Distance of a Point from a Plane

Let us see about distance of a point from a plane,
In mathematics, a plane is placed at distance from a point of any horizontal, two-dimensional smooth surface. A plane is the two dimensional have a distance of a point that means zero-dimensions, a line that is known as one-dimension and a space  is known as three-dimensions. A lot of mathematics may be performed in the surface of plane, particularly in the areas of geometry, trigonometry, graph theory and graphing.

Properties of plane:

• Two planes may be parallel or both intersect in a line.
• A line may be parallel to a plane, intersects in point.
• Two lines at a 90 degree angle to the same plane should be parallel to each other.

• Two planes at a 90 degree angle to the similar line should be parallel to each other.

Point - Plane distance:

Let us see about distance of point from plane:
Given a plane,


and a point X0 = (xo,y0,z0),the normal to the plane is given by,

and a vector from the plane to the point is given by,

Projecting W onto V gives the distance D from the point to the plane as,

This equation is known as the distance of a point from a plane.

Examples:

1)      Find the distance from the point P = (2, 2, 4) for the plane 2x +2y + 3z + 4 = 0.
Solution:
We use formula from the distance of a point from a plane.
From the above equation we substitute for the plane A = 2, B = 2, C = 3, D = 4. From the point P, we substitute x₁ = 2, y₁ = 2, and z₁ = 4.
Plane's distance of a point is,


2) Find the distance from the point P = (2, 3, 5) for the plane x - y + z + 5 = 0.
Solution:
We use formula from the distance of a point from a plane.
From the above equation we substitute for the plane A = 1, B =- 1, C = 1, D = 5. From the point P, we substitute x₁ = 2, y₁ = 3, and z₁ = 5.
Plane's distance of a point is,

Frequency Distribution Histograms

A frequency distribution can be defined as the number of observations falling into each of several ranges of values. It can be represented as frequency tables, histograms. Either the percentage of observations or the actual number of observations falling in each range can be represented by Frequency Distribution. If the Frequency distribution shows the percentage of observation then it is said to be Relative frequency distribution.

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In statistics Frequency distribution plays a huge role and it can be represented either in a tabular or graphical format and it displays the number of observation within a given interval.

A frequency distribution can be represented by a histogram or pie chart. In case of large data sets, the stepped graph is approximated by the smooth curve of a distribution function which is known as a density function.

Frequency distribution mostly used for assigning probabilities and summarizing large data sets.
It is a method of representing unorganized data e.g. to show results of an income of people for a certain region, election, sales of a good within a certain period, etc.

Frequency distribution in histograms

Histograms:
The histogram is nothing but a summary graph representing a count of the data’s falling in various ranges. The groups of data are known as classes; here in histogram they are called as classes.
       
In mathematical sense, generally a histogram can be defined as a function X_i that counts the number of observations or data’s that fall into each of the bins. The graph of a histogram is similar to the one way to represent a histogram.

If,
n = number of observations and
k = total number of classes then, the histogram meets the following criteria.
                n= `sum_(i=1)^k` X_i

Histograms can be defined as follows,

• In statistics, it is a pictorial representation of table frequencies, shown as bars. The diagram shows what proportion of cases fall into each of several categories. Simply Histogram is a form of data binning.
• Histogram is a bar chart representing the frequency distribution of values along a spectrum of possible values.
• Histograms are mainly used for density estimation.

Sample Problems

Univariate frequency tables:
The lists ordered by quantity which shows the number of times each value appears is known as Univariate frequency distributions.

Example 1:
If 200 people rate a product assessing their agreement with a statement on the product on which ‘A’ indicates strong agreement and ‘E’ strong disagreement, the frequency distribution of their responses can be given as follows:

Number	Degree of agreement	Rank
60	Strongly agree	A
40	Agree somewhat	B
30	Not sure	C
40	Disagree some what	D
30	Strongly Disagree	E

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Example 2:
The weight of the students in a class could be represented into the following frequency table:

Weight range (Kg's)	Number of students	Cumulative number
40 -50	25	25
50-55	35	60
55-60	15	75
60-65	24	99
65-70	10	109

 

How Draw an Ellipse

In geometry, an ellipse (from Greek ellipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant. (Source : Wikipedia)

Draw an ellipse using string and 2 pins

Constructions – instruments defined as Euclidean structure is not a correct and ellipse is used to draw the ways of laws, significant is not a accuracy of the arithmetical, but if you are careful, you can closed to fairly. It is a little bits called the "Gardener's Ellipse", because scale of great is well effort to using a cable and stakes, to lay out elliptical flower beds in proper gardens.

