Fraction Conversion Table

Fraction:
A fraction is a number that can represent part of a whole. The earliest fractions are reciprocals of integers: ancient symbols represent one part of two, one part of three, one part of four, and so on. A much later development was the common fractions which are used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

Fraction can be converted into to forms as shown below,

• Fraction conversion into decimal
• Fraction conversion into percentage.

Through this article you can learn about fraction conversion table, how to make conversion of fraction into decimal and how to make conversion of fraction into percentage.

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Example for fraction conversion table:

Fraction conversion table:

         We can express the conversion of fraction in terms of table format.This table is known as fraction conversion table.The following are the two examples for fraction conversion table.

                       

Fig(i) Fraction conversion table 1                                      Fig(ii) Fraction conversion table 2

The fractions that contains the denominator as a  power of  10 called as decimal fractions.
Ex:
0.1 = 1/10
0.01 = 1/100
0.001 = 1/1000

Problems on fraction conversion table:

Pro 1: Convert the fraction 45 / 90
Sol:
Given, 45 / 90 into decimal.
To find the decimal,divide the numerator by the denominator ,
             _________
         90 )  45      (
45 cannot be divided by 90.
So Multiply 45 by 10 and add one decimal place in quotient,
             _________
         90 )  450     ( 0.5
                450     
                 0        
So the decimal of fraction 45/90 is 0.5
Ans: 45/90 = 0.5
Pro 2: Convert the fraction 76 / 43 into decimal.
Sol:
Given, 76 / 43 as decimal
To find the decimal,divide the numerator by the denominator ,
             _________
         43)  76     ( 1.76
               43     
                33      
33 cannot be divide by 43. So multiply 33 by 10 and one decimal place in quotient,
            _________
         43)  76     ( 1.7
               43     
               330    
               301   
                 290
                 258
                   38
The decimal for 76 / 43 = 1.76
Ans:76 / 43 = 1.76

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Pro 3: Convert the fraction 6 / 5 into decimal.
Sol:
Given, 6 / 5 as decimal
To find the decimal,divide the numerator by the denominator ,
           _________
        5 )  6     ( 1
              5     
              1   
1 cannot be divide by 5. So multiply 1 by 10 and one decimal place in quotient,
            _________
           5 )  6     ( 1.2
                 5     
                 10
                 10  
                   0 
The decimal for 6/5 = 1.2
Ans: 6/5 = 1.2

Venn Diagram Example of Disjoint

Let use see about venn diagram example of disjoint.Venn diagrams also called as the set diagram.These are diagrams that show all theoretically possible logical relations between finite groups of set. Venn diagrams were considered approximately 1880 by John Venn. They are used to teach elementary set theory, and show simple set relationships in logic, probability and statistics.


venn diagram example of disjoint - Notations


Curly  braces - {...} - are used to phrase.
These braces can be used in various ways.
For example:

• List the elements of a set: {-3, -2, -1, 0, 1, 2, 3,4}

• Describe the elements of a set: {integers between -3 and 3 inclusive}

• Use an identifier (the letter x for example) to symbolize a typical element, a '|' symbol to stand for the axiom such that', and then the rule or rules that the identifier must follow: {x | x is an integer and |x| < 5}

The Greek letter ∈ is used as follows:

• ∈ means 'is an element of ...'. For example: 3 ∈ {positive integer}

• ∉ means 'is not an element of ...'. For example: Washington DC ∉ {European capital cities}

• The set is a finite: {British citizens}

                      Or

•  infinite: {6, 12, 21, 24, 35, ...}

Sets are usually be represented using upper case letters: A, B,X,Z ...

venn diagram example of disjoint - Example

The following is the diagram representation of disjoint in venn diagram.
Two sets are equally exclusive also called disjoint. If do not have any elements in common and need not together contain the universal set.
The following venn diagram represents the disjoint sets.

