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

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• Which of the following is the best approximation for y? 1/2 - 1/3 + 1/6 - 1/10 + 1/12 - 1/14 + 1/16 0.31 1. Get the Ballpark ! 3/6 - 2/6 + 1/6 = 1/3 = 0.333 2. -1/10 + 1/12 = a little less 3. -17714 + 1/16= a little less  = 0.31
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• Snapshot Ballpark for Geometry Always Start with a snapshot ballpark: 1) Compare the target value (length/area/angle) to a given value in the question - how many times does one fit into the other? 2) Quickly eliminate out all answers that are clearly out of the ballpark. If more than 1 answer is left standing - choose whether to: → Proceed to finer POE considerations → Proceed to a direct geometric calculation The benefit of a snapshot ballpark is a quick elimination of dangerous answer choices which represent common calculation errors and careless mistakes. Snapshot ballparking is faster and safer.
• In the circle above, the shaded region is a semicircle - half of the area of the circle. If the shaded region's area is 4π, which of the following is the best approximation of the area of the entire circle? Start with a snapshot ballpark: The given value is the semicircle's area = 4π. The target value is the area of the circle, which is twice as big = 8π. Ballpark π=3+, so 8π is 8*3+=24+. Only one answer choice fits that description, and that is D 25.
• Ballpark Numbers to Rember √2 is approximately 1.4. √3 is approximately 1.7.  √6Π is approximately 7.1  Π  is approximately 3.1
• Work Order Critical Reasoning QATLS = Question stem, Argument, Think, Look, Scan
• Critical Reasoning Questions Work Order1. Read the Question stem first.2. Read the Argument and map it to its components (i.e., premise, conclusion).3. Think of a possible answer/direction to the question.4. Go over the answer choices; Look for one that is similar to the one you thought of yourself. 5. Scan the remaining answer choices to make sure there isn't a better choice. DO NOT READ CONTENT BUT STRUCUTURE OF CONTENT 
• Boldface-Type Questions: A premise might be also called evidence support reason basis factual possibility piece of information generality condition general opinion rule assumption postitive statement  fact  inconvenience / capability / 
• Boldface-Type Questions: A conclusion might be also called position objection opinion explanation claim prediction judgement inference
• Boldface Type Questions Breaking down the argument(s) into Premises and Conclusion and correctly  Crott out distractors A premise may also be called evidence, support, reason, basis A conclusion may also be called position, objection, explanation, claim, prediction ... so keep your eyes peeled. Good luck!
• "0 is a non-negative number" Positive: > 0 Non-negative: ≥ 0 Check for 0
• "Which of the following is NOT always true?" The sum of two distinct non-positive numbers is negative. The product of two distinct non-positive numbers is positive. The sum of two distinct non-negative numbers is positive. The product of two non-negative numbers is non-negative. Any x2 is non-negative. The product of two distinct non-positive numbers is positive. This is the correct answer because negative x zero = zero. For example: (-3) x 0 = 0. Therefore, the product of two distinct non-positive numbers is NOT always positive.
• Integer Definition 1) A real number is a fancy name for "number". When a GMAT question uses "real numbers", the number can be anything: positive, negative, integer or fraction. 2) An Integer is a non-fraction or a non-decimal. 3) By the above definition, zero is an integer.
• Which of the following is NOT even? -4 - 2 - 0.5 Even: any integer that is divisible by 2 with no remainder. Examples: 2, 4, 14, -6. AND 0 Odd: Any integer that is not Even - not divisible by 2 with no remainder REMAINDER
• If y=−m2, which of the following must be true? I) y is negative. II) m is non-negative. III) If m is negative then y is negative. Variables in the answer choices? Make the algebra go away - plug in good numbers and eliminate.  0 -2 2 0.5 CHECK FOR EACH AND POE! Only lll) is true!
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• If m=4−(4−x)2, what is the greatest possible value of m? 4
• So is there nothing to be done with x3+x5? Just leave it as is? For example,  x3 + x5 can be rewritten as  x3+ x3∙x2 x3(1+x2) Extracting a common factor out of an addition or subtraction of powers will solve the GMAT problem at hand 99% of the time.
• Scientific Notation a×10n as the product of two numbers: a (the digit term) and -> number of significant figures  10n (the exponential term). -> exponent n is Anzahl der Nullen A positive exponent => Decimal point is shifted that number of places to the right. A negative exponent => decimal point is shifted that number of places to the left. Maintain the balance! If one goes up (↑) by 10, the other must go down (↓) by the same magnitude, and vice versa.
