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# 114 Handy Formulae for Quantitative Aptitude Problems

## Author: Sagar Sonker

ℵ Numbers
ℵ H.C.F & L.C.M of Numbers
ℵ Surds & Indices
ℵ Percentage
ℵ Profit & Loss
ℵ Ratio & Proportion
ℵ Partnership
ℵ Chain Rule
ℵ Time & Work
ℵ Pipes & Cisterns
ℵ Time And Distance
ℵ Trains
ℵ Boats & Streams
ℵ Alligation or Mixture
ℵ Simple Interest
ℵ Compound Interest
ℵ Logarithms
ℵ Area
ℵ Volume & Surface Area
ℵ Stocks & Shares
ℵ True Discount
ℵ Banker’s Discount

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

Numbers

## 2. A number is divisible by 3, if the sum of its digits is divisible by 3

3. A number is divisible by 4, if the number formed by its last two digits is divisible by 4

4. A number is divisible by 8, if the number formed by its last three digits is divisible by 8

## 6. A number is divisible by 11, if, starting from the RHS,

(Sum of its digits at the odd place) – (Sum of its digits at even place) is equal to 0 or 11x

7. (a + b)2 = a2 + 2ab + b2

8. (a - b)2 = a2 - 2ab + b2

## Dividend = Quotient × Divisor + Remainder

15. An Arithmetic Progression (A. P.) with first term ‘a’ and Common Difference ‘d’ is given
by:
[a], [(a + d)], [(a + 2d)], … … …, [a + (n - 1)d]

## nth term, Tn = a + (n - 1)d

Sum of first ‘n’ terms, Sn
= n/2 (First Term + Last Term)

16. A Geometric Progression (G. P.) with first term ‘a’ and Common Ratio ‘r’ is given by:
a, ar, ar2, ar3, … … …, arn-1

## nth term, Tn = arn-1

Sum of first ‘n’ terms Sn = [a(1 - rn)] / [1 - r]

## 19. (13 + 23 + 33 + … … … + n3) = [n2(n + 1)2] / 4

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

## Surds & Indices

21. am × an = a(m + n)

22. am / an = a(m - n)

23. (ab)m = a mb m

24. (a / b)m = an / bn

25. a0 =1

26. = a1/n

27. = (a1/n)n
=a

28. =

29. = /

30. ( )m =

31. =

Percentage

## 33. To express a / b as a percent, we have a / b = (a / b × 100) %

34. If ‘A’ is R% more than ‘B’, then ‘B’ is less than ‘A’ by
OR
If the price of a commodity increases by R%, then the reduction in consumption, not
to increase the expenditure is

{100R / [100 + R] } %

35. If ‘A’ is R% less than ‘B’, then ‘B’ is more than ‘A’ by
OR
If the price of a commodity decreases by R%, then the increase in consumption, not to
increase the expenditure is

{100R / [100 - R] } %

36. If the population of a town is ‘P’ in a year, then its population after ‘N’ years is

P (1 + R/100)N

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

37. If the population of a town is ‘P’ in a year, then its population ‘N’ years ago is

P / [(1 + R/100)N]

## Profit & Loss

38. If the value of a machine is ‘P’ in a year, then its value after ‘N’ years at a depreciation of
‘R’ p.c.p.a is

P (1 - R/100)N

39. If the value of a machine is ‘P’ in a year, then its value ‘N’ years ago at a depreciation of
‘R’ p.c.p.a is

P / [(1 - R/100)N]

## 41. The equality of two ratios is called a proportion. If a : b = c : d, we write a : b :: c : d and

we say that a, b, c, d are in proportion.
In a proportion, the first and fourth terms are known as extremes, while the second and
third are known as means.

## 43. Mean proportion between a and b is

44. The compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf)

## 51. If a / b = c / d, then, (a + b) / (a - b) = (c + d) / (c - d), which is called the componendo &

dividendo.

52. Variation: We say that x is directly proportional to y if x = ky for some constant k and we
write, x α y.

53. Also, we say that x is inversely proportional to y if x = k / y for some constant k and we
write x α 1 / y.

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

Partnership

54. If a number of partners have invested in a business and it has a profit, then

Chain Rule

## 55. The cost of articles is directly proportional to the number of articles.

56. The work done is directly proportional to the number of men working at it.

57. The time (number of days) required to complete a job is inversely proportional to the
number of hours per day allocated to the job.

58. Time taken to cover a distance is inversely proportional to the speed of the car.

## Time & Work

59. If A can do a piece of work in n days, then A’s 1 day’s work = 1/n.

60. If A’s 1 day’s work = 1/n, then A can finish the work in n days.

## 61. If A is thrice as good a workman as B, then:

Ratio of work done by A and B = 3 : 1,
Ratio of times taken by A & B to finish a work = 1 : 3

## Pipes & Cisterns

62. If a pipe can fill a tank in ‘x’ hours and another pipe can empty the full tank in ‘y’ hours
(where y > x), then on opening both the pipes, the net part of the tank filled in 1 hour is

(1/x – 1/y)

## Time And Distance

63. Suppose a man covers a distance at ‘x’ kmph and an equal distance at ‘y’ kmph, then
average speed during his whole journey is

## [2xy / (x + y)] kmph

Trains

64. Lengths of trains are ‘x’ km and ‘y’ km, moving at ‘u’ kmph and ‘v’ kmph (where, u > v) in
the same direction, then the time taken y the over-taker train to cross the slower train is

## [(x + y) / (u + v)] hrs

66. If two trains start at the same time from two points A and B towards each other and after
crossing they take a and b hours in reaching B and A respectively.

