GMAT (Fach) / Economist Prep - Quant (Lektion)

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• Absolute Value The distance from zero on the number line. A positive number is already in the same form as that numbers absolute value.
• Factors Positive integers that divide evenly into an integer. Factors are equal to or smaller than the integer in question. 12 is a factor of 12, as are 1, 2, 3, 4, and 6.
• Greatest Common Factor Greatest Common FACTOR refers to the largest factor of two (or more) integers. Factors will be equal to or smaller than the starting integers. The GCF of 12 and 30 is 6 because 6 is the largest number that goes into both 12 and 30.
• If m<0, which of the following must be true? m ≤ 1 But you might say, “m cannot be equal to 1.” You are right! That doesn’t mean that m≤1 isn’t true though. It is important to remember that a variable can only have one singular value at a time. When you see the inequality m<0, that means that m is negative. It doesn’t mean that m is every negative number. Similarly, saying that m≤1 doesn’t mean that m is all of the numbers less than or equal to 1.
• Work-Order: DS-Questions Type: Determine whether it is a value question or a yes/no question. Issue: Figure out the GMAT knowledge field that is tested (e.g averages, quadrilateral area formulas etc.)  Missing piece: While focusing on the question stem, determine what is the piece of information that's needed to answer the question asked.
• Yes/No Data Sufficiency Questions: Figure out the issue before you dive deeper into the statements. The issue in Yes/No Data Sufficiency means which numbers yield a Yes or a No. The issue is also an important part in solving Must Be questions. While Plugging In you must find numbers that do not contradict the statements. Regard the statements as facts that cannot be broken. Only then should you plug in the numbers you used in the question stem. Ask yourself every time you get a Yes or No: "Is it always true, for any number?" This is completely equivalent to Must Be questions. Remember: only a definite Yes/No is Sufficient. Answering a definite "Yes" or "No" Means Sufficient. If the answer is sometimes "Yes" and sometimes "No", it means Maybe which means Insufficient. After that, follow the Data Sufficiency FlowChart to get the final answer.
• If a and b are positive integers, is ab < 6? (1) 1 < a + b < 7 (2) ab = a + b According to Stat. (2), ab = a + b the only possible pair of positive integers that satisfies this equation is (2, 2) i.e. a=2 and b=2 (2)(2) = 2 + 2 --> 4 = 4 based on this ab < 6 Definite answer, so Stat.(2)->S->B.
• If a and b are integers, what is the value of a−b? (1) a·b=1 (2) a+b=2 A Correct. The only integers a and b for which Stat.(1) is true are a=b=1 and a=b=-1. In both cases a-b=0. Therefore, Stat.(1)->S->AD. Stat.(2): plug in a=b=1, so that a-b=0. Then plug in a=2 b=0, so that a-b=2. The value of a-b cannot be determined, so Stat.(2)->IS->A.
• If 3<x<6<y<10, then what is the greatest possible positive integer difference of x and y? "the greatest possible...difference" they have to be as far apart as possible from each other. But who said the numbers you use have to be integers? The question asks for an integer difference, not necessarily for integer numbers. For instance x=3.5 and y=9.5. The greatest possible integer difference is 6.
• Dividend÷Divisor = Quotient. The quotient is the number of times the divisor divides into the dividend Dividend÷Divisor=Quotient. When dividing two integers, the quotient refers only to the integer part of the result  7 ÷ 5=1 (2) ---> The quotient is 1, while the remainder is 2.
• Math Fundamentals: Order of Operations Parentheses Exponents Multiplication (together with Division, from left to right) Division (together with Multiplication, from left to right) Addition (together with Subtraction, from left to right) Subtraction (together with Addition, from left to right)
• PEMDAS Purple Elephants May Destroy A School: PEMDAS
• Consecutive Integers Consecutive Integers: integers that follow in sequence, each number being 1 more than the previous number.  Consecutive Even Integers: A set of even integers with a distance of 2 between each member and the following/preceding member. Examples: 2, 4, 6... Consecutive Odd Integers: A set of Odd integers with a distance of 2 between each member and the following/preceding member. Examples: 1, 3, 5...
• Remainder The distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend. Note that when 8 is divided by 3, the remainder - the distance between 8 and the nearest multiple of 3 (which is 6) - is still 2.