Set of all elements of A is also known as difference of set A-B, which do not go to B. In the set planner form, the difference set is :   

 

Example problem for disjoint set.
A={2,3,4,1,8,9}   B={2,3,4,1,10,12} What is the A-B and B-A?
Solution:
A-B=?
Given A={2,3,4,1,8,9}
            B={2,3,4,1,9,10,12}
Here all elements of A  an available in B except 9.
So the A-B is 9.
B-A=?
Here all elements of A  an available in A except 12.
So the B-A is 12.

Line Intercept Sampling

In this article, we will discuss about the line intercept sampling. Sampling means method of selecting sample. The line intercepts have two types, one is x intercept and next one is y intercept. X intercept means that, the point crosses the x -coordinates or axis and y intercept means point crosses the y- axis of the line. The slope intercepts form of line y = mx + b, where m is slope of the line, and b - the y intercept. Let us learn about the line intercept sampling example problems are given below.

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Example problems for line intercept sampling:

Example problem 1:

Find the x intercept of the line, 11.5x + 13.5y = 27

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the x- intercept Plug y = 0 in the equation

Here x intercept, so y = 0

11.5x + 13.5(0) = 27

11.5x = 27

Divide by 11.5 on both sides.

` (11.5x)/(11.5)` = ` (27)/(11.5)`

After simplify this, we get

x = 2.34

x intercept = (2.34, 0)

Example problem 2:

Find the y intercept of the line, 4x + 24y = 72

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the y- intercept Plug x = 0 in the equation

Here y intercept, so x = 0

4(0) + 24y = 72

0 + 24y = 72

24y = 72

Divide by 24 on both sides.

`(24y)/(24)` = ` (72)/(24)`

After simplify this, we get

y = 3

y intercept = (0, 3).

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More example problems for line intercept sampling:

Example problem 3:

Find the y intercept of the line, 5x + 30y = 120

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the y- intercept Plug x = 0 in the equation

Here y intercept, so x = 0

5(0) + 30y = 120

0 + 30y = 120

30y = 120

Divide by 30 on both sides.

`(30y)/(30)` = ` (120)/(30)`

After simplify this, we get

y = 4

y intercept = (0, 4)

The above examples are helpful to learn of line intercept sampling.

Alternating Current Generator

The household electricity that we used is based on the principle of Alternating Current. One advantage of Alternating Current is that it does not induce fatal shock. Also it has an added advantage of easy voltage amplification. In this article we shall discuss the Alternating Current Generator which is often abbreviated as A.C. Generator.

Understanding Alternating Exterior Angles is always challenging for me but thanks to all math help websites to help me out. 

Introduction to A.C. generator

The phenomenon of electromagnetic induction has been technologically exploited in many ways. An exceptionally important application is the generation of alternating current (A.C.).

Principle of ac generator

One method to induce an emf or current in a loop is through a change in the loop orientation or a change in it’s effective area. As the coil rotates in a magnetic field B , the effective area of the loop (the face perpendicular to the field) is AcosӨ, where Ө is the angle between A and B. This method of producing a flux change is the principle of operation of a simple ac generator.

A. C. Generator : Construction and Working

It consists of a coil mounted on a rotor shaft. The axis of rotation of the coil is perpendicular to the direction of the magnetic field. The coil called the armature is mechanically rotated in the uniform magnetic field by some external means . The rotation  of the coil causes the magnetic flux through it to change, so an emf is induced in the coil . The ends of the coil are connected to an external circuit by means of slip rings and brushes.

Working
When the coil is rotated with the constant angular speed of w , the angle Ө between the magnetic field vector B and the area vector A of the coil at any instant t is Ө=wt(assuming Ө=0⁰at t=0). As a result, the effective area of the coil exposed to the magnetic field lines changes with time, and the flux at any time is ф_B=BAcosӨ=BAcoswt From faraday’s law, the induced emf for the rotating coil of N turns is then,  E=-N×dф_B/dt = -NBA×dcoswt∕dt thus, the instantaneous value of the emf is  E= NBAwsinwt.Where NBAw is the maximum value of the emf,which occurs when sinwt= +1 or -1. If we denote NBAw as E₀, then E=E₀sinwt since the value of the sine function varies between +1 or -1, the sign, or polarity of the emf changes with time. 

