Quants One Stop Guide

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Quantitative Aptitude

-A quick reference


Arithmetic

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 ½ = 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 fi rst/only even Prime Number All Prime numbers are 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 Remainde r {…,-3,-1,1,3…} Prime Greater than 1, with exactly two integer factors/divisors {2,3,5,7,11,….} Composite Any Number except 1 that is not Prime {4,6,8,9,10..}

Consecutive Set of Numbers with Fixed interval {1,2,3,4,….} Distinct Numbers with Di fferent Values 2 and 5


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 opposi te 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 Operation Addition or Multiplication can be regrouped in any order. Distributive Law of Operation Factors can be dis tributed across the terms being added/subtracted/multiplied/divided.

When the sum or difference is in the Denominator, no dis tribution is applicable Divisibility Tests

Tests Description

Divisibility Test for 2 If Unit’s Digi t is divisible by 2 or is a multiple of 2 Divisibility Test for 3 Sum of all digits is divisible by 3 or is a multiple of 3 Divisibility Test for 4 Number made by Ten’s and Unit’s Digi t is divisible by 4 or is a multiple of 4 Divisibility Test for 5 If Unit’s Digi t is equal to ‘0’ or ‘5’

Divisibility Test for 6 If i t is divisible both by 2 and 3. Divisibility Test for 8 Last three digi ts are divisible by 8. Or if it’s divisible by 2 thrice Divisibility Test for 9 Sum of the digits is divisible by 9 or multiple of 9 Divisibility Test for 10 If las t Digi t is ‘0’ Divisibility Test for 12 If i t 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’ consecutive integers is always divisible by n, or is a multiple of ‘n’

If there is one even Integer 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 largest 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 Even Even + 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


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 i ts lowest terms Step-2: Next, divide the numerator by denominator

For example, 1/10 0.10 Converting Decimals to Fractions

Step-1: Fi rs t Eliminate the decimal point, and wri te(Right to decimal Point) i t as the numerator of the resulting fraction Step-2: Next, divide i t by 1 followed by as many zeroes as the number of places to the right of the decimal point of the given number, and

wri te that as the denominator of the resulting fraction Step-3: Simplify the resulting fraction to i ts lowest terms by dividing the numerator and denominator by i ts GCF

For example, 0.10 10/100 1/10

Proper Fraction a/b, where a<b ; Improper Fraction a/b, where a>b ; Mixed Fraction a(b/c)

A (b/c) ((c*a) + b)/c Additive Inverse Negative of the Number

Multiplicative Inverse Reciprocal 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 52, there are 2 factors . Here 5 is the base and 2 is the exponent.

For any number “a”: a n = a*a*a*a*…… “n” number of times = b i .e., nth root of b is a n √b = a Square of any posi tive number or square of i ts negative will always be positive

n0 =

1, where n # 0


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/a n

a m/n = n √ a m (nth root 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 . a m * a n = a (m + n)

When multiplying expressions with the same exponent, MULTIPLY the bases a n * a n = (ab) n

Multiply

Subtract When dividing expressions with the same base, SUBTRACT the exponents a m / a n = a (m - n)

When dividing expressions with the same exponent, DIVIDE the bases a n / a n = (a/b) n

Divide

Same Base Same Exponent

Format of Scienti fic notation a.bcde * 10 (n), where a ,b,c,d,e are any posi tive numeric digi ts , such that, 0<a<10, and n is the number of

places the decimal point is moved, which can be negative, positive or zero.

Radicals

Exponents and Radicals are opposi te operations Root of a number or an expression. a √b Denotes a th root of b, where a is the Index, b is the Radicand a 1/n = n √a (m √n)m = n Square root of negative numbers is not defined. Negative Numbers do not have real numbers as their roots

If √x yields an integer, then ‘x’ is a perfect square If √x yields a non-integer, then ‘x’ is an imperfect square

n√(a * b) =

n√a *

n√b

Common Square roots

X 2 3 5 6 7 8 11 2√X 1.41 1.73 2.24 2.45 2.65 2.83 3.32

( a* b) √ (n * m) = a√n * b√m √(a / b) = √a / √b √a * √b = √(a * b) √a / √b = √(a / b) √(a + b) # √a + √b

√(a - b) # √a - √b √(a + b) # √a + √b √a- √b # √(a - b)

