Quants One Stop Guide

5/22/2018 Quants One Stop Guide

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E-mail: [email protected]

Quantitative Aptitude-A quick reference


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Arithm

Types Description Example

Real Numbers All numbers on Number Line

Rational Numbers Any number that can be represented in the form a/b, where a & b are integers

Integers All Whole Numbers, without a fractional or Decimal Part 5

Common Decimals/fractions All Numbers , with a fractional or Decimal Part 0.555, 0.567

Terminating For a/b, when remainder equals 0 = 0.5

Non-Terminating For a/b, when remainder never comes to 0 0.777.

Pure Recurring Decimals in which all figures after decimal point Recur 0.99999.

Mixed Recurring Decimals in which only some figures after decimal point Recur 0.31222

Irrational Non-Terminating & Non-Repeating 2 = 1.414213

Integers

All Integers are Numbers, but all Numbers are not Integers 0 and 1 are not Prime Numbers

2 is the first/only even Prime Number All Prime numbers a re Positive

Absolute Value of n = |n| = Distance between 0 and n on the number line . For example, |-2| = 2

Types of Integers

Types Description Example

Whole or Counting All +ve numbers {0,1,2,}

Positive or Natural Greater than 0 {1,2,3,}

Negative Lesser than 0 {..,-3,-2,-1}

Even Divided by 2 with 0 as Remainder {,-2,0,2,4,}

Odd Divided by 2 with 1 as Remainder {,-3,-1,1,3}

Prime Greater than 1, with exactly two integer factors/divisors {2,3,5,7,11,.} Composite Any Number excep t 1 that is not Prime {4,6,8,9,10..}

Consecutive Set of Numbers with Fixed interval {1,2,3,4,.}

Distinct Numbers with Different Values 2 and 5


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Arithmetic Operations

Addition, Subtraction, Multiplication and Division

Subtracting a number is same as adding i ts opposite

Dividing by a number is the same as multiplying i ts opposite Dividend = (Divisor * Quotient) + Remainder

Order of Operation

PEMDAS :Parentheses Exponents Multiplication Division Addition Subtraction

The operations of multiplication and division must be performed in order from left to right

The operations of multiplication and division must be performed before those of addition and subtraction

Laws of Operation

Commutative Law of Operation Addition or Multiplication can be performed in any order without changing the result

Associative Law of OperationAddition or Multiplication can be regrouped in any order. Distributive Law of OperationFactors can be distributed across the terms being added/subtracted/multiplied/divided.

When the sum or difference is in the Denominator, no distribution is applicable

Divisibility Tests

Tests Description

Divisibility Test for 2 If Units Digi t is divisible b y 2 or is a multiple of 2

Divisibility Test for 3 Sum of all di gits is divisible b y 3 or is a multiple of 3

Divisibility Test for 4 Number made by Ten s and Units Digit is divisibleby 4 or is a multiple of 4

Divisibility Test for 5 If Units Digit is equal to 0 or 5

Divisibility Test for 6 If it is di visible both by 2 and 3.

Divisibility Test for 8 Last three digits are divisible by 8. Or if its divisible by 2 thrice

Divisibility Test for 9 Sum of the digits is divisible b y 9 or multiple of 9

Divisibility Test for 10 If last Digit is 0

Divisibility Test for 12 If it is Divisible by 3 and 4

The Product of n consecutive integers is always divisible by n, or is a multiple of n The Sum of n consecutiveintegers is always divisible by n, or is a multiple of n

If there is one e ven I nteger in a Consecutive series , the Product of the series is divisible by 2

If there are two even Integer in a Consecutive series, the Product of the series is divisible by 4 If a is divisible by b, then a is also divisible by all the factors of b

Greatest Common Factor

GCF of two or more numbers is the la rgest integer that is a factor of both numbers . For Example, 6 is the GCF of 12 and 18.

Methods for Determining Prime Numbers:Test all the prime numbers that fall below the approximate square of the given number

Least Common Multiple

Smallest common multiple of all the given numbers

Adding and Subtracting with Odd and Even Numbers

Tasks Description

Even + Even or Odd + Odd Sum and Difference is EvenEven + Odd Sum and Difference is Odd

Sum/Difference of two Even Even

Sum/Difference of two Odd Even

Sum/Difference of Even and Odd Odd


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Multiplying and Dividing with Odd and Even Numbers

Tasks Description

Even * Even , Even/Even Even

Odd * Odd, Odd/Odd Odd

Even * Odd, Even/Odd, Odd/Even Even

Sum of any two Primes will be Even If sum of two primes is Odd, then one of the number must be 2

Product of any two numbers a and b = GCF * LCM

Fractions and Decimals

Converting Fractions to Decimals

Step-1: Reduce the fraction to its lowest terms Step-2: Next, divide the numerator by denominator

For example, 1/100.10

Converting Decimals to Fractions

Step-1: First Eliminate the decimal poin t, and write(Right to decimal Point) i t as the numerator of the resulting fraction

Step-2: Next, divide it by 1 followed by as many zeroes as the number of places to the right of the decimal point of the given number,

write that as the denominator of the resulting fraction

Step-3: Simplify the resulting fraction to its lowest terms by dividing the numerator and denominator by its GCF

For example, 0.1010/1001/10

Proper Fraction a/b, where ab ; Mixed Fraction a (b/c)

A (b/c)((c*a) + b)/c

Additive In verseNegative of the Numbe r

Multiplicative InverseReciprocal of the Number

Quotient of any given number and its negative is -1

How to Simplify Fractions

Method-1: To Reduce a fraction to lowest terms, divide the numerator and denominator by their G.C.F

Method-2: Cancel all common factors of numerator and denominator until there is no common factor other than 1

A fraction is said to be in its lowest terms when the G.C.F of the numerator and denominator is 1

Addition of Fractions:

With Common Denominators: (a/c) + (b/c) = (a + b)/c

With Different Denominators: (a/b) + (b/d) = ((a*d) + (b*c))/(b*d) Same logic holds for subtracting Fractions too

Exponents (a ^ n)

An Exponent is a number that tells how many times the base is a factor. For example, in 5

2

, there a re 2 factors. Here 5 is the base andthe e xponent.

