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FORMULA LIST:
Arithmetic Progression (A.P.):
An A.P. is of the form a, a+d, a+2d, a+3d, …
where a is called the ‘first term’ and d is called the ‘common difference’
1. nth term of an A.P. tn = a + (n-1)d
2. Sum of the first n terms of an A.P. Sn = n/2[2a+(n-1)d] or Sn = n/2(first term + last term)
Geometrical Progression (G.P.):
A G.P. is of the form a, ar, ar2, ar3, …
where a is called the ‘first term’ and r is called the ‘common ratio’.
1. nth term of a G.P. tn = arn-1
2. Sum of the first n terms in a G.P. Sn = a|1-rn|/|1-r|
Permutations and Combinations :
1. nPr = n!/(n-r)!
2. nPn = n!
3. nP1 = n
1. nCr = n!/(r! (n-r)!)
2. nC1 = n
3. nC0 = 1 = nCn
4. nCr = nCn-r
5. nCr = nPr/r!
Number of diagonals in a geometric figure of n sides = nC2-n
Tests of Divisibility :
1. A number is divisible by 2 if it is an even number.
2. A number is divisible by 3 if the sum of the digits is divisible by 3.
3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
4. A number is divisible by 5 if the units digit is either 5 or 0.
5. A number is divisible by 6 if the number is divisible by both 2 and 3.
6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
7. A number is divisible by 9 if the sum of the digits is divisible by 9.
8. A number is divisible by 10 if the units digit is 0.
9. A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of
its digits at even places, is divisible by 11.
H.C.F and L.C.M :
Quantitative
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H.C.F stands for Highest Common Factor. The other names for H.C.F are Greatest Common
Divisor (G.C.D) and Greatest Common Measure (G.C.M).
The H.C.F. of two or more numbers is the greatest number that divides each one of them
exactly.
The least number which is exactly divisible by each one of the given numbers is called their
L.C.M.
Two numbers are said to be co-prime if their H.C.F. is 1.
H.C.F. of fractions = H.C.F. of numerators/L.C.M of denominators
L.C.M. of fractions = G.C.D. of numerators/H.C.F of denominators
Product of two numbers = Product of their H.C.F. and L.C.M.
PERCENTAGES :
1. If A is R% more than B, then B is less than A by R / (100+R) * 100
2. If A is R% less than B, then B is more than A by R / (100-R) * 100
3. If the price of a commodity increases by R%, then reduction in consumption, not to increase the
expenditure is : R/(100+R)*100
4. If the price of a commodity decreases by R%, then the increase in consumption, not to decrease
the expenditure is : R/(100-R)*100
PROFIT & LOSS :
1. Gain = Selling Price(S.P.) – Cost Price(C.P)
2. Loss = C.P. – S.P.
3. Gain % = Gain * 100 / C.P.
4. Loss % = Loss * 100 / C.P.
5. S.P. = (100+Gain%)/100*C.P.
6. S.P. = (100-Loss%)/100*C.P.
Short cut Methods:
1. By selling an article for Rs. X, a man loses l%. At what price should he sell it to gain y%?
(or)
A man lost l% by selling an article for Rs. X. What percent shall he gain or lose by selling it
for Rs. Y?
(100 ? loss%) : 1st S.P. = (100 + gain%) : 2nd S.P.
2. A man sold two articles for Rs. X each. On one he gains y% while on the other he loses y%.
How much does he gain or lose in the whole transaction?
In such a question, there is always a lose. The selling price is immaterial.
Formula: Loss % =
Common loss or gain% 2
%
10
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3. A discount dealer professes to sell his goods at cost price but uses a weight of 960 gms. For
a kg weight. Find his gain percent.
