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### cat-quant-cheat-sheet

1. 1. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 1 Arithmetic Fundamentals Table of Squares Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 20 25 Square 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 3 4 4 5 9 10 1 Decimals 0.1 0.2 0.25 0.3 0.33 0.4 0.5 0.6 0.66 0.75 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
2. 2. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 2 Divisibility Rule Number Rule Example 2 If last digit is 0,2,4,6, or 8 22, 30, 50, 68, 1024 3 If sum of digits is divisible by 3 123 is divisible by 3 since 1 + 2 + 3 = 6 (and 6 is divisible by 3) 4 If number created by the last 2 digits is divisible by 4 864 is divisible by 4 since 64 is divisible by 4 5 If last digit is 0 or 5 5, 10, 15, 20, 25, 30, 35, 2335 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 9 621 is divisible by 9 since 6 + 2 + 1 = 9 (and 9 is divisible by 9) 10 If last digit is 0 10, 20, 30, 40, 50, 5550 Logarithms Definition: π = πππ π π β π = π π Example: πππ π ππ = π β ππ = π π Properties πππ π π = π πππ π π = π πππ π π π = π ππππ π π = π πππ π π π = ππππ π π πππ π π Γ π = πππ π π + πππ π π πππ π π π = πππ π π β πππ π π Special Exponents Reciprocal π π π = πβπ Power 0 π π = π Power 1 π π = π Operations Involving Exponents Multiplication π π Γ π π = π(π+π) Division π π Γ· π π = π(πβπ) Power (π π ) π = π πΓπ Roots π π π = π ππ = ( π π ) π
3. 3. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 3 Progressions Arithmetic Progression: π π‘π term of an Arithmetic Progression π π = π π + π β π π Sum of n terms of an arithmetic expression π π = π π π + π π π = π ππ π + π β π π π The first term is π1, the common difference is π, and the number of terms is π. Geometric Progression: π π‘π term of a Geometric Progression π π = π π π πβπ Sum of n terms of a geometric progression: π π = π(π π+π β π) π β π The first term is π1, the common ratio is π, and the number of terms is π. Infinite Geometric Progression Sum of all terms in an infinite geometric series = π π (πβπ) π€ππππ β π < π < π Roots of a Quadratic Equation A quadratic equation of type ππ± π + ππ± + π has two solutions, called roots. These two solutions may or may not be distinct. The roots are given by the quadratic formula: βbΒ± b2β4ac 2a where the Β± sign indicates that both βbβ b2β4ac 2a and βb+ b2β4ac 2a are solutions Common Factoring Formulas 1. π π β π π = π β π Γ π + π 2. π π + πππ + π π = (π + π) π 3. π π β πππ + π π = (π β π) π 4. π π + πππ π + ππ π π + π π = (π + π) π 5. π π β πππ π + ππ π π β π π = (π β π) π Binomial Theorem The coefficient of π₯(πβπ) π¦ π in (π₯ + π¦) π is: πΆπ π = π! π! (π β π)! Applies for any real or complex numbers x and y, and any non-negative integer n.
4. 4. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 4 Summary of counting methods Order matters Order doesnβt matter With Replacement If π objects are taken from a set of π objects, in a specific order with replacement, how many different samples are possible? π π N/A Without Replacement 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? ππ π = 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Άπ π = π! π!(πβπ)! Probability The probability of an event A, π π΄ is defined as π π΄ = ππ’ππππ ππ ππ’π‘πππππ  π‘πππ‘ ππππ’π ππ ππ£πππ‘ π΄ πππ‘ππ ππ’ππππ ππ ππππππ¦ ππ’π‘πππππ  Independent Events: If A and B are independent events, then the probability of A happening and the probability of B happening is: π π΄ πππ 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΄ πππ 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΄ πππ π΅) π(π΄)
5. 5. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 5 Geometry Fundamentals The sum of angles around a point will always be 360 degrees. In the adjacent figure π + π + π + π = πππΒ° Vertical angles are equal to each other. In the adjacent figure π = π and π = π The sum of angles on a straight line is 180Λ. In the adjacent figure sum of angles and b is 180 i.e. π + π = πππΒ° When a line intersects a pair of parallel lines, the corresponding angles are formed are equal to each other In the figure above π = π When a line intersects a pair of parallel lines, the alternate interior and exterior angles formed are equal to each other In the adjacent figure alternate interior angles a=b and alternate exterior angles c=d Any point on the perpendicular bisector of a line is equidistant from both ends of the line. In the figure, k is the perpendicular bisector of segment AB and c=d, e=f
