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Quantitative Ability Formula Sheet CAT and Management Entrance Tests
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Arithmetic Fundamentals Table of Squares Number 1 2 3 4 Square
5
6
7
8
9
10
11
12
13
14
15
16
20
25
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 400 625
Table of Cubes Number
2
3
4
5
10`
20
100
Cube
8
27
64
125
1000
8000
1000000
Commonly used Decimal, Percent and Fractions(Less than 1) Percent
10% 20% 25% 30% 33% 40% 50% 60% 66% 75% 80% 90% 100%
Fractions
1 10
2 10
1 4
3 10
1 3
2 5
1 2
3 5
2 3
Decimals
0.1
0.2
0.25
0.3
0.33
0.4
0.5
0.6
3 4
0.66 0.75
4 5
9 10
1
0.8
0.9
1
Commonly used Decimal, Percent and Fractions (Greater than 1) Percent
100%
125%
133.33%
150%
200%
Fractions
1
5 4
4 3
3 2
2
Decimals
1
1.25
1.33
1.5
2.0
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Divisibility Rule Number Rule 2 If last digit is 0,2,4,6, or 8 3 If sum of digits is divisible by 3 4
Example 22, 30, 50, 68, 1024 123 is divisible by 3 since 1 + 2 + 3 = 6 (and 6 is divisible by 3) 864 is divisible by 4 since 64 is divisible by 4
5
If number created by the last 2 digits is divisible by 4 If last digit is 0 or 5
6
If divisible by 2 & 3
522 is divisible by 6 since it is divisible by 2 & 3
9
If sum of digits is divisible by 9
10
If last digit is 0
621 is divisible by 9 since 6 + 2 + 1 = 9 (and 9 is divisible by 9) 10, 20, 30, 40, 50, 5550
5, 10, 15, 20, 25, 30, 35, 2335
Operations Involving Exponents Multiplication
ππ Γ ππ = π(π+π)
Special Exponents
Division
ππ Γ· ππ = π(πβπ)
Reciprocal
(ππ )π = ππΓπ
Power Roots
π
ππ =
π
π
ππ = ( π)π
π = πβπ π π
Power 0
ππ = π
Power 1
ππ = π
Logarithms Definition: π = ππππ π β π = ππ
Properties
Example: ππππ ππ = π β ππ = ππ
ππππ π = π ππππ π = π ππππ ππ = π πππππ π = π ππππ ππ = πππππ π ππππ π Γ π = ππππ π + ππππ π ππππ
π = ππππ π β ππππ π π
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Progressions Arithmetic Progression: ππ‘π term of an Arithmetic Progression
Geometric Progression: ππ‘π term of a Geometric Progression
ππ = ππ + π β π π
ππ = ππ π πβπ
Sum of n terms of an arithmetic expression
Sum of n terms of a geometric progression:
ππ =
π ππ + ππ π πππ + π β π π
= π π
ππ =
The first term is π1 , the common difference is π, and the number of terms is π.
π(ππ+π β π) πβπ
The first term is π1 , the common ratio is π, and the number of terms is π.
Infinite Geometric Progression Sum of all terms in an infinite geometric series =
ππ (πβπ)
π€ππππ β π < π < π
Roots of a Quadratic Equation A quadratic equation of type ππ± π + ππ± + π has two solutions, called roots. These two solutions may or may not be distinct. The roots are given by the quadratic formula: the Β± sign indicates that both
βbβ b 2 β4ac 2a
and
βb+ b 2 β4ac 2a
βbΒ± b 2 β4ac 2a
where
are solutions
Common Factoring Formulas 1. 2. 3. 4. 5.
ππ β ππ = π β π Γ π + π ππ + πππ + ππ = (π + π)π ππ β πππ + ππ = (π β π)π ππ + ππππ + πππ π + ππ = (π + π)π ππ β ππππ + πππ π β ππ = (π β π)π
Binomial Theorem The coefficient of π₯ (πβπ) π¦ π in (π₯ + π¦)π is: π ππΆ
=
π! π! (π β π)!
Applies for any real or complex numbers x and y, and any non-negative integer n.
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Quantitative Ability Formula Sheet CAT and Management Entrance Tests
Summary of counting methods With Replacement
Without Replacement
Order matters
Order doesnβt matter
If π objects are taken from a set of π objects, in a specific order with replacement, how many different samples are possible? ππ Permutation Rule: If π objects are taken from a set of π objects without replacement, in a specific order, how many different samples are possible?
