GMAT (Fach) / Quantitative - Fractions & Percents (Lektion)
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- THEORY ROOTS: EVEN ROOTS: HIDE THE SIGN => more than 1 solution ODD ROOTS: KEEP THE SIGN => only 1 solution NEVER SEPERATE OR COMBINE THE SUM OR DIFFERENCE OF 2 ROOTS
- Triplets Adam, Bruce and Charlie enter a race. There are nine competitors in the race. If every competitor has an equal chance of winning and three medals will be awarded, what is the probability that at least 2 of them will win a medal? Probability = Specified outcome / all possible outcomes 1. Total number of possible outcomes for the triathlon ANAGRAM GRID 9 competitors, 3 will win, 6 will not 2. Number of ways 3 medals can be awarded = 9!/3!6! = 84 3. Determine number of instances when at least 2 will win a metal 3 brothers win = 3!/3! * 6!/6! = 1 2 brothers win = 3!/2!*1! * 6!/1!/5! = 3*6 = 18 4. 18 + 1 = 19 cases, so the probability is 19/84
- te expression √2+ √2+ √2+... extends to an infinite number of roots. Which of the following most closely approximates the value of the expression? ALGEBRAIC SOLUTION: the infinite expression is nested within itself: x = √2+ (√2+ √2+.....) = √2+x Solve for x as follows: x = √2+x <=> (x-2)(x+1)=0 x = 2 or x=-1 ---> so x=2
- The probability that event M will NOT occur is 0.8 and the probability that event R will NOT occur is 0.6. If event M and R cannot both occur, what is the probability that either M or R will occur? P(M) = M WILL occur = 0.2 P(R) = R WILL occur = 0.4 P(M AND R)= M AND R BOTH occur = 0 P ( EITHER M OR R) = P(M) + P(R) - P(M UND R) = 0.2+0.4-0 = 0.6
- 4 extra large subs of the same size were ordered by m students, where m>4. 3 of the subs were evenly divided among the students. Since 4 students did not want any of the 4th sub, it war evenly divided among the remaining students. If C ate one piece from each of the four subs, the amount to sandwich that she ate would be what fraction of a whole extra-large sub? Each piece was 1/m of a Sub the 4th Sub was: 1/m-4 of a Sub Carol hence ate 3*1/m + 1/m-4 = 3(m-4)+m/m(m-4) = 4m-12/m(m-4) Trick hier: Auf den selben Nenner bringen: Linker Zähler mal (m-4) und Rechter Zähler mal m, Nenner Produkt von beiden
- In country C, the unemployment rate among construction workers dropped from 16% in 1992 to 9% in 1996. The number of workers in 1996 was 20% greater than in 1992. What was approx. the percentage change in the number of unemployed workers over this period? X = number of unemployed workers in 1992 = 0.16N Y = number of unemployed workers in 1996 = 0.09 * 1.2N Percentage change = Y-X / X * 100% -> Change in Value / Original Value 0.09 * 1.2N - 0.16 N / 0.16 = -30%
- A pharma company received 3 Mio in royalties from the first 20 Mio sales of the generic equivalent of its product and then 9 mil royalties on the next 108 million in sales. By what percent did the ratio of royalties to sales decrease from the first 20 Mil in sales to the next 108 Mio in Sales? The ratio of royalities to sales for the first 20 Mio in sales is 3/20 The ratio of royalities to sales for the next 108 million in sales is 9/108 = 1/12 The change is Percentage change = change in value /( original value) 1/12 - 3/20 // 3/20 = -0.44*100 approx. 45% decrease
- A researcher plans to identify each participant in a medial experiment with a code consisting of either a single letter or a pair of distinct letters written in an alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? TRY OUT: 3 letters = a,b,c => there are 6 codes: a,b,c,ab,ac,bc,cb 4 letters = a,b,c,d, => there are 10 codes 5 letters = at least 12 codes.. so the least number is 5 letters
- Parallel line testing 1. Calculate the slope of the graph: y1-y2/x1-x2 2. Choose the graph with the same slope ! This is the parallell graph
- There are 8 team in a certain league and each team plays each of the other teams exactly once. If the game is played by 2 teams, what is the total number of games played? RESEMBLE : SCATTER TABLE 1. There are 8*8 teams playing = 64 2. Since no team needs to play itself, substract the diagonal (8) = 56 3. Since two team play each game, divide by 2 = 56/2 = 28
- A bar over some sequence of digits in a decimal indicates that the sequence repeats indefinitely. What is the value of (10^4 - 10^2)*(0.0012)? (104 - 102)*(0.00‾12) DISTRIBUTE AND SIMPLIFY 10000* 0.0012 - 100*0.0012 12.12 - 0.12 = 12
- At a loading dock, each worker on the night crew loaded 3/4 as many boxes as each worker on the day crew. If the night crew hat 4/5 as many workers as the day crew, what fraction of all the boxes loaded by the 2 crews did the day crew load? Workers on night crew will load 3/4 D* 4/5 = 3/5 D (as many boxes ad the day crew work) Total loaded by the day and night crews: 1*3/5 = 1 + 3/5 = 8/5 The fraction of all boxes loaded is hence 1/ 8/5 = 5/8
- Of 300 subject who participated in an experiment using VR therapy to reduce their fears, 40% experienced S, 30% experiences D and 75% experiences V. If all of the subjects experiences at least one effect and 35% experienced exactly 2, how many experienced only 1 effect? A = experience only 1 effect B = experienced 2 C = experienced 3 A+B+C = 300 B = 300*0.35 = 105 A+C = 195 effect S = 0.4*300 = 120 effect V = 0.3*300 = 90 effect D = 0.75*300 = 225 A+2B+3C = 435 because each participant who experienced only one of the effects is counted once, those who experienced 2 is counted twice and those who experience all three is counted three times!
