GMAT (Subject) / Quant (Lesson)

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• area of a triangle area of a triangle - 1/2 × base × height
• ratio of any circle's circumference to its diameter π = 3.1415926535897932384626433832795028841971
• √2 approx. 1.4
• √3  approximately 1.7
• Real Numbers All numbers in the GMAT quantitative section are Real numbers. A real number is a fancy name for just any ol' number as we know it:  a real number could be positive (such as 4) or negative (-4), a fraction (2.5) or an integer (-10,000). Many of the GMAT algebra problems revolve specifically around integers. Questions involving integers usually test your understanding of algebraic concepts, rather than your ability to merely calculate a certain value. Therefore, some of the more challenging questions you will meet on the test will come from this group. 
• What is an Integer an integer is simply a non-fraction or a non-decimal. As we said, Integers are real numbers, and can be positive (1, 2, 3, 1573) or negative (-1, -3, -10523)
• Negative/positive Numbers By definition, negative means "less than zero", or "to the left of zero on the number line". Apply this definition to non-negative, and realize that non-negative means "not less than zero", or "not to the left of zero in the number line"
• Even and Odd: Definition Even: any integer that is divisible by 2 with no remainder (remainder =0). Examples: 2, 4, 14. Odd: The formal definition of an odd integer is an integer that is divisible by 2 with a remainder of 1. For simplicity's sake, we'll say that an odd integer is simply one that is NOT even i.e. not divisible by 2 with no remainder. Examples: 1, 3, 5, 7, 9
• Power x^5: The x is called the "base". The 5 is called the "exponent".
• Powers - Special Cases Four special powers you want to remember: 1) Anything raised to the power of "1" equals itself. 2) 1 raised to any power equals 1. 3) Any number (except zero) raised to the power of zero equals 1. 4) Zero raised to any power (except zero) equals zero.
• Powers - multiplying powers of the same base add the exponents
• Powers - raising a power to another power multiply the exponents
• Powers - Even and Odds An even power is always positive, whether the base is positive or negative. 24 is positive = 16. (-2)4 is also positive. Even though the base is negative, (-2)4 = (-2)·(-2)·(-2)·(-2) = 16. Remember: any pair of minus times minus become plus. An odd power retains the base's original sign.  If the base is positive, raising it to an odd power will give a positive result as well. e.g. 23 = +8 if the base is negative, raising it to an odd power will give a negative result. e.g. (-2)3 = (-2)·(-2)·(-2) = -8
• Powers - Minus within/outside of brackets Note that whether the minus sign is inside or outside the parentheses is important! (-2)4 means that the minus sign is part of the base and is affected by the exponent. e.g. (-2)4 = (-2)·(-2)·(-2)·(-2) = 16 -24 means that the minus sign is not part of the base - First apply the power, then apply the minus sign. In this case, -24 = -(2·2·2·2) = -(16) = -16.
• Powers - addition and subtraction 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) 2) If you're not sure that you factored the expression correctly, check that re-expanding the brackets does return to the original expression. 3) Like terms (same base and same exponent) can always be added/subtracted: 3a2 + 2a2 = 5a2
• Powers - Scientific notation Scientific notation is the way in which scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0043, we write 4.3 x 10-3. So, how does this work? Think of a×10n as the product of two numbers: a (the digit term) and 10n (the exponential term). In scientific notation, the digit term indicates the number of significant figures in the number. The exponentialterm only places the decimal point. The exponent of 10 is the number of places the decimal point must be shifted to give the number in long form: A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.
• Identifying exponential equations - an equation with variables in the exponents. 1) Bring both sides of the equation to the same base(s). 2) Ignore the bases and equate the exponents. 3) Solve for the needed variables
• Raising power to power (reverse) (am)n = am·n x10 = x5∙2 = (x5)2
• Powers - multiplying When multiplying powers of the same base, add the exponents. an·am = an+m This rule may also be applied in reverse. To do so, first rewrite the exponent as an addition. Next, apply the parts of the addition as exponents in two powers multiplied by one another, each with the same base as the original power. For example: x4 = x3+1 = x3∙x x7 = x5+2 = x5∙x2 This is also the proper way to treat an exponent which already includes an addition sign: xY+5 = xy∙x5 y3n+1 = y3n∙y 3x+1 = 3x∙31 = 3x⋅3
• Roots Roots are simply the opposite of powers. A root is signified by the "√" sign, called the radical. A plain radical √ signifies the second root (also known as square root) of a number; a  3√ is the third root (a.k.a cube root), 4√ is the fourth root etc. Roots are a slightly less intuitive concept than powers. A root asks a question: Which number, when raised to the root's power, will equal whatever is under the radical sign? For example, Take 3√8, which we know is also equal to 2. To find out why, ask yourself the question: "which number, when raised to the power of 3 (cube root), will equal 8?" Put another way, a root asks you to fill the blank in the following equation: ☐3 = 8. Now, since 23 = 8,  we know that the answer is 2, or 3√8=2. Another example is √4. √4 asks the same question - "which number, when raised to the second power, will equal 4?" the answer is, of course, 2, which is why √4=2.
