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# Category Archives: maths

## Exponent

– where 2 is the base & 6 is the exponent, and read as “two raised to the power six”

Negative integral exponent of a rational number when a is the non zero R.N

,

(there is and multiplication inverse)

where m > n

Express the numbers in exponantal form

where

e.g

**Square**

The square of a number is the product of the number with the number itself

e.g. , ,

1,4,9,16,25……… are called perfect square.

**To find the perfect square**– factorize the given number, if their factors are in pair, then the number is perfect square,

e.g. is a perfect square.

**Number perfect square**.

No perfect square will end with 2,3,7,or 8 at the unit place.

A number having 0,1,4,5,6,9, in the unit place may or may not be a perfect number

**(1)** If the number has 1 or 9 at the unit place than its square and with 1

**(2)** The square of a number which has 4 or 6 at the units place will and in 6

**(3)** A number ending in an odd number of zero is never a perfect square.

E.g. 640, 5000, 44000

Between the square of the numbers n and n+1 there are 2n non perfect square numbers

, ,

, ,

, ,

The square of an even number is always an even number & the square of an odd number is always an odd number.

,

The secure of a natural number (expert ) is either or multiple of 3 or exceeds a multiple of 3 & 41

Similarly- with 4

The square of natural number ending with five follows a delimit pattern

52 = (0×1) hundred + 25 = 25

152 = (1×2) hundred + 25 = 225

252 = (2×3) hundreds + 25 = 625

352 = (3×4) hundred + 25 = 1225

452 = (4×5) hundred + 25 = 2025

The sum of first nodal natural numbers in n2

Sum of first odd number = 1 =12

Sum of first two odd number = 1+3 = 4 = 22

Sum of first three odd number = 1+3+5 = 9 = 32

The sum of rfirst four oddd number = 1+3+5+> = 16 =42

Look at this pattern whose number include only one.

12 = 1 ___________(= 12)

112 = 121 (1+2+1 = 4 = 22)

1112 = 12321 (1+2+3+2+1 = 9 = 32)

11112 = 1234321 (1+2+3+4+3+2+1 = 16 = 42)

111112 = 121 (1+2+3+4+5+4+3+2+1 = 25 = 52)

Square of these numbers is serially overdraft 1,2,3 …………..equal to the number of its digit & decreases vice verse ………3,2,1

The sum of the digit of their product is also a perfect square.

72 = 49 pattern- when number of digit = 1 than number of 4 = n, & number of 8 = n time +1

672 = 4489

6672 = 444889

666722 = 44448889

**Pythagorean triplet**

In right angled triangle:-

For any numbewr m>1, (2m, m2-1, m2 +1) is a Pythagorean triplet

If 3, 4, 5

M=2 22-1, 2×2, 22+1

**Square root**

The square root of the number , is that number, which when multiplied by itself , gives the number as the product.

√x×√x=x we denote the square root of x, by √x , square root is a inverse process, of square.

2×2 = 4 & √(4 )= √(2×2) = 2

Note:- if a number has a natural numbewr as square root then its units digit must be 0,1,4,5,6 or 9. Negative numbers have no square root in the system of natual numbers

e.g √25= ≠5

to find the square root by factorisation method

√16= √(4×4)=4

The number being the perfect square, will have one or more pairs of ther prime factor, write one factor from each pair & multiplied these factors , the product will be the square root of the number e.g√81= √(3×3×3×3)=3×3=9

To find square root by successive substraction method

The sum of the first n odd natural numbers is n2

This method is useful to find the square root of smaller natural numbers.

81-1 = 80

80-3 = 77

77-5 = 72

72-7 = 65

65-9 = 56

56-11 = 45

45-13 = 32

32-15 = 17

17-17 = 0

Square root by division method

e.g.

to find the number of digit in the square root

√((81) ̅ ) = 1 digit = 9

√(2 ̅(25) ̅ )=2 digit=25

√((20) ̅(25) ̅ )=2 digit=48

√(2 ̅8 ̅(224) ̅ )=3 digit=168

Square root of rational numbers(fraction)

√(a/b)= √a/√b

Square root of decimal

We see that 0.2 ×0.2 = 0.04 ∴ √(0.(04) ̅ ) = 0.2 number of digit 1 & root of 4 = 2

Approximate value of square root

We get the square of that number is multiplied by itself x×x = x2

Similarly if a number is multiplied by itself 3 times we get cube of that number x×x×x = x3

Perfect cube:- a natural number is said to be a perfect cube if it is the cube or a natural number.

e.g 13 = 1, 23 = 8, 33 = 27 thus 1,8,27 are perfect cube.

