Ordinal Variable Example

Introduction ordinal variable example:

          Ordinal variable is a variable which is mainly used to give order or to rank the particular data. There can be no any arithmetic operation performed on an ordinal variable like a normal mathematical variables. The important use of Ordinal variables is comparison of data to rank them according to its performance. The ranking given using ordinal variables is also known as level of data.

Ordinal Variable Example Explanation:

      Ordinal variable is sometimes known as discrete data. Our ranking given to data by ordinal variable must be in meaningful way. For example,

• Agree / disagree
• Maximum/ minimum

 

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          Ordinal variable is plays a vital role in surveying and we can do logical operations on ordinal variables to compare the data. 

Ordinal Variable Example:

 Example 1:

We can give the ranking order for the ordinal variable as follows,

1 = Excellent

2 = Good

3 = Satisfactory

4 = Low

5 = Poor

By the above exposed values, the data are analyzed.

For example, we are taking the labor’s performance on the work wage. This can be given as; if the work of the labor is above 10 points means then the ranking will be 1. If the work of the labor is above 8 points means, then the ranking will be 2. And If work level is above 5 means the rank will be 3. Likewise below 5 means low and below 3 means very poor. This is what the ordinal variable is performed.

Representing Ordinal variable example in the table format,

1	2	3	4	5
Excellent	good	ok	low	Poor

The performance level of the labors can be calculated by using one of the five stages. This ordinal variable table will be more helpful while analyzing individual performance also

Ordinal variable Example 2:

We can also use ordinal variable in medical research by taking the survey of patients according to our requirement.

Introduction to parallel and perpendicular lines worksheet

Introduction to parallel and perpendicular lines worksheet
The equation of a straight line is an algebraic condition which is satisfied by every point on it.

The equation of a straight line is an algebraic condition which is satisfied by every point on it.

The various forms of straight lines

The equation of a straight line in various forms

Normal form of an equation is x cos a + y sin ß = p
Intercept form is x / a + y / b = 1
Symmetric form (x - x ) :cos ? = (y - y) :sin ?
General form of an equation ax + by +c = 0 (a2 + b2 ? 0 )
Parametic form x = x1 + r cos ? ; y= y1 + r sin ?

Parallel and Perpendicular Lines Worksheet : Slope

The slope of a straight line is represented in various forms from different forms of the equations

If the given vertices are (x1,y1) and (x2 ,y2 ) then the slope m is taken as y2 - y1 / x2 - x1

The slope of an equation can be represented as m = dy / dx where the derivatives are applied

If the line is in the form of a tanget equation y = mx +c then the slope will be equal to 'm'.

If the straight line makes an angle  theta with x axes then tan theta is called the slope.

If theta is acute, then tan theta is positive.

If theta is obtuse, then tan theta is negative.

If two lines are parallel, then they have same slope

If two lines are perpendicular, then m2 = -1/m1

where m2 is slope of second line

          m1 is slope of first line.

The slope concept for parallel and perpendicular lines are useful in  solving problems in worksheet.

Problems on Parallel and Perpendicular Lines Worksheet

Find the equation of line passing through ( 0,0) and parallel to   y = 3x + 2

Solution:  Given equation  is    y  = 3x +2

                slope of line  = m1  = 3

 As the lines where parallel  m1 = m2  = 3

Equation of line passing through (0,0)

is y = m2 x + c

As the line passes through origin , replace x and y values by 0,0 to find value of c

0 = 3(0) +c

c = 0

So equation of line parallel to y = 3x + 2 is   y = 3x

2) Find equation of line passing through (1,1) and perpendicular to y  = x + 3

Solution)  Given equation :  y = x + 3

           m1 =      Slope of line =  1

As lines where perpendicular , product of their slopes are equal to -1

                                               m1 *m2 = -1                   ( * represents multiplication symbol)

                                                   m2 = -1

Equation of line :   y  = m2 x + c

                           y = -x  +c

as the equation passes through (1,1) replace x and y by 1 , 1

                   1 = -1 + c

                     c = 2

Equation of required line is   y  = m2 x  + c

                                         y = -x + 2

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Introduction to angles of a pentagon

Introduction to angles of a pentagon:

              A pentagon is a closed two dimensional figure that is the union of line segments in a plane. A pentagon has the five sides and five angles.