For any number a: an= a*a*a*a* n number of times = bi.e., n

throot of b is a

nb = a

Square of an y positive number o r square of i ts negative will always be positive

n0 =

1, where n # 0


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Any number raised to the negative power equals the reciprocal of that same number or expression raised to the absolute value of the

power indicated, which results in a fraction with a numerator of 1. a-n

= 1/an

am/n

=n a

m(nth roo t of a raised to the power of m)

Table for Combining Exponents

Same Base Same Exponent

Add When multiplying expressions with the same base,

ADD the exponents.

am

* an= a

(m + n)

When multiplying expressions with the same exponent,

MULTIPLY the bases

an

* an

= (ab)n

Multiply

Subtract When dividing expressions with the same base,SUBTRACT the exponents

am

/ an

= a(m - n)

When dividing expressions with the same exponent,DIVIDE the bases

an/ a

n= (a/b)

n

Divide

Same Base Same Exponent

Format of Scientific notationa.bcde * 10(n)

, where a,b,c,d,e are any positive numeric digits, such that, 0


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Alge

(Xm

Yn)/ (X

pY

q) = X

m-pY

n-q

(Xm

+ Yn)/Z = (X

m)/Z + (Y

n)/Z

Factor a x2+ bx + c = 0 into the following two factors . (p+q) (r+s); such that:

First term of the trinomial p * r

Last term of the trinomialq * s

Middle term of the trinomial

(ps) + (qr)

Eight steps to solve equations (To be followed in same order)

Step-1: Get rid of fractions and/or decimals by multiplying each term of both sides by the LCD. (Apply only if equation has Decimal/Fractions)

Step-2: Get rid of all the parentheses using distributive law. (Apply only if equation has parentheses)

Step-3: Combine Like Terms on both sides (Apply only if Like Terms Exist)

Step-4: Isolate all the terms with variable expressions on one side by addition or subtraction , and then combine them (Apply only if Variables ex

on both sides)

Step-5: Isolate all the terms with numerical expressions on the other side of the equation by addition or subtraction, and combine them (Apply

if numerical e xpressions are o n both sides)

Step-6: Get rid o f the radical signs if there are an y, by squaring both sides of the equation (Apply only if equation has radicals)

Step-7: Get rid of the exponents if there are any, by taking the root of both the sides by the same number (Apply only if equation has exponents

Step-8: Multiply and/or Divide both sides by the coefficient of the variable (Apply only if equation has co -efficient)

Six Steps to Solve Linear Equations

Step-1: Multiply one or both the equations by the same or different numbers so that the coefficient of one of the variables are of same absolut

value but of o pposite signs

Step-2: Add the resulting equations

Step-3: Now, one of the variables will be eliminated by cancelling out to zero; hence new equations with only one variable results out.

Step-4: Solve this new linear equation with one variable by following the above 8 steps

Step-5: This will result in a value of one of the variables; substitute this value into either one of the original equations , which will result in new

equation with the other variable

Step-6: Solve this equation and find the value of other variable

Quadratic Equations roots/solutions

X = 1/2a [-b + (b2

4ac)] and X = 1/2a [-b - (b2

4ac)]

If (b24ac) > 0, then (b

24ac) will be two distinct real number roots

If (b24ac) < 0, then there exists no solu tion or real roots

If (b24ac) = 0, then (b

24ac) will be zero. And expression has only one real root or solution

Sum of R oots -b/a

Product of R ootsc/a Axis of symmetry-b/2a

Solving Quadratic Equations

Step-1: If required manipulate the equation by grouping, such that, all the terms are set on one s ide of equation and othe r side is zero in such a

way that it can be factored and put into the standard form: ax2+ bx + c = 0

Step-2: Combine the Like terms on the nonzero side of the equation

Step-3: Factor the left side of the equation into linear binomial expression factorsStep-4: After breaking the equation into linear factors, set each linear factor equal to zero

Step-5: Solve for both the mini equations, the two resulting values is the solution set


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Applicat

To

From

Fraction (1/2) Decimal (0.50) Percent (50%)

Fraction (1/2) Not Applicable Step-1: Divide the numerator by

Denominator

Ex: 1 / 2 = 0.50

Step-1: Multiply the Fraction b y 1

Step-2: Simplify and insert % sign

Ex. 1/2 = (1/2) * 100 = 50 %

Decimal (0.50) Step-1: Drop the decimal point by

dividing it by 1 plus add as many zeroes

as the number of places to the right of

the decimal point.

Step-2: Simplify.

Ex: 0.5050 / 100 = 1/2

Not Applicable Step-1: Move the Decimal Point tw

places to the ri ght

Ex: 0.5050 %

Percent (50%) Step-1: Drop the percent sign, next

divide the percent number by 100.

Step-2: Simplify.

Ex: 50 %50/100 = 1/2

Step-1: Move the percents decimal

point two places to the left.

Ex: 50%0.50

Not Applicable

Percents:

What is a % of b? Problem Set-Up: x = (a/100)*b

a is what percent of b? Problem Set-Up: a = (x/100)*b

What % of a is b? Problem Set-Up: b = (x/100)*aa is b% of what number? Problem Set-Up: a = (b/100)*x

a% of what number is b? Problem Set-Up: b = (x/100)*a

Percent Changes:

Percent Change(Actual Change/Original Value) * 100 %

Percent Increase((New ValueOriginal Value)/ (Original Value)) * 100 %

Percent Decrease((Original ValueNew Value)/(Original Value)) * 100 %

To Increase a numbe r by K%, multiply it by (100% + K%)

To Decrease a number by K%, multiply it by (100 - K%)

If a number is the result of increasing another number by K%, then, to find the original number, divide by (100% + K%)

If a number is the result of decreasing another number by K%, then, to find the original number, divide by (100% - K%)

Successive Percent Changes

Appl y the following steps when two or more series of subsequent percent changes a re applicable:

Step-1: Compute the first percentage change on the original base. If the original base is not given, assume it to be 100

Step-2: Add/Subtract the first percent change from the base of 100 to find the value after first percent change, also known as the intermediate

value.

