50 Keys To CAT Arithmetic, Algebra, Geometry and Modern Mathematics A collection of 50 very important formulae for Quantitative Ability Section of CAT from the pioneers of online CAT training ▪ EASY ▪ EFFECTIVE ▪ PERSONALISED 1. Averages 2. Mean For two numbers a and b, 3. Percentage Change For two successive changes of a% and b%, 4. Interest Amount = Principal + Interest © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 1 5. Population Formula [Here, P = Original population, P’ = population after n years, r% = rate of annual change] 6. Depreciation Formula [Here, P = original value, P’ = final value after n years, r% = rate of annual depreciation] 7. Growth Absolute Growth = Final Value – Initial Value [Here, S. A. G. R. = Simple Annual Growth Rate, A. A. G. R. = Average Annual Growth Rate and C. A. G. R. = Compound Annual Growth Rate] 8. Profit and Loss Profit = SP − CP Loss = CP − SP © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 2 9. False Weights If an item is claimed to be sold at cost price, using false weights, then the overall percentage profit is given by 10. Discount Discount = Marked Price − Selling Price Buy x and Get y Free If articles worth Rs. x are bought and articles worth Rs. y are obtained free along with x articles, then the discount is equal to y and discount percentage is given by Successive Discounts When a discount of a% is followed by another discount of b%, then 11. Ratios If a : b = c : d, then a : b = c : d = (a + c) : (b + d) If a < b, then for a positive quantity x, © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 3 If a > b, then for a positive quantity x, 12. Proportions 13. Successive Replacement Where x is the original quantity, y is the quantity that is replaced and n is the number of times the replacement process is carried out. © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 4 14. Alligation Rule The ratio of the weights of the two items mixed will be inversely proportional to the deviation of attributes of these two items from the average attribute of the resultant mixture. Alligation Cross: 15. Time, Speed and Distance Important Conversion Factors: 16. Average Speed If the distance is constant, then average speed is given by harmonic mean of two speeds: © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 5 If the time is constant, then average speed is given by arithmetic mean of two speeds: 17. Relative Speed For Trains For Boats and Streams 18. Time and Work 19. Application of H.C.F. The greatest natural number that will divide x, y and z leaving remainders r1, r2 and r3, respectively, is the H.C.F. of (x − r1), (y − r2) and (z − r3) 20. Application of L.C.M. The smallest natural number that is divisible by x, y and z leaving the same remainder r in each case is the L.C.M. of (x, y and z) + r 21. H.C.F. and L.C.M. of Fractions © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 6 22. Properties of Surds 23. Laws of Indices If a and b are non-zero rational numbers and m and n are rational numbers, then © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 7 24. Laws of Logarithms 25. Binomial Theorem If n is a natural number that is greater than or equal to 2, then according to the binomial theorem: 26. Roots of Quadratic Equation The two roots of the quadratic equation, ax2 + bx + c = 0 are given by: © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 8 27. Algebraic Formulae (a + b) (a − b) = a2 − b2 (a + b)2 = a2 + 2ab + b2 (a − b)2 = a2 − 2ab + b2 (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a − b)3 = a3 − 3a2b + 3ab2 − b3 a3 + b3 = (a + b)(a2 − ab + b2) a3 − b3 = (a − b)(a2 + ab + b2) 28. Arithmetic Progression 29. Geometric Progression 30. Harmonic Progression © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 9 31. Sum of Important Series Sum of first n natural numbers Sum of the squares of the first n natural numbers Sum of the cubes of the first n natural numbers 32. Factorial n! = 1 × 2 × 3 × … × (n − 1) × n n! = n × (n − 1)! 33. Permutations 34. Combinations Important Properties: © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 10 35. Partition Rule Number of ways of distributing n identical things among r persons when each person may get any number of things = 36. Probability 37. Pythagoras Theorem For right triangle ABC © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 11 For acute triangle ABC For obtuse triangle ABC © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 12 38. Area of Triangle When lengths of the sides are given When lengths of the base and altitude are given © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 13 When lengths of two sides and the included angle are given For Equilateral Triangle © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 14 For Isosceles Triangle 39. Apollonius Theorem If AD is the median, then: AB2 + AC2 = 2(AD2 + BD2) © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 15 40. Angle Bisector Theorem If AD is the angle bisector for angle A, then: 41. Area of Quadrilateral For Cyclic Quadrilateral If lengths of one diagonal and two offsets are given © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 16 If lengths of two diagonals and the included angle are given For Trapezium For Parallelogram © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 17 For Rhombus For Rectangle For Square © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 18 42. Polygons Number of Diagonals The sum of all the interior angles The sum of all the exterior angles 43. Area of Regular Hexagon © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 19 44. Circle Circumference Area Length of Arc Area of Sector Or, Perimeter of Sector © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 20 45. Ellipse If semi-major axis (OD) = a and semi-minor axis (OA) = b, Perimeter of the ellipse Area of the ellipse 46. Trigonometric Ratios For a right triangle, if P is the length of perpendicular, B is the length of base, H is the length of hypotenuse and is the angle between base and hypotenuse, © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 21 47. Distance between Points Distance between two points A (x1, y1) and B (x2, y2) is given by 48. Right Prism Lateral Surface Area (L.S.A.) = Perimeter of base × height Total Surface Area (T.S.A.) = L.S.A. + 2 × Area of base Volume (V) = Area of base × height © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 22 Cuboid L.S.A. = 2(lh + bh) T.S.A. = 2(lh + bh + lb) Volume (V) = lbh Cube L.S.A. = 4a2 T.S.A. = 6a2 Volume (V) = a3 © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 23 Cylinder Curved Surface Area (C.S.A.) = 2πrh T.S.A. = 2πrh + 2πr2 Volume (V) = πr2h 49. Right Pyramid T.S.A. = L.S.A. + Area of base © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 24 Cone C.S.A. = πrl T.S.A. = πrl + πr2 Frustum of a cone © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 25 50. Sphere C.S.A. = 4πr2 T.S.A. = 4πr2 Hemisphere C.S.A. = 2πr2 T.S.A. = 3πr2 © 2008 Enabilon Learning Pvt. Ltd. All rights reserved. 26 Personalised Online CAT preparation EASY   Prepare at your own pace – don’t be constrained by the pace of the class  Anytime, anywhere, on any computer - no installation required Prepare at your convenience – don’t be bothered by unsuitable classroom timings, travel difficulties, hectic work schedule EFFECTIVE     16 National Online Mock CATs; Instant access to your national percentile Comprehensive study material – 95 lessons with solved examples Detailed solution for every question in the tests and practice exercises Effective instructional strategy for the CAT aspirant; by a team that created international award winning internet-based educational products PERSONALISED   Study the concepts YOU need to with Personalised Study Guides  Improve YOUR accuracy, speed, question selection with Detailed Analysis Reports  Personalised help from content experts – Ask a Doubt Practice the concepts YOU are weak in with Personalised Post-Assessment Tests © 2008 Enabilon Learning Pvt. 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