√a * √a = a √(a 2 * b) = a √b

(√a + √b)2 = a + b + 2 √(a * b) (√a - √b)2 = a + b - 2 √(a * b) (√a + √b) (√a - √b) = a -b

√n (√a + √b) = √n √a + √n √b


Algebra

(Xm Yn)/ (Xp Yq) = X m-p Y n-q (X

m + Y

n)/Z = (X

m)/Z + (Y

n)/Z

Factor ax2 + bx + c = 0 into the following two factors . (p+q) (r+s); such that: Firs t term of the trinomial p * r Last term of the trinomial q * 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 i f equation has Decimal/Fractions)

Step-2: Get rid of all the parentheses using dis tributive 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 exis ts 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 only if numerical expressions are on both sides) Step-6: Get rid of the radical signs if there are any, by squaring both sides of the equation (Apply only if equation has radicals ) Step-7: Get rid of the exponents i f 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 absolute value but of opposi te 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 s teps

Step-5: This will result in a value of one of the variables ; substitute this value into ei ther 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 (b2 – 4ac) > 0, then √ (b2 – 4ac) will be two distinct real number roots If (b2 – 4ac) < 0, then there exis ts no solution or real roots If (b2 – 4ac) = 0, then √ (b2 – 4ac) will be zero. And expression has only one real root or solution Sum of Roots -b/a

Product of Roots c/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 side of equation and othe r side is zero in such a way that i t can be factored and put into the s tandard 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 factors Step-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


Applications

To→

From ↓

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

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

Ex: ½ 1 / 2 = 0.50

Step-1: Multiply the Fraction by 100 Step-2: Simplify and insert % sign.

Ex. 1/2 = (1/2) * 100 = 50 % Decimal (0.50) Step-1: Drop the decimal point by

dividing i t by 1 plus add as many zeroes as the number of places to the right of

the decimal point.

Step-2: Simplify. Ex: 0.50 50 / 100 = 1/2

Not Applicable Step-1: Move the Decimal Point two

places to the right Ex: 0.50 50 %

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 percent’s 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)*a “a” 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 Value – Original Value)/ (Original Value)) * 100 %

Percent Decrease ((Original Value – New Value)/(Original Value)) * 100 %

To Increase a number by K%, multiply i t by (100% + K%) To Decrease a number by K%, multiply i t 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

Apply the following s teps when two or more series of subsequent percent changes are applicable: Step-1: Compute the fi rst percentage change on the original base. If the original base is not given, assume i t to be 100 Step-2: Add/Subtract the fi rs t percent change from the base of 100 to find the value after fi rs t percent change, also known as the intermediate value.

Step-3: Compute the second percent change on the value of fi rst percent change Step-4: Add/Subtract the second percent change from value after the fi rst percent change to find the final percent change

Example problem: If the price of an i tem raises by 10% one year and by 20% the next, what’s the combined increase?

Percent Discounts Original Price Sale Price + Discount Amount

Original Price (Discount Amount/Discount %) * 100 Original Price Sale Price / (100% + Markup %)


New Price Original Price (100 % + Mark-up %) or Original Price (100 % - Mark-up %) Sale Price Original Price – Discount Amount Discount Amount Original Price – Sale Price

Discount % (Rate of Discount) ((Original Price – Selling 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 . It is the cost that the seller pays or incurs to procure or produce an i tem. Selling Price: Amount that a seller sells an i tem for, which may include a profi t (mark -up) or loss (mark-down) or neither (break-even price) Break-Even Price: Nothing but the Cost price

Mark-up or Profi t Selling price – cost price Selling Price Cost Price + Profi t

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

New Price Original Price + Ma rk-up (Increase) Mark-down or Loss Cost price - Selling price Selling Price Cost Price - Loss Original Price Sale Price/ (100% - Mark-down %) New Price Original 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 + (interest 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 Interest Rate is compounded annually, divide i t by 1

If the Interest Rate is compounded semi -annually, divide i t by 2 If the Interest Rate is compounded quarte rly, divide i t by 4 If the Interest Rate is compounded bi -monthly, divide i t by 6

If the Interest Rate is compounded monthly, divide i t by 12

The Di fference 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 are the mathematical relationship between two or more things . Ratios are nothing but another form of fractions . “Perce nt” is a ratio in which the second quanti ty is 100.