How to draw ellipses

Consider an ellipse, in two positions, each one called a focus. The position F1 and F2 are shown in diagrams. The calculation of the distance to the centre point is stable, If you get any position on the ellipse. Around the ellipse position is drag and observes point in every distance to the focus of the different point and their calculation is stable. The sum of the double distances is resolute by the size of the ellipse. The major axis of the duration is same to the sum of these distances.

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Properties of an ellipse: How to draw an ellipse

Centre:
The ellipse of the point which is medium of the row segment between the two foci of the axes. The connecting of the major and minor axes.


Major / minor axis:
The diameters of the ellipse are greatest and shortest. Both the measure of the major and minor axis of the generator lines is to be same.



Semi-major / semi:
The distance from the middle to the extreme and nearby position on the minor axis of ellipse.


Foci /Focus points:
The two points of the position is defining the ellipse.

Real Size Ruler

Let us know about real size ruler.Ruler is an instrument, using this instrument we can draw the line, those lines are straight and also very neat, size of the ruler depends on the separation (they are separated in cm and inches).In real life we are using the ruler in many places.Type s of rulers are

• Desk ruler
• Practical ruler.

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Desk ruler-real size ruler:

Desk ruler is used for three purposes and size of the ruler is12 inches or 30 cm

• Drawing
• The diagram should be straight and
• Cutting the things into straight line

Practical ruler –real size ruler:
Practical  ruler is used for measuring the distance,  all the carpenter use this kind of ruler The lenght of the ruler is 5 meter or 2 meter(metal tape measure)
Real size ruler diagram:
The diagram of the ruler is shown below, (separation in both inches and centimeter)
 

Using the ruler we can draw the line and also measure the line

Example problem-real size ruler

Example 1:
Draw the 5cm line using the real size desk ruler?
Solution:
The above diagram shows the how the desk ruler look like, we can construct desk ruler in both the plastic and wood. Using this ruler we can draw the line with 5cm



Mark 0(starting point to the ruler) and mark 5 on ruler (ending point),then draw the straight line .we get line as 5 cm length.

 Example 2:
Draw 11cm using the real size desk ruler?
Solution:
Desk ruler has the value from 0 to 20 , using the desk ruler we can draw a straight line ,mark 0 on the ruler (we call as starting point ) and mark 11 on the ruler (we call as ending point ) then connect those two points we get the straight line with the measure of 11cm.

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Rulers in different units

Rulers come in different units like:

• Fractional Inch
• Decimal Inch
• Metric

Metric rulers are fairly easy to read. They deal with centimeters and millimeters only.

Steps:

1. Decide if we have a metric or an English ruler. Metric rulers have numbers every centimeter, and English have them every inch. Centimeters are much smaller than inches, so metric rulers will have a lot more numbers printed on them. If we still have trouble telling what type of ruler you have, centimeters are normally about as wide as one fingertip, and inches are about as wide as two.
2. Read an English ruler using fractions of an inch. The distance between any two large numbered lines is 1 inch. The large unnumbered line that is halfway between them is 1/2 inch. The smaller (but still prominent) line between the 1/2 mark and the numbered inch line is 1/4 inch. The tiny little lines between all of the more prominent lines are 1/16 inch.
3. Observe the much simpler metric rulers. The distance between any two large numbered lines is 1 cm. The prominent line between any two numbered lines is 1/2 cm. The small lines between the 1/2 mark and the numbered centimeter mark are 1/10 cm, otherwise known as a millimeter.
4. Record distances by the name of the line that it most closely matches. If the length of an object goes to one mark past the halfway mark on your ruler then it will be 9/16 inch on an English ruler or 6/10 cm (or 6 mm) on a metric ruler.