N/A
π ππ
=
π! (π β π)!
Combination Rule: If π objects are taken from a set of π objects without replacement and disregarding order, how many different samples are possible? π ππΆ
=
π! π!(πβπ)!
Probability The probability of an event A, π π΄ is defined as π π΄ =
ππ’ππππ ππ ππ’π‘πππππ π‘πππ‘ ππππ’π ππ ππ£πππ‘ π΄ πππ‘ππ ππ’ππππ ππ ππππππ¦ ππ’π‘πππππ
Independent Events: If A and B are independent events, then the probability of A happening and the probability of B happening is: π π΄ πππ π΅ = π(π΄) Γ π(π΅) Dependent Events: If A and B are dependent events, then the probability of A happening and the probability of B happening, given A, is: π π΄ πππ π΅ = π(π΄) Γ π(π΅ | π΄) Conditional Probability: The probability of an event occurring given that another event has already occurred e.g. what is the probability that B will occur after A π π΅π΄ =
π(π΄ πππ π΅) π(π΄)
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Quantitative Ability Formula Sheet CAT and Management Entrance Tests
Geometry Fundamentals The sum of angles around a point will always be 360 degrees.
Vertical angles are equal to each other. In the adjacent figure π = π and π=π
In the adjacent figure π + π + π + π
= πππΒ°
The sum of angles on a straight line is 180Λ.
When a line intersects a pair of parallel lines, the corresponding angles are formed are equal to each other
In the adjacent figure sum of angles and b is 180 i.e. π + π = πππΒ°
When a line intersects a pair of parallel lines, the alternate interior and exterior angles formed are equal to each other
In the figure above π=π
Any point on the perpendicular bisector of a line is equidistant from both ends of the line.
In the adjacent figure alternate interior angles a=b and alternate exterior angles c=d
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In the figure, k is the perpendicular bisector of segment AB and c=d, e=f
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Triangle Properties 1. Sum of all internal angles of a triangle is 180 degrees i.e. π + π + π = πππΒ° 2. Sum of any two sides of a triangle is greater than the third i.e. π + π > π or π + π > π or π + π > π 3. The largest interior angle is opposite the largest side; the smallest interior angle is opposite the smallest side i.e. if π© > πͺ β π>π 4. The exterior angle is supplemental to the adjoining interior angle i.e. π + π = πππΒ°. Since π + π + π = πππΒ° it follows that π = π + π 5. The internal bisector of an angle bisects the opposite side in the ratio of the other two sides. In the BD
AB
adjoining figure DC = BC
Pythagoras Theorem Commonly Used Pythagorean Triples
ππ = ππ + ππ
Height, Base
Hypotenuse
3,4 or 4,3
5
6,8 or 8,6
10
5, 12 or 12,5
13
7, 24 or 24,7
25
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Special Right Triangles
The lengths of the sides of a 45ο°- 45ο°90ο° triangle are in the ratio of π: π: π
The lengths of the sides of a 30ο°- 60ο°- 90ο° triangle are in the ratio of π: π: π
Test of Acute and Obtuse Triangles If ππ < ππ + ππ then it is an acute-angled triangle, i.e. the angle facing side c is an acute angle.
If ππ > ππ + ππ then it is an obtuse-angled triangle, i.e. the angle facing side c is an obtuse angle.
Polygons Sum of interior angles of a quadrilateral is 360Λ π + π + π + π = 360Β°
Sum of interior angles of a pentagon is 540Λ π + π + π + π + π = 540Β°
If n is the number of sides of the polygon then, sum of interior angles = (n - 2)180Β°
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Properties of a Circle The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc.
Every angle subtended at the circumference by the diameter of a circle is a right angle (90Λ).
In the adjacent figure angle π = ππ
In a cyclic quadrilateral, the opposite angles are supplementary i.e. they add up to 180Λ.
The angles at the circumference subtended by the same arc are equal.
In the figure above angle π+π= πππΒ° and π + π
= πππΒ°
In the adjacent figure angle π=π
A chord is a straight line joining 2 points on the circumference of a circle.
