Math Study Material for SSC and Banking Exam

1
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CHAPTER – 1
AVERAGE
Average Formulae: - Add up all the numbers, than divide by how many numbers there are.
Sum of observations
Average
Number of observations

Find the average speed
 If a person travels a distance at a speed of x Km/hr and the same distance at a
speed of y km/hr then the average speed during the whole journey is given by:-
2xy
x y
 If a person covers A km at x km/hr and B km at y km/hr and C km at z km/hr, then the
average speed in covering the whole distance is:
A B C
A B c
x y z
 
 
 When a person leaves the group and another person joins the group in place of that
person then-
o If the average age is increased,
Age of new person = age of separated present + (increase in avg. x total no. of
persons)
o If the average age is decreased.
Age of new person = age of separated person - (decreases in avg. x total no of
persons).
o When a person joins the group – in case of increase in avg.
Age of new member = previous average + (increase in avg. x number of members
including new members).
o In case of decrease in average
Age of new members = previous avg. – (decrease in avg. x number of members
including new member).
 When the number of terms is odd, the average will be the middle term.
 When the number of terms is even the Avg. will be the avg. of two middle terms.
 Arithmetic Progression:- a, a + d, a + 2d, a + 3d,…………..

2
First term = a
Difference = d
nth
term =  �� = a +(n-1) x d
 Sum of ‘n’ terms
2
n
Sn  [2a + (n-1) x d]
 Sum of first ‘n’ natural numbers
(1+2+3+--------+n) = ½ n(n+1)
 Sum of ‘n’ odd no.
+ + 5 + ⋯ … … + � − = n2
' 'sum of n even no
(2+4+6+…………+2n) = n (n+1)
= sum of first squares of ‘n’ natural no.
2 2 2 2 1 1
(1 2 3 .........t ) x x ( 1)x(2 1)
6 6
n n n n n      
Sum of first cubic of ‘n’ natural no.
3 3 3 1
(1 2 .........t ) ( 1)}.
2
n n n

   

****##****

3
CHAPTER-2
COMPOUND INTEREST
Compound Interest:- Where interest is calculated on both the amount borrowed and any
previous interest. Usually calculated one or more times per year.
formulae:-
When The Interest is compounded annually
Amount after ‘n’ years = A = P(1 )
100
nR

Compound interest = 4
(1 ) [(1 ) 1]
100 100
nR R
p p p    
If P = principal
R = rate percent
Time = number of years
C. I = compound interest
A = amount
If the rate of interest differs from year to year i.e. R, in the first year, R2 in the second year,
R3 in the third year, Then,
31 2
(1 )(1 )(1 )
100 100 100
RR R
A    
 When the principal changes every year, we say the interest is compound annually. Then,
(1 )
100
nR
A P 
 When the principal changes as per every six months we say that the interest is
compounded half yearly or semi – annually. Then,
22(1 )
100
n
R
A P 
- Quarterly
44(1 )
100
n
R
A P 
****##****

4
CHAPTER – 3
SIMPLE INTEREST
SIMPLE INTEREST:- Define:- simple interest is interest paid any the original principal not
on the interest accrued.
Formula:-
P X R X T
.
100
S I 
S.I – Simple Interest
R = Rate percent
T = Time in years

5
CHAPTER – 4
BOAT AND STREAM
Boat and Stream
Formulae:-
 If the boat speed is x km/h in downstream y km/h in upstream, then
Speed of boat in still water is
( )
2
x y
speed of stream =
( )
2
x y
 If speed of boat is x km/h and speed of stream is y km/h, then
Boat speed in downstream = x + y km/hr
(both speeds are added)
Boat speed in upstream = x-y km/hr
(y is subtracted from x)
Because boat speed is in opposite direction to speed of stream).
If the speed of boat or person in still water is x and speed of stream is y and the boat has to
cover a distance ‘d’ km Then time taken in down stream 1
tan
( )
( )
d dis ce
T Time
x y speed
  

Time taken in upstream 2
( )
d
T
x y


Total time taken in going down stream and upstream 1 2 [ ]
d d
T T T
x y x y
   
 
***##****

6
CHAPTER – 5
H.C.F AND L.C.M
H.C.F and L.C.M:- Highest common factor and lowest common factor
Formulae:-
When solving H.C.F and L.C.M questions with fractions These formulas are very helpful
1.
. .
. .
. .
H C F of Numerators
H C F
L C M of Numerators