Eccentricity of 1

Conic sections:
Conic sections are formed when a right cone is intercepted by a plane.The shape so formed depends on the angle at which the conic is cut.
Consider the below figure of right cone intercepted by plane and we can see the different shapes formed.



Conic sections:
Conic sections are formed when a right cone is intercepted by a plane.The shape so formed depends on the angle at which the conic is cut.
Consider the below figure of right cone intercepted by plane and we can see the different shapes formed.



The first picture formed is known as parabola
Second one is ellipse
The third is hyperbola.

explanation to eccentricity of 1

 Eccentricity of 1  :-
Eccentricity is the measure of the deviation of a conic from a circular path .The parabola is the conic haing eccentricity as 1.

Definition of parabola :-
Parabola is locus of points whose distance from fixed line, called directrix, and a from a fixed point ,called focus,is equal. Vertex is a point where parabola changes its direction .The distance from focus to vertex and vertex to directrix is equal.

The main compenents of Parabola are :-
i)Vertex
ii) axis
iii) focus
iv) directrix
Consider the parabola given below

Description  of paraboala :-
i)Point F is called the focus of parabola
ii)Point "O" is called as the vertex of parabola.
iii)C ,D are the points on the parabola whoce distance fron focus and directrix are equal .
iv) Line AB is called directric fo parabola
v) Line ox is called the axis of parabola.

i) Y² = 4aX                                                                      

This parabola opens towards right side ie + ve x axis having  vertex at  (0,0). and focal distance "a" and focal point (a,0) . This curve transforms to (Y-k)² = 4a(X-h) when vertex shifted to a point  (h,k). with focus shifting to (a+h ,k)

ii) Y²= -4aX

This parabola opens towards  left side ie -ve  x axis having  vertex at  (0,0). and focal distance "a" and focal point (-a,0).This curve transforms to (Y-k)² = - 4a(X-h) when vertex shifted to a point  (h,k) and focus shifts to (h-a ,k)

iii) X²= 4aY
This parabola opens towards  top  side ie +ve  y axis having  vertex at  (0,0). and focal distance "a"and focal point (0,a).This curve transforms to (X-h)² = 4a(Y-k) when vertex shifted to a point  (h,k) and focus shifts to (h,k+a).

iv) X²=-4aY
This parabola opens towards down side ie -ve  y axis having  vertex at  (0,0). and focal distance "a" and focal point (0,-a).This curve transforms to (X-h)² = - 4a(Y-k) when vertex shifted to a point  (h,k-a).

examples to eccentricity of 1

Ex 1:
Given the equation of parabola Y² = 8X find the focus of parabola
solution:
Comparing with standard form of equation
Y² = 4aX
we get  4a =8
=> a=2
so focus of parabola is (a,0) = (2,0)

Ex 2:
find focus of parabola   y =  1/4 x²
solution :
y =  1/4 x² 
=> x²= 4y
=> a= 1 on comparing with standard form
so focus of parabola is (0,1) as axis of symmetry is +ve y axis

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Ex 3:
Given a parabolic equation Y²= -16X-32
Find i)  Vertex of parabola
ii) Focus of parabola

Solution:
Given Y²= -16X-32
=>  Y²= -16(X+2)
=>  Comparing with standard equation (Y-k)² = - 4a(X-h)
(h, k) =(-2,0)
a=4
i) vertex of parabola is (h,k) =(-2,0)
ii)focus of parabola is (-6,0) as parabola is opening towards left side.

Draw isosceles Triangle

The triangle is a closed geometric shape that contains three sides. There are several types of triangles in the geometry world. We can divide the triangles according to their angles and sides. We can define the isosceles triangle by using its sides.