• Perfekt Squares - LEARN!!! A perfect square is an integer with an integer square root 1=12=43=94=165=256=367=498=649=8110=10011=12112=14413=16914=19615=22516=25617=289
• Multiplying Roots with the same Base Since roots are simply fractional powers, they follow the rules of powers with regards to arithmetic operations. When multiplying roots with the same base, simply convert the root to power form and add the exponents √a·∜a = a½·a¼ = a½+¼ = a¾ = ∜a3
• Dividing Roots with the same Base Since roots are simply fractional powers, they follow the rules of powers with regards to arithmetic operations. When Dividing roots with the same base, simply convert the root to power form and subtract the exponents. For example: √a / ∜a = a½-¼ = a¼ = ∜a
• Rasing Roots to a Power Since roots are simply fractional powers, they follow the rules of powers with regards to arithmetic operations. When raising a root to another power, simply convert the root to power form and multiply the exponents. For example: (√a)2 = (a½)2 = a½·2 = a1 = a
• Quotient A quotient is the result of a division between two numbers. For instance, when 10÷2=5, the quotient is 5.
• Division - Variables The dividend in the preceding example is 10, while the divisor is 2. And so,  Dividend÷Divisor=Quotient. In other words the quotient is the number of times the divisor divides into the dividend.
• Remainder When dividing two integers, the quotient refers only to the integer part of the result . For instance, 7 ÷ 5=1 (2). The quotient is 1, while the remainder is 2.
• Fraction A fraction is defined as part/whole. The number on top is called numerator, and the one on the bottom is the denominator
• Set questions Set = Group of objects. Sample space = All the relevant objects from which it is possible to select a set.
• Complementary Sets Der Rest des sample space Complementary[A] = Not A Complementary set Complementary to set A: Is the set of all items in the sample space that are not included in set A. Mathematical phrasing: Not A. Everyday language phrasing: Everything but A. In special cases set B can be complementary to set A if everyday knowledge denotes that B is the only Not Athat exists - boys/girls, smokers/non-smokers etc. Notice that opposites doesn't necessarily mean complementary: tall is not complementary to short, happy is not complementary to sad, etc. When it is stated in the question that every item in the sample space belongs to either set A or set B, and no other option exists, then for this question set B is complementary to set A.
• Basics of Must be Questions Let's review the basics of Must be questions: Must Be Questions look for the one answer that is always true, for any number. The correct approach to these problems is to Plug in more than once and try to "break" the problem - try to find an example for which the question is NOT true. The basic steps for Must Be Questions: Try to figure out the issue of the problem. That's how you'll know what to Plug In, especially, while plugging in more than once. Here are a few rules of thumb to get you going;If the question contains multiplication and divisibility, try to plug in positive vs. negative numbers.If the question contains exponents and roots, try to plug in integers vs. fractions.Plug In good numbers first, then POE.If several answers remain, ask yourself for each answer choice: Is it always true, for any number? Plug In again using DOZEN F, then POE. Keep Plugging In until only one choice remains.Remember the DOZEN F: Different (e.g. even vs. odds, prime vs. multiple, fractions vs. integers, positive vs. negative, etc.)OneZeroEqual numbers for different variablesNegativesFractions
• Sequences A sequence is a set of numbers that follow a certain RULE. In fact, sequences can be viewed as a subset of functions questions. Examples: 1) An = n+2 The nth term in this sequence is determined by the RULE of n+2. The value of a term in this sequence isdetermined solely by its place in the sequence n. The first term (n=1) is A1=1+2=3. The second term (n=2) is A2=2+2=4. 1) Place dependent - The value of a term An is determined solely by its place in the sequence n. e.g. An = n+2 2) Term dependent - The value of a term An is determined by the value of the some other term, such as the preceding term An-1. e.g. An = An-1+4 3) Combination - The value of a term An is determined both by the value of the preceding term An-1 AND by its place n in the sequence - both are needed to calculate the value of An. e.g. An=An-1 - n + 2
• Percent Change Percent Change = (Difference/Original Value) * 100
• Probability Probability = Number of wanted Outcomes / Total Number of Possible Outcome Probability of one means something MUST happen. Examples: What is the probability of getting a result of "1" when rolling a regular six-sided die? = 1/6 What is the probability of getting a "2" on a single roll of a regular six sided die? = 1/6 What are the chances of getting a number within the range of 1-6 inclusive, when rolling a six-faced die? = 1
• Two-Part Analysis questions Some Two-Part Analysis questions will be purely verbal based. When solving these questions, first try to determine whether the answer choices are interdependent (i.e., they rely on each other) or independent(i.e., each answer can be found independently).  Interdependent questions may include answer choices that reflect a cause/effect, a precondition/condition, an if/then, an assumption/supporting fact, or a true/false dichotomy.  Independent questions may include dichotomies involving increases/decreases, strengthening/weakening, sacrifices/gains, or what is true/false.  You can use strategies that you have already learned from the Critical Reasoning and Reading Comprehension sections.