2.
. .
. .
. . Denanenators
L C M of Numerators
L C M
H C F of

3. Product of two numbers:- Product of two numbers = product of their H.C.F and L.C.M

7
CHAPTER – 6
TIME AND DISTANCE
tanDis ce
Speed
Time

Conversion of km/hr to m/s
1 kmph =
5
18
m/s
Conversion of m/s to km/hr
1 m/s =
18
5
km/hr
 If the ratio of speed of train A and B is a : b, then the ratio of time taken by them to cover
the same distance = b : a.
 If a man covers a certain distance at x km/hr and an equal distance at y km/hr. Then the
avg. speed during the whole journey is.
2
/
( )
xy
km h
x y
 The time taken by a train in passing a pole or standing man is the same as the time
taken by the train to cover a distance equal to its own length
 The time taken by a train of length ‘L’ meters in passing a stationary object of length ‘B’
meters is equal (L+B) m.
 If two trains are moving in the same direction at U m/s and V m/s where U>V, then their
relative speed will be equal to the difference of their speeds (U-V) m/s.
 If two trains are moving in the opposite direction at um/s and Vm/s then their relative
speed will be equal to the sum of their speed (U+V) m/s.
 If two trains of length ‘a’ meters and ‘b’ meters and moving in the same directions at U
m/s and Vm/s respectively, then
 The time taken by the faster train to cross the slower train is:-
(a + b) / (u-v) sec.
 If two trains of length ‘a’ meters and ‘b’ meters are moving in the opposite direction at U
m/s and V m/s respectively, then the time taken by the faster train to cross the slower
train is
a b
u v


sec.

8
CHAPTER – 7
PERMUTATION AND COMBINATION
Formulae:
1. Factorial Notation:-
Let ‘n’ be the positive integer. Then, factorial n, dented n: is defined as:
Factorial n, denoted n: is defined as:
1
0 ( 1)( 2).........3. 2. 1n n n n  
2. Number of Permutations:-
Number of all permutations of ‘n’ things, taken r at a time, is given by :
1
0
1
0
P ( 1)( 2)..........( 1)
( )
n
r
n
n n n n r
n r
     

3. Permutation: - The different arrangements of a given no. of things by taking some or all
at a time, are called permutations.
4. Combination:- Each of the different groups or selections which can be formed by taking
some or all of a number of objects is called combination.
5. Number of Combination:-
The number of all combinations of ‘n’ things,
Taken r at a time is:-
1
0
1
0( 1)( )
n
r
n
C
r n r



9
CHAPTER – 8
TIME AND WORK
Formulae:-
1. Work from days:
If A can do a piece of work in ‘n’ days, then A’s 1 day’s work =
1
n
2. Days from work:
If A’s 1 day’s work =
1
n
Two A can finish the work in n day.
3. Ratio:
If A is Thrice as good a work man as then, Ration of work done by A due B = 3 : 1
Ration of times taken by A and B to finish a work = 1 : 3.

10
CHAPTER – 9
PIPES AND CISTERN
Inlet:- A pipe connected with a tank or a cistern or a reservoir, that tills it, is known as an
inlet.
Outlet:- 1. A pipe connected with a tank or a cistern or a reservoir, emptying it, is known as
an outlet.
1) if a pipe can fill a tank in x hours, two; part filled in 1 hour
1
x

2) if a pipe can empty a tank in y hours, then: part emptied in 1 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
(where y>x), then an opening both the pipes, then the net part filled in 1 hour =
1 1
( )
x y
 
4) if a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours
where (x > y), then an opening both the pipes, then the net part emptied in 1 hour
1 1
( )
y x
 

11
CHAPTER – 10
MIXTURE AND ALLEGATION
Mixture and Allegation:-
1. Allegation:- it is the rule that enables us to find the ratio in which two or more
ingredients at two given price must be mixed to produce a mixture of desired price.
2. Mean Price:- The cost of a unit quantity of the mixture is called the mean price.
3. Rule of allegation:-
It two ingredients are mixed, then
. Pr
( )=( )
Pr .
Quantity of Cheaper C P of dearer Mean ice
Quantity of dearer Mean ice C P of Cheap



We present as under:-
.
( )
E P of a unit Quantity
c
.
( )
E P of unit Quantity
of dearer d
C.P of a unit quantity Mean price
(m)
(Cheaper Quantity) : (Dearer Quantity) ( )d m ( )m c
= (d-m) : (m-c).
4. Suppose a Container Contains x of liquid from which y units are taken out an replaced by
water.
After ‘n ’ operations, the Quantity of pure liquid = (1 )ny
x
x
 
  
units.