Formula: Gain % =
RATIO & PROPORTIONS:
1. The ratio a : b represents a fraction a/b. a is called antecedent and b is called consequent.
2. The equality of two different ratios is called proportion.
3. If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.
4. In a : b = c : d, then we have a* d = b * c.
5. If a/b = c/d then ( a + b ) / ( a ? b ) = ( d + c ) / ( d ? c ).
TIME & WORK :
1. If A can do a piece of work in n days, then A’s 1 day’s work = 1/n
2. If A and B work together for n days, then (A+B)’s 1 days’s work = 1/n
3. If A is twice as good workman as B, then ratio of work done by A and B = 2:1
PIPES & CISTERNS :
1. If a pipe can fill a tank in x hours, then part of tank filled in one hour = 1/x
2. If a pipe can empty a full tank in y hours, then part emptied in one hour = 1/y
3. If a pipe can fill a tank in x hours, and another pipe can empty the full tank in y hours, then on
opening both the pipes,
the net part filled in 1 hour = (1/x-1/y) if y>x
the net part emptied in 1 hour = (1/y-1/x) if x>y
TIME & DISTANCE :
1. Distance = Speed * Time
2. 1 km/hr = 5/18 m/sec
3. 1 m/sec = 18/5 km/hr
4. Suppose a man covers a certain distance at x kmph and an equal distance at y kmph. Then, the
average speed during the whole journey is 2xy/(x+y) kmph.
PROBLEMS ON TRAINS :
1. Time taken by a train x metres long in passing a signal post or a pole or a standing man is equal
to the time taken by the train to cover x metres.
2. Time taken by a train x metres long in passing a stationary object of length y metres is equal to
the time taken by the train to cover x+y metres.
3. Suppose two trains are moving in the same direction at u kmph and v kmph such that u>v, then
their relative speed = u-v kmph.
4. If two trains of length x km and y km are moving in the same direction at u kmph and v kmph,
where u>v, then time taken by the faster train to cross the slower train = (x+y)/(u-v) hours.
5. Suppose two trains are moving in opposite directions at u kmph and v kmph. Then, their relative
speed = (u+v) kmph.
Error
*100 %
True value – Error
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6. If two trains of length x km and y km are moving in the opposite directions at u kmph and v
kmph, then time taken by the trains to cross each other = (x+y)/(u+v)hours.
7. 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, then A’s speed : B’s speed =
(?b : ?
SIMPLE & COMPOUND INTERESTS :
Let P be the principal, R be the interest rate percent per annum, and N be the time
period.
1. Simple Interest = (P*N*R)/100
2. Compound Interest = P(1 + R/100)N ? P
3. Amount = Principal + Interest
LOGORITHMS :
If am = x , then m = logax.
Properties :
1. log xx = 1
2. log x1 = 0
3. log a(xy) = log ax + log ay
4. log a(x/y) = log ax – log ay
5. log ax = 1/log xa
6. log a(xp) = p(log ax)
7. log ax = log bx/log ba
Note : Logarithms for base 1 does not exist.
AREA & PERIMETER :
Shape Area Perimeter
Circle ? (Radius)2 2?(Radius)
Square (side)2 4(side)
Rectangle length*breadth 2(length+breadth)