• The reciprocal or the inverse of a fraction is that fraction flipped over; top to bottom and bottom to top. i.e., the reciprocal of 2/5 is 5/2.
• Fractions The greater the denominator, the smaller the fraction (under the same numerator). e.g. 1/2 > 1/3 > 1/4
• Sequences: Consecutive Integers Consecutive Integers: integers that follow in sequence, each number being 1 more than the previous number. Examples:  1, 2, 3, 4...; -10, -9, -8,... ; -1, 0, 1,...
• Consecutive Integers Consecutive Integers: integers that follow in sequence, each number being 1 more than the previous number. Examples:  1, 2, 3, 4...; -10, -9, -8,... ; -1, 0, 1,... 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...
• Math Fundamentals: The Remainder When dividing 5 by 3, the result can be viewed in many different ways, depending on what the question asks: 1) A Fraction: 5/3 2) A decimal: 1.67 3) A quotient and an Integer remainder: 5 divided by 3 gives a quotient of 1 with a remainder of 2. A remainder is what's left after the arithmetic action of division. Remainder is defined as the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend.
• Math Fundamentals - Remainder 2 Where plugging in is difficult to use because the question requires large numbers, use the following equation: For any integer i divided by another integer d i = quotient·d + remainder Thus, if when i is divided by 5 the remainder is 3, i can be expressed as the equation: i=5x+3  x being the quotient. This form also supports plugging in: plug in values for x, and you will find the corresponding values of i: If x=0   -->     i=5⋅0+3 = 3 If x=1   -->     i=5⋅1+3 = 8 If x=2   -->     i=5⋅2+3 =13 etc. To sum up: 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. e.g. When dividing by 5, the highest possible remainder is 4.Identifying remainder problems in the GMAT - the question uses the word remainder (Duh!).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 + remainderThus, if dividing i by 5 leaves a remainder of 3, i can be expressed as the equation: i=5x+3  x being the quotient.
• Divisible without remainder In the GMAT, the default meaning of the term divisible means divisible without remainder (i.e. a remainder of zero). For example, if x is divisible by 5, this means that x is divisible by 5 with no remainder. Note that this also means that x is a multiple of 5 - since there's no distance between x and the nearest multiple of 5.
• Factorials are a necessary concept for dealing with Combinations and Permutations problems.  Factorials are marked by a "!" sign. n! means "n multiplied by all consecutive integers less than n down to 1": n·(n-1)·(n-2)·...·1. For example: 5! = 5·4·3·2·1 0! = 1! = 1
• The rules of arithmetic using Odd and Even are important in many GMAT integers questions. Here are the rules for even and odd numbers in positive integer powers: Evenpositive integer (even or odd) = Even (e.g. 22=4; 23=8) Oddpositive integer (even or odd) = Odd (e.g. 32=9; 33=27) Bottom line: for powers (with a positive integer exponent), the base rules . If the base is even, the result is even. If the base is odd, the result is odd. 
• Even and Odd: Division Rules no rules, except: If there is one general rule that can be used for a specific case of division, it is this: Odd/Even = Fraction An odd number is not divisible by 2. An even number is divisible by 2, and will therefore be a multiple of 2. Therefore, an odd integer will never be divisible by an even integer, and the result will be a fraction.
• Even and Odd: Rules of Subtraction and Addition The rules of arithmetic using Odd and Even numbers are important in many GMAT integers questions. Here are the rules for adding / subtracting even and odd numbers: Even ± Even = Even (e.g. 2+2=4; 4-2=2) Even ± Odd = Odd (e.g. 2+1=3; 2-1=1) Odd ± Odd = Even (e.g. 1+1=2; 3-1=2)
• Even and Odd: Rules of Multiplication Even × Even = Even (e.g. 2×2=4) Even × Odd = Even (e.g. 2×3=6) Odd × Odd = Odd (e.g. 1×1=1)
• Integers: Factor = Divisor Definition of a factor: A factor is a positive integer that divides evenly into another integer. Another way of looking at the (slightly confusing) sentence above: A factor of n is a positive integer that n is divisible by with no remainder. So if 6 is divisible by 1, 2, 3 and 6, these positive integers are all factors of 6. You may be familiar with the word divisor - for the purpose of the GMAT, factors and divisors are quite similar, with a slight difference: a divisor can be positive or negative, while a factor is always positive. Remember this: a factor is a positive divisor.  In the above example, 1, 2, 3, and 6 are all divisors, as well as factors, of 6.
• Factors vs. Multiples It is sometimes easy to confuse factors and multiples, especially under the "fog of war" of the test, with questions whizzing past your head. Factors and multiples are essentially opposite terms: Factors of a number are positive integers that evenly divide into that number. Multiples of a number are formed by multiplying that number by any integer.
• Factors: Example If x is a factor of 6 then 6 is a multiple of x.  Example: 2 is a factor of 6, which means that 6 is a multiple of 2 - 6=2⋅3.

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