Step-3: Compute the second percent change on the value of first percent change

Step-4: Add/Subtract the second percent change from value after the first percent change to find the final percent change

Example problem: If the price of an item raises by 10% one year and by 20% the next, whats the combined in crease?

Percent Discounts

Original PriceSale Price + Discount Amount

Original Price(Discount Amount/Discount %) * 100

Original PriceSale Price / (100% + Markup %)


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New PriceOriginal Price (100 % + Mark-up %) or Original Price (100 % - Mark-up %)

Sale PriceOriginal PriceDiscount Amount

Discount AmountOriginal PriceSale Price

Discount % (Rate of Discount)((Original PriceSelling Price)/Original Price) * 100(Discount Amount/Original Price) * 100

Percent Mark-Ups/Downs

Cost Price: Amount that costs the seller without any profit or loss. I t is the cost that the seller pays or incu rs to procure or produce an item.

Selling Pri ce: Amount that a seller sells an i tem for, which ma y include a profit (mark-up) or loss (mark-down) or neither (break-even price)Break-Even Price: Nothing but the Cost price

Mark-up or ProfitSelling pricecost price

Selling PriceCost Price + Profit

Original Price or Cost PriceSale Price/ (100% + Mark-Up %)

New PriceOriginal Price + Ma rk-up (In crease)

Mark-down or LossCost price - Selling price

Selling PriceCost Price - Loss

Original PriceSale Price/ (100% - Mark-down %)

New PriceOriginal Price - Mark-Down (Decrease)

Percent Interests

Simple Interest:

Interest = Principal * Rate * Time (In Years).

Before applying any of these formulas, make sure the units of each measure are in accordance.

Compound Interest:

Final Balance(Principal ) * (1 + (interes t rate/c))(time) (C)

Where, C = Number of times compounded annually; time = Number of years

Dividing the Interest Rate by the Number of Periods in a year:

If the In terest Rate is compounded annually, divide it by 1

If the Interest Rate is compounded semi -annuall y, divide i t by 2 If the Interest Rate is compounded qua rte rly, divide it by 4

If the Interest Rate is compounded bi -monthly, divide i t by 6

If the Interest Rate is compounded monthly, divide it by 12

The Difference between Simple Interest and Compound Interest: Simple Interest is computed only on the principal; and compound interest is

computed on the principal as well as any interest al ready earned.

Ratios

Ratios a re the mathematical relationship between two or more things. Ratios are nothing but another form of fractions . Perce nt is a ratio in

which the se cond quanti ty is 100.


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Terms of Ratio

The Two numbers in the ratio a re called the terms of the ratio

1st

Term called the antecedent; 2nd

Term called the consequent

Terms of Ratio must be the in the same unit

Real Number value of each part of Ratio (nth

part / (Total parts))*Whole

Combining Ratios by Multiplying Ratios

Step-1: Multiply both the given ratios so that the common terms cancel out, i.e., the se cond term of the first ratio cancel first term of second ra

Step-2: Once the terms they have in common cancel out; combine the ratio as two-part or multiply the cancelled terms to write it as 3 part ratio

For Example; If the Ratio of a to b is 6:5 and b to c is 2:1, what is the ratio of a: b: c?

By Multiplying Ratios:

a / b & b/c 6/5 & 2/1(a/b) * (b*c)(6/5) * (2/1)12/5a:c = 12:5

Now Multiply both can celled bs to get the middle part of the ratio = 5 * 2 = 10. Now, a: b: c = 12:10:5

Laws of Proportion

If , a :b = c:d or a/b = c/d, then following a re true:

ad = bc

b/a = d/c a/c = b/d

(a + b)/b = (c + d)/d (a - b)/b = (c - d)/d

Direct Proportions

Two Quantities x and y, are said to be directly proportional if they satisfy a relationship of the form x = ky, where k is a non zero constant

Different Types of Direct Proportions are:

Money SpentQuantity Bought WeightQuantity

HeightShadow

Actual SizeMap Scale GasolineMiles

TimeWages

Indirect Proportions

Two Quantities x and y, are said to be indirectly proportional if they satisfy a relationship of the form x = k/y, where k is a non zero constant

Different Types of Indirect Proportions are:

WorkersTime

SpeedTime Monthly InstallmentsLoan Period

MembersTime Period for Suppliers


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How to figure out if two Quantities vary directly or inversely? Answering one of the following questions would get the result.

Question1: Will an in crease in one q uantity lead to an increase or decrease in the other quantity?

If it leads to Increase, then the two quantities vary directly

If it leads to Decrease, then the two quantities vary inversely

Question2: Will a decrease in one q uantity lead to a decrease or an increase in the other q uantity?

If it leads to decrease, then the two quantities vary directly If it leads to increase, then the two quantities vary inversely

Compound Proportions

When two ratios that have three or more parts, are in the same proportion, it is called a compound proportion


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Geom

Geometry

Geometry is the study of Shapes (both fla t and curved). Mathematics of the prope rties, measurements, and relationships of points , lines , an

surfaces, and solids

PerimeterMeasurement of the distance all the way round any closed 2-D figure or Object Sum of measure o f all the lengths of all i ts side

AreaCertain amount of region covered or Occupied by 2-D or 3-D closed figuresMeasure of the space inside a flat figure

Square Units (Unit2)Units of measure used to measure the area of any 2-D or the Surface area of any 3-D figures .