Properties:-

The cube of even numbers are even & odd number are odd.

In a perfect cube each prime numbers appears three times in its prime factorization √27= √(3×3×3)=

Cube of negative numbers is negative

Cube of number ending with 0,1,4,5,6 & 9 also end with the same digit ending with 8 will end with similarly cubes of number ending 3 &7 will end with 7 & 3 respectively.

Smallest number:- some numbers are expressed as the same of two square & sum of two cubes also.

c.g. :-

1729 = 1728 + 1 = 123 +1

1729 = 1000 + 729 = 103 + 93

4104 = 8 + 4096 = 23 + 163, 4104 = 729 + 3375 = 92 + 153

13832 = 5832 + 8000 = 183 + 203, 13832 = 8+13824 = 23 + 243

Cube root of decimal numbers

To find the cube root of aq decimal numbers, write the number in the form of p/q and then find their cube root

## Sets Identity function

The function that associates each real number to it self is called the identity function and denoted by I

I(x) = x : x R

The domain and range of the identity function are both equal to R. the graph is a straight line passing through the origin and inclined at an angle of with x axis

F(x) = x

F(1) = 1

F(2) = 2

Modulus Functions:-

If f(x) = |x| = {x when x ≥ 0,

-x when x < 0}
Is called the modulus function or absolute value function the domain of the modulus function is the set of all real number. & range is the set of all non negative real number
= {x R : x ≥ 03}

(1) If x ≥ 0 the graph coincide. With the graph of

the identity functional (g= x)

If x < 0 it is coincide to the line y = -x

**Properties of Modulus Function**

1) For any real number x

= |x|

2) a,b is a positive real number, then

≤ |x| ≤ a -a ≤ x ≤ a

≥ |x| ≥ a x ≤ -a or x ≥ a

< |x| < a -a < x < a
> |x| > a x < -a or x > a

≤ ≤ a≤ |x| ≤ b x [-b, -a] u [a,b]

< < a ≤ |x| ≤ b [-b, -a] u [a,b]

3) for real number x and y

|x+y| = |x| + |y| (x≥0 and y≥ 0) or (x< 0 and y< 0) |x-y| = |x| - |y| (x≥0 and |x|≥ |y|) or (x≤ 0 , y≤ 0 and |x| ≥ |y|) |x± y| ≤ |x| + |y| |x± y| ≥ |x| - |y|

**Greatest Integer Function (Floor Function)**

f(x) = [x] for all xεR or ⌊x⌋ is called greatest integer function

for any real number x the smallest [x] to denote the greatest integer less than or equal to x

Domain of the greatest integer function is the set of R to all real number and the range is the set fo Z of all integers as it’s attains only integer value.

e.g. [2.75] = 2, [3]= 3, [0.74] = 0, [-7.45] = -8 etc

Properties :- if n is integer and x is a real number between n and n+1 then

[-n] = -[n]

[x + k] = [x] + k for any integer k

[-x] = -[x] -1

[x] + [-x] =

[x] – [-x] =

[x] ≥ k = x ≥ k where k z

[x] ≤ k = x < k where k z

[x] > k = x ≥ k +1 where k z

[x] < k = x < k where k z

[x+y] = [x] + [y+x-[x] for all x,y R

= [n x], n N

**Smallest Integer Function**

f(x) = ⌈x⌉ for all x εR

Is called the smallest integer function or the ceiling function.

The domain of the smallest integer function is the set of R of all real number and its range is the set z 0 < all integers e.g :- ⌈4.75⌉=5, ⌈-7.2⌉ = -7,⌈5⌉=5, ⌈.75⌉=1 etc Properties of Smallest Integer Function ⌈-n⌉= -⌈n⌉ ,n z

⌈-x⌉= -⌈x⌉+ 1,x R-z

⌈x+n⌉= ⌈x⌉+ n,xεR-z and n z

⌈x⌉+ ⌈-x⌉ =

⌈x⌉- ⌈-x⌉ =

**FRACTIONAL PART FUNCTION**

The function f(x) = {x} for all xεR

The symbol {x} denote the fractional part of or decimal part of x.