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

## Boats & Streams

69. If the speed of a boat in still water is u km/hr and the speed of the stream is v hm/hr,
then:
Speed downstream = (u + v) km/hr.
Speed upstream = (u - v) km/hr.

70. If the speed downstream is a km/hr and the speed upstream is b km/hr, then:

## Speed in still water = ½ (a + b) km/hr.

Rate of stream = ½ (a - b) km/hr.

Alligation or Mixture

71. Alligation: It is the rule that enables us to find the ratio in which two or more ingredients at
the given price must be mixed to produce a mixture at a given price.

72. Mean Price: The cost price of a quantity of the mixture is called the mean price.

## 75. .: (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c)

Simple Interest

76. Let Principle = P, Rate = R% per annum and Time = T years. Then,

a. S.I. = ( P × R × T ) / 100
b. P = ( 100 × S.I. ) / ( R × T ),
c. R = ( 100 × S.I. ) / ( P × T ),
d. T = ( 100 × S.I. ) / ( P × R ).

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

Compound Interest

77. Let Principle = P, Rate = R% per annum and Time = T years. Then,

## I. When interest is compounded Annually,

Amount = P (1 + R/100)N

## II. When interest is compounded Half-yearly:

Amount = P (1 + R/2/100)2N

## III. When interest is compounded Quarterly:

Amount = P (1 + R/4/100)4N

78. When interest is compounded Annually, but the time is in fraction, say 3⅞ years.

## Then, Amount = P (1 + R/100)3 × (1 + ⅞R/100)

79. When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd, and 3rd year
respectively,
Then, Amount = P (1 + R1/100) (1 + R2/100) (1 + R3/100)

## Present Worth = x / (1 + R/100)n

Logarithms

81. Logarithm: If a is a positive real number, other than 1 and am = x, then we write m = loga x
and say that the value of log x to the base a is m.

## e. loga (xp) = p loga x

f. loga x = 1 / logx a

## g. loga x = logb x / logb a

= log x / log a (Change of base rule)

## h. When base is not mentioned, it is taken as 10

i. Logarithms to the base 10 are known as common logarithms
j. The logarithm of a number contains two parts, namely characteristic and
mantissa. The integral part is known as characteristic and the decimal part is
known as mantissa.

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

## I. Case 1: When the number is greater than 1.

In this case, the characteristic is one less than the number of digits
in the left of decimal point in the given number.
II. Case 2: When the number is less than 1.
In this case, the characteristic is one more than the number of
zeroes between the decimal point and the first significant digit of
the number and it is negative.

e.g.
Number Characteristic
234.56 2
23.456 1
2.34 0
0.234 -1
0.0234 -2
0.00234 -3

Area

83. Rectangle:

84. Square:

## 85. Area of 4 walls of a room

= 2 (length + breadth) × height

86. Triangle:

## e. Radius of circumcircle of an equilateral triangle of side a = a /

87. Parallelogram/Rhombus/Trapezium:

## a. Area of a parallelogram = Base × Height

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

## b. Area of a rhombus = ½ × (Product of diagonals)

c. The halves of diagonals and a side of a rhombus form a right angled triangle with
side as the hypotenuse.

= Ө/360 × πR2

89. Cuboid:

## Let length = l, breadth = b & height = h units Then,

a. Volume = (l × b × h) cu units

## b. Surface Area = 2 (lb + bh + hl) sq. units

c. Diagonal = units

90. Cube:

## Let each edge of a cube be of length a. Then,

a. Volume = a3 cu units

## b. Surface Area = 6a2 sq. units

c. Diagonal =( × a) units

91. Cylinder:

92. Cone:

## Let radius of base = r & height = h. Then,

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

93. Sphere:

94. Hemi-sphere:

## 96. When stock is purchased, brokerage is added to the cost price.

97. When the stock is sold, brokerage is subtracted from the selling price.

## a. at par, if S.P. is Rs. 100 exactly;

b. above par (or at premium), if S.P. is more than Rs. 100;
c. below par (or at discount), if S.P. is less than Rs. 100.

99. By ‘a Rs. 800, 9% stock at 95’, we mean a stock whose face value is Rs. 800, annual
interest is 9% of the face value and the market price of a Rs. 100 stock is Rs. 95.

True Discount

100. Suppose a man has to pay Rs. 156 after 4 years and the rate of interest is 14% per
annum. Clearly, Rs. 100 at 14% will amount to Rs. 156 in 4 years. So, the payment of
Rs. 100 now will clear off the debt of Rs. 156 due 4 years hence. We say that:

## Sum due = Rs. 156 due 4 years hence;

Present Worth (P.W.) = Rs. 100;
True Discount (T.D.) = Rs. (156 - 100)
= (Sum due) – (P.W.)

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114 Handy Formulae for Quantitative Aptitude Problems
Author: Sagar Sonker

## 102. Amount = (P.W.) + (T.D.)

103. Interest is reckoned on R.W. and true discount is reckoned on the amount

## a. P.W. = (100 × Amount) / (100 + [R × T])

= (100 × T.D.) / (R × T)

## b. T.D. = (P.W.) × R × T / 100

= ([Amount] × R × T) / (100 + [R × T])

## P.W. = Amount / (1 + R/100)T

Banker’s Discount

105. Banker’s Discount (B.D.) is the S.I. on the face value for the period from the date on
which the bill was discounted and the legally due date.

106. Banker’s Gain (B.G.) = (B.D.) – (T.D.) for the unexpired time

107. When the date of the bill is not given, grace days are not to be added

= S.I. on T.D.
= (T.D.)2 / P.W.

110. T.D. =

## 114. T.D. = (B.G. × 100) / (Rate × Time)

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