1. Area of a triangle = 1/2*Base*Height or
2. Area of a triangle = ? (s(s-(s-b)(s-c)) where a,b,c are the lengths of the sides and s = (a+b+c)/2
3. Area of a parallelogram = Base * Height
4. Area of a rhombus = 1/2(Product of diagonals)
5. Area of a trapezium = 1/2(Sum of parallel sides)(distance between the parallel sides)
6. Area of a quadrilateral = 1/2(diagonal)(Sum of sides)
7. Area of a regular hexagon = 6(?3/4)(side)2
8. Area of a ring = ?(R2-r2) where R and r are the outer and inner radii of the ring.
VOLUME & SURFACE AREA :
Cube :
Let a be the length of each edge. Then,
1. Volume of the cube = a3 cubic units
2. Surface Area = 6a2 square units
3. Diagonal = ? 3 a units
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Cuboid :
Let l be the length, b be the breadth and h be the height of a cuboid. Then
1. Volume = lbh cu units
2. Surface Area = 2(lb+bh+lh) sq units
3. Diagonal = ? (l2+b2+h2)
Cylinder :
Let radius of the base be r and height of the cylinder be h. Then,
1. Volume = ?r2h cu units
2. Curved Surface Area = 2?rh sq units
3. Total Surface Area = 2?rh + 2?r2 sq units
Cone :
Let r be the radius of base, h be the height, and l be the slant height of the cone. Then,
1. l2 = h2 + r2
2. Volume = 1/3(?r2h) cu units
3. Curved Surface Area = ?rl sq units
4. Total Surface Area = ?rl + ?r2 sq units
Sphere :
Let r be the radius of the sphere. Then,
1. Volume = (4/3)?r3 cu units
2. Surface Area = 4?r2 sq units
Hemi-sphere :
Let r be the radius of the hemi-sphere. Then,
1. Volume = (2/3)?r3 cu units
2. Curved Surface Area = 2?r2 sq units
3. Total Surface Area = 3?r2 sq units
Prism :
Volume = (Area of base)(Height)
ALGEBRA :
1. Sum of first n natural numbers = n(n+1)/2
2. Sum of the squares of first n natural numbers = n(n+1)(2n+1)/6
3. Sum of the cubes of first n natural numbers = [n(n+1)/2]2
4. Sum of first n natural odd numbers = n2
5. Average = (Sum of items)/Number of items

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Table of Contents
? Numbers
? H.C.F & L.C.M of Numbers
? Surds & Indices
? Percentage
? Profit & Loss
? Ratio & Proportion
? Partnership
? Chain Rule
? Time & Work
? Pipes & Cisterns
? Time And Distance
? Trains
? Boats & Streams
? Alligation or Mixture
? Simple Interest
? Compound Interest
? Logarithms
? Area
? Volume & Surface Area
? Stocks & Shares
? True Discount
? Banker?s Discount
? Copyright Notice
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Numbers
1. A number is divisible by 2, if its unit?s place digit is 0, 2, 4, or 8
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
5. A number is divisible by 9, if the sum of its digits is divisible by 9
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
9. (a + b)2 – (a – b)2 = 4ab
10. (a + b)2 + (a – b)2 = 2(a2 + b2)
11. (a2 ? b2) = (a + b)(a – b)
12. (a3 + b3) = (a + b)(a2 – ab + b2)
13. (a3 ? b3) = (a – b)(a2 + ab + b2)
14. Results on Division:
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]
17. (1 + 2 + 3 + ? ? ? + n) = [n(n + 1)] / 2
18. (12 + 22 + 32 + ? ? ? + n2) = [n(n + 1)(2n + 1)] / 6
19. (13 + 23 + 33 + ? ? ? + n3) = [n2(n + 1)2] / 4
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H.C.F & L.C.M of Numbers
20. Product of two numbers = Their H. C. F. ? Their L. C. M.
Surds & Indices
21. am ? an = a(m + n)
22. am / an = a(m – n)
23. (ab)m = ambm
24. (a / b)m = an / bn
25. a0 = 1
26. = a1/n
27. = (a1/n)n
= a
28. =
29. = /
30. ( )m =
31. =
Percentage
32. To express x% as a fraction, we have x% = x / 100
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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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]
40. Selling Price = [(100 + Gain%) ? Cost Price] / 100
= [(100 – Loss%) ? Cost Price] / 100
Ratio & Proportion
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.
42. Product of extremes = Product of 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)
45. a2 : b2 is a duplicate ratio of a : b
46. : is a sub-duplicate ration of a : b
47. a3 : b3 is a triplicate ratio of a : b
48. a1/3 : b1/3 is a sub-triplicate ratio of a : b
49. If a / b = c / d, then, (a + b) / b = (c + d) / d, which is called the componendo.
50. If a / b = c / d, then, (a – b) / b = (c – d) / d, which is called the dividendo.
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 ? y.
53. Also, we say that x is inversely proportional to y if x = k / y for some constant k and we
write x ? 1 / y.
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Partnership
54. If a number of partners have invested in a business and it has a profit, then
Share Of Partner = (Total_Profit ? Part_Share / Total_Share)
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
65. Time taken to cross each other 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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Then, A?s speed : B?s speed = ( : ).