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The emf has its extremum value when Ө=90⁰ or Ө=270, as the change of flux is greatest at these points. The direction current changes periodically and therefore the current is called alternating current.
The modern ac generator with a typical output capacity of 100 MW is a highly evolved machine.

Polynomials gcf Calculator

Polynomials gcf calculator is one of the interesting topics in mathematics. It is the process of performing greatest common factor for the given polynomial expression. It is the sums of a finite number of monomials are called as polynomial. Polynomial has more than one term and it has a constant value for the given each term, for that variable power of integral is raised to more than two.

Example for Polynomial expression is a2 – 26a – 28.

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Definition of Greatest common factor calculator:

Greatest common factor (gcf):

Greatest common factor is defined as the process of the highest number which divides more than two numbers or terms exactly.

Steps to find the Greatest common factor:

Step 1:

Given polynomial expression can be arranged in the order of powers

Step 2:

Each term in the given expression can be factored.

Step 3:

Find the common factors in each terms

Step 4:

Take the greatest common factor

Step 5:

Simplify the each term.

Example problem for polynomials gcf calculator:

Some example problems for polynomials gcf calculator are,

Example 1:

Find the gcf for the given Polynomial expression 3x2 – 9x

Solution:

Step 1:

3x2 – 9x

Step 2:

3 . x . x – 9 . x

Step 3:

In the given expression, common factors in the each term is 3x

3 . x . x – 3 . 3 . x

Step 4:

Take the common term outside

3x ( x – 3)

Step 5:

Solution to the given polynomial gcf is 3x (x – 3)

Example 2:

Find the gcf for the given Polynomial expression 5x2y5 – 20x4y3

Solution:

Step 1:

6x2y5 – 24x4y3

Step 2:

6 . x . x . y . y . y . y . y – 24 . x . x . x .x . y . y . y

Step 3:

In the given expression, common factors in the each term is 6x2y3

6 . x . x . y . y . y – 6 . 4 . x . x . x . x . y . y . y

Step 4:

Take the common term outside

6x2y3 ( y2 – 4x2)

Step 5:

Solution to the given polynomial gcf is 6x2y3 ( y2 – 4x2)

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Example 3:

Find the gcf for the given Polynomial expression 15x3y4 – 45x5y6

Solution:

Step 1:

15x3y4 – 45x5y6

Step 2:

15 . x . x . x . y . y . y . y  – 45 . x . x . x .x . x . y . y . y . y . y . y

Step 3:

In the given expression, common factors in the each term is 15x3y4

15 . x . x . x . y . y . y . y – 15 . 3 . x . x . x . x . x . y . y . y . y . y . y

Step 4:

Take the common term outside

15x3y4 ( y – 3x2  y2)

Step 5:

Solution to the given polynomial gcf is 15x3y4( y – 3x2  y2 )

How to add Trinomials

Trinomials, a function is in the structure of ax2+bx+c =0 (where a≠0, b, c are constants). Quadratic function or quadratic equation also called  trinomials. An algebraic expression which has 3 terms known as trinomials. The product of two binomials gives a trinomial and there are two solutions for the given trinomial. To sum trinomial first it needs to combine the liked terms and then solve it. Since, the trinomial is a combination of two terms and the sum of trinomials also has three terms.

Examples problem for add trinomials:

1. Add trinomials for given function is 2x + 3y + 4z and 6x + 4y +2z.

   Solution:
   Given functions is,
    2x + 3y + 4z and 6x + 4y +2z,

Step 1:
    Write the given trinomials as,
                       = 2x + 3y + 4z + 6x + 4y +2z

Step 2:
    Then combine the terms like as,
                       = 2x + 6x + 3y + 4y + 4z + 2z

Step 3:
    Then add the given trinomials, and we get the answer,
             Answer = 8x +7y + 6z.