Area of 2-D FiguresMeasure of the number of square units that completely fills the region on the surface area of the figure

Area of a Flat Surfacebase * altitude

Surface Area of 3-D figuresSum of the total areas of all the 2-D outer surfaces of the 3-D object Sum of the a reas of ea ch of the so

surfa ces or faces.

Volume Certain amount of space covered, occupied, enclosed inside 3-D closed figures.Are of i ts base times its depth or height.

Cubic Uni tsUni t of measure used to measure the volume of an y 3-D object Multiply the area of one of the bases of the solid by the heig

the solidarea of base * height

Lines

Point: Identify specific location in space, b ut is not an object by itself. Represented b y a small do t (.)

Line: 1-D straight path that has no e ndpoints. Minimum of two points required making a line and there is no maximum number of points on a

Practicall y i t is impossible to draw a line sin ce line drawn would have some fixed length and wid th. The symbol () written on top of two le

represents the line.

Ray: Part of line that begins at one labeled fixed endpoint and extends infinitely from that point in the other direction. Its like a half line.

Line Segment: Its a Finite, segment or part of a line with two labeled fixed endpoint. The Symbol () written on top of two letters represe

line segmen t

Types of Lines

Perpendicular Lines:Two lines that intersect each other to form four angles of equal measure, and each has a measure of 900

Parallel Lines: Lines that remain apart, and maintain an equal and constant distance between each other and never intersect each oth

extended infinitely in ei ther di rection

Transversal Lines:A line that intersect two or more parallel lines.

Angles

Angles are formed by intersection or union of two lines, line segments, or rays. Angles are measured in counterclockwise.

Sides:Si des of the angle are two lines, rays , or line segments .

Vertex:Point of intersection at which two sides meet or disconnect. (Note: Vertex Singular, VerticesPlural)

Degree:Unit of angular measure. (Note: 10

= 60(Minutes) and 1

= 60

(Seconds)

Types of Angles

Zero Angle:An Angle whose measure is e xactly 00.

Acute Angle:Angle whose measure is greater than 00but less than 90

0.

Right Angle:An Angle whose measure is exactly 900.

Obtuse Angle: An Angle whose measure is greater than 900and less than 180

0.

Straight Angle:An Angle, whose measure is exactly 1800, forming a s traigh t line.

Reflex Angle:An Angle, whose measure is greater than 1800and less than 360

0. Sum of angles around a point is 360

0. An Angle is formed when

line segments extend from a common pointCongruent Angle:Congruent Angles are angles of equal measure. I f two angles have the same degree, they are said to be cong ruent.

Angle Bisector: A line or line segment bisects an angle as i t splits the angle into two smaller and equal angles .


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Types of Pair of Angles:

Adjacent Angles:Pair of two angles that share a common vertex and a common side

Complimentary Angle:Pair of two adjacent angles that make up a right angle, i.e. whose degree measurements exactly adds up to 900

Supplementary Angle:Pair of two adjacent angles that make up a straight angle, i.e. whose degree measurements exactly adds up to 1800.

Polygons:

Polygon is a geometric figure in a plane that is composed of and bounded by three or more straight line segments, called the sides of the polygo

Parts of Polygon

Side: Sides are the line segments

Angle:Intersection of two sides results in an angle of the pol ygon

Vertex:The point of in tersection of line segmen ts or e ndpoints of two adjacent sides

Diagonal:Line segment inside the polygon connecting two nonadjacent vertices or whose endpoints are vertices is called diagonal of the polygo

Altitude: Any line segment that starts from one of i ts vertices and e nds on one of i ts sides in such a manner that it is pe rpendicular to that side.

Types of Polygon

Equilateral Polygon:All sides are of equal measure

Equiangular Polygon:Al l angles are of equal measure

Regular Polygon:Equal Sides and Equal Angles

Irregular P olygons:Unequal sides and unequal angles

Types of Polygons based on number of sides or angles

Types Description

Triangle 3 si ded polygon

Quadrilateral 4 si ded polygon

Pentagon 5 si ded polygon

Hexagon 6 si ded polygon

Heptagon 7 si ded polygon

Octagon 8 si ded polygon

Nonagon 9 si ded polygon

Decagon 10 sided polygon

Dodecagon 11 sided polygon

N-gon N- sided polygon

Sum of Angles of Polygon:

By using Formula

Sum of the measures o f ninterior angles in a polygon with n sides(n-2) * 1800

Degree measure of each interior angle of a regular polygon with n sides ((n-2) * 1800)/n

By Diving Polygon

From any vertex, draw diagonals, and divide the polygon into as many non-overlapping adjacent triangles as possible.

Count the number of triangles formed

Since there is a total of 1800in the angles of each triangle, multiply the number of triangles by 180

0the p roduct will be the sum o

angles in the polygon

Any Polygons can be divided into Triangles in two different ways:

By drawing all diagonals emanating from any one given vertex to all other nonadjacent vertices or,

By drawing all diagonals connecting all the opposite vertices


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To divide polygons into triangles, quadrilaterals would need one diagonal; pentagons would need two diagonals; hexagons would need

diagonals; heptagons would need two diagonals; octagons would need two diagonals;

Sum of Ex terior Angle of a Polygon3600/ n; Measure of an exterior angle + Measure of an interior angle in polygon = 180

0

Perimeter of PolygonSum of all sides; Perimeter of Regular Polygon Length of side * Number of Sides

Area of a regular polygon * Apothem * perimeter; ApothemLine Segment from center of polygon perpendicular to any side of polygon

Radius of Regular PolygonA Line segment connecting any vertex of a regular polygon with the center of the polygon

Triangles

Triangle is a 3-Sided Polygon

Parts of Triangle

Sides: Line Segment connecting vertices of two angles of the triangle.

Angle:Formed by in tersection or union of any two of i ts sides.

