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50 Keys To CATArithmetic, 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
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1. Averages
2. Mean
For two numbers a and b,
3. Percentage Change
For two successive changes ofa% and b%,
4. Interest
Amount = Principal + Interest
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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. GrowthAbsolute 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 LossProfit = SP CP
Loss = CP SP
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9. False WeightsIf an item is claimed to be sold at cost price, using false weights, then the overall percentage
profit is given by
10. DiscountDiscount = Marked Price Selling Price
Buy xand GetyFree
If articles worth Rs. x are bought and articles worth Rs. y are obtained free along with x
articles, then the discount is equal toyand discount percentage is given by
Successive Discounts
When a discount ofa% is followed by another discount ofb%, then
11. RatiosIfa : b = c : d, then a : b = c : d=(a + c) : (b + d)
Ifa < b, then for a positive quantityx,
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Ifa > b, then for a positive quantityx,
12. Proportions
13. Successive Replacement
Where xis the original quantity,yis the quantity that is replaced and n is the number of
times the replacement process is carried out.
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14. Alligation RuleThe 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:
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If the time is constant, then average speed is given by arithmetic mean of two speeds:
17. Relative SpeedFor Trains
For Boats and Streams
18. Time and Work
19. Application of H.C.F.The greatest natural number that will dividex,yandzleaving 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 byx,yandzleaving the same remainder rin
each case is the L.C.M. of (x,yandz) + r
21. H.C.F. and L.C.M. of Fractions
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22. Properties of Surds
23. Laws of IndicesIfa and b are non-zero rational numbers and m and n are rational numbers, then
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24. Laws of Logarithms
25. Binomial TheoremIfn is a natural number that is greater than or equal to 2, then according to the binomial
theorem:
26. Roots of Quadratic EquationThe two roots of the quadratic equation, ax2 + bx+ c = 0 are given by:
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27. Algebraic Formulae(a + b) (ab) = a2b2
(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
(ab)3 = a3 3a2b + 3ab2b3
a3 + b3 = (a + b)(a2ab + b2)
a3b3 = (ab)(a2 + ab + b2)
28. Arithmetic Progression
29. Geometric Progression
30. Harmonic Progression
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31. Sum of Important SeriesSum of firstn natural numbers
Sum of the squares of the firstn natural numbers
Sum of the cubes of the firstn natural numbers
32. Factorialn! = 1 2 3 (n 1) n
n! = n (n 1)!
33. Permutations
34. Combinations
Important Properties:
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35. Partition RuleNumber of ways of distributing n identical things among rpersons when each person may
get any number of things =
36. Probability
37. Pythagoras TheoremFor right triangle ABC
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For acute triangle ABC
For obtuse triangle ABC
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38. Area of TriangleWhen lengths of the sides are given
When lengths of the base and altitude are given
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When lengths of two sides and the included angle are given
For Equilateral Triangle
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For Isosceles Triangle
39. Apollonius Theorem
If AD is the median, then:
AB2 + AC2 = 2(AD2 + BD2)
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40. Angle Bisector Theorem
If AD is the angle bisector for angle A, then:
41. Area of QuadrilateralFor Cyclic Quadrilateral
If lengths of one diagonal and two offsets are given
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If lengths of two diagonals and the included angle are given
For Trapezium
For Parallelogram
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For Rhombus
For Rectangle
For Square
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42. PolygonsNumber of Diagonals
The sum of all the interior angles
The sum of all the exterior angles
43. Area of Regular Hexagon
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44. Circle
Circumference
Area
Length of Arc
Area of Sector
Or,
Perimeter of Sector
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45. Ellipse
If semi-major axis (OD) = a and semi-minor axis (OA) = b,
Perimeter of the ellipse
Area of the ellipse
46.
Trigonometric RatiosFor a right triangle, ifPis the length of perpendicular, B is the length of base, His the length
of hypotenuse and is the angle between base and hypotenuse,
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47. Distance between PointsDistance 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
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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
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Cylinder
Curved Surface Area (C.S.A.) = 2rh
T.S.A. = 2rh+ 2r2
Volume (V)= r2h
49. Right Pyramid
T.S.A. = L.S.A. + Area of base
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Cone
C.S.A. = rl
T.S.A. = rl+ r2
Frustum of a cone
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50. Sphere
C.S.A. = 4r2
T.S.A. = 4r2
Hemisphere
C.S.A. = 2r2
T.S.A. = 3r2
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