Terms of Ratio The Two numbers in the ratio are 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 second term of the fi rs t ratio cancel fi rst term of second ratio

Step-2: Once the terms they have in common cancel out; combine the ratio as two-part or multiply the cancelled terms to wri te i t 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) 12/5 a :c = 12:5 Now Multiply both cancelled “b’s” 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 are 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 Quanti ties x and y, are said to be directly proportional i f they satis fy a relationship of the form x = ky, where k is a non zero constant Different Types of Direct Proportions are:

Money Spent – Quanti ty Bought Weight – Quanti ty Height – Shadow

Actual Size – Map Scale Gasoline – Miles Time – Wages

Indirect Proportions

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

Different Types of Indirect Proportions are:

Workers – Time Speed – Time Monthly Installments – Loan Period Members – Time Period for Suppliers


How to figure out if two Quantities vary directly or inversely? Answering one of the following questions would get the result. Question – 1: Will an increase in one quantity lead to an increase or decrease in the other quanti ty?

If i t leads to Increase, then the two quantities vary di rectly If i t leads to Decrease, then the two quantities vary inversely

Question – 2: Will a decrease in one quantity lead to a decrease or an increase in the other quantity?

If i t leads to decrease, then the two quanti ties vary di rectly If i t leads to increase, then the two quanti ties vary inversely

Compound Proportions

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


Geometry

Geometry Geometry is the s tudy of Shapes (both flat and curved). Mathematics of the properties, measurements , and relationships of points , lines , angles, surfaces, and solids

Perimeter Measurement of the distance all the way round any closed 2-D figure or Object Sum of measure of all the lengths of all its sides Area Certain amount of region “covered” or “Occupied” by 2-D or 3-D closed figures Measure of the space inside a flat figure Square Units (Unit

2) 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 Figures Measure of the number of square units that completely fills the region on the surface area of the figure Area of a Flat Surface base * alti tude Surface Area of 3-D figures Sum of the total areas of all the 2-D outer surfaces of the 3-D object Sum of the areas of each of the solid’s surfaces or faces.

Volume Certain amount of space “covered”, “occupied”, “enclosed” inside 3-D closed figures . Are of i ts base times i ts depth or height. Cubic Units Unit of measure used to measure the volume of any 3-D object Multiply the area of one of the bases of the solid by the height of the solid area of base * height

Lines

Point: Identify specific location in space, but is not an object by i tself. Represented by a small dot (.) Line: 1-D s traight path that has no endpoints. Minimum of two points required making a line and there is no maximum number of points on a line. Practically i t is impossible to draw a line since line drawn would have some fixed length and width. The symbol () wri tten on top of two letters represents the line. Ray: Part of line that begins at one labeled fixed endpoint and extends infini tely from that point in the other di rection. It’s li ke a half line. Line Segment: It’s a Finite, segment or part of a line with two labeled fixed endpoint. The Symbol (—) wri tten on top of two letters represents a line segment Types of Lines Perpendicular Lines: Two lines that intersect each other to form four angles of equal measure, and each has a measure of 90 0

Parallel Lines: Lines that remain apart, and maintain an equal and constant dis tance between each other and never intersect each other if 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: Sides of the angle are two lines , rays , or line segments . Vertex: Point of intersection at which two sides meet or disconnect. (Note: Vertex Singular, Vertices Plural ) Degree: Unit of angular measure. (Note: 10 = 60 ‘(Minutes) and 1 ‘= 60 “(Seconds) Types of Angles Zero Angle: An Angle whose measure is exactly 00. Acute Angle: Angle whose measure is greater than 00 but less than 900.

Right Angle: An Angle whose measure is exactly 900.

Obtuse Angle: An Angle whose measure is greater than 900 and less than 1800. Straight Angle: An Angle, whose measure is exactly 1800, forming a s traight line.

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

0. Sum of angles around a point is 360

0. An Angle is formed when two

line segments extend from a common point Congruent Angle: Congruent Angles are angles of equal measure. If 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 .