      The triangle that has two equal or congruent sides in its measure is called as isosceles triangle. We can say that the equilateral triangle is also as an isosceles triangle,since the equilateral triangle has three equal sides . while we draw a isoscles triangle we have to see that two sides triangle must be equal.

draw Isosceles Triangle:

• when we draw a isoceles triangle, angles are congruent i.e., an angles of each sides are equal.
• when we draw a isoceles triangle diagonals of an isosceles triangle must be congruent.
• The measurement of the adjacent angles is giving the 180 degree. x = y = 180^o

Formulas for Isosceles triangle:

• Hight,  h = √(b² - `(1/4)` a²)
• Perimeter of Isosceles Triangle = A + B + C
• Area of isosceles triangle A = (b* h)/2

Example problems:

1) Find the area of isosceles triangle with the base and height are 5 cm and 7 cm.

solution:

We can find the area of given problem using the following formula:

Area A= (b*h)/2

Substitute the values of b and h into the above formula. Then we get,

         A= (5*7)/2

 Here multiplying to the values of 5 and 6 then dividing by 2.

  Then we get the final solution.   

 Answer A=17.5 cm²

2) Find the perimeter of Isosceles triangle that has side, Side 1 =10 cm, Side 2 = 10, Side 3 = 7 cm.

Solution:

   Given, Side 1 =10 cm, Side 2 = 10, Side 3 = 7 cm.

  Perimeter of Isosceles triangle   = Side 1+ Side 2+ Side 3

                                                         = 10 + 10 + 7

  Perimeter of Isosceles triangle  = 27 cm

3) Find the height of the isosceles triangle of the base a = 7 cm and equal sides b = 11 cm.

Solution:

The height of the isosceles triangle is given by the formula:

             h = √(b² - `(1/4)` a²)

substitute the a = 7cm and b = 11 cm. in the formula,

             h = √(11² - (1/4)*7² )

             h = √(121 - `(49/4)` )

             h = √(121 - 12.25 )

             h = √(108.75) = 10.428 ≈ 10.4 cm

Height of the isosceles triangle = 10.4 cm

Straight Triangle

In geometry, a triangle is the fundamental shapes. The polygon with 3 corners and 3 sides are the line segments. The corner otherwise called as the vertices. The sides otherwise called as the edges. A triangle with corners A, B and C is represented by ∆ABC.Here we will see about the triangle types and formulas.

Classification of Straight Triangles

The straight triangle has three types. The names are as follows,

• Equilateral Triangle
• Isosceles Triangle
• Scalene triangle

Equilateral triangle
     Here all three sides are the equal length. An equilateral triangle is also a normal polygon with every angles calculate 60 degree. The equilateral triangle is as follows,
               
Isosceles triangle
     These triangle two sides only the equal length. The figure is as follows,
                 
Scalene triangle
Here all sides are the different type. The three angels are also various in measures. This triangle figure is as follows,

           

Classified Based by the Internal Angels

• A triangle does not have an angle that calculates 90 degree is known as the oblique triangles.
• It has all interior angles calculating less than 90 is known as the acute triangle or acute-angled triangle.
• A straight triangle has one angle that calculates more than 90 degree is known as obtuse triangle or obtuse-angled triangle.

Formula
Triangle area is A=(1/2).b.h
Base is denoted by b and height is denoted by h

Formula for angles
     Sin(q)=opposite/Hypotenuse
     Cos(q)=Adjacent/Hypotenuse
     Tan(q)=Opposite/Adjacent

Properties

• Angle sum property
• Exterior angle property
• Triangle Inequality property
• Pythagoras theorem

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Examples

1)Find the area of straight triangle with the base 5 cm and the height 7cm.
Solution
     Given b=5 cm and h=7 cm.
     Formula A=(1/2).b.h
                    =(1/2).5 cm.7cm
                    =(1/2).35cm²
                    =17.5cm²

2)what is the obtuse triangle with the base 4 inches and the height 6 inches.
Solution
     Given b=4 in and height h=6 in
     Formula A=(1/2).b.h
                    =(1/2).(4 in).(6 in)
                    =(1/2).24 in²
                    =12in²

Solve Regular Polygons

If all sides of the polygon is same length in all of its sides and all of its angles are equal, then the polygon is known as regular polygon.  The regular polygon cannot be a concave polygon. Most regular polygons are convex or star. We will see about Solve regular polygons in this article.