              Pentagon can be classified as

•  Convex pentagon
• Concave pentagon

             For  a given pentagon, we can measure the following angles of Pentagon.

•  Interior angle
• Exterior angle

Interior Angles of a Pentagon:

             An angle located within a closed figure or the angle formed inside a polygon by two adjacent sides is referred to as an Interior angle.   

             For finding the interior angles of a pentagon, start with one vertex and join it to all other vertices to form three triangles inside the pentagon.

            A pentagon has 5 sides and 5 vertices and three triangles are formed by connecting the vertices, i.e., number of triangles inside a pentagon is two less than the sides of the pentagon. So, the sum of all interior angles in a polygon is given by,

                                      Sum of all interior angles = 180 ( n – 2 )

                     For a pentagon, number of sides n = 5 and the sum of all interior angles are

                                                                                    = 180 ( n – 2 )

                                                                                    = 180 ( 5 – 2 )

                                                                                    = 180 * 3

                                                                                    = 540 degrees.

                     Each interior angle of a pentagon  = 540 / 5

                                                                                   = 108 degrees.

Ex 1: Find the number of sides of pentagon if the sum of interior angles of pentagon is 540 degrees.

Sol: Step 1: Sum of interior angles of pentagon = ( n - 2) x 180°, n is number of sides.

                        540°   =  ( n - 2 ) x 180°

         Step 2:  Divide by 180° to each side.

                        540°/180° = ( n - 2 ) x 180° / 180°

                             3           =  n - 2

         Step 3: Add 2 to each side.

                            3 + 2      = n - 2 + 2

                                5         =  n

Therefore, nmber of sides for pentagon is five.

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Exterior Angles of a Pentagon:

            An exterior angle of a pentagon is an angle formed by two sides of the pentagon which normally shares a common vertex. In a vertex, two exterior angles can be formed.

           In each vertex the angle formed by extending a side is equal to 180 degree

                                       Supplementary angle  = interior angle + exterior angle

           For a n sided polygon,

                                    Sum of all exterior angles, = 180 n – 180 ( n - 2 )

                                                                                     = 180 n – 180 n + ( - 180  x  – 2 )

                                                                                     = 0 + 360

                                                                                     = 360 degrees. 

                      So,The sum of exterior angles of a pentagon = 360 degrees

                      In a pentagon, for 5 vertices 5 exterior angles can be formed.      

                       Each exterior angle of a pentagon = 360 / 5 = 72 degrees.

Factoring quadratic equation into binomials

Factoring quadratic equation into binomials:
    Remember how we factorized polynomials of the second degree? How do we factorize `x^2 + 7x + 12`?
12 = 1 x 12             1 + 12 = 13
12 = 2 x 6                2 + 6 =8

12 = 3 x 4      3 + 4 = 7

`x^2 + 7x +12` = `x^2 +(3 + 4)x + (3 x 4)`
                               = `(x+3)(x+4)`
There is another way of doing such factoring quadratic binomials. Supppose we want to find the factors of p(x) = `x^2 + 3x -10`. We know that if x - a factor of p(x), then p(a) = 0. In other words, a is a solution of the equation p(x) = 0;m that is, the equation `x^2+3x -10 =0` .
We know how to solve the quadratic equation `x^2 + 3x -10 = 0`. Using the quadratic formula

Factoring Quadratic Binomial Formula:

   The quadratic equation `ax^2 + bx + c = 0` of  factoring  formulae:

                  `x` ` =` ` (-b +-sqrt(b^2 - 4ac))/2`  

In the above problems,

                 x = -3 `+-` ` sqrt(3^2 - 4 .1 . (-10))/2`    =  (-3 `+-` `sqrt(59)` ) / 2 

                                                                             = (-3 `+-` 7)/ 2

                                                                             = 2 or -5

   Thus if we write p(x) = `x^2 + 3x -10 `, then p(2) = 0 and p(-5) = 0, so that by the Factor Theorem x-2 and x-(-5) = x + 5 are factors of p(x). By actual multiplication, we do find

                               (x - 2) (x +5) = `x^2 + 3x -10`

Examples for Factoring Quadratic Binomials:

Example 1:

Factoring the quadratic equation `6x^2-11x+3`into binomials.