• If 6^x > 216^2, what is the smallest integer value of x? 7
• If k=5^3, and k^x=5, what is the value of x? Now ignore the bases and equate the exponents -->    3x = 1 -->    x = 1/3 Hence, this is the correct answer.
• If x and y are integers, and 4^x·5^8=2^3x·5^x+y what is the value of y? Now the bases are the same on both sides, so we can equate the exponents: --> 2x = 3x Isolate x: move the 2x to the other side of the equation by subtracting 2x from both sides: --> 2x-2x = 3x-2x --> x = 0 Plug in x=0 into second equation: --> 8 = x+y --> 8 = 0+y --> 8 = y
• Roots Multiplication / Division As long as the root is equal, multiplying and dividing roots with different bases involves combining the two bases under the same root. Example (multiplying roots with different base, same root): √2·√8 = √(2·8) = √16 = 4 Example (dividing powers with different base, same root): √75 / √3 = √(75 / 3) = √25 = 5
• Roots Addition and Subtraction There's not much you can do if faced with a situation involving adding or subtracting roots with different bases, except beware of the traps, which will definitely be laid by our friends at GMAC: As we've seen in our Powers - Do's and Don'ts lesson, this is a big no-no. The same goes for roots.  Adding √2 and √3 does NOT magically reach a result of √(2+3) = √5.  The same goes for subtraction: √7 - √4 ≠ √3
• Reasons for Plugging-In ALWAYS when there are variables in the answer choices, use plugging in numbers! Being extra careful when GMAC messes around with different units in the same problem (e.g. Dollars vs. Cents). Not trying to approach algebraic problems with variables in an algebraic way. Use numbers instead. That way you can be 150% sure with your answer choices. Don't miss opportunities to PLUG IN: use this technique when variables are around. Choose Good Numbers: integers, positive, small, and fit the problem (For example, if the problem asks about one-tenth of x, choose a number that's divisible by 10).
• positive and negative roots If the root sign is already there to begin with, it signifies only the positive root It's true that every positive number has both a positive and a negative square root. That's why if x2 = 9, then x may be either 3 or -3. However, as a mathematical convention, a root sign always denotes the positive root. Thus, √9 denotes only  3, and not -3. That's why in this case √64 denotes positive 8 only.
• "Molly Malone drives a wheelbarrow in streets broad and narrow. If she earns £x per hour, then how many pence will she earn in y minutes? (£1=100 pence)" - what would be a good PLUG-IN for y? After choosing a comfortable number for y, choose a comfortable number for x, such as x=1. And so, y=60, x=1
• (√512)×(√2048)=? Look for a pattern that will allow you to break up these big roots into smaller, more manageable chunks.     Divide 2048 by 512 to find out if it is a multiple of 512. Cancel out the root sign and solve the rest to find the answer. 2048/512 = 4 i.e. √512 x √2048 --> √512 x √(512 x 4) --> √(512 x 512 x 2 x 2) --> √(5122 x 22) --> 512 x 2 --> 1024
• If 2a=b, and the average (arithmetic mean) of a and b is half the average of b and c, which of the following must be true? arithmetic mean, or just mean, it means just that -  average Variables in the answer choice? plug in good numbers and eliminate. Begin with the variable which is easier to use to calculate the others. If a=2, then b = 2a = 4. The average of 2 and 4 is 3, so the average of b and c is 2·3=6. Find c using the average formula: (b+c) / 2 = 6         /·2 -->    b+c = 12 Plug in b=4 to get -->    c=12-4 = 8. So, the variables you're plugging in are a=2, b=4 and c=8
• The Median-Concept The median is the middle number in a set of numbers, arranged in ascending or descending order. If the number of elements is odd, the median is the middle number. If the number of elements is even the median is the average of the middle two elements. Arrange the elements in ascending order, like so: {3, 7, 7, 14, 26, 42, 111}.
• The Average Pie In GMAT averages problem solving questions, when any two parts of the average formula are given, you can and should find the third part. In Data Sufficiency problems involving averages, two parts of the average formula are needed to determine sufficiency. It is more convenient to use the average formula in the average pie format (or better yet, the plumber's butt). The relations between the different parts of the formula are shown in the average pie
• Percent change formula Words like percent increase, percent decrease, percent more, percent less can help you identify percent change problems. percentage change = change / original If the question asks about percent more or percent increase, the original is the smaller number. If the question asks about percent less, percent decrease, the original is the larger number.
• "...there are 25 percent as many Kingfishers as Paddyfield-Warblers in the reserve" => 25% of the Paddyfield Warblers is the number of Kingfishers (25% of 28 is 7).