Regular polygon examples:
Few examples of Regular polygons are,
                      Square                                 Equilateral triangle                  Pentagon                                    
               

                        Hexagon

                    

To solve regular polygons formulas:

     The following formulas are used to solve regular polygons,

• Interior angle of each side = `(180(n-2))/n ` degrees 

• Exterior angle of each side = `360/n ` degrees 

• Diagonal=` (n(n-3))/2` 

• Area = n x area of triangle 

      = ½ * (apothem * perimeter)  (Or)

• Area  = ½ *( n* s*r) ( or )

    A= `(s2 n)/ (4tan (pi/n))`
    And there is lot of formulas for area.
 Where, n = number of sides, s = side length, r = radius or apothem

Solve regular polygons Example problems:

Example 1:
The regular hexagon has the apothem 7 cm and side is 5 cm. Calculate the area.
Solution:
Now, let us solve regular polygons using the first formula (above mentioned)
By the formula,
Area = (½) * (apothem) *(perimeter)

Perimeter of hexagon = Length of the side * Number of side
                                    = 5 * 6
                                    = 30cm
   Area of the hexagon = (1/2) * 7 * 30
                                     = 105cm²
Example 2:
The octagon has the apothem of 9 cm and the side length is 6 cm. Find its area.
Solution:
Given, n=8, s=6,  r = 9
    Let us solve regular polygons using the second formula of area (above mentioned)
           By the formula
                                    Area = (½) * n * s * r
                                            = (1/2) * 8 * 6 *9
                                            = 216cm²
Example 3:
A regular pentagon has the side of 5 inches. Determine its area.
Solution:
Given,
N= 5(pentagon), s= 5 inches
We can use the third formula now,
                                      Area= `(s^2 n)/ (4tan (pi/n))`
                                       = ` (5^2 * 5) / ( 4tan(pi/5))`

                                      = `125/ 2.906`
                                     =  43.01inches²   


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Example 4:
 An equilateral triangle has the side of 5inches length. Find its area and perimeter.
Solution:
Length = a= 5 inches
By formula,
     Area of equilateral triangle =   `sqrt(3)/4` a²  
                                                     = `sqrt(3)/4` * 25
                                                     = 10.825 sq.inches
    Perimeter of the triangle= a+b+c
                                               = 5+5+5

                                               = 15 inches.

Surface Normals

Let us see about surface normals. A surface normal also called the simple normal, to a flat surface is a vector that is vertical to that surface.A line normal to a flat, the normal factor of the force, the normal vector, etc. The concept of routine generalizes to orthogonality.A normal is used for computer graphics.

Calculating a surface normals

For a convex polygon such as a triangle, a surface usual will be calculated as the vector cross product of two (non-parallel) edges of the polygon.
For a plane given by the equation ax + by + cz + d = 0.
For a plane is represent the following equation r = a + αb + βc.
where a is a vector to get onto the level surface and b and c are non-parallel vectors lying on the plane, the normal to the plane defined is given by b × c .
For a hyperplane in n+1 dimensions equation given from following equation r = a₀ + α₁a₁ + α₂a₂ + ... + α_na_n,
where a₀ is a vector to get onto the hyperplane and a_i for i = 1, ... , n are non-parallel vectors two-faced on the hyperplane, the normal to the hyperplane can be approximated by (AA^T + bb^T) ^{− 1}b where A = [a₁, a₂, ... , a_n] and b is a random vector in the space not in the linear span of a_i.

Cross product partial derivation


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In surface normal,a surface S is given completely   as the set of points (x,y,z) it satisfying F(x,y,z) = 0, so the normal point (x,y,z) on the plane.

For a surface S given clearly as a function f(x,y) of the independent variables x,y (e.g., f(x,y) = a₀₀ + a₀₁y + a₁₀x + a₁₁xy).

The normals can be identified in at least two equivalent ways.

The first getting its contained from F(x,y,z) = z − f(x,y) = 0, from the normal follows readily.

Angle Sum Theorem

In a triangle the sum of all the three interior angles will be equal to 180^o.  If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles. Here we are going to see about the angle sum theorem.