12
CHAPTER – 11
PARTNERSHIP
Partnership:- When two or more than two persons runs a business jointly, They are called
partners and the deal is known as partnership.
1. Ration of Divisions of Gains:-
(i) When investments of all the partners are for the same time, the gain or loss is
distributed among the partners in the ration of their investments.
Suppose A and B invest Rs x and Rs. Respectively for a year in a business, then
at the end of the year.
(A’s share of Profit) : (B’s share of profit) = x:y
(ii) When investments are for different time period then equivalent capitals are
calculated for a unit of time by taking (capital x number of units of time). Now gain
or loss is divided in the ratio of these capitals suppose A invests Rs x for p
months then, B (A’s share of forfeit) : (B’s share of profit) = x p : y q.

13
CHAPTER – 12
NUMBER SYSTEM:-
Formulas:-
1. 11,2,3..........O a are called digits.
2. 10, 11, 12,……….are called number.
3. Natural number (N):- Counting numbers are called natural numbers.
Ex:- 1, 2, 3,…………..etc. are all natural numbers, minimum natural number 1 and
maximum ∞.
Whole numbers:- if W is the set of whole numbers, then we write = {0, 1, 2,……….} The
smallest whole number 0.
Integer:- if I is the set of integers, then we write I = {-3, -2, -1, 0, 1, 2, 3………}
Rational No. – Any number which can be expressed in the form of plq. When p and q are
both integers and g o are called rational numbers.
E. g. =
3 7
, ,5, 2
2 9


There exists infinite number of rational number between any two rational numbers.
Irrational numbers:- Non-recurring and non-termination decimals are called irrational
numbers. These number cannot be expressed in the form of
p
q
.
E.g. = 3, 5, 29..........
Real numbers:- Real numbers include both rational and irrational numbers.
In General the number of n digit number are 1
9 10n
X 
 Sum of first ‘n’ natural numbers
1 +2+ 3+ ……..+n
( 1)
2
n n 

 Sum of the squares of the first n natural no 2 2 2 2 ( 1)(2 1)
1 2 3 ..........
6
n n n
n
 
  
Sum of cube of the first n natural numbers –

14
2
3 3 3 3 ( 1)
1 2 3 ..........
2
n n
n
 
      
 Rules of divisibility:-
Divisibility by 2: A number is divisible by 2 when the digit at ones place is 0, 2 4, 6 or 8.
Eg:- 3582, 460,……….
Is divisibity by 3:- A number is divisible by 3 when sum of all digits of a number is
divisible by 3.
Eg:- 435 = 4+5+3=12
12 is divisible by 3. So, 435 is also divisible by 3.
 Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
Eg:- 10, 25, 60.
 Divisibility by 6: A number is divisible by 6, if it is divisible by both 2 and 3.
48, 24, 108.
 Divisibility by 7:- A number is divisible by 7 when the difference between twice the digit
at ones place and the number formed by other digits is either zero or divisible by 7.
Eg:- 658.
65-2 X 8 = 65 – 16 = 49
As 49 is divisible by 7 the number 658 is also divisible by 7
 Divisibility by 8:- A number is divisible by 8.
If a number formed by the last 3 digits of the number is divisible by 8.
Eg:- if we take the number 57832. The last three digits form 832. Since, the number 832
is divisible by 8. The number 57832 is also divisible by 8.
 Divisibility by 9: A number is divisible by if the sum if all the digits of number is divisible
by 9. Eg: - 6 & 4 = 6+8+4=18
If is divisible by 9 so, 684 is also divisible by 9.
 Divisibility by 10:- A number is divisible by 10, if its last digit is o eg. 20, 180, 350,
 Divisibility by 11:- A number is divisible by 11. When the diffevncle between the sum of
its digits in odd places and in even places is either O or divisible by 11
30426
3+4+6=13
0+2=2
13-2=11
As the difference is divisible by 11 the numbers 30426 is also divisible by 11.
 Division on numbers:-
In a sum of division, we have four quantities:-
They are (i) dividend (ii) Divisor (iii) Quotient (iv) Remainder
Relation:-
(a) Dividend = Divisor X Quotient + Remainder

15
(b) Divisor = (Dividend – Remainder)  Quotient
(c) Quotient = (Dividend – Remainder)  Divisor

16
CHAPTER – 13
SERIES:
Series:
Arithmetic Sequence:- if a sequence of values follows a pattern of adding a fixed amount
from are term to the next, it is referred to as an aritumatic sequence. The number is added to
each term is constant.
Formula:-
To find any term of an aritumatic sequence 1 ( 1) xna a n d  
To find the sum of a certain number of tens of an arithmatic sequences:-
1( )
2
n
n
n a a
S