Vertex:Point-of-Intersection of the sides of the triangle

Degree:Unit of Angular Measure

Terms Used in Triangles

Base:One of the three sides

Altitude:Perpendicular distance from a vertex to its opposite side. For Acute Triangle , al ti tude falls inside the triangle; For Obtuse triangle , al tfalls outside the triangle; for right triangle, altitude is one of the legs that is perpendicular to the base

Acute Triangle Right Triangle Obtuse Triangle

Median: Line Segment connecting one of the vertices of the triangle to the midpoint of the opposite side

Perpendicular Bisector: Line Segmen t that bisects and is perpendicular to one o f the sides of the triangle.Angle Bisector:Li ne se gment containing one of the sides o f the triangle to the opposite vertex bisecting that angle into two halves, that is , it bis

one of the angles of the triangle into two equal angles

Midline:Line Segment that connects the midpoints of any two sides of the triangle.

Sum of the measures of all three interior angles = 1800

Sum of the measures of all three exterior angles = 3600

If two triangles share a common angle, then the sum of other two angles are equal

Largest angle of the triangle is always opposite to the longest side.

Smallest angle of the triangle is always opposite to the smallest side

Angles with same measure a re opposi te sides with same length

Sum of two sides > 3rd

Side

Difference of two sides < 3rd

Side

Sum of two sides > 3rd

side > Difference of two sides

Exterior Angle + Adjacent Interior Angle = 1800

Exterior Angle = sum of measure of two opposite interior angles

Exterior Angle > either of opposite interior Angles


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Types of Triangles

Equilateral:All 3 sides are equal i n length and all 3 angles are equal in measure

Isosceles:At least two sides are of equal length and two angles opposite to these sides measures equally.

Scalene: None of i ts sides are equal in length an d none of the angles are equal in measure

Acute: All 3 angles are acute angles

Obtuse:On e of the angles is an obtuse angle

Right:One of the interior angles is a right angle

Isosceles Right: One of the angles is a right angle and the other two angles are equal in measure exactly 450ea ch.

Pythagoras Theorem

Square of the length of the hypotenuse = Sum of the Squares of the lengths of the other two sides.

For any positive number x, there is a right triangle whose sides are in the ratio 3x, 4x, and 5x. Such triangles a re known as Pythagorean Triples

In a 450

- 450

- 900 triangle, also known as Isosceles right triangle, the lengths o f the sides a re in the constan t ration of x : x : x2, where x i

length of each leg. The Diagonal of a Square divides the square into two equal isosceles right triangles.

In a 300- 60

0- 90

0triangle, the sides a re in the constant ratio of x : x3 : 2x, where x is the length of the shorter leg

Trigonometric Ratios

SineOpposite/Hypotenuse (SOH)

CosineAdjacent/Hypotenuse (CAH)TangentOpposite/Adjacent (TOA)

Height of the equilateral Triangle3x

Perimeter of TrianglesSum of all sides

Area of Triangle * (base * height)

Are of Isosceles Triangle * leg2

Area of Equilateral Triangle(S23)/4, where S is the side of the equilateral triangle

Conditions of Triangle Congruency

Two Triangles are congruent if two pairs of corresponding sides and the corresponding included angles are equal

Two Triangles are congruent if two pairs of corresponding angles and the corresponding included sides are equal

Two Triangles are congruent if all 3 pairs of corresponding sides of two triangles are equalTwo right triangles that have any two equal corresponding sides

In an Isosceles triangle, the altitude to the third side divides the original triangle into two congruent triangles

Conditions of Triangle Similarity

Two Triangles are Similar, if all 3 pairs of corresponding angles are equal

Two Triangles are Similar, if all 3 pairs of corresponding sides has the same ratio

Quadrilaterals

Type of Polygon with exactly four sides and four angles


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Parts of Quadrilaterals

Sides: Length is the measure of the longer side; Width is the measure of the shorter side

Diagonals: Line Segments connecting any two non-subsequent vertices

Alti tude: Perpendicular distance between two parallel sides

Angles: Sum of the measures of 4 interior Angles = Sum of the measures of 4 exterior angles = 3600

Types of Quadrilateral:

Square; Rectangle; Parallelogram; Rhombus; Trapezoid

Quadrilateral Type Area Perimeter Others

Square Side ; Diagonal

Side + Side +Side + Side = 4S Side = Diagonal / 2

Rectangle Length * Width 2(Length + Width) Width2

= Diagonal2Length

2

Parallelogram Base * Height 2(Length + Width)

Rhombus Base * Height; * (Diagonal1+ Diagonal2) Side + Side +Side + Side = 4S All Rhombuses are Parallelograms

Trapezoid * (Base1+ Base 2) * Height Base1+ Base2 + Side1+ Side2 BasePair of Parallel Sides

SidesPair of non-Parallel Sides

Circles

A Circle is a closed linear figure that consists of a set or series of all the points in the same plane that is all located a t the same distance from

fixed point.

Parts of Circle:

Radius: Distance between center of circle and any point on the circle. Half of Diameter

Diameter: Distance between any two points on the circle passing through the center. Twice the Radius

Chord: Line Segment joining two points on the ci rcle . Diameter is the longes t chord in the ci rcle. A diameter that is pe rpendicular to a chord bi

the chord into two congruent halves.

Inscribed Triangles

Triangles Inscribed in Semicircle: A Triangle inscribed in a semicircle is always a right triangle . Any right triangle inscribed in a circle

have one of its sides coincide with the diameter of the circle, thus splitting the circle in two semicircles

Triangles formed b y two Radii : Any Triangle formed at the center of a ci rcle by connecting the endpoints of any two radii alwa ys resuan Isosceles triangle.

Secant: Any Line or Line Segment that cuts through the circle by intersecting the circle at any two points.