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

Complimentary Angle: Pai r of two adjacent angles that make up a right angle, i .e. whose degree measurements exactly adds up to 90 0 Supplementary Angle: Pai r of two adjacent angles that make up a s traight angle, i .e. whose degree measurements exactly adds up to 180 0. Polygons:

Polygon is a geometric figure in a plane that is composed of and bounded by three or more s traight line segments , called the sides of the polygon Parts of Polygon

Side: Sides are the line segments Angle: Intersection of two sides results in an angle of the polygon Vertex: The point of intersection of line segments or endpoints of two adjacent sides

Diagonal: Line segment inside the polygon connecting two nonadjacent vertices or whose endpoints are vertices is called diagonal of the polygon. Al ti tude: Any line segment that starts from one of i ts vertices and ends on one of i ts sides in such a manner that i t is perpendicular 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 Polygons: Unequal sides and unequal angles

Types of Polygons based on number of sides or angles

Types Description

Triangle 3 sided polygon Quadrilateral 4 sided polygon Pentagon 5 sided polygon

Hexagon 6 sided polygon Heptagon 7 sided polygon Octagon 8 sided polygon

Nonagon 9 sided 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 of “n” interior 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 1800 in the angles of each triangle, multiply the number of triangles by 1800

the product will be the sum of the

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


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

Sum of Exterior Angle of a Polygon 3600 / n; Measure of an exterior angle + Measure of an interior angle in polygon = 1800

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

Area of a regular polygon ½ * Apothem * perimeter; Apothem Line Segment from center of polygon perpendicular to any side of polygon Radius of Regular Polygon A 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 intersection 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 opposi te side. For Acute Triangle, al ti tude falls inside the triangle; For Obtuse triangle , al titude falls outside the triangle; for right triangle, al ti tude 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 opposi te side Perpendicular Bisector: Line Segment that bisects and is perpendicular to one of the sides of the triangle. Angle Bisector: Line segment containing one of the sides of the triangle to the opposite vertex bisecting that angle into two halves , that is , it bisects 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 opposi te to the longest side. Smallest angle of the triangle is always opposite to the smallest side

Angles with same measure are opposi te sides with same length Sum of two sides > 3rd Side Di fference 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 opposi te interior angles Exterior Angle > ei ther of opposi te interior Angles


Types of Triangles Equilateral: All 3 sides are equal in length and all 3 angles are equal in measure

Isosceles: At least two sides are of equal length and two angles opposi te to these sides ’ measures equally. Scalene: None of i t’s’ sides are equal in length and none of the angles are equal in measure Acute: All 3 angles are acute angles Obtuse: One 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 450 each. 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 are known as Pythagorean Triples

In a 450 - 450 - 900 triangle, also known as Isosceles right triangle, the lengths of the sides are in the constant ration of x : x : x√2, where x is the

length of each leg. The Diagonal of a Square divides the square into two equal isosceles right triangles. In a 30

0 - 60

0 - 90

0 triangle, the sides are in the constant ratio of x : x√3 : 2x, where x is the length of the shorter leg

Trigonometric Ratios Sine Opposite/Hypotenuse (SOH)

Cosine Adjacent/Hypotenuse (CAH) Tangent Opposite/Adjacent (TOA)

Height of the equilateral Triangle √3x

Perimeter of Triangles Sum of all sides Area of Triangle ½ * (base * height)

Are of Isosceles Triangle ½ * leg2

Area of Equilateral Triangle (S2√3)/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 equal Two right triangles that have any two equal corresponding sides In an Isosceles triangle, the alti tude 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


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 Al ti tude: Perpendicular dis tance 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 Side2; ½ Diagonal 2 Side + Side +Side + Side = 4S Side = Diagonal / √2 Rectangle Length * Width 2(Length + Width) Width2 = Diagonal2 – Length2

Parallelogram Base * Height 2(Length + Width) Rhombus Base * Height; ½ * (Diagonal1 + Diagonal2) Side + Side +Side + Side = 4S All Rhombuses are Parallelograms Trapezoid ½ * (Base1 + Base2) * Height Base1 + Base2 + Side1 + Side2 Base Pai r of Parallel Sides

Sides Pai r of non-Parallel Sides

Circles A Ci rcle 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 one fixed point.

Parts of Circle:

Radius : Distance between center of ci rcle and any point o n the ci rcle. Half of Diameter Diameter: Distance between any two points on the ci rcle passing through the center. Twice the Radius Chord: Line Segment joining two points on the ci rcle. Diameter is the longest chord in the ci rcle. A diameter that is pe rpendicular to a chord bisects

the chord into two congruent halves .

Inscribed Triangles

Triangles Inscribed in Semicircle: A Triangle inscribed in a semici rcle is always a right triangle. Any right triangle inscri bed in a ci rcle must

have one of i ts sides coincide with the diameter of the ci rcle, thus splitting the ci rcle in two semici rcles Triangles formed by two Radii : Any Triangle formed at the center of a ci rcle by connecting the endpoints of any two radii always results in

an Isosceles triangle.