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Angle sum theorem:

Angle sum theorem of a triangle is equal to two right angles, i.e., 180 degrees
Given:
ABC is a triangle
To Prove
Angle A + Angle B + Angle ACB = 180^o
Produce BC to D. Through C draw CE || BA.

Proof of angle sum theorem of a Triangle:

Statement	Reason
1. Angle A = Angle ACE	Alternate angles angles BA is parallel to CE
2. Angle B = Angle ECD	Corresponding angles BA is parallel to CE
3. Angle A + angle B = Angle ACE + Angle ECD	statements (1) and (2)
4. Angle A + angle B  = Angle ACD	statement (3)
5. Angle A + Angle B + Angle ACB = Angle ACD + Angle   ACB	adding Angle ACB to both sides
6. But Angle ACD + Angle ACB = 180o	linear pair
7. Angle A + Angle B + Angle ACB = 180 °	statements (5) and (6)

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Corollary of Theorem on Angle theorem of a Triangle

If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles.
Given:
In Triangle ABC, BC is produced to D.
To Prove Corollary of Sum of Angles of Triangle:
Angle ACD = Angle A + Angle B



Proof:

Statement	Reason
1. Angle ACB + Angle ACD = 180o.	linear pair
2. Angle A + Angle B + Angle ACB = 180o	sum of the angles of a triangle = 180
3.  Angle ACB + Angle ACD = Angle A + Angle B + Angle ACB	statements (1) and (2)
4. Angle ACD = Angle A + Angle B Reason	statement (3); Angle ACB is common

Hard Fraction Problems

Hard fraction problems:

This articles discusses hard fraction problems solving. A fraction is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (`1/2` , `5/8` , `3/4` etc.) and which consist of a numerator and a denominator. (Source – Wikipedia)

Hard fraction problems solve for proper fraction, improper fraction and complex fraction problems.

Example for Hard fraction problems:

Example 1:
Subtract the fractions `4/5` – `3/4`

Solution:
The denominator (bottom number) is different so we have to take least common denominator (lcd).
LCD = 5 x 4 = 20
`(4 xx 4)/ (5 xx 4)` = `16/20` and `(3 xx 5)/ (4 xx 5)` = `15/20`
`16/20 ` – `15/20`
The denominators are equals
So subtracting the numerator directly = `(16-15)/20`
Simplify the above equation we get = `1/20`
Therefore the final answer is` 1/20`

Example 2:
Subtract the mixed fractions for given fractions,` 4 7/5` – `5 8/5`

Solution:
The given two mixed fractions are `4 7/5` – `5 7/5`
We need convert to mixed fraction to improper fraction `27/5` – `33/5`
The same denominators of the two fractions, so
                                         = `27/5` – `33/5`
Subtract the numerators the 27 and 33 = 27 - 33 = - 6.
The same denominator is 5.
                                         = `-6/5`
The subtract fraction solution is -`6/5` .

Example 3:
Multiply the mixed fractions for given two fraction,` 4 2/4` x `5 2/6`

Solution:
The given two mixed fractions are `4 2/4` x `5 2/6`
We need convert to mixed fraction to improper fraction `18/4` x `32/6`
Multiply the numerators the 18 and 32 = 18 x 32 = 576.
Multiply the denominators the 4 and 6 = 4 x 6 = 24
                                         =` 576/24`
The multiply fraction solution is 24

Example 4:
Convert `16/ (5/4)` to a simple fraction and reduce.

Solution:
The given complex fraction `16/ (5/4)`
Can be written as `16/1 -: 5/4`
First we have to take the reciprocal of the 2^nd number, and then multiply with the second one
Reciprocal of `5/4 ` is `4/5`
`16/1` x `4/5`
Multiply the numerator and denominator
`(16 xx 4) / (1 xx 5)` = `64/5`
Therefore complex fraction solution is `64/5`

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Practice for hard fraction problems:

Problem 1: Convert `5/ (5/4)` to a simple fraction and reduce.
Solution: 4.
Problem 2: Adding the mixed fractions for given two fraction,` 4 2/3` + `5 2/3`
Solution: `31/3`
Problem 3: Subtract the mixed fractions for given fractions,` 4 7/3` – `5 8/3`

Solution:`-4/3`

Written Translation

Introduction:
In word problem we can use the English words for translating words into mathematical expressions. In the translation we can use the math equation. This is used in the math problem to convert in the fairly simple problems. But we can’t do the figuring problem. This translating method is help to solve the word problems.

