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present

Shortcuts, Formulas & Tips

For MBA, Banking, Civil Services & Other
Entrance Examinations

Vol. 1: Number System & Arithmetic

Glossary Absolute value: Absolute value of x (written as |x|) is
the distance of 'x' from 0 on the number line. |x| is
Natural Numbers: 1, 2, 3, 4….. always positive. |x| = x for x > 0 OR -x for x < 0

Whole Numbers: 0, 1, 2, 3, 4…..

Tip: The product of ‘n’ consecutive natural
Integers: ….-2, -1, 0, 1, 2 ….. numbers is always divisible by n!

Rational Numbers: Any number which can be expressed Tip: Square of any natural number can be written
as a ratio of two integers for example a p/q format in the form of 3n or 3n+1. Also, square of any
where ‘p’ and ‘q’ are integers. Proper fraction will have natural number can be written in the form of 4n or
(p<q) and improper fraction will have (p>q) 4n+1.

Factors: A positive integer ‘f’ is said to be a factor of a Tip: Square of a natural number can only end in 0,
given positive integer 'n' if f divides n without leaving a 1, 4, 5, 6 or 9. Second last digit of a square of a
remainder. e.g. 1, 2, 3, 4, 6 and 12 are the factors of 12. natural number is always even except when last
digit is 6. If the last digit is 5, second last digit has
Prime Numbers: A prime number is a positive number to be 2.
which has no factors besides itself and unity.
Tip: Any prime number greater than 3 can be
Composite Numbers: A composite number is a number written as 6k 1.
which has other factors besides itself and unity.
Tip: Any two digit number ‘pq’ can effectively be
Factorial: For a natural number 'n', its factorial is written as 10p+q and a three digit number ‘pqr’
defined as: n! = 1 x 2 x 3 x 4 x .... x n (Note: 0! = 1) can effectively be written as 100p+10q+r.

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Laws of Indices Last digit of an
n(Right)
1 2 3 4 Cyclicity
a(Down)
0 0 0 0 0 1
( ) 1 1 1 1 1 1
2 2 4 8 6 4
( ) 3 3 9 7 1 4
√
4 4 6 4 6 2
5 5 5 5 5 1
6 6 6 6 6 1
( ) 7 7 9 3 1 4
√ 8 8 4 2 6 4
9 9 1 9 1 2

Tip: If am = an, then m = n Tip: The fifth power of any number has the same units
m m
Tip: If a = b and m ; place digit as the number itself.
Then a = b if m is Odd
Or a = b if m is Even

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HCF and LCM Factor Theory
For two numbers, HCF x LCM = product of the two. If N = xaybzc where x, y, z are prime factors. Then,

HCF of Fractions = Number of factors of N = P = (a + 1)(b + 1)(c + 1)

– – –
LCM of Fractions = Sum of factors of N =

Number of ways N can be written as product of two

Tip: If a, b and c give remainders p, q and r factors = P/2 or (P+1)/2 if P is even or odd respectively
respectively, when divided by the same number H,
then H is HCF of (a-p), (b-q), (c-r) The number of ways in which a composite number can
be resolved into two co-prime factors is 2m-1, where m is
Tip: If the HCF of two numbers ‘a’ and ‘b’ is H, then, the number of different prime factors of the number.
the numbers (a+b) and (a-b) are also divisible by H.
Number of numbers which are less than N and co-prime
Tip: If a number N always leaves a remainder R when
to ( ) ( )( )( ) {Euler’s Totient}
divided by the numbers a, b and c, then N = LCM (or a
multiple of LCM) of a, b and c + R.

Relatively Prime or Co-Prime Numbers: Two positive Tip: If N = (2)a(y)b(z)c where x, y, z are prime factors
integers are said to be relatively prime to each other if Number of even factors of N = (a)(b+1)(c+1)
their highest common factor is 1. Number of odd factors of N = (b+1)(c+1)

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Divisibility Rules Algebraic Formulae
A number is divisible by:
a3 ± b3 = (a ± b)(a2 ab + b2). Hence, a3 ± b3 is divisible
2, 4 & 8 when the number formed by the last, last two, by (a ± b) and (a2 ± ab + b2).
last three digits are divisible by 2,4 & 8 respectively. an - bn = (a – b)(an-1 + an-2b+ an-3b2 + ... + bn-1)[for all n].
3 & 9 when the sum of the digits of the number is Hence, an - bn is divisible by a - b for all n.
divisible by 3 & 9 respectively.
an - bn = (a + b)(an-1 – an-2b + an-3b2 ... – bn-1)[n-even]
11 when the difference between the sum of the digits in
Hence, an - bn is divisible by a + b for even n.
the odd places and of those in even places is 0 or a
multiple of 11. an + bn = (a + b)(an-1 – an-2b + an-3b2 + ... + bn-1)[n-odd]
6, 12 & 15 when it is divisible by 2 and 3, 3 and 4 & 3 Hence, an + bn is divisible by a + b for odd n.
and 5 respectively. a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - ac - bc)
7, if the number of tens added to five times the number Hence, a3 + b3 + c3 = 3abc if a + b + c = 0
of units is divisible by 7.
For ex., check divisibility of 312 by 7, 13 & 19
13, if the number of tens added to four times the
number of units is divisible by 13.  For 7: 31 + 2 x 5 = 31 + 10 = 41 Not divisible
19, if the number of tens added to twice the number of  For 13: 31 + 2 x 4 = 31 + 8 = 39 Divisible.
units is divisible by 19.  For 19: 31 + 2 x 2 = 31 + 4 = 35 Not divisible.