 , Where nS is the sum of in terms ( th
n Partial sum) 1a is the first term, na is
the th
n term

17
CHAPTER – 14
RATIO AND PROPORTION
What is Ratio:- A ratio is a relations ship between two numbers by division of the same
kind. The ration of a or b is written as a : b = a/b, in ratio a : b, we can say that a as the first
term or antecedent and b the second term or consequent
What is Proportion: - The idea of proportions is that two ration are like equal.
If a : b : = c : d, we write a : b : : c : d
Ex:-
3 1
15 5

A and d called extremes, where as b and C called mean terms.
Proportion of Quantities:-
The four quantities like a, b, c, d we can express in a : b = c : d
Tune a : b : : c : d = (a x d) = (b x c)
If there is given three quantities like a, d, c of same like then we can say it proportion of
continued.
a : d = d : c, d is called mean term,
a and c are called extremes.

18
CHAPTER-15
PROBLEMS ON AGES:-
1. If the current age is x, the ‘n’ times the age is nx.
2. If the current age is x, then age n years / later = x + n
3. If the current age is x, then age n years ago = x – n
4. The ages in a ratio a : b will be a x and b x
5. If two current age is x, then
1
x
of the age is
x
n

19
CHAPTER – 16
ALGEBRA
Basic algebra:- One should be well versed in basics of algebra.
Certainly, we should know about same questions based an equations and their roots.
Polynomials:- An expression in terms of same variables is called polynomial.
For e.g.:- ( ) 2 5f x n  is a polynomial in variable x.
2
( ) 5 3 4g y y y   is a polynomial in viable
The expression f (x) =
1
2
1 1 0.......
n
x
nanx a a x a

     is called a polynomial in variable x
where n be a positive integer and 0 1............,a a be constants.
Degree of a poly nominal:-
The exponents of the highest degree term in a polynomial is known as its degree.
Linear polynomial:-
A polynomial of degree one is called a linear polynomials In general f (x) = a x + b, where
1
a o is a linear polynomial.
Quadratic Polynomial:- A polynomial of degree two is called a quadratic polynomials., In
general
2
,ax bx c  where 1
a ois a Quadratic polynomial.
Cubic Polynomial:- A polynomial of degree 3 is called a cubic polynomial in general.
3 2 1
( ) 9 ( )f x x bx x d a o    is a cubic polynomial
Remainder Theorem:- Let f (x) be a polynomial of a degree greater than or equal to are and
a be any real number, if f(x) is divisible by (x-a), then the remainder is equal to f (a).
Factor Theorem:- Let f(x) be a polynomial of degree greater than or equal to one and a be
any real number such that f(a)=0, then (x-a) is a factor of f (x).
Useful formulae:-
1)
2 2 2
( ) 2x y x y xy   
2)
2 2 2
( ) 2x y x y xy   
3)
2 2
( ) ( )( )x y x y x y   
4)
3 3 3
( ) 3 ( )x y x y xy x y    
5)
3 3 3
( ) 3 ( )x y x y xy x y    
6)
3 3 2 2
( ) ( ) ( )x y x y x y xy     
7)
3 3 2 2
( ) ( ) ( )x y x y x y xy     
8)
2 2 2 2
( ) 2( 3 )x y z x y z xy yz x       
9)
3 3 3 2 2 2
( 3 ) ( )( )x y z xyz x y z x y z xy yz zx         

20
10) If 0,x y z   then
3 3 3
3 ....x y z xyz  

21
CHAPTER – 17
PROFIT AND LOSS
Cost Price:- The price, at which an article is purchased, is called its cost price, abbreviate
as C.P.
Selling Price:- The price, at which an article is sold, is called its selling Prices, abbreviated
as S.P.
Profit or Gain:
If S.P. is greater than C.P, the seller is said to have a profit or gain.
Loss:- If S.P. is less than C.P, the seller is said to have incurred a loss.
Important formulae:-
1. Gain = (S.P) – (C.P)
2. Loss = (C.P) – (S.P)
3. Loss or gain is always reckoned an C.P
4. Gain% =
x 100
( )
.
Gain
C P
5. Loss%=
x 100
( )
.
Loss
C P
6. Selling Price:-
(100 %)
. [ x . ]
100
Gain
S P C P