1. Working clearly will help you think clearly, and
2. Figuring out what you require will help you convert your final answer back into English.

Written translation - Translation methods:

• Addition
• Subtraction
• Multiplication
• Division
• Power

Written translation - Addition
Words that mean adding

words that mean adding	example
the sum of	the sum of a number and 6	y + 6
more than	6 more than a number	y + 6
plus	a number plus two	y + 2
added to	6 added to a number	y + 6
increased by	a number increased by 3	y + 3

Written translation - Subtraction

Words that mean subtracting

words that mean subtracting	example
less than	6 less than a number	y - 6
less	6 less a number	6 - y
minus	6 minus a number	6 - y
subtract from	subtract 6 from a number	y - 6
reduced by	a number reduced by 6	y - 6
decreased by	a number decreased by 6	y - 6
diminished by	6 diminished by a number	6 - y

Written translation - Multiplication

Words that mean multiplying

Words that mean multiplying	example
multiply	multiply a number by 3	3y
multiplied by	a number multiplied by 3	3y
times	4 times a number	4y
double	double a number	2y
twice	twice a number	2y
triple	triple a number	3y

Written translation - Division

Words that mean dividing

Words that mean dividing	example
divided by	a number divided by 6	y/6
divide into	divide 6 into a number	y/6
the quotient of	the quotient of a number and 3	y/3
half of	half of a number	y/2
one third of	one third of a number	y/3

Power

Word that mean rising to the n^th power

words that mean raising to the nth power	example
a number raised to the nth power	a number raised to the fifth power	y6
raise a number to the nth power	raise a number to the fifth power	y6
the nth power of a number	the fifth power of a number	y6
squared	a number squared	y2
cubed	a number cubed	y3

Binomial Theory

In algebra, a binomial theory is nothing but the study of binomial. The binomial is defined as the polynomial with sum of two monomial terms. Some of the binomial term is given as,The binomial also consist of distributions, coefficient, variables etc.some of the binomial terms are given as,

3x + 1 = 0

4x² + 5 = 0

9x³ + 15 = 0

The difference of binomial a2 − b2 is the product of two other binomials:

a2 − b2 = (a + b) (a − b).



Properties of binomial:

A sum of binomial coefficients of an exponent (a + b) n is equal to 2 n.

(1 + 1) n = 2 n

A sum of binomial coefficients of even term is equal to a sum of binomial coefficients of odd terms, and it is equal to

2n-1

The product of a pair of linear binomials (ax + b) and (cx + d) is:

(ax + b)(cx + d) = acx2 + adx + bcx + bd.

Examples

Example1: find the value of 5²- 3² and prove it?

Solution:

The difference of binomial a2 − b2 is the product of two other binomials:

a2 − b2 = (a + b) (a − b).

52 − 32 = (5 + 3) (5 − 3).

= (8)(2)

= 16.      ---------- (1)

Proof:

a2 − b2 = 52 − 32

= 25 – 9

= 16       ---------- (2)

From 1 and 2 it proved.

Example 2: find the value of (3x + 1) and (2x + 3)?

Solution:

The product of a pair of linear binomials (ax + b) and (cx + d) is:

(ax + b)(cx + d) = acx2 + adx + bcx + bd.

(3x + 1)(2x + 3) = 3*2x2 + 3*3x + 1*2x + 1*3.

= 6x2 + 9x + 2x + 3.

Therefore, (3x + 1) (2x + 3) = 6x2 + 9x + 2x + 3

Example 3: Prove that (a + b) ³ = 8?

Solution:

From the property of binomial,

A sum of binomial coefficients of an exponent (a + b) n is equal to 2 n.

(1 + 1) n = 2 n

(a + b) ³ = (1 + 1) 3 = 2 3

= 8

Hence it is proved.