Tangent

Line Tangent to a Circle: Any line or Line Segment outside the circle that intersects or touches the circle at exactly one point on the circumferen

Two Circles tangent to each other: If two circles intersect or touch exactly at one point

Point-of-Tangency: The point common to a circle and a tangent to the circle or two circles

Radius of a circle is Perpendicular to its Tangent; Two Tangents to a Circle are equal

Line of Centers: Line passing through the Centers of two or more circles

Sector: Portion of a Circle bounded by two radii and an arc

Degree Measure of a Circle: 3600

Types of Circles

Full; Semi; Quarter; Concentric


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Types of Angles in Circle

Central angle: An Angle whose vertex lies exactly at the center point of the circle and i ts two sides are the radii of the ci rcle

Inscribed Angle: An Angle whose vertex lies at any point on the circle itself and the two sides are chords of the circle

Circumference of a Circle = Perimeter of the Circle = Total distance around the circle

Circumference * Diameter2**Radius

Arc of Circle: Part or Portion of the Circumference of the Circle. It consists of two endpoints on a circle and all the points between them

Arc MeasureCentral Angle:

Arc Degree MeasureDegree Measure of the Central Angle that in tercep t i t.

Arc Length Measure(Degrees of Central Angle/3600) * Circumference

Arc MeasureInscribed Angle:

Arc Degree Measure (Degree Measure of the Cen tral Angle that intercept it)

Arc Length Measure((2 * Degrees o f Central Angle)/3600))* Circumference

Arc MeasureIntersecting ChordsEqual in degrees to one-half of the sum of its intercepted arcs

Arc MeasureInterse cting Secants/TangentsEquals Degrees to one-half the dif ference of i ts intercepted arcs.

Perimeter of Sector of CircleArc Measure + (2 * Radius)

Area of Full Circle*radius2Area of Sector of Circle(Degrees of Central Angle/360

0) * *radius

2

Solid Geometry

Study of Shapes and figures that are drawn in more than one plane

Terms used in Solids

VertexPoint at its corners where the edges meet

EdgeLine Segments that connect the vertices and fo rm the sides of ea ch face of the solid.

FacePolygons that form the outside boundaries of the solid

Types of Solids

Rectangular SolidsSolids with rectangula r or square faces. For Example, BrickTypes of Rectangular solidsCubes, Rectangular Prisms

Circular SolidsSolids with Circular or Conical Faces. For Example,Ice-Cream cones

Types of Circulare SolidsCylinders, Cones, Spheres, Pyramids, Tetrahedrons

Surface Area of Rectangular Solids

Area of Front and Back Faces 2(Length * Height)

Area of Top and Bottom Faces 2(Length * Width)

Area of Right and Left Faces2(Width * Height)

Total Surface AreaSum of the a rea of the six ou tside rectangular faces 2(LH + LW + WH)

Volume of Rectangular SolidsLength * Breadth * Height

Diagonal (Length2+ Widht2 + Height2)


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Types Surface Area Volume Diagonal Others

Cube 6 * Side2 Side

3 Side * 3

Cylinder (Area of Top and Bottom Circular Bases) + (Lateral Surface

Area)

( 2**Radius2

)+ (2**Radius * Height)

*Radius2*Height Use only Lateral Surface Area w

its a hollow cylinder to calcu

Surface Area

Cone Area of Circular Base + Lateral Surface Area

(*Radius*Slant Height) + * Radius2

(1/3)* *

Radius2*Height

Sphere 4**Radius

(4/3)**Radius

Coordinate Geometry

Study of geometric figures and properties on the coordinate place using algebraic principles

Coordinate PlaneXY-Plane

Coordinate Axis:

X-AxisAbscissaHorizontal Number line, which goes left and right

Y-AxisOrdinateVertical Number Line, which goes up and down

Coordinate Points (X, Y)(X-Coordinate, Y-Coordinate)

Parts of Coordinate Plane

1stQuadrantTop rightNorth-East(+X, +Y)

2nd

QuadrantTop leftNorth-West(-X, +Y)

3rd

QuadrantBottom LeftSouth-West(-X, -Y)

4th

QuadrantBottom RightSouth East(+X, -Y)

Origin(0, 0)

Distance between any two given points, A(x1, y1) and B(x2, y2) ((x1- x2)2+ (y1- y2)

2)

Mid-Point between two Axes((x1+ x2)/2, (y1+ y2)/2)

Intercepts of Line

Poi nt at whi ch a line i ntercepts the coordinate axes

X-InterceptValue of X-Coordinate of the point at which the line intersects the x-axis

Y-InterceptValue of Y-Coordinate of the point at which the line intersects the y-axis

Slope of Line

Step-1: Pick any two points on the line a(x1, y1) and b(x2, y2) that lie on the line

Step-2: Next find the Rise and the Run

RiseAmount the line raises verticallyy1y2 RunAmount the line runs horizontally x1x2Step-3: Finally, divide the Rise by the Run

Slope Intercept Formy = mx + b


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Applications of Coordinate Geometry

Categories Description

Finding Slope and Y-Intercept of Line from its equation Put the Equa tion in Standard Form y = mx + b Identify the m-term and b-term

Finding Equation of Line from its S lope & One-Point Find the Y-Intercept (b) by substituting the slope and the coordinates in

general equation

Apply the formula y y1 = m( x x1), where m is the slope, and (x1,

the given coordinate

Finding Y-Intercept of Line Passing through two points Find the slope (m)by using slope formula m = ((y1y2)/(x1x2))

Find the Y-Intercept by substituting the slope and one of the gcoordinates in the general equation; y = mx + b

Finding the Equation of Line Passing through two Points Find the slope using Slope formula

Find the y-Intercept (b) of the line by substituting either (x, y) in general f

Find the equation of the line y plugging the values in general formFinding the Equation of Line from One-Point and Y-

Intercept

Find the Value of another Coordinate from y-Intercept

Find the Slope using Slope formula

Find the equation of the line by plugging the values in general formFinding Point-Of- Intersection of Two lines Find the slope using co-ordinates

Find the equation of each line by substituting one of the coordinates

slope in general equation

Find the point of in tersection of lines by equating the equation of both

and solve for x and y by substitution method

Finding Equation of Perpendicular Bisectors Find the slope using Slope formula

Find the slope of the perpendicular bisector (Negative reciprocal or Slope Find the midpoint of the line, which is also a point in the perpendic

bisector

Find the y-intercept of the perpendicular bisector by substituting slope a

intercept in the general equation

Find the equation of the perpendicular bisector by substituting the slope

y-Intercept in the general equation


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Word Probl

Apply the following steps to solve any type of word problems:

Read the question and determine what all information is giventhese are the given, and are known as known quantities.

