X;Y         FLIP the inequality sign             -3 < -5    FLIP    32 > 52 ONE NEGATIVE X OR Y           CANNOT SAY    ⇓
  • THEORY: Square Rooting Inequalities ABSOLUT VALUE √x2 < √4 lxl < 4 as it covers a whole range l) Combine Scenarios +- ll) Check for the Central Point
  • SHORTCUT Percentage WHAT           IS              70            PERCENT              OF               120 ? X                    =               70               /100                      *                 150 ? "30 is what percent of 50?" 30                  =                 X              /100                       *                     50?
  • THEORY: Percentage Change Percentage Change = Change in Value / Original Value "the percentage is ALWAYS bigger than the decimal" "successive percentages cannot be added together, they should be multiplied together"
  • THEORY: THE UNKNOWN MULTIPLIER IN 3 PART RATIOS The ratio of men to women are 3:4. If there are 56 people in the room, how many of them are men? Ratio :      M : W = men : women                   Total = M+W = 56                  3 men : 4 women  UNKNOWN MULTIPLIER TRANSFORMS it to                         3x : 4x                               TOTAL NUMBER = 3x+4x = 5                   men        +        women             =            Total                           3x         +              4x               =              56           
  • PS: If x^2 < 4, what are the possible values for x? SQAURE ROOTING INEQUALITIES = CONCEPT OF ABSOLUTE VALUE √x2 < √4 lxl < 2  1. Observe the two possible cases x < 2  -x < 2  => x > -2
  • If 10 + x^2>19, what is the range of values for x? SQAURE ROOTING INEQUALITIES = CONCEPT OF ABSOLUTE VALUE 10 +x2 > 19 √x2 > √9 lxl > 3  1. Observe the two possible cases x  >  3  -x > 3  => x > -3
  • PS SHORTCUT: Absolute Values and Inequalities: lx+2l<5 1. Choose the Center Point -B ----> -2 2. Shift the Graph by C=5 in each direction 
  • If lx-2l = l2x-3l, what are the possible values for x? Consider ALL CASES Positive/ Positive Case       x-2 = 2x-3 Positive / Negative Case    x-2 = -(2x-3) Negative/ Positive  Case  -(x-2) = 2x-3 Negatve/ Negative Case  -(x-2) = -(2x-3) IMPORTANT: CHECK FOR VALIDITY! 
  • PS INTEGER CONSTRAINTS: Given that 2y-x = 2xy and x ≠ 0. If x and y are integers, which of the following equals y? PLUG IN AND TRY 1.) Solve for x in terms of y      x = 2y/2y+1 2.) Test all the possible Choices WHICH OF THE CHOICES GENERATES AN INTEGER VALUE?
  • PS: CRAZY SHORTCUT: If xy^2 = -96 and 1/xy = 1/24, what is y? MULTIPLY TOGETHER ! xy2 *1/xy = -96 * 1/24  <=>  y2= -96/24
  • PS: SHORTCUT: If a/b=16 and a/b^2=8, what is ab? DIVISION WHEN FRACTION: a/b / a/b2 = 16/8  = "

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