7. Selling Price:-
(100 %)
. [ x . ]
100
Loss
S P C P


Cost Price (C.P)
8. Cost Price (C.P)
100
. [ x . ]
(100 %)
C P S P
Gain


9. Cost Price (I.P)
100
. [ x . ]
(100 %)
C P S P
Loss


10. When a person sells two similar items, one at a gain at say x%, and the o three at a loss
of x%, then free seller always incurs a loss given by:
2 2& %
%( ) ( )
10 10
Common Loss and Grain x
Loss 
11. If a trader professor to sell his goods at C.P, but uses false weights, then
% ( x100)%
( ) ( )
Error
Gain
True value error

 

22
CHAPTER – 18
DISCOUNT
Formula:-
Discount = List price – selling price
Selling price = list price – discount
Selling price = list price – discount
List price = selling price + discount
Rate of discount = discount % x100
Pr
Discount
List ice

List price = selling Price x
100
(100 %)discount


Selling price = list price x
(100)
=
(100-discount%)

23
CHAPTER – 19
MENSURATION
It Cautions L- Parts:-
2-D- 2 dimensional figures.
1. Square:- ‘a’ is side of a square all side are equal
Area = 2
a
Perimeter = 4 a
Diagonal = 2a
2
2
2
d
a 
2. Rectangle:-
Opposite sides are equal.
Area = l x b
L = length
B= breadth
Perimeter = 2 (l + b)
3. A equilateral triangle:-
All sides are equal.
Area =
2
3
4
a
Perimeter = 3a
3
2
xa
- height
Isosceles triangle and Right angled triangle:-
B= base
H= height
The, Area =
1
2
x b x h
4. Scalene triangle:- All sides are different to each outer.
C semi perimeter:-
2
a b c
S
 

Area = ( )( )( )s s a s b s c  
5. Area of Triangle-
1
x
2
D X base height

24
( )( )( )s s a s b s c    
is is perimeter.
4
abc
D
R

R = Circumradius
D= r x s
r= inradius, s = remiperimeter
1 1 1
sin sin sin
2 2 2
D ab bc ab    
6. Parallelogram:- Area = sinab 
= 2 (a+b)
Area = b x h
Area = a x h
7. Rhombus:- Parameter:- 4 a
Area = 1 2
1
x x
2
d d
2 2
1 2
1
2
a d d 
1d and 2d are two diagonals in Rhombus – both diagonals cut at 0
90
8.Trapezium:-
Area =
1
2
(sum of parallel sides) x altitude
Area =
1
2
x (a + b) x h
H= height
A, b are two parallel sides
9. Hexagon:-
Hexagon have sex sides and if cautions six equilateral triangles.
So, the area = 23
6x x
4
a

25
Because the area of an equilateral triangle perimeter = 6 a
10. 3 – D = 3 dimensional figures:-
Prism Pyramid Sphere
Cylinder Cone Hemisphere
Cube Other base
Cuboid Pyramid
Outer base prism frustum
Tetrahedral
11. Prism:- Surface area:- Perimeter of base x height
Pyramid – Surface area =
1
x
2
Parimeter of base x slant height
Volume =
1
x
3
Area of base x height
12. Cuboid:-
6 – faces
8 – corner
12 – edges
Vol. – l x b x h
Diagonal =
2 2 2
l b h 
Curved Surface area = 2 (bh + hl)
Total surface area = C.S.A + 2 lb
2 (lb + bh + hl)
We are not count the base and Top in curved surface area a
13. Cube:- Volume =
3
a
Diagonal = 3a
Total Surface area = 2
6a
Cylinder:-
Vol:- 2
r h
Curved surface area = 2 xTTr h
2
. . . . 2T S A C S A r  
2
2 2rh r  
2 ( )r r h 

26
14. Cone:-
2 2
l r h 
Slant height.
Vol:- 21
x
3
r h
C.S.A = rl
2
. .T S A rl r  
( )r r l 
15. Sphere:- Vol:-
34
3
r
2
. . 4C S A r 
16. Hemisphere – Vol: -
32
3
r
2
. . 3C S A r 
17. Frustum of a cone:-
Vol:- 2 2
x (R )
3
h
r Rr

 
2 2 2
= h ( )l R r  
Curved Surface area = l (R + r)
Total Surface area:- (area of the base) + (area of the Top) + (curved surface area)
=
2 2
{ ( )}R r l R r   
=
2 2
[ ( )]R r l R r   
18. Probability:- it’s a possibility of any towing.
Probability =
l n .
Number of favourable out comes
Tota o of out comes
If tossed a coin, then we have two possibility either head or tail.
2-possibilities
If we tossed two coins times then Possibility {TH, HT, HH, TT}
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Math Study Material for SSC and Banking Exam

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Math Study Material for SSC and Banking Exam