• the greatest possible remainder The greatest possible remainder is one less than the divisor. The remainder can never be equal to, or greater than the divisor Dividing 10 by 5 does not give a remainder of "5" - 10 itself is the nearest multiple of 5 that is closest to itself, giving a remainder of zero (i.e. no remainder).
• Remainder Problem of GMAT Remainder is the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend. The greatest possible remainder is one less than the divisor =>  when dividing by 5, the highest possible remainder is 4. GMAT problems involving remainders can usually be easily solved by Plugging in numbers that fit the problem. Try plugging in the remainders themselves: When X is divided by 5, the remainder is 3 - plug in X=3.  when plugging in is difficult to use, use the following formula: for any integer i divided by another integer d i = quotient·d + remainder i=5x+3  x being the quotient.
• C/D = 9,75 -> When dividing positive integer C by positive integer D, the remainder is 3. Which of the following could be the value of C? The issue of this question is correctly interpreting the remainder. If C/D=9.75, then      C = 9.75·D -->  C = 9·D + 0.75·D Realize that 0.75D is the remainder Use the fact the 0.75D = 3 to find D Use D to find C.
• When m is divided by 9, the remainder is 2. When m is divided by 13, the remainder is 8. If 1 < m < 200, what is the greatest possible value of m? Divisors: 9 and 13 Start plugging in to try to find high numbers that satisfy both statements. If you do so, start from 190, the highest number within the range that is 8 more than a multiple of 13, and see if it also satisfies the other requirement. The nearest multiple of 9 to 164 is 18×9=162, leaving a remainder of 2. The nearest multiple of 13 to 164 is 12×13 = 156, leaving a remainder of 8.
• Finding a Range in Scatterplots To find the range, subtract the highest value minus the lowest value.
• Definitions of prime numbers "A prime is a natural number with exactly two distinct factors: 1 and itself". 0 is NOT prime 1 is NOT prime 2 is SMALLEST PRIME NUMBER!
• If a, b and c are distinct prime numbers, and a+b=c, which of the following must be true? I) c>a+1 II) b is odd. III) a>2 I only 2 ist die kleinste PRIME Number
• Adding & Substracting Multiples Multiple of n ± multiple of n = MUST be a multiple of n. multiple of n ± NOT multiple of n = CANNOT be a multiple of n. NOT multiple of n ± NOT a multiple of n - May or May NOT be a multiple of n
• Greatest Common Divisor & Prime Numbers Since prime numbers are divisible only by themselves and 1, any two primes will only have a G.C.D of 1. However, note that the reverse isn't necessarily true: The fact that two integers have a G.C.D of 1 does not necessarily indicate that they are Primes. This is a common misconception, easily disproved by example: 8 and 9 have a G.C.D of 1, even though neither of them is Prime.
• Decimal Digits Place Names _ _ _ . _ _ _  Hundreds Tens Units  _____ Tenths Hundreths Thousandth
• Which of the following sums of fractions equals 0.123? Thus, 0.123 can be literally translated into 1 tenth plus 2 hundredths plus 3 thousandths or 1/10 + 2/100 + 3/1000
• Rounding Numbers Rounding a number is a way of making it simpler. Rounding to the nearest tenth, you should end up with nothing less than tenths. Decide how to round according to the digit in the next decimal place. If the digit in the next decimal place is: 0-4, inclusive: leave the number as is, canceling out all other digits to the right. (i.e., 12.341 rounded to the nearest tenths is 12.3, because the digit in the hundredths place is 4) 5 or more: increase the digit you are rounding for by one, canceling out all other digits to the right. (i.e, 12.361 rounded to the nearest tenths is 12.4, because the digit in the hundredths place is 6) Do not make the mistake of rounding in "steps" - i.e. round 4.3149 to 4.315, and then to 4.32. Instead, look directly at the next digit after the digit you're rounding to, and decide according to the ranges above.
• Is it true that x is equal to the value of x rounded to the tenths place? (1) x is equal to the value of x rounded to the hundredths place. (2) x is equal to the value of x rounded to the units place. Stat. (2) is sufficient: x rounded to the nearest unit has nothing smaller than units, i.e. x is an integer. Therefore, x must have nothing smaller than tenths, and the answer is "Yes". Stat.(2)->Yes->S->B.
