Math-Book-GMAT-Club

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AlgebraScopeManipulation of various algebraic expressionsEquations in 1 & more variablesDealing with non‐linear equationsAlgebraic identitiesNotation & AssumptionsIn this document, lower case roman alphabets will be used to denote variables such as a,b,c,x,y,z,wIn general it is assumed that the GMAT will only deal with real numbers ( ) or subsets of such as Integers (), rational numbers ( ) etc.Concept of variablesA variable is a place holder, which can be used in mathematical expressions. They are most often used for twopurposes :(a) In Algebraic Equations : To represent unknown quantities in known relationships. For e.g. : "Mary's age is 10more than twice that of Jim's", we can represent the unknown "Mary's age" by x and "Jim's age" by y and then theknown relationship is(b) In Algebraic Identities : These are generalized relationships such as, which says for any number,if you square it and take the root, you get the absolute value back. So the variable acts like a true placeholder,which may be replaced by any number.Basic rules of manipulationA. When switching terms from one side to the other in an algebraic expression + becomes ‐ and vice versa.E.g.B. When switching terms from one side to the other in an algebraic expression * becomes / and vice versa.E.g.C. you can add/subtract/multiply/divide both sides by the same amount.E.g.D. you can take to the exponent or bring from the exponent as long as the base is the same.Egg 1.Egg 2.It is important to note that all the operations above are possible not just with constants but also with variablesthemselves. So you can "add x" or "multiply with y" on both sides while maintaining the expression. But what youneed to be very careful about is when dividing both sides by a variable. When you divide both sides by a variable‐ 23 ‐GMAT Club Math Bookpart of GMAT ToolKit iPhone App
(or do operations like "canceling x on both sides") you implicitly assume that the variable cannot be equal to0, as division by 0 is undefined. This is a concept shows up very often on GMAT questions.Degree of an expressionThe degree of an algebraic expression is defined as the highest power of the variables present in the expression.Degree 1 : LinearDegree 2 : QuadraticDegree 3 : CubicDegree 4 : Bi‐quadraticExample: the degree is 1the degree is 3the degree of x is 3, degree of z is 5, degree of the expression is 5Solving equations of degree 1 : LINEARDegree 1 equations or linear equations are equations in one or more variable such that degree of each variable isone. Let us consider some special cases of linear equations :One variableSuch equations will always have a solution. General form isand solution isOne equation in Two variablesThis is not enough to determine x and y uniquely. There can be infinitely many solutions.Two equations in Two variablesIf you have a linear equation in 2 variables, you need at least 2 equations to solve for both variables. The generalform is :Ifsatisfy the secondthen there are infinite solutions. Any point satisfying one equation will alwaysIfthen there is no such x and y which will satisfy both equations. No solutionIn all other cases, solving the equations is straight forward, multiply equation (2) by a/d and subtract from (1).More than two equations in Two variablesPick any 2 equations and try to solve them :Case 1 : No solution ‐‐> Then there is no solution for bigger setCase 2 : Unique solution ‐‐> Substitute in other equations to see if the solution works for all othersCase 3 : Infinite solutions ‐‐> Out of the 2 equations you picked, replace any one with an un‐picked equation andrepeat.More than 2 variablesThis is not a case that will be encountered often on the GMAT. But in general for n variables you will need at leastn equations to get a unique solution. Sometimes you can assign unique values to a subset of variables using lessthan n equations using a small trick. For example consider the equations :In this case you can treat as a single variable to get :These can be solved to get x=0 and 2y+5z=20‐ 24 ‐GMAT Club Math Bookpart of GMAT ToolKit iPhone App
Page 2 and 3: For the latest version of the GMAT
Page 4 and 5: Number TheoryDefinitionNumber Theor
Page 6 and 7: • 1 (and ‐1) are divisors of ev
Page 8 and 9: other digits: 21‐(9+8+6+9)=‐11,
Page 10 and 11: Sum of n first positive integers:Su
Page 12 and 13: Converting Decimals to Fractions•
Page 14 and 15: Operations involving the same bases
Page 16 and 17: PERCENTDefinitionA percentage is a
Page 18 and 19: Absolute ValueDefinitionThe absolut
Page 20 and 21: satisfied (‐15 is not within (‐
Page 22 and 23: FactorialsDefinitionThe factorial o
Page 26 and 27: There is a common misconception tha
Page 28 and 29: RemaindersDefinitionIf and are posi
Page 30 and 31: A. 2B. 4C. 8D. 20E. 45divided by yi
Page 32 and 33: Example #10 (hard)When positive int
Page 34 and 35: Word Problems OverviewThe Following
Page 36 and 37: Key words: less than (implies subtr
Page 38 and 39: Distance/Speed/Time Word ProblemsWh
Page 40 and 41: In the next section we will learn h
Page 42 and 43: Time(3) = Time(1) + Time(2) ‐‐
Page 44 and 45: 3*x = 5*(x ‐ 20) ‐‐> x = 50 m
Page 46 and 47: Work Word ProblemsWhat is a ‘Work
Page 48 and 49: Example 3:Working together, printer
Page 50 and 51: Advanced Overlapping Sets ProblemsS
Page 52 and 53: the writing club. If 6 students sig
Page 54 and 55: two of the courses and 1 employee h
Page 56 and 57: Consider this problem to be an over
Page 58 and 59: Median The median of a triangle is
Page 60 and 61: • The midsegment is always parall
Page 62 and 63: Equilateral triangle all sides have
Page 64 and 65: • A right triangle where the angl
Page 66 and 67: Then the triangles ABC, CHB and CHA
Page 68 and 69: Quadrilateral A polygon with four '
Page 70 and 71: Squares A 4‐sided regular polygon
Page 72 and 73: • Area ‐ The usual way to calcu
Page 74 and 75:
• A circle is the shape with the
Page 76 and 77:
ChordA line that links two points o
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angle.• An inscribed angle is exa
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chords.‐ This becomes the theorem
Page 82 and 83:
Coordinate GeometryDefinitionCoordi
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As you can see, the distance formul
Page 86 and 87:
Lines in Coordinate GeometryIn Eucl
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The equation of a straight line tha
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Positive slopeHere, y increases as
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A horizontal line has a slope of ze
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Solution: We first find the slope o
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To answer, we must find the slope o
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Example #2Q: Find the point of inte
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ParabolaA parabola is the graph ass
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Standard DeviationDefinitionStandar
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Standard deviation of is . Decrease
Page 106 and 107:
ProbabilityDefinitionA number expre
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Therefore, the probability of getti
Page 110 and 111:
2) Reversal combinatorial approach:
Page 112 and 113:
Combinations & PermutationsDefiniti
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1. The total number of arrangements
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is the first termDefining Propertie
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Solution(1) a1=8, does not tell us
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CuboidA cube is the 3‐D generaliz
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ConeA cone is a 3‐D object obtain
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Volume =Surface Area w/o base =Surf
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(2) : Sum of sides = . Not sufficie
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Math-Book-GMAT-Club

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