2. Add trinomials for given function is 8x - 2y - 9z and 3x + y +2z.

    Solution:
    Given functions is,
     8x - 2y - 9z and 3x + y +2z,

Step 1:
    Write the given trinomials as,
                       = 8x - 2y - 9z + 3x + y +2z

Step 2:
    Then combine the terms like as,
                   = 8x + 3x - 2y + y - 9z + 2z

Step 3:
    Then add the given trinomials, and we get the answer,
            Answer = 11x - y - 7z.

Examples problem for add trinomials:

3. Add trinomials for given function is 3x² + 6y² + 4z² and x² + 5y² + 6z².
    Solution:
    Given functions is,
    3x² + 6y² + 4z² and x² + 5y² + 6z²,
Step 1:
    Write the given trinomials as,
                       = 3x² + 6y² + 4z² + x² + 5y² + 6z²
Step 2:
    Then combine the terms like as,
                       = 3x² + x² + 6y² +5y² + 4z² + 6z².
Step 3:
    Then add the given trinomials, and we get the answer,
            Answer = 4x² + 11y² + 10z².

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4. Add trinomials for given function is x + y + 4z and 6x + 4y +2z.
    Solution:
    Given functions is,
    x + y + 4z and 6x + 4y +2z,
Step 1:
    Write the given trinomials as,
                        = x + y + 4z + 6x + 4y +2z
Step 2:
    Then combine the terms like as,
                        = x + 6x + y + 4y + 4z + 2z
Step 3:
    Then add the given trinomials, and we get the answer,
             Answer = 7x + 5y + 6z.

Sine Cosine Cotangent

Sine:

          A trigonometry functions of an angle. The sine of an angle theta shortened as sin theta In a right angled triangle is the ratio of the side opposite angle to the hypotenuse. This definition applies only of angles between 0 to 90 (0 and `pi/2 ` radians).

Sin `theta` = Opposite / hypotenuse  = `(BC)/(AC)`

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Cosine:

A trigonometry function of an angle.The cosine of an angle `theta` abbreviated as cos `theta` In a right angled triangle is the ratio of the side adjacent to the hypotenuse.

Cos `theta` = Adjacent / hypotenuse = `(BC)/(AC)`

cotangent:

A trigonometry function of an angle.The cotangent of an angle `theta` ( cot  `theta` ) in a right angled triangle is the ratio of the side adjacent to it to the opposite side.
Cot `theta` = adjacent / opposite = `(BC)/(AB)`

Example problem for sin

Find the measure of the length of other sides and also find the sin function values for the given right angle triangle.

we want to find the length of side c, the hypotenuse.

Here, we know that side a has a length of 8 and side b has a length of 6.
To find the length of side c, we can use the Pythagorean Theorem which says that c²=a²+b², or
Substitute in that a=8 and b=6, we find that:

c = √ (( 8²) + (6²))

  = √ (64 + 36)

  = √ 100

c = 10 m

So the value of x is found as x = 10 m
Now we have to find the value of `theta` . we can use the sin function to find the value of `theta`
Sin `theta`     = Opposite / hypotenuse  
          = 6/10
          = 0.6
Sin `theta`  = 0.6
  `theta` = sin^-1 (0.6)
`theta`  = 37<sup>o


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Example problem for cosine, cotangent:

Find the cosine and cotangent function of the given right angled triangle.

Solution:
Here we have to find the cosine and cotangent of the given right angled triangle
Cosine`theta` = Adjacent / hypotenuse
               =  4 /  5

= 0.8

cos `theta`  = 0.8
`theta` = cos^-1 (0.8)

      = 36^o
Cotangent `theta` = adjacent / opposite =  4/3

= 1.33

cot`theta`  = 1.33
`theta` = cot^-1 (1.33)