• Absolute Value: Equations Example: |x-2| = 5 1) Copy the equation without the absolute value sign: x-2 = 5 -->    x = 7 2) Copy the equation without the absolute sign, and put a negative sign on the part without the absolute value. x-2 = -5 -->    x = -3 The 2 solutions of x are x1=7 and x2=-3. 3) check your work - plug in x1=7 and x2=-3 into the original equation |x-2| = 5 |7-2|=5 |-3-2| = |-5| = 5
• Simultaneous Inequalities Adding the inequalities 1) First, line up the inequalities so that the sign goes in the same direction. 2) Add the inequalities, much as you would a set of simultaneous equations. Always add! It is better to manipulate the inequalities by multiplying by -1 until they reach the state where adding will get you where you need to go. Don't forget to flip the sign. Plugging in the extremes  Focus on the highest and lowest possible values of the inequalities. Plug in those to reach the desired result proposed by the question. This is especially effective in cases where adding inequalities doesn't help.
• The two perfect square trinominals: (a+b)2 = a2+2ab+b2 (a-b)2 = a2-2ab+b2
• The difference of squares: (a+b)(a-b) = a2 - b2
• Recycled quadratics I      (a+b)2 = a2+2ab+b2 II     (a-b)2 = a2-2ab+b2 III    (a+b)(a-b) = a2 - b2
• The discriminant (b2-4ac)
• If the discriminant is..  positive -> (b2-4ac > 0) then x1 and x2 will have different values -> 2 different solutions. equal to zero -> (b2-4ac = 0), then x1=x2 =-b / 2a -> only one solution. negative -> (b2-4ac < 0), then the quadratic has no real solutions ( since you can't take the square root of a negative number without resorting to imaginary numbers (which are thankfully not tested on the GMAT )
• The number of solutions (discriminant testing) b2-4ac > 0 - 2 different solutions. b2-4ac = 0 - one solution. b2-4ac < 0 - no (real) solutions.
• Factoring quadratics Factoring a quadratic equation is a matter of transforming the quadratic from the form of x2+bx+c=0 into the form of (x+e)(x+f)=0.  The Factoring process goes as follows: 1) Your first order of business is finding e and f. If a=1 (which it almost always will in GMAT problems) think of two integers e and f which satisfy the following conditions: a) Their product is equal to c. b) Their sum is equal to b.
• Factoring quadratics Factoring a quadratic equation is a matter of transforming the quadratic from the form of x2+bx+c=0 into the form of (x+e)(x+f)=0.  The Factoring process goes as follows: 1) Your first order of business is finding e and f. If a=1 (which it almost always will in GMAT problems) think of two integers e and f which satisfy the following conditions: a) Their product is equal to c. b) Their sum is equal to b.
• Factoring quadratics Factoring a quadratic equation is a matter of transforming the quadratic from the form of x2+bx+c=0 into the form of (x+e)(x+f)=0.  The Factoring process goes as follows: 1) Your first order of business is finding e and f. If a=1 (which it almost always will in GMAT problems) think of two integers e and f which satisfy the following conditions: a) Their product is equal to c. b) Their sum is equal to b.
• recycled quadratics; difference of squares: (a+b)(a-b) = a2-b2 1) Factored form: (a+b)(a-b) a. Identification: Two sets of parentheses, one with the sum of two terms, the other with the difference of the same two terms. 2) Expanded form: a2 - b2 a. Identification - recognize the pattern: Look for the difference of even powers (square in particular) Example: is x2 - 1  a recycled quadratic?
• The Difference of squares 1) Factored form: (a+b)(a-b) a. Identification: Two sets of parentheses, one with the sum of two terms, the other with the difference of the same two terms. Example: (x+5)(x-5) 2) Expanded form: a2 - b2 a. Identification: Recognize the pattern:Look for the difference of even powers (square in particular). Example: is x2 - 1  a recycled quadratic? x2 - 1 = (x+1)(x-1)
• Which of the following is the reciprocal of (3−√8)? Recall that if a·b=1 then a and b are reciprocals. Eliminate any negative answers. Multiply the remaining answer choices by (3−√8). The answer choice whose product with (3−√8) is 1 is the correct answer. Checking D you get (3+√8) × (3−√8). That should ring a bell. The numbers fit in the quadratic formula (a−b)(a+b)=a2−b2, so (3+√8) × (3−√8) = 32−(√8)2 = 9−8 = 1.