The domain of the fractional part function is the set of R of real number and range is the set [0,1]

f(x) = {x} = x – [x] : x εR

e.g. 3.45 = 0.45, [2.75] = 0.25, [-0.55] = 0.45, [3] = 0, [-7] = 0 etc

**SIGNUM FUNCTION**

If f(x) = .3 × .3 =.09

f(x) =

The domain of the signum function is the set of R (all real number) and the domain is the set

(-1,0,1)

EXPONENTIAL FUNCTION:-

F (x) = where a> 0 and a ≠ 1

If a > 1

If y = f(x) =

We observe that –

If 0 < a<1
Y = f(x) = decrease with the increase in x.

And y>0 : x ε R

if then graph

## Set Relations

A relation (“ is Religion of”) between SET A to SET B is a sub set of A×B

Where ‘R ≤ A×B

If SET A & SET B is not Vol D SET & If (a,b) R read as a is related to b by relation R’ if (a, b) R then a ×b (a is not related to b & y any relation R

**Total Number Of Relation**: If SET A and SET B not empty finite sets consisting of m and n elements then A×B consists of mn ordered pairs and total no. of sub sets of A×B is

Among these relations the void relation & the universal relation A×B are trivial relations from A to B.

**Domain And Range Of A Relation**– relation to SET A to SET B all first components or co ordinates of the ordered pairs belonging to R is called the domain while the set of all second components or co-ordinates of the ordered pair in the R is called the range of R.

**Relation On A Set**– A non raid SET- A to relation is set (SET A) = A×A is called a relation on SET A

Inverse relation:- If SET A and SET B two sets and R is relation between them then inverse of R denoted by , is a relation from B to A

= {(b,a): (a,b) R}

**Functions**: a relation F from SET A to SET B i.e a subset of A×B is called function L or a mapping (or a map) from A to B if

(i) For each a A there exists bε B such that (a,b) f

(ii) (a,b) f and (a,c) f b=c

A non void set f of A×b is a function from A to B if each elements of A appears in some ordered pairs in f and no two ordered pairs in f have the first element.

If (a,b) f then b is called the image of a under f.

e.g. set A= (2,3,4)

set B = (3,4,5)

f1, f2, f3 is three sub set of A×B as below.

f= {(1,2,3), (3,4), (4,5)}

f2 = {(2,3), (2,4) (3,4) (4,5)}

f3= {(2,4), (3,4)}

here f1 is a function from A to B

f2 is not a function from A to B because 2 A has two images 3 and 4 in B.

f3 is not a function A to B because 4 A has no image in B.

if a function F is expressed as the set of ordered pair, the domain of f is the set of all first component(elements ) of member s of f & the range of f is the set of second components of member of f.

**Function As A Correspondence**

If A and B is non empty sets, then a function f from set A to set B is a rule or method or correspondence. Which associate elements of set A to set B such that

(1) All elements of Set A are associated to elements in set B.

(2) An element of set A is associated to a unique element in set B.

If ‘ f ’is a function from a set A to set B then we write f:A B or A B and read as “f is a function from A to B”.

If an elements a A is associated to an elements bε B then B is called the f image of a’ or image of a under f’ or the value of the function f at a’

A is called the pre image of B under function f & write b = f(a)

Description of a function:- if f:A B be a function such that the set A consists of a finite number of elements than f(x) described by listing the values it attains at different paints of its domain

**Domain Co Domain And Range Of A Function**:

IF f :A B , then the set A is known as the domain of f and the set B is known as the co domain of f. The set of all of image of elements of A is known as the range of f or image set of A under f and is denoted by f(A)

F(A)= {f(x): x A} = range of f or f(A) ≤ B

e.g. Set A= {-2,-1,0,1,2,} and B= {0,1,2,3,4,5,6}

Consider a rate f(x) =

then f(-2) = = 4

f(-1) = = 1

f(0) = = 0

f(1) = = 1

f(2) = = 4

as above each elements of A is associated to a unique elements of B so, is given by f(x) = is a function

Here domain (f) = A {-2,-1,0,1,2}

Range (f) = {0,1,4}

**EQUAL FUNCTION**

Two functions f & g are said to be equal, if

(1) Domain of f = domain of g.

(2) Codomain of f = co-domain of g

(3) F(x) = g(x) for every x beginning to their common domain then function f = function g

**Real Function** : if domain and co domain are sub set of :

(A) (B)

The set R of all real number , function are called real function.

