50 keys to cat from testfunda

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50 Keys To C ATArithmetic, Algebra, GeometryandModern MathematicsA collection of 50 very important formulaefor Quantitative Ability Section of CAT fromthe pioneers of online CAT training

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1. Averages

2. Mean

For two numbers a andb,

3. Percentage C hange

For two successivechanges ofa% andb%,

4. Interest

Amount = Principal + Interest


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5. Population Formula

[Here, P= Original population,P n years,r% = rate of annual change]

6. Depreciation Formula

[Here, P= original value,P n years,r% = rate of annual depreciation]

7. G rowthAbsolute G rowth = Final Value Initial Value

[Here, S . A. G . R. = S imple Annual Growth Rate, A. A. G . R . = Average Annual G rowth Rateand C. A. G . R. = C ompound Annual G rowth Rate]

8. Profit and LossProfit = SPC P

Loss = CPS P


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9. False WeightsIf an item is claimed to besold at cost price,using false weights,then the overall percentage

profit is given by

10. DiscountDiscount = Marked PriceS elling Price

Buy xand GetyFree

If articles worth Rs.x are bought and articles worth Rs.y are obtained free along withx

articles, then the discount is equal toyand discount percentage is givenby

S uccessive Discounts

When a discount ofa% is followed by another discount ofb%, then

11. RatiosIfa : b = c : d, thena : b = c : d=(a + c) : (b + d)

Ifa < b, thenfor a positive quantityx,


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Ifa > b, thenfor a positive quantityx,

12. Proportions

13. S uccessive Replacement

Where xis the original quantity,yis the quantity that is replacedand n is the number of

times the replacement process iscarried 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 averageattribute of the resultant

mixture.

Alligation C ross:

15. Time, S peed and Distance

Important C onversion Factors:

16. Average S peed

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 S peedFor Train s

For Boats and S treams

18. Time and Work

19. Application of H .C.F.The greatest natural number that will dividex,yandzleaving remaindersr1, r2 andr3,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 remainderrineach 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 S urds

23. Laws of IndicesIfa andb are non-zero rational numbers and m andn 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 binomialtheorem:

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

(ab)2 = a22ab + b2

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(ab)3 = a33a2b + 3ab2b3

a3 + b3 = (a + b)(a2ab + b2)

a3b3 = (ab)(a2 + ab + b2)

28. Arithmetic Progression

29. G eometric Progression

30. Harmonic Progression


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31. S um of Important S eriesS um of first n natural n umbers

S um of the squares of the first n natural numbers

S um of the cubes of the first n natural numbers

32. Factorialn (n1) n

n! = n (n1)!

33. Permutations

34. C ombinations

Important Properties:


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35. Partition RuleNumber of ways of distributing n identical things amongrpersons 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 length s of one diagonal and two offsets are given


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If length s 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. C ircle

C ircumference

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 hypotenuseand 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 S urface Area (L.S.A.) = Perimeter of base height

Total S urface Area (T.S.A.) = L.S.A. + 2 Area of base

Volume (V) = Area of base height


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C uboid

L.S.A. = 2(lh + bh)

T.S.A. = 2(lh + bh + lb)

Volume (V) = lbh

C ube

L.S.A. = 4a2

T.S.A. = 6a2

Volume (V) = a3


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C ylinder

C urved S urface Area (C.S.A.) rh

T.S.A. rh r2

Volume (V) r2h

49. Right Pyramid

T.S.A. = L.S.A. + Area of base


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C one

C.S.A. rl

T.S.A. rl r2

Frustum of a cone


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50. S phere

C.S.A. = 4r2

T.S.A. = 4r2

Hemisphere

C.S.A. = 2r2

T.S.A. = 3r2


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