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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