A radius that is perpendicular to a chord bisects the chord into two equal parts and vice versa. In the adjacent figure PW=PZ
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Area and Perimeter of Common Geometrical Figures Rectangle
Square
Triangle
Area π¨ = π Γ π Perimeter π· = π Γ (π + π) Circle
Area π¨ = ππ Perimeter π· = ππ Equilateral Triangle
Area π¨ = π Γ π Γ π
Area π¨ = π
ππ Perimeter π· = ππ
π
Parallelogram
Area π¨ = π Γ π Perimeter π· = π Γ (π + π)
π
Trapezoid
π
Area π¨ = π Γ ππ Perimeter π· = ππ Altitude π = π Ring
π π
Area π¨ = π (ππ + ππ )
π π
π
Sector
π
Area π¨ = π
(πΉ β π )
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π in degrees Area π¨ = π
ππ Γ
π πππ
Length of Arc π³ = π
π« Γ Perimeter π· = π³ + ππ
π πππ
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Volume and Surface Area of 3 Dimensional Figures Cube
Sphere
Right Circular Cylinder
π
Volume π = π Γ π Γ π Surface Area π¨ = π(ππ + ππ + ππ) Pyramid
Volume π = π π
ππ
Volume π = π
ππ π‘ Surface Area π¨ = ππ
π«π‘
Right Circular Cone
Frustum
Surface Area π¨ = ππ
ππ
π
Volume π = π π©π
ππ
Surface Area = π¨ = π© + π B is the area of the base π is the slant height
π
Volume π = π π
ππ π‘ Surface Area π¨ = π
π(π + π)
Coordinate Geometry Line: Equation of a line π = ππ + π
Volume π π = π π
π‘(π« π + π«π + ππ )
π βπ
Slope of a line π = ππβππ π
π
In the adjoining figure, c is the intercept of the line on Y-axis i.e. c=2, m is the slope
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The slopes of parallel lines are equal In the adjoining figure, the two lines are parallel to each other i.e. ππ = ππ
Midpoint between two points (ππ , ππ ) and (ππ , ππ ) in a x-y plane:
ππ , ππ =
ππ β ππ ππ β ππ , π π
Horizontal and Vertical line Equation of line parallel to x-axis (line p) with intercept on y-axis at (0, b) is π = π. Equation of line parallel to y-axis (line q) with intercept on x-axis at (a,0) is π = π
Quantitative Ability Formula Sheet CAT and Management Entrance Tests
The slopes of perpendicular lines are opposite reciprocals of one another. In the adjoining the two lines are perpendicular to each other βπ
i.e. ππ = π
π
Distance between two points (ππ , ππ ) and (ππ , ππ ) in a x-y plane:
π
=
(ππ β ππ )π + (ππ β ππ )π
Equation of a circle with center (h,k) and radius r
(π β π)π + (π β π)π = ππ
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Trigonometry Basics Definition: Right Triangle definition for angle ΞΈ such that 0 < ΞΈ < 90Β°
Pythagorean Relationships
Tan and Cot
π ππ2 π + πππ 2 π = 1
π‘πππ =
π πππ , πππ π
πππ‘π =
πππ π π πππ
π‘ππ2 π + 1 = π ππ 2 π 2
2
πππ‘ π + 1 = ππ π π
ππππππ‘ π πππ = , ππ¦πππ‘πππ’π π πππ π = π‘πππ =
πππ π , ππ¦πππ‘πππ’π π ππππππ‘ , πππ π
1 ππ ππ = π πππ π πππ =
πππ‘π =
1 πππ π
1 π‘πππ
Half Angle Formulas
Even/Odd
1 π ππ2 π = (1 β cos 2π ) 2
sin βπ = βπ πππ
π‘ππ2 π =
(1 β cos 2π ) (1 + cos 2π )
Sin
0Β°
30Β°
45Β°
60Β°
90Β°
0
1 2
1
3 2
1
3 2
1
1 2
0
1
1
cosβ‘ (π + 2ππ) = cos π β cosβ‘ (ΞΈ + 180Β°) = πππ π tan π + ππ = tan π β tan ΞΈ + 90Β° = tanΞΈ
tan βπ = βπ‘πππ
sin 2π = 2π ππππππ π, cos 2π = πππ 2 π β π ππ2 π
Periodic Formula: Where n is an integer sin π + 2ππ = sin π β sin π + 180Β° = π πππ
cos βπ = πππ π
1 πππ 2 π = 1 + cos 2π 2
Cos
Tan
1 0
2 2
3
β
3
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