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

Line Tangent to a Ci rcle: Any line or Line Segment outside the ci rcle that intersects or touches the ci rcle at exactly one point on the ci rcumference

Two Circles tangent to each other: If two ci rcles intersect or touch exactly at one point Point-of-Tangency: The point common to a ci rcle and a tangent to the ci rcle or two ci rcles

Radius of a ci rcle is Perpendicular to i ts Tangent; Two Tangents to a Ci rcle are equal

Line of Centers : Line passing through the Centers of two or more ci rcles Sector: Portion of a Ci rcle bounded by two radii and an arc Degree Measure of a Ci rcle: 3600

Types of Circles Full ; Semi; Quarter; Concentric


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

Inscribed Angle: An Angle whose vertex lies at any point on the ci rcle i tself and the two sides are chords of the ci rcle Ci rcumference of a Ci rcle = Perimeter of the Circle = Total distance around the ci rcle

Ci rcumference ∏ * Diameter 2*∏*Radius Arc of Ci rcle: Part or Portion of the Circumference of the Circle. It consists of two endpoints on a ci rcle and all the points between them Arc Measure – Central Angle:

Arc Degree Measure Degree Measure of the Central Angle that intercept i t. Arc Length Measure (Degrees of Central Angle/3600) * Ci rcumference

Arc Measure – Inscribed Angle:

Arc Degree Measure ½ (Degree Measure of the Central Angle that intercept it) Arc Length Measure ((2 * Degrees of Central Angle)/360

0))* Ci rcumference

Arc Measure – Intersecting Chords Equal in degrees to one-half of the sum of i ts intercepted arcs Arc Measure – Intersecting Secants/Tangents Equals Degrees to one-half the difference of i ts intercepted arcs. Perimeter of Sector of Ci rcle Arc Measure + (2 * Radius )

Area of Full Ci rcle ∏*radius2

Area of Sector of Ci rcle (Degrees of Central Angle/3600) * ∏*radius2

Solid Geometry

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

Terms used in Solids

Vertex Point at i ts corners where the edges meet Edge Line Segments that connect the vertices and form the sides of each face of the solid.

Face Polygons that form the outside boundaries of the solid Types of Solids Rectangular Solids Solids with rectangular or square faces . For Example, Brick Types of Rectangular solids Cubes, Rectangular Prisms

Ci rcular Solids Solids with Circular or Conical Faces . For Example,Ice-Cream cones Types of Ci rculare Solids Cylinders , 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 Faces 2(Width * Height) Total Surface Area Sum of the area of the six outside rectangular fa ces 2(LH + LW + WH)

Volume of Rectangular Solids Length * Breadth * Height Diagonal √ (Length2 + Widht2 + Height2)


Types Surface Area Volume Diagonal Others

Cube 6 * Side2 Side3 Side * √3

Cylinder (Area of Top and Bottom Circular Bases) + (Lateral Surface Area) ( 2*∏*Radius 2 )+ (2*∏*Radius * Height)

∏*Radius2*Height Use only Lateral Surface Area when

i t’s a hollow cylinder to calculate Surface Area

Cone Area of Ci rcular Base + Lateral Surface Area (∏*Radius*Slant Height) + ∏ * Radius 2

(1/3)*∏ * Radius2*Height

Sphere 4*∏*Radius2

(4/3)*∏*Radius3

Coordinate Geometry

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

Coordinate Plane XY-Plane Coordinate Axis :

X-Axis Abscissa Horizontal Number line, which goes left and right Y-Axis Ordinate Vertical Number Line, which goes up and down Coordinate Points (X, Y) (X-Coordinate, Y-Coordinate) Parts of Coordinate Plane

1st Quadrant Top right North-East (+X, +Y) 2nd Quadrant Top left North-West (-X, +Y)

3rd

Quadrant Bottom Left South-West (-X, -Y) 4th Quadrant Bottom Right South East (+X, -Y)

Origin (0, 0)

Dis tance 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 Point at which a line intercepts the coordinate axes

X-Intercept Value of X-Coordinate of the point at which the line intersects the x-axis Y-Intercept Value 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