• Hidden Plugging In - Percents If the question asks about percents, plug in 100 or a multiple of 100 Substitute difficult words with simpler ones to understand what to do ("...there are 25 percent as many Kingfishers as Paddyfield-Warblers in the reserve")
• Sets Set = Group of objects. Sample space = All the relevant objects from which it is possible to select a set. Complementary set = The set of all items in the sample space that do NOT belong to set A. Concept of MECE Only possible answers when sets are MECE Can the sample space include any more options besides A and B?
• How to set a table: A/Not A Table 1. Draw an empty 3×3 table.2. Place one set (Set A) as the the title of the first row.3. Title the second row as the complementary set (Not A) and the third row as the Total.4. Repeat the process in the columns for the other set in the question.5. Insert the data in the stem question to the proper cells.6. Mark the cell that denotes the requested value with a question mark.7. Calculate the value of the empty cells based on the rule that every row/column in the table fulfills the summation X+Y=Z.
• Must Be Questions: Use DOZEN F numbers Different (e.g. even vs. odds, prime vs. multiple, fractions vs. integers, positive vs. negative, etc.) One Zero Equal numbers for different variables Negatives Fractions Different: even => try odd a multiple of x => try a prime small (e.g. x=3) => huge numbers (e.g. x=10,000) One  :  special case often overlooked.Zero : ditto. Equal: ditto  Negatives : The average test taker automatically thinks of variables X and Y as positive integers. Fractions
• If -2<x<2, which of the following must be non-negative? "If it fails - POE. If it doesn't, make it fail!" Your mission in "Must Be" questions should be to try and disqualify the answer choices, then POE, POE, POE, until only one remains. Your target is non-negative, and the issue here is negative vs. non-negative. Plug in numbers that fit -2<x<2, then POE; Plug in x=0.5 to POE B. Plug in x=-1.5 to POE A and C. Plug in x=1.5 to POE E.
• ZERO is a multiple of ZERO is a multiple of all integers.
• If a is an odd integer and m=(a+2)(a+3), which of the following must be true? (I) m is a multiple of 2 (II) m is a multiple of 3 (III) m is a multiple of 4 Although statements I and II are seemingly correct, ask yourself for each one: "Is it always true, for any number?" Plug in a DOZEN F number, for instance a=-1. Now, m=2, and statement (II) fails. POE B and D. The correct answer is A. [You may suggest Plugging In a=-3 to make m=0. This is perfectly fine and doesn't invalidate the solution. In fact, 0 is a multiple of all integers. A multiple of n means that the number can be written as an integer times n. Since 0 = 0•n, 0 is a multiple of all integers.]
• Data Sufficiency Question stems can never be answered without additional data; Consider the following stems: Your mission (should you choose to accept it) is to decide if the data in the statements enable you to answer the question. Consider the following Data Sufficiency problem: 1. What is the value of x? (1) 3x+4=10 (2) Blah...Blah...Blah... 2. We were just teasing, but you're right. The value of x is irrelevant so don't waste time on figuring it out. Remember, all you have to do is decide whether you can answer the question, but not really answer it. 3. The "Blah... Blah...Blah..." in statement (2) is not a mistake. That is our special way of telling you to focus on each statement alone, starting with statement (1). Statement (1)-> Sufficient    -> AD Statement (1)->Insufficient   -> BCE
• Which of the following must be non-negative? Plug in a=0.95 to POE A. Plug in a=0 to POE B. Plug in a=-0.95 to POE C. Plug in a=-0.5 to POE E.
• If a and b are integers, and a·b≠0, and ab<0, then which of the following must be true? I) a<0 II) b<0 III) b is odd. For a power to be negative, the base must be negative and the exponent must not change the base to positive, i.e. the exponent has to be odd. Note that II is not necessarily true: (-2)3=-8 is an example of a negative power with a positive b. Remember this difference between even and odd powers: An even power is always positive, whether the base is positive or negative. An odd power retains the base's original sign.
• Addition: EVEN & ODD Operations Addition : Function             Result even + even        =eveneven + odd          =oddodd + odd            =even
• Multiplication Even&Odd Operations MultiplicationFunction         Resulteven x even    =eveneven x odd      =evenodd x odd        =odd
• Substraction: Even&Odd Operations E - E  =   E O - O =   E E - O =   O
• Division Even&Odd Operations E / E = EON E / O = EN O / E = N O / O = ON even / even         anything (even, odd, or not an integer)even / odd           even or not an integerodd / even           not an integerodd / odd             odd or not an integer
• Even Integers that are divisible by 2.
• Odd Integers that are not divisible by 2.
• Is the number 0 even or odd? The result is an integer, so the number 0 is divisible by 2. As a result, the number 0 is even.

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