6. 6. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 6 Triangle Properties 1. Sum of all internal angles of a triangle is 180 degrees i.e. π + π + π = πππΒ° 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© > πͺ β π > π 4. The exterior angle is supplemental to the adjoining interior angle i.e. π + π = πππΒ°. Since π + π + π = πππΒ° 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 adjoining figure BD DC = AB BC Pythagoras Theorem π π = π π + π π Commonly Used Pythagorean Triples 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
7. 7. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 7 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 π π < π π + π π then it is an acute-angled triangle, i.e. the angle facing side c is an acute angle. If π π > π π + π π 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°
8. 8. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 8 Properties of a Circle The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc. In the adjacent figure angle π = ππ Every angle subtended at the circumference by the diameter of a circle is a right angle (90Λ). In a cyclic quadrilateral, the opposite angles are supplementary i.e. they add up to 180Λ. In the figure above angle π + π = πππΒ° and π + π = πππΒ° The angles at the circumference subtended by the same arc are equal. 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
9. 9. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 9 Area and Perimeter of Common Geometrical Figures Rectangle Area π¨ = π Γ π Perimeter π· = π Γ (π + π) Square Area π¨ = π π Perimeter π· = ππ Triangle Area π¨ = π π Γ π Γ π Circle Area π¨ = ππ π Perimeter π· = πππ Equilateral Triangle Area π¨ = π π Γ π π Perimeter π· = ππ Altitude π = π π π Trapezoid Area π¨ = π π π (π π + π π) Parallelogram Area π¨ = π Γ π Perimeter π· = π Γ (π + π) Ring Area π¨ = π(πΉ π β π π ) Sector π in degrees Area π¨ = ππ π Γ π πππ Length of Arc π³ = ππ« Γ π πππ Perimeter π· = π³ + ππ
10. 10. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 10 Volume and Surface Area of 3 Dimensional Figures Cube Volume π = π Γ π Γ π Surface Area π¨ = π(ππ + ππ + ππ) Sphere Volume π = π π ππ π Surface Area π¨ = πππ π Right Circular Cylinder Volume π = ππ π π‘ Surface Area π¨ = πππ«π‘ Pyramid Volume π = π π π©π Surface Area = π¨ = π© + ππ π B is the area of the base π is the slant height Right Circular Cone Volume π = π π ππ π π‘ Surface Area π¨ = ππ(π + π) Frustum Volume π = π π ππ‘(π« π + π«π + π π ) Coordinate Geometry Line: Equation of a line π = ππ + π In the adjoining figure, c is the intercept of the line on Y-axis i.e. c=2, m is the slope Slope of a line π = π πβπ π π πβπ π
11. 11. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 11 The slopes of parallel lines are equal In the adjoining figure, the two lines are parallel to each other i.e. π π = π π The slopes of perpendicular lines are opposite reciprocals of one another. In the adjoining the two lines are perpendicular to each other i.e. π π = βπ π π Midpoint between two points (π π, π π) and (π π, π π) in a x-y plane: π π, π π = π π β π π π , π π β π π π Distance between two points (π π, π π) and (π π, π π) in a x-y plane: π = (π π β π π) π + (π π β π 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 Equation of a circle with center (h,k) and radius r (π β π) π + (π β π) π = π π
12. 12. The smart way to learn online http://www.snapwiz.co.in Quantitative Ability Formula Sheet CAT and Management Entrance Tests Visit http://www.snapwiz.co.in for free mock CAT tests Page 12 Trigonometry Basics Definition: Right Triangle definition for angle ΞΈ such that 0 < ΞΈ < 90Β° π πππ = ππππππ‘ ππ¦πππ‘πππ’π π , ππ ππ = 1 π πππ πππ π = πππ π ππ¦πππ‘πππ’π π , π πππ = 1 πππ π π‘πππ = ππππππ‘ πππ π , πππ‘π = 1 π‘πππ Pythagorean Relationships π ππ2 π + πππ 2 π = 1 π‘ππ2 π + 1 = π ππ2 π πππ‘2 π + 1 = ππ π2 π Tan and Cot π‘πππ = π πππ πππ π , πππ‘π = πππ π π πππ Half Angle Formulas π ππ2 π = 1 2 (1 β cos 2π ) πππ 2 π = 1 2 1 + cos 2π π‘ππ2 π = (1 β cos 2π ) (1 + cos 2π ) Even/Odd sin βπ = βπ πππ cos βπ = πππ π tan βπ = βπ‘πππ sin 2π = 2π ππππππ π, cos 2π = πππ 2 π β π ππ2 π Periodic Formula: Where n is an integer sin π + 2ππ = sin π β sin π + 180Β° = π πππ cosβ‘(π + 2ππ) = cos π β cosβ‘(ΞΈ + 180Β°) = πππ π tan π + ππ = tan π β tan ΞΈ + 90Β° = tanΞΈ 0Β° 30Β° 45Β° 60Β° 90Β° Sin 0 1 2 1 2 3 2 1 Cos 1 3 2 1 2 1 2 0 Tan 0 1 3 1 3 β