- 7 pieces of rope have an average arithmetic mean of 68 centimeters and a median of 84 centimeters. If the length of the longest piece of the rope is 14 centimeters more than 4 times the length of the shortest piece of the rope, what is the maximum possible length of the longest piece? a b c d e f and g are the pieces of rope listed from least to greatest. d= 84 g= 4a+14 a, a, a, 84, e, f, 4a+14 e and f are right to the median, so they must be greater than 84 (a, a, a, 84, 84, 84, 4a+14) /7 = 68 ---> a=30
- The letters D,G,I,I and T can be used to form 5-letter strings such as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one another letter? There are 6 ways to select the locations of the 2 occurences of the letter I and this number can be determined by listing all such ways as shown below, where the symbol * is used in place of the letter D;G;T 1. LOCATIONS OF LETTER l = 6 I**I* , I*I**, I***I, *I*I*, **I*I, *I**I ALTERNATIVELY: 5! / 3!2! -4 = 6 because -4 ways in which the 2 selected locations are adjacent 2. LOCATIONS OF LETTER D;G;T = 3! 3. It follows that there are 6*6 = 36 ways to form a new word
- A number line contains 3 points R,S and T, whose coordinates have absolute values r,s, and t. Which of the following equals the average of the coronets of the Points R,S,T if R is to the left of 0 on the number line? Because R is to the elft of 0 on the number line, the coordinate of R is negative. It is given that r is the absolute value of the coordinate of R and so the coordinate of R is -r. - Because points S and T are to the right of 0 on the number line, their coordinates are positive. It is given that s and t are the absolute vales of the coordinates of S and T, so the aritmethic mean of the coordinates of R,S and T is s+t-r / 3
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- Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m ? If P and E are Price and Earning of the Share, then (1+k/100)P and (1+k/100)E are the incrases, respectively. The percentage increase of the ratio of P to E is therefore ratio after increase - ratio before the increase / ratio before the increase ((1+k/100)P/(1+k/100)E // P/E -1) * 100 Final result: 100 (k-m) / 100+m ‰
- What is the 4√16 4th root of 16? Since 4√ is an even root, both 2 or -2 are possible answers; However, an even root means "only the positive solution", so 2 is the answer. Thus, if a GMAT question presents the expression 5×4√16, it will equal 5×2 = 10, and not have two optional positive or negative values.
- Dividing Powers with the Same Base In order to divide powers of the same base, subtract the exponents: an / am = an-m the power 3x-2 could be transformed into the fraction 3x / 32.
- Suppose we need to rewrite m^3 as a fraction of two powers. As a preliminary step, rewrite the exponent as a subtraction. Any two numbers with a difference equal to the original exponent will do. In our case, 3 could be replaced by '5-2'. (Of course, you can use any other pair of numbers with a difference of 3). ---> m5-2 REWRITE: m5 / m2 technique is especially needed when the problem introduces powers with a minus sign in the exponent you need to be able to break down to a fraction i.e. break down 2x-1 ----> 2x / 21 = 2x / 2.
- Adding and subtracting powers with the same base: 1) Adding and subtracting powers with the same base: DON'T: add or subtract the exponents Example: x3 + x5 ≠ x8 Do: extract the greatest common factor. Example: x3 + x5 = x3(1+x2)
- A perfect square A perfect square is an integer with an integer square root. The following table lists the first 17 perfect squares - memorize these by heart. Integer (a) Perfect square (a2)1 12 53 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 22516 25617 289
- The number 91 can be written as the sum of squares of three integers. Which of the following could be the difference between the largest and smallest integers of the three? 92=81, which leaves 10 more to get to 91. 32=9, which leaves 1 more to get the 10. 1 is 12, so the three numbers are 1, 3, and 9, and so the difference between the greatest and the smallest is 9 -1 = 8