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Remainder / Modular Arithmetic Example: [ ] [ ]

[ ] [ ] [ ] [ ]  [ ] [ ] [ ]

[ ] [ ] [ ] [ ]  [ ]

Case 1 – When the dividend (M) and divisor (N) have a {Such that 3x+5y=1}
factor in common (k)  Valid values are x = -3 and y = 2

 [ ] [ ] [ ]  [ ]

Example: [ ] [ ]
Case 3 – Remainder when ( )
Case 2 – When the divisor can be broken down into is divided by ( ) the remainder is ( )
smaller co-prime factors.

 [ ] [ ] {HCF (a,b) = 1}
Tip: If f(a) = 0, (x-a) is a factor of f(x)
 Let [ ] [ ]

 [ ] {Such that ax+by = 1}

Continued >>

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Remainder Related Theorems Fermat’s Theorem:
If N is a prime number and M and N are co-primes
Euler’s Theorem:
 [ ]
Number of numbers which are less than N =
and co-prime to it are  [ ]

 ( ) ( )( )( ) Example: [ ] [ ]
If M and N are co-prime ie HCF(M,N) = 1
Wilson’s Theorem
( )
 [ ] If N is a prime number
( )
 [ ]
Example: [ ] ( )
 [ ]
 ( ) ( )( )( ) Example: [ ] [ ]

 ( ) Tip: Any single digit number written (P-1) times is

divisible by P, where P is a prime number >5.
 [ ] [ ]
Examples: 222222 is divisible by 7
 [ ] [ ] [ ] 444444….. 18 times is divisible by 19

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Base System Concepts Converting from base ‘n’ to decimal

Decimal  (pqrst)n = pn4 + qn3 + rn2 + sn + t

Binary Hex
0 0000 0 Converting from decimal to base ‘n’
1 0001 1 # The example given below is converting from 156 to
2 0010 2 binary. For this we need to keep dividing by 2 till we get
3 0011 3 the quotient as 0.
4 0100 4 2)156 0
2)78 0
5 0101 5 2)39 1
2)19 1
6 0110 6 2)9 1
7 0111 7 2)4 0
2)2 0
8 1000 8 2)1 1
0
9 1001 9
10 1010 A Starting with the bottom remainder, we read the
11 1011 B sequence of remainders upwards to the top. By that, we
12 1100 C get 15610 = 100111002
13 1101 D
14 1110 E Tip: (pqrst)n x n2 = (pqrst00)n
15 1111 F
(pqrst)n x n3 = (pqrst000)n

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Averages Tip: AM GM HM is always true. They will be
equal if all elements are equal to each other. If I have
Simple Average = just two values then GM2 = AM x HM

Tip: The sum of deviation (D) of each element with

Weighted Average = respect to the average is 0
 ( ) ( )
Arithmetic Mean = (a1 + a2 + a3 ….an) / n
( ) ( )
Geometric Mean =
Tip:

Harmonic Mean =

For two numbers a and b

Median of a finite list of numbers can be found by
 AM = (a + b)/2 arranging all the observations from lowest value to
 GM = √ highest value and picking the middle one.
 HM =
Mode is the value that occurs most often

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Percentages
Fractions and their percentage equivalents: Tip: r% change can be nullified by % change in
another direction. Eg: An increase of 25% in prices can
Fraction %age Fraction %age be nullified by a reduction of [100x25/(100+25)] =
1/2 50% 1/9 11.11% 20% reduction in consumption.
Tip: If a number ‘x’ is successively changed by a%,
1/3 33.33% 1/10 10%
b%, c%...
1/4 25% 1/11 9.09%
 Final value = ( )( )( )
1/5 20% 1/12 8.33%
Tip: The net change after two successive changes of
1/6 16.66% 1/13 7.69%
a% and b% is ( )
1/7 14.28% 1/14 7.14%
1/8 12.5% 1/15 6.66%

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Interest Growth and Growth Rates

Amount = Principal + Interest Absolute Growth = Final Value – Initial Value

Simple Interest = PNR/100 Growth rate for one year period =

–
n x 100
Compound Interest = P(1+ ) –P
n –
Population formula P’ = P(1 ) SAGR or AAGR = x 100
n
Depreciation formula = Initial Value x (1 – ) –
CAGR= ( ) –1

Tip: SI and CI are same for a certain sum of money

Tip: If the time period is more than a year, CAGR <
(P) at a certain rate (r) per annum for the first year.
AAGR. This can be used for approximating the value
The difference after a period of two years is given by
of CAGR instead of calculating it.