Domain of real function:- real functions are described. By providing the general for mald for finding the image of elements in its domain.

Range Of Real Function:- the range of a real function of a real variable. Is the set of all real value fallen by f(x) at paints in its domain.

Constant Function:- if K is a fixed real number

Then f(x) = k (x R)

The graph of a constant function f(x) = K is a straight line. Parallel of x – axis above or below of x axis according positive or negative value of K, if k = 0 the straight line is coincident of x axis

## Law of Algebra of sets

**Law of Algebra of sets**:

a) Impotent law- A A=A and A A = A

b) Identity Law – A = A and A u=A

c) Commutative law- A B = B A and A B = B A

d) Associative law- (A B) C = A (B C)

e) Distributive law- A (B C) = (A B)∩(A C)

– A (B C)= (A B) (A C)

Some Use Full Theorem

If A and B are any two sets then

a) A-B = A

b) B-A=B

c) A-B=A A B =

d) (A-B) B = A B

e) (A-B) B =

f) A B

g) (A-B) (B-A) = (A B) – (A B)

Some Important Result on Number of Elements In SETS

If A,B and C are finite sets and u be the universal set then.

1. n(A B) = n(A) + n(B) – n(A B)

2. n(A B) = n(A) + n(B) + n(B) A,B are adjoint non vaid sets.

3. n(A-B) = n(A) – n(A B)

i.e n(A-B) + n(A B) = n(A)

4. n(AB) = number of elements which belong to exactly one of A or B

= n(A-B)(B-A)

= n(A-B) +n(B-A) ……………. [(A-B) and (B-A) are disjoint]

= n(A)- n(A B) + n(B) – n(A B)

= n(A) + n(B) – 2n(A B)

5. n(A B C) =

n(A) + n(B) + n(c) – n(A∩B) –n(B C)-n(A C) + n(A B C)

6. number of elements in exactly two of the sets A,B,C

= n(A B) + n(B C) + n(C A)- 3n(A B C)

7. number of elements in exactly on cot the sets A,B,C

= n(A) + n(B) + n(c) – 2n(A B) –2n(B C)-2n(A C) + 3n(A B C)

8. n( ) = n((A∩B)’) = n(u) – n(A B)

**Certesian Product Of Sets**

**Ordered Pair**– an ordered pair consists of two objects or elements in a given fixed order

If A&B are any two sets then by an order pair of elements are (a,b) Where a A and b B

The position of a paint in two dimensional plane intercession coordinate is represented by an order pair (-1,5) Where x R and Y r

**CARTESIAN PRODUCT OF SETS**: if A and B are two non empty sets the set of all ordered pairs (a,b) such as a a A and b B is called Cartesian product of the set A and B and is denoted by AB

A×B = {a,b}: a A and b B}

Example if A= (x,y) and B= (2,3,4) find A×B, B×A, A×A

A×B = {(x,z), (x,3), (x,4), (y,z), (y,3), (y,4)}

B×A= {(z,x), (z,y), (3,x), (3,y), (4,x), (4,y)}

A×A= {(x,x), (x,y), (y,x), (y,y)}

As above (A×B) (B×A) =

Graphical Representation Of Cartesian Product Of Sets

If A = {2,3,4}

B = {3,4}

A×B = {(2,3) (2,4) (3,3) (3,4), (4,3), (4,4)}

To represent A×B graphical, draw to line perpendicular to each X & Y axis and then draw these pairs.

As above n(A×B) = n(A), n(B) = 2×3 = 6 Pair

A×B = A = , B =

A×A×A= {(a,b,c) : a,b,c A}

(a,b,c) is called an ordered triplet

(i) A× (BUC) = (A×B)U (A×C)

(ii) A×(B∩C) = (A×B)∩ (A×C)

(iii) A×(B-C) = (A×B)- (A×C)

(iv) A×B = B×A A=B

(v) A≤B then A×A ≤ (A×B) (B×A)

(vi) A≤ B then A×C ≤ B×C

(vii) A≤B and C≤D than A×C ≤ B × D

(viii) For any set Four A,B,C,D

(A×B)∩ (C×D) = (A∩C) × (B∩D)

For any set A and B

(A×B)∩ (B×A) = (A∩B) × (B∩A)

(ix) For any three set A,B,C

A × (B’UC’) = (A×B) ∩(A×C)

A× (B’∩C’) = (A×B)U (A×C)

(x) If any two non empty sets have n elements in common. Then A×B and B×A have elements in common.