Read the ques tion and interpret whats being asked or, what needs to be sol ved, or what informa tion you need to know the answer othe questionthese are the quantities you are seeking, and they are known as the Unknown quantities

Name the Unknown quanti ties by sele cting variables, such as x, y, z, etc.

Determine the relationships between the knowns and unknowns, that is, the variables and the other given quantities in theproblem, and connect them using arithmetic problems, su ch as (+), (-) , etc. and write them as algebraic expressions.

Using these variables and the rela tionships between the known and unknown quantitiesform algeb raic equations by applying the

appropriate mathematical formulas

Solve the algebraic equations to find the value of the unknown(s), and plug that value in other relationships or equations tha t invol ve

this variable in order to find any other unknown quantities, if there are any.

Basic Coin Conventions to be known:

1 Dollar100 Cents; 1 Half Dollar50 Cents ; 1 Quarter25 Cents; 1 Dime10 Cents; 1 Nickel5 Cents

Apply the following steps to solve Age problems:

Assign a different letter (Variable) for each persons age

Establish relationships between the ages of two or more in the problem

Transform these relationships into algebraic equations Solve the equations and determine the unknowns

Important Note in Age Problems:

Years Agomeans you need to subtract

Years from nowmeans you need to add

Rate of Work or Quantity:

RateAmount of work done per time unit

Work Problem tips:

Greater the rate of work

fas ter you work

sooner the job is done

Lesser the rate of workslower you workslower the job is done

Greater number of workerslesser the time required to finish the job

Lesse r number of workersgreater the time required to finish the job If it takes k workers 1 hour to do a particular job, then each worker does 1/k of the job in an hour o r works @ 1/k of the job per hou

If it takes k workers m hours to do a pa rticular job, then each worker does 1/k of the job in an hour or works @ 1/(mh) o f the job phour

Work Problem Formula1/x + 1/y = 1/zInverse of the time it would take everyone working together equals the sum of the inverses of the

time it would take each working individually.

DistanceRate * Time

Cos t per UnitTotal Cost of the Mixture/Total Weight of the Mixture

Mixture of Weaker and Stronger Solutions ProblemWeaker (DesiredStronger) = s (StrongerDesired), sAmount of 1st

+ 2nd


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Unit of Measures

US Customary System

1 yd (Yard)3 ft (Feet)36 Inches

1 Ton2000lbs (Pounds)

1 lb16 oz (Ounces)

1 Gallon4 qt(Quart)8 pt(Pint)16 c(Cup)128 fl oz(Fluid Ounce)256 tbsp(Table Spoon)

1 sq yd9 sq ft1296 sq in

Metric System:

millimeans one thousandths

centimeans one thousandths

decimeans one tenths

Basic Standard unitmeans one

Deka- or Deca-means tens

Hectormeans hundreds

Kilo-means thousands

US Customary and Metric System

US units Metric System Metric Units US Units

1 in 2.54 cm 1 cm 0.39 in

1 yd 0.9144 m 1 m 1.1 yd

1 mi 1.6 km 1 km 0.6 mi

1 lb 0.4545 kg 1 kg 2.2 lbs

1 lb 454 gm 1 l tr 1.056 fluid quart

1 oz 28 gm

1 MT (Metric Ton) 1.1 t (Ton)

1 fl oz 29.574 ml

1 fluid quart 0.9464 ltr

1 gallon 3.785 ltr

1 ton 2000 lbs

1 lb 16 oz

1 sq yd 9 sq f t

1 yd 3 ft

1 yd 36 in

Time Measures

1 Millennium/Century 10 Decades/100 Years

1 Year 12 Months/52 Weeks/365 Days

1 Day 24 Hours

1 Hour 60 Minu tes

1 Minute 60 Seconds

A.MAnte Meridianbefore Noon; P.MPost MeridianAfter noon

As we travel eastSun rises earlie rand therefore clock is aheadAs we travel westSun rises laterand therefore clock is behind


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From East to West

EST (Eastern Standard Time)1 hour ahead of CST (Central Standard Time) 2 Hours ahead of MST (Mountain Standard Time)3 Hours a

of PST (Pacific Standard Time)

Temperature Conversion

Celsius = (5/9) (Farenheit-32); Fahrenhei t = (9/5) Celsius + 32;

Freezing Point = 320F; Boiling Point = 212

0F; Normal Body Temperature = 98.6

0F


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Logic & S

Simple Counting

Involves figuring out how many integers are between any two given integers

Rule # 1: When exactly one Endpoint is inclusivesubtract the two values

Rule # 2: When both Endpoints are inclusivesubtract both values, and then add 1

Rule # 3: When neither Endpoint is inclusive subtract the two val ues, and then subtract 1

Fundamental Principle of Counting If two jobs need to be completed and there a re m ways to do the first job, and n ways to do the seco

job , then there are m * n ways to do one job followed by the other. This can be extended to any number of events.

Factorials: Factorial of n is the number of ways that the n elements of a group can be o rdered.

n! = n * (n-1) * (n-2) * * 2 * 1until the last term becomes 1; 0! = 1

Permutations

Permutation is the Number of ways in which a set of terms or elements can be arranged in order or sequentially. Also known as a selection

process in which objects a re selected one by one in a certain predefined order

Factorials are involved in solving permutations or counting number of ways that a set can be ordered.