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Profit and Loss Mixtures and Alligation
– Successive Replacement – Where a is the original
%Profit / Loss = quantity, b is the quantity that is replaced and n is the
number of times the replacement process is carried out,
In case false weights are used while selling, then
( )
% Profit = ( )
Alligation – The ratio of the weights of the two items
– mixed will be inversely proportional to the deviation of
Discount % = x 100 attributes of these two items from the average attribute
of the resultant mixture

Tip: Effective Discount after successive discount of  =

a% and b% is (a + b – ). Effective Discount when

you buy x goods and get y goods free is x 100.

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Ratio and Proportion
Tip: If a/b = c/d = e/f = k
Compounded Ratio of two ratios a/b and c/d is ac/bd,
Duplicate ratio of a : b is a2 : b2
 =k
Triplicate ratio of a : b is a3 : b3
Sub-duplicate ratio of a : b is a : b
Sub-triplicate ratio of a : b is ³ a : ³ b  =k
Reciprocal ratio of a : b is b : a

Componendo and Dividendo  = kn

If
Four (non-zero) quantities of the same kind a,b,c,d are
said to be in proportion if a/b = c/d. Given two variables x and y, y is (directly) proportional
to x (x and y vary directly, or x and y are in direct
The non-zero quantities of the same kind a, b, c, d.. are variation) if there is a non-zero constant k such that y =
said to be in continued proportion if a/b = b/c = c/d. kx. It is denoted by
Proportion Two variables are inversely proportional (or varying
a, b, c, d are said to be in proportion if inversely, or in inverse variation, or in inverse
a, b, c, d are said to be in continued proportion if proportion or reciprocal proportion) if there exists a
non-zero constant k such that y = k/x.

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Time Speed and Distance Tip: Given that the distance between two points is
Speed = Distance / Time constant, then

1 kmph = 5/18 m/sec; 1 m/sec = 18/5 kmph  If the speeds are in Arithmetic Progression,
then the times taken are in Harmonic
SpeedAvg = = Progression
 If the speeds are in Harmonic Progression, then
the times taken are in Arithmetic Progression
If the distance covered is constant then the average
speed is Harmonic Mean of the values (s1,s2,s3….sn)
For Trains, time taken =
 SpeedAvg =
For Boats,

SpeedUpstream = SpeedBoat – SpeedRiver

 SpeedAvg = (for two speeds)
SpeedDownstream = SpeedBoat + SpeedRiver
If the time taken is constant then the average speed is
Arithmetic Mean of the values (s1,s2,s3….sn) SpeedBoat = (SpeedDownstream + SpeedUpstream) / 2

 SpeedAvg = SpeedRiver = (SpeedDownstream – SpeedUpstream) / 2

 SpeedAvg = (for two speeds)

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For Escalators,The difference between escalator  Time for 1st meeting at the starting point =
problems and boat problems is that escalator can go LCM ( )
either up or down.
Two people are running on a circular track of length L
Races & Clocks with speeds a and b in the opposite direction
Linear Races  Time for 1st meeting =
 Time for 1st meeting at the starting point =
Winner’s distance = Length of race
LCM ( )
Loser’s distance = Winner’s distance – (beat distance + Three people are running on a circular track of length L
start distance) with speeds a, b and c in the same direction
Winner’s time = Loser’s time – (beat time + start time)  Time for 1st meeting = LCM ( )
 Time for 1st meeting at the starting point =
Deadlock / dead heat occurs when beat time = 0 or beat
LCM ( )
distance = 0

Circular Races Clocks To solve questions on clocks, consider a circular

track of length 360 . The minute hand moves at a speed
Two people are running on a circular track of length L of 6 per min and the hour hand moves at a speed of ½
with speeds a and b in the same direction per minute.
 Time for 1st meeting = Tip: Hands of a clock coincide (or make 180 ) 11 times in every
12 hours. Any other angle is made 22 times in every 12 hours.
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Time and Work
If a person can do a certain task in t hours, then in 1 Tip: If A does a particular job in ‘a’ hours, B does the
hour he would do 1/t portion of the task. same job in ‘b’ hours and ABC together do the job in
A does a particular job in ‘a’ hours and B does the same ‘t’ hours, then

job in ‘b’ hours, together they will take hours

 C alone can do it in hours
A does a particular job in ‘a’ hours more than A and B  A and C together can do it in hours
combined whereas B does the same job in ‘b’ hours
 B and C together can do it in hours
more than A and B combined, then together they will
take √ hours to finish the job.
Tip: If the objective is to fill the tank, then the Inlet
pipes do positive work whereas the Outlet pipes do
Tip: A does a particular job in ‘a’ hours, B does the
negative work. If the objective is to empty the tank,
same job in ‘b’ hours and C does the same job in ‘c’
then the Outlet pipes do positive work whereas the
hours, then together they will take hours.
Inlet Pipes do negative work.

Tip: If A does a particular job in ‘a’ hours and A&B

together do the job in ‘t’ hours, the B alone will take
hours.

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