67. x kmph = (x ? 5/18) m/sec.
68. y metres/sec = (y ? 18/5) km/hr.
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 = ? (a + b) km/hr.
Rate of stream = ? (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.
73. Rule of Alligation: If two ingredients are mixed, then:
74. We represent the above formula as under:
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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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 3? years.
Then, Amount = P (1 + R/100)3 ? (1 + ?R/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)
80. Present worth of Rs. x due n years hence is given by:
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.
82. Properties of Logarithms:
a. loga (xy) = loga x + loga y
b. loga (x/y) = loga x – loga y
c. logx x = 1 (i.e. Log of any number to its own base is 1)
d. loga 1 = 0 (i.e. Log of 1 to any base is 0)
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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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
III. For mantissa, we look through the log table.
IV. Antilog: If log x = y, then antilog y = x.
Area
83. Rectangle:
a. Area of a rectangle = (length ? breadth)
b. Perimeter of a rectangle = 2 (length + breadth)
84. Square:
a. Area of square = (side)2
b. Area of a square = ? (diagonal)2
85. Area of 4 walls of a room
= 2 (length + breadth) ? height
86. Triangle:
a. Area of a triangle = ? ? base ? height
b. Area of a triangle = , where
s = ? (a + b + c), and a, b, c are the sides of the triangle
c. Area of an equilateral triangle = / 4 ? (side)2
d. Radius of incircle of an equilateral triangle of side a = a / 2
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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b. Area of a rhombus = ? ? (Product of diagonals)
c. The halves of diagonals and a side of a rhombus form a right angled triangle with
side as the hypotenuse.
d. Area of trapezium = ? ? (sum of parallel sides) ? (distance between them)
88. Circle/Arc/Sector, where R is the radius of the circle:
a. Area of a circle = ?R2
b. Circumference of a circle = 2?R
c. Length of an arc = ?/360 ? 2?R
d. Area of a sector = ? (arc ? R)
= ?/360 ? ?R2
Volume & Surface Area
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:
Let radius of base = r & height (or length) = h. Then,
a. Volume = (?r2h) cu. units
b. Curved Surface Area = (2?rh) sq. units
c. Total Surface Area = 2?r(r + h) sq. units
92. Cone:
Let radius of base = r & height = h. Then,
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a. Slant height, l = units
b. Volume = (? ?r2h) cu. units
c. Curved Surface Area = (?rl) sq. units
d. Total Surface Area = ?r(r + l) sq. units
93. Sphere:
Let the radius of the sphere be r. Then,
a. Volume = (4/3 ?r3) cu. units
b. Surface Area = (4?r2) sq. units
94. Hemi-sphere:
Let the radius of the sphere be r. Then,
a. Volume = (2/3 ?r3) cu. units
b. Curved Surface Area = (2?r2) sq. units
c. Total Surface Area = (3?r2) sq. units
Stocks & Shares
95. Brokerage: The broker?s charge is called brokerage.
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.
98. The selling price of a Rs. 100 stock is said to be:
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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101. T.D. = Interest on P.W.
102. Amount = (P.W.) + (T.D.)
103. Interest is reckoned on R.W. and true discount is reckoned on the amount
104. Let rate = R% per annum & time = T years. Then,
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])
c. Sum = ([S.I.] ? [T.D.]) / ([S.I.] ? [T.D.])
d. (S.I.) ? (T.D.) = S.I. on T.D.
e. When the sum is put at compound interest, then
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
108. B.D. = S.I. on bill for unexpired time
109. B.G. = (B.D.) ? (T.D.)
= S.I. on T.D.
= (T.D.)2 / P.W.
110. T.D. =
111. B.D. = (Amount ? Rate ? Time) / 100
112. T.D. = (Amount ? Rate ? Time) / (100 + [Rate ? Time])
113. Amount = (B.D. ? T.D.) / (B.D. ? T.D.)
114. T.D. = (B.G. ? 100) / (Rate ? Time)
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114 Handy Formulae for Quantitative Aptitude Problems
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