• 6^4−4^4=? Recycled quadratic III: (a+b)(a-b) = a2-b2 --> 64−44= (62)2 - (42)2 = (62−42)×(62+42)  =  (36−16)×(36+16) = 20×52 = 1040
• 3√(14^2-13^2)=? 3√(142-132) = 3√((14-13)(14+13)) = 3√((1)(27)) = 3√(27) = 3√(33) = 3
• Which of the following CANNOT be true (for any real x)? When can't a quadratic equation be true for any real x? when it has no real solution. This happens when the discriminant (b2-4ac) is negative.
• Different Multiplier son ratios To combine different ratios you must equate the number representing the member common to both ratios. Write the ratios one on top of the other and expand/reduce the ratios to equate the common member.  Example: Turn this ratio pair Camels : Goats : Sheep 1st ratio         ( 5    :   2 ) 2nd  ratio                  ( 1    :    3 ) Into this comparable ratio pair: Camels : Goats : Sheep 1st ratio         ( 5    :   2 ) 2nd  ratio                  ( 2    :    6 ) So that the Goats have the same ratio units in both ratios.
• In McDonald's farm, the ratio of the number of goats to the number of sheep is half the ratio of the number of cows to the number of goats in his farm. If the ratio of the number of goats to the number of sheep is 1:3, then what is the ratio of the number of sheep to the number of cows in the farm? Break the problem into two steps: First, find the cows:goats ratio. The goats:sheep ratio (1:3) is half the cows:goats ratio, hence, the cows:goats ratio is double the goats:sheep ratio. This means that the ratio of cows:goats is 2:3. Next, write the two ratios in the following diagram to see what's going on:      Cows : Goats: Sheep 1st ratio                       ( 1    :   3 ) 2nd  ratio          ( 2    :    3 ) In order to find the ratio of sheep to Cows, combine the two ratios: find a quantity that is common to the two (goats). Expand / reduce the ratios to equate the number representing the common quantity.
• In a certain salad, the ratio, by weight, of potatoes to boiled eggs is double the ratio, by weight, of boiled eggs to green peas. If the weight of the eggs in the salad is half the weight of the potatoes, then what is the ratio, by weight, of potatoes to green peas in the salad? Break the problem into two steps: First, find the eggs:peas ratio. Since the weight of the eggs is half the weight of potatoes, the potatoes:eggs ratio is 2:1. The ratio of potatoes to eggs is double the eggs:peasratio, hence, the eggs:peas ratio is half the potatoes:eggs ratio. This means that the ratio of eggs:peas is 1:1.
• A candy box contains only marshmallows and pralines at a ratio of 2:3. A hungry hippo sneaks in and steals 2 marshmallows, leaving the marshmallows and pralines at a new ratio of 4:9. How many pralines are in the box? Notice the essential ingredients provided by this type of question: 1) An original ratio - 2:3 2) A change (through addition or subtraction) -2 marshmallows 3) A new ratio - 4:9 In order to find the real number of Pralines, we have to find the multiplier. Usually, we'd put the data in a ratio box and find the multiplier by dividing the real numbers presented in the question with the corresponding ratio in the same line. However, the presence of two different ratios makes this difficult. Organize the ratios in a diagram:                            M        P Original ratio         2   :    3 change                                 --->     -2 marshmallow New ratio              4   :    9
• DS: Ratio Changes Addition and Subtraction In Data Sufficiency questions involving ratio changes, a start point ratio end point ratio a real quantity in between are sufficient to answer the question with no calculations needed. In Problem Solving you have to go through all the work.
• PS: Ratio Changes with Addition and Subtraction 1) Compare the two ratios 2) Expand / reduce by multiplier                                 M        P  Original ratio          2×3=6  : 3×3=9 change                        -2                    -->       -2 marshmallow New ratio                        4   :    9 3) Use the multiplier to find the required quantity. 

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