Rise Amount the line raises vertically y1 – y2

Run Amount the line runs horizontally x1 – x2

Step-3: Finally, divide the Rise by the Run Slope Intercept Form y = mx + b


Applications of Coordinate Geometry

Categories Description

Finding Slope and Y-Intercept of Line from its equation

Put the Equation in Standard Form y = mx + b Identify the m-term and b-term

Finding Equation of Line from its Slope & One-Point Find the Y-Intercept (b) by substi tuting the slope and the coordinates in the general equation

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

Finding Y-Intercept of Line Passing through two points Find the slope (m)by using slope formula m = ((y1 –y2)/(x1 – x2)) Find the Y-Intercept by substituting the slope and one of the given

coordinates 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 substi tuting ei ther (x, y) in general form

Find the equation of the line y plugging the values in general form Finding 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 form Finding Point-Of-Intersection of Two lines Find the slope using co-ordinates

Find the equation of each line by substi tuting one of the coordinates and slope in general equation

Find the point of intersection of lines by equating the equation of both lines

and solve for x and y by substi tution 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 perpendicular

bisector Find the y-intercept of the perpendicular bisector by substi tuting slope and y-

intercept in the general equation Find the equation of the perpendicular bisector by substi tuting the slope and

y-Intercept in the general equation


Word Problems

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

Read the question and determine what all information is “given” these are the given, and are known as “known quantities”. Read the question and interpret what’s being asked or, what needs to be solved, or what information you need to know the answer of

the question these are the quanti ties you are seeking, and they are known as the “Unknown quantities”

Name the Unknown quanti ties by selecting variables, such as x, y, z, etc. Determine the relationships between the “knowns” and “unknowns”, that is, the variables and the other given quanti ties in the

problem, and connect them using arithmetic problems, such as (+), (-), etc. and wri te them as algebraic expressions. Using these variables and the relationships between the known and unknown quanti ties form algebraic 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 that involves

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

Basic Coin Conventions to be known:

1 Dollar 100 Cents ; 1 Half Dollar 50 Cents ; 1 Quarter 25 Cents ; 1 Dime 10 Cents ; 1 Nickel 5 Cents

Apply the following steps to solve Age problems:

Assign a different letter (Variable) for each person’s 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 Ago” means you need to subtract “Years from now” means you need to add

Rate of Work or Quantity:

Rate Amount 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 work slower you work slower the job is done Greater number of workers lesser the time required to finish the job Lesser number of workers greater the time required to finish the job If i t takes ‘k’ workers 1 hour to do a particular job, then each worker does 1/k of the job in an hour or works @ 1/k of the job per hour If i t takes ‘k’ workers ‘m’ hours to do a particular job, then each worker does 1/k of the job in an hour or works @ 1/(mh) o f the job per

hour Work Problem Formula 1/x + 1/y = 1/z Inverse of the time i t would take everyone working together equals the sum of the inverses of the time i t would take each working individually.

Dis tance Rate * Time

Cost per Unit Total Cost of the Mixture/Total Weight of the Mixture Mixture of Weaker and Stronger Solutions Problem Weaker (Desired – Stronger) = s (Stronger – Desi red), s Amount of 1st + 2nd


Unit of Measures US Customary System

1 yd (Yard) 3 ft (Feet) 36 Inches 1 Ton 2000lbs (Pounds) 1 lb 16 oz (Ounces)

1 Gallon 4 qt(Quart) 8 pt(Pint) 16 c(Cup) 128 fl oz(Fluid Ounce) 256 tbsp(Table Spoon) 1 sq yd 9 sq ft 1296 sq in Metric System: milli means one thousandths

centi means one thousandths deci means one tenths Basic Standard unit means one

Deka- or Deca- means tens Hector means 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 l tr

1 gallon 3.785 l tr 1 ton 2000 lbs 1 lb 16 oz

1 sq yd 9 sq ft 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 Minutes 1 Minute 60 Seconds

A.M Ante Meridian before Noon; P.M Post Meridian After noon As we travel east Sun rises earlier and therefore clock is ahead As we travel west Sun rises later and therefore clock is behind


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 ahead

of PST (Pacific Standard Time) Temperature Conversion

Celsius = (5/9) (Farenheit-32); Fahrenheit = (9/5) Celsius + 32; Freezing Point = 320 F; Boiling Point = 2120 F; Normal Body Temperature = 98.60 F


Logic & Stats

Simple Counting Involves figuring out how many integers are between any two given integers Rule # 1: When exactly one Endpoint is inclusive subtract the two values