Permutationm P nm! / (m-n)!m * (m-1) * (m-2) * (m-n+1)

Where, m Number in the larger group; nnumber being a rranged

If there are m different terms/elements in a set, and there a re k available or empty spots, then there are p dif ferent ways o f arranging the

given by the formulap = m! / k!

Combinations

Combination is the number of ways of choosing a given number of elements from a set, where the order of elements does not matter. For insta

AB and BA counts as two different permutations, but only as 1 combination

Combinationm C nm! / n! (m-n)! (m * (m-1) * (m-2) * (m-n+1))/n! = m P n / n!

Where, m Number in the larger group; nnumber being chosen

Probability

ProbabilityP (E)Number of Favorable Outcomes/ Total number of possible Outcomes

Probability in all cases is always between 0 and 1

If two or more events constitute all the possible outcomes, then the sum of their probabilities is 1

Probability of Event that will not happen = 1Probability of Event that will happen

If A and B are independent events, then to determine the probability that even t A and event B will BOTH together occur: M ULTIPLY the

probabilities of two individuals together

If A and B are independent events, and that they are mutually exclusi ve, then to determine the probabili ty that event A o r event B will occur:

ADD the probabilities of two individuals togethe r. Two Events are said to be mutuall y e xclusive if the occurrence of one event will rule out the

other

If A and B are independent events, and that they are mutually non-exclusive, then to determine the probability that even t A or event B willoccur: ADD the p robabilities of two individuals together and then SUBTRACT the probability that both events occur together. Two Events are sa

to be mutually non-exclusive if the occurrence of one event will not rule out the other


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Dependent Events:Two Events are said to be dependent, if the outcome of one event affects the probability of another event. For example,

picking a card from a fair deck of cardswith each card we pick, the total possible events for the next event will be 1 less than the one before t

P (A and B) = P (A) * P (B|A); where P (B|A) is the conditional probability of B given A

Sets

A Set is a collection of well defined things or items called elements or members of the set

Finite Set: If a set contains only a finite number of elements

Infinite Set: If a set contains infinite number of elementsSubset: If all the elemen ts of one se t S, are also elements of another set T; then the fi rst set S, is a Subset of T

Venn DiagramsGraphically represents se ts

Union Set: The set consisting of all the elements that e xist in ei ther one or all of the sets what we get when we merge two or more sets

Intersection Set: The set of elements that are common in different sets involved

SequenceA series, list, collection, or group of numbers that follows a specific pattern

PatternA series of numbers or objects whose sequence is determined by a particular rule

Arithmetic Sequence: If d is the common difference and a is the first term of an arithmetic progression, then the nth term of the arithmeti c

progression will be = a + (n-1)d.

Geometric Sequence: If a1 is the fi rst term, and r is the common ratio between consecutive terms of a geometric progression, and a nis the nterm, then the n

thterm will bean= a1r

n-1

Sum of n terms in a Geometric Sequence(arna)/(r-1), when r # 1

Harmonic SequenceSequence of fractions in which the numerator is 1, and the denominators form an arithmetic sequence

Arithmetic MeanMeanAverageTotal Sum of all terms / To tal number of terms

Sum of consecutive termsMean of Consecutive Terms * Number of Consecutive Terms

Where, Mean of Consecutive Terms(First Term + Last Term) / 2; Number of terms (Last termFirst Term) + 1

Sum of Existing term + Missing Term = Sum of all terms

Weighted MeanNumber of times a quantity o r term occursSum of Products / Sum of WeightsSum / Frequency

MedianMiddleWhen there are n terms, the median is the value of ((n+1)/2) thterm

ModeSet of Data that occurs mos t frequently

QuartilesDivides data into equal quarters or four equal parts

RangeLargest termSmallest Term

Standard DeviationDistance or the gap between the ari thmetic mean and the set of numbers

Apply the following steps to calculate the Standard Deviation of a set of n numbers:

Find the Average (Arithmetic Mean) of the set

Find the differences between that average and each value of the numbers in the set

Square each of the differences Find the average of squared differences by summing the squared values and dividing the sum by the number of values

Take the positive square root of that average


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Statistics Graphs

Graph Type How to Read?

Tables and charts Look for a specific unit on the row heading Then ma tch that row with the corresponding unit on the column heading

Pictographs Look at the specific row

Then compute i ts value based on the conversion factor given in the key. Each symbol rep resents a fixednumber of items as indicated in the key

Single Line Graph Look for a specific time period on the horizontal axis

Match the height of the point on the line with the number on the vertical axis which is the actual quantit

for that specific time period In order to find a specific numerical value of a particular point on the line from a line graph, find the corr

point on the line and move horizontally across from that point on the line to the value on the scale on th

left.

The vertical distance from the bottom of the graph to the point on the line is the value of that point A line that slopes up from left to right, shows an increase in the quantity during that time period

A line that slopes down from left to right, shows a decrease in the quantity during that time period

Double Line Graph Look for a specific time period on the horizontal axis Match the point on the line with the number on the vertical axis which is the actual quantity of that spec

variable for that specific time period

Single Bar Graph Look for a bar label or specific time period on the horizontal axis Match the height of the bar with the number on the vertical axis which is the actual quantity for that spe

bar or time period

In order to find a specific numeric value of a particular bar from a ba r graph, find the correct bar

Move horizontally across from the top of the bar that points on the line to the value on the scale on the The vertical distance from the bottom of the graph to the point on the line is the value of that point

Double Bar Graph Look for a specific time period on the horizontal axis Match the height of each of the bars with the number on the vertical axis which is the actual quantity of

specified variable for that specific time period

Scatter Plot Graphs Look for the specific Quantity on the horizontal axis and the 2nd

quantity in the vertical axis

The Point of Intersection of these two values is the point that represents those two quantities

Circle Graphs/Pie Charts Look at a specific sector and then identify the category and the quantity it represents To find the value of a particular piece of the pie, multiply the appropriate percent by value of the whole

Quants One Stop Guide

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