Solution:

Given:

    `6x^2-11x+3`

To factor the above quadratic equation, we put the values for a, b, and c into the quadratic formula.

like this:

Calculating these expressions through, we get that

                                                  One root or solution is: 1.500
                                                  and the other root or solution is: 0.333
                                                  These answers have been rounded to 3 digits after the decimal point.

 Therefore, the required binomials are `(x - 1.500)` and `(x - 0.333)` 

Example 2:

Factoring the quadratic equation `x^2 -2x-1 =0` into binomials.

Solution:

To factor the above quadratic equation, we put the values for a, b, and c into the quadratic formula

like this:

Calculating these expressions through, we get that

One root or solution is: 2.414
and the other root or solution is: -0.414
These answers have been rounded to 3 digits after the decimal point.

Therefore the required binomials are `(x - 2.414)`  and `(x + 0.414)` .

Introduction to math answer generator

Introduction to Math Answer Generator:

            The generator in the sense, a author who solves the set of math problems and delivers the answer. Here we consider the generator produces algebra problems with detailed answer. Algebra is the main branch of mathematics use to find out the unknown variables with the help of known variables. In math, Algebra gives the basic ideas to solve the expressions, equations etc. In algebra the answer generator delivers the stepwise solution to problems.

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Examples for Math Answer Generator:

Ex 1: Simplify the math expression 3(a -4) + 2b - 3(a -b -5) + 4
Sol :   Given the algebraic expression

3(a -4) + 2b - 3(a -b -5) + 4
Multiply factors.

= 3a - 12 + 2b -3a + 3b + 15 + 4
Grouping the terms

= 5b + 7
Ex 2:   Find the distance between the points below
 (1,3) and (7,11).
Sol :  The distance d between points (-1, -3) and (-7, -11) is given by

d = sqrt [(1 - 7) 2 + (3 - 11) 2]
Simplify.

d = sqrt (36 + 64) = 10
Ex 3 : Find the x intercept of the graph of the equation.
                      3x - 5y = 6
Sol :   Given the equation

3x - 5y = 9
To find the x intercept we set y = 0 and solve for x.

3x - 0 = 9
Solve for x.

x = 9 / 3 =3
The x intercept is at the point (3, 0).
Ex 4 : Determine the slope of the line
         8x - 8y = 16
Sol :  Given the line

8x - 8y = 16
Write the above equation in slope intercept form y = mx + b and identify the value of m the slope.

-8y = -8x + 16

y = x - 2
The slope is given by the coefficient of x, which is 1.
         Slope = 1
       
Practice Problems for Math Generator:

1) Find the x intercept of the graph of the equation.
             6x - 8y = 3
Ans : The x intercept is at the point (0.5, 0).

2) Evaluate f (4) - f (3)
            f(x) = 6x + 4
Ans :   6

LEARNING ABOUT DIFFERENCE QUOTIENTS

The Difference Quotient is an important topic of study under Calculus. It is defined as the function difference divided by the point difference.
It is given by the formula :

Delta F(P) / Delta P
= F(P+ Delta P)- F(P) /Delta P
= Delta F (P +Delta P) / Delta P

Where, Delta P is the difference between 2 points.
Difference quotient is also often known as Newton's Quotient.

Students need to learn the difference between Difference Quotient Derivative and as a Divided Difference.

Students can also learn about Higher order Difference quotients like the:

• Second order
• Third order
• Nth order

SOLVING EQUATIONS IN ALGEBRA

Algebra is a branch of Math which deals with study of rules of operations and concepts related to them like variables, equations, expressions etc,. Equations are an important topic of study under algebra.
Equations are values or expressions separated by an ' = ' (equal to) sign. In Algebra, equations consist of variables. Equations are of different types based upon the exponential value to the 'degree' to which they are raised to. They are :

• Linear Equations
• Differential Equations
• Quadratic Equations
• Radical Equations etc,.

Students can learn to solve simpler Linear Equations first. They are called 2 Step equations. They can first learn to Solve 2 step equations. These, as the name suggests, can be solved in 2 simple steps.

• First step would involve grouping of Variables and constants
• Second would be determining the value of the variable.

Students can also learn solving more complex equations like Solve 5x equations are involved. There might be 2 algebraic expressions forming a second or third degree equation to be solved.