Rule # 2: When both Endpoints are inclusive subtract both values , and then add 1 Rule # 3: When nei ther Endpoint is inclusive subtract the two values, and then subtract 1 Fundamental Principle of Counting If two jobs need to be completed and there are “m” ways to do the fi rs t job, and “n” ways to do the second 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 ordered.

n! = n * (n-1) * (n-2) * ………… * 2 * 1 until 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 are 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. Permutation m P n m! / (m-n)! m * (m-1) * (m-2) …… * (m-n+1) Where, m Number in the larger group; n number being arranged If there are “m” different terms/elements in a set, and there are “k” available or empty spots, then there are “p” di fferent ways of arranging them, given by the formula p = 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 instance,

AB and BA counts as two different permutations , but only as “1” combination Combination m C n m! / n! (m-n)! (m * (m-1) * (m-2) …… * (m-n+1))/n! = m P n / n!

Where, m Number in the larger group; n number being chosen Probability Probability P (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 probabil i ties is 1 Probability of Event that will not happen = 1 – Probabili ty of Event that will happen If ‘A’ and ‘B’ are independent events , then to determine the probability that event A and event B will BOTH together occur: MULTIPLY the probabilities of two individuals together

If ‘A’ and ‘B’ are independent events , and that they are mutually exclusive, then to determine the probabili ty that event A o r event B will occur: ADD the probabilities of two individuals together. Two Events are said to be mutually e xclusive i f 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 event A or event B will occur: ADD the probabilities of two individuals together and then SUBTRACT the probability that both events occur together. Two Events are said

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


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 fai r deck of cards – with each card we pick, the total possible events for the next event will be 1 less than the one before that.

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 i tems called elements or members of the set Finite Set: If a set contains only a fini te number of elements Infini te Set: If a set contains infini te number of elements Subset: If all the elements of one set S, are also elements of another set T; then the fi rs t set S, is a Subset of T

Venn Diagrams Graphically represents sets

Union Set: The set consisting of all the elements that exist 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 di fferent sets involved

Sequence A series , lis t, collection, or group of numbers that follows a specific pattern Pattern A 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 fi rs t term of an arithmetic progression, then the nth term of the ari thmetic progression will be = a + (n-1)d.

Geometric Sequence: If “a 1” is the fi rs t term, and “r” is the common ratio between consecutive terms of a geometric progression, and a n is the nth term, then the n th term will be a n = a1rn-1 Sum of “n” terms in a Geometric Sequence (ar n – a )/(r-1), when r # 1

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

Ari thmetic Mean Mean Average Total Sum of all terms / Total number of terms Sum of consecutive terms Mean of Consecutive Terms * Number of Consecutive Terms

Where, Mean of Consecutive Terms (Fi rs t Term + Last Term) / 2; Number of terms (Last term – Fi rs t Term) + 1

Sum of Existing term + Missing Term = Sum of all terms Weighted Mean Number of times a quanti ty or term occurs Sum of Products / Sum of Weights Sum / Frequency Median Middle When there are “n” terms, the median is the value of ((n+1)/2) th term Mode Set of Data that occurs most frequently

Quartiles Divides data into equal quarters or four equal parts

Range Largest term – Smallest Term

Standard Deviation Dis tance or the gap between the ari thmetic mean and the set of numbers

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

Find the Average (Ari thmetic 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 di fferences by summing the squared values and dividing the sum by the number of values Take the posi tive square root of that average


Statistics – Graphs

Graph Type How to Read?

Tables and charts Look for a speci fic unit on the row heading Then match that row with the corresponding unit on the column heading

Pictographs Look at the speci fic row Then compute i ts value based on the conversion factor given in the key. Each symbol represents a fixed

number of i tems as indicated in the key Single Line Graph Look for a speci fic 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 quanti ty 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 correct point on the line and move horizontally across from that point on the line to the value on the scale on the 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 quanti ty during that time period A line that slopes down from left to right, shows a decrease in the quanti ty during that time period

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

variable for that speci fic 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 quanti ty for that speci fic

bar or time period In order to find a specific numeric value of a particular bar from a bar 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 left 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 speci fic 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 quanti ty of that

specified variable for that specific time period Scatter Plot Graphs Look for the specific Quanti ty on the horizontal axis and the 2nd quanti ty in the vertical axis

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

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

Documents

Quants One Stop Guide