(xi) If A is non empty set and A×B = A×C

Then B=C

## Venn Diagram

first of all a swiss mathematician euler gave an idea to represent a set by the points in a closed curve. Later on British mathematician john venn brought this idea to practice. Such the diagrams drawn to represent sets are called venn ealer diagrams or venn diagram.

In venn diagrams the universal set u is represended by points within a rectangle and its sub sets are represented by the point in closed curve (usually circle) with in the rectangle.

SOME TYPES OF REPRESENTATION

U (rectangle) universal Set

A- Set

B- Sub set of A

are show common element s of set A & set B

intersecting area =

suppose u = {1,2,3,4,5,6,7,8,9,10}

Set A = {2,3}

Set B = {6,7,8}

**OPERATIONS ON SETS**

1) **Union of sets**:

A & B be two sets the union of A & B is the set of all those elements which belong either to A or B or the both A & B.

= {x:x A or x B} (A union B)

: x A B x A or x B

And x A B xA and xB

As above A B = B (if A B)

A B = A (if B A)

e.g if A = {1,2,3}

B = {1,3,5,7}

Then A B = {1,2,3,5,7}

2) **Intersection of sets**: A & B be two sets the inter section of A & B is the set of all those elements that belong to both A & b and denoted by A B

Thus A B = {x:xεA and xεB}

Or x A B x A and x B

Or A B ≤ A and A∩B ≤ B

e.g if Set A = {1,2,3}

Set B = {1,3,5,7}

Then A B = {1,2,3,5,7}

3) **Disjoint set**: Two set A & B are said to be disjoint if A B=

If A B then A and B are said to be intersecting or over capping sets.

4) **Difference of sets**:

**Symmetric difference of two sets**:- if A and b be two set s. then symmetric difference of set A and B is A b = (A-B) (B-A)

= {x:x A B}

e.g. Set A = {1,2,3,4,6}

Set B = {2,3,4,5,6,7}

Then A B = (A-B) (B-A)

= {1,} (5,7)

A B = {1,5,7}

5) **complement of a set**– If u is the universal set & A is a set such that A U than the complement of set A with respect to set u = or or U-A

Thus = = { x u : x A}

e.g- if u = {1,2,3,4,5,6}

A= {3,4}

= {1,2,5,6}

Thus =

= u

= A

= u

=

## SETS

A set is a well defined collection of objects.

The collection member is called number, object or element .

We shall denote sets by capital alphabets eg. A,b, ——- &

The elements by the small alphabets e.g- a,b,c,—————- z

Some reserve letters for these sets.

N: for the set of natural numbers {1,2,3,4,5,6,……………….}

Z: for the set of integers {-3,-2,-1,0,1,2,3…………………….}

: for the set of all positive integers.

Q: for the set of all rational numbers.

T- the set of all irrational number

(x:x = , m,n e z, ) (x:xe r and )

: for the set of all positive rational number.

R: for the set of all real numbers.

: for the set of all positive real numbers.

C: for the set of all complete numbers.

Then NCZCQCR

TCR AND

**Description of sets**:- Set is often described in the following two ways.

1) ROSTER METHOD :- a set is described by listing elements,separated by commas, within fraces { } e.g the set of vowels of English alphabets described as (a,e,i,o,u).

2) SET SUICDER METHOD:- in this method a set is described by a characterizing property

P(x) of elements x in such a case the set is described .

{x:P(x) holded} or {x/P(x) holder} is read as

The set of all x such that P(x) holds

e.g:- A= (0,1,2,3,4,8,16,………………..) is written as A = { : xe}

**TYPES OF SETS**

**EMPTY SET**:- if it has no elements, said empty or void or nail set & denoted by

in roster method is denoted by {}

builder method = {xeR : = -z}

**SINGLE TON SET** a set consisting of a single element e.g {5} is a single ton set.

{x:xEN} and =9 } is a single ton set equal to {3}

**FINITE SET**: A finite set it is either empty set or its element can be listed or counted. By natural number (1,2,3,……….) the process of listing terminates at a certain natural number n(A) and is called ordinal number or order of a finite set

e.g. set A = (1,2,3,………………….99) Natural number less then 100

**INFINITE SET **: A set whose element cannot be listed by the natural number 1,2,3 …………… n for only natural number n is called an infinite set

Example

set of all paints in a plane.

{ex er : 0

set A = (1,2,3) set B = {a.b,c,}

Here n(A) = n(B)

**EQUAL SETS**: if every elements of set A is a number of set Band every elements of set B is a number of set A, are called equal sets

Set A = {1,2,5,6}

Set B = {5,6,2,1}

**SUB SETS**: if every elements of A is a an element of B, then A is called sub set of B and denoted by

“A is sub set of B”

Thus if

The symbole stands for “implies”

If A is not a sub set of B we write

**SUPER SET** : If A is subset of B then B is a super set of A

Note:- 1) every set is a subset of itself

2) empty set is subset of every set

3) above 1 and 2 are called improper sub set

4) A subset A of a set B is called a proper sub set of B

If AB then B is super set of A

If A=B then A≤B and

5) the total number of subsets of a finite set containing n elements is 2n

A finite set A containing n elements each sub set of A have r elements

Then

element is then the total number of sub set of A

= + + ………………………….. + (binomial theorem )

**INTERVAL OF SUB SETS OF R**

1) **CLOSED INTERVELS** – on real line a & b two given real numbers such that a** is called closed interval & written [a b]
**

E.g. is the set of all real numbers laying between -1 & 2 including -1 and 2 , clearly it is a infinite sub set of R

2) **OPEN INTERVALS** – on real line, a & b two given real number such that a < b , then the set of all real number x such that a < x < b is called open interval & is denoted by (a,b) or ]a,b[

3) **SEMI OPENED OR SEMI CLOSED INTERVAL**: on real line, a & b two given real number such that a < x ≤ b or a ≤ x < b is called semi open or semi closed interval & denoted by (a,b] or [a,b) or ]a,b], [a,b[
Note:- the number b-a is called the length of any of the intervals.
(a,b) [a,b] [a,b) (,b]
The interval (0, ) denoted by set R+ of all non negative real number

The interval (- ,0) denoted by set R- of all negative real number

The interval (- ,) denoted by set R of all real number

**UNIVERSAL SET**:a set that contains all sets under consideration i.e. it is a super set of each of the given set, is called universal set & is denoted by U Thus a set that contains all set in a given contex is called the universal.

e.g A= {1,2,3}, B = {2,4,5,6}, c= {1,3,5,7,}

then u = {1,2,3,4,5,6,7}

e.g. when we are using intervals on real line, the set R of real numbers is taken as the universal set.

**POWER SET**: the collection or family of all sub sets of a is called the power set & is denoted by p(A) = {s:s CA}

Let set A = {1,2,3}

Then the sub set are-

, {1}, {2}, {3},{1,2}, {1,3}, {2,3}, {1,2,3}

= P(A) (a set having n elements has 2n sub set s)

( sub sets)

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## Logarithms

It is a passive real number other than

We write

4 then we can write

then we can write

**Properties of logarithms**

Example so we can write

Example in logarithm form

Logarithms to the base 10 are known as common logarithms. When base is not mentioned it is taken as 10.

Characteristic – when the number is greater than 1 – the characteristic is one less from the number of digit in the left of the decimal point in the given number.

̅.41 = 3

When the number is less than 1 one more than the number of zero between the decimal point & the 1st significant digit of the number & it is negative.

0.518 = -1 0.0347 = etc.

Log table-

## Simple and compound interest

**Simple interest **

P= principal amount

R = rate/unit

T = time/year month day etc.

**Compound interest **

P= principal amount

R= rate/unit

T= time unit

Where rate are difference for different unit or time.

Than

Present worth of Rs x lac in year hence is given by present worth=

## Alligation Or Mixture

Alligation- the rate to find the ratio in which two or more ingredients at the given price.must be mixed to produce a mixer of a desired price.

Mean price- the cost price of a unit remobilty of the mix are is called mean price.

Rule of allegation

cheaper quantity: dearer quantity= (d-m): (m-c)

A container of x unit capacity of liquid, when we taken out some unit say x replaced by water, when we do the same operation n times.

quantity of pure liquid=

## Boat and Stream

if the speed of a boat in still water is u km/h & the speed of the streams of v km/h than-

speed towards downstream-

Speed towards up stream-

if the speed of boat in downstream u km/h and in upstream v km/h

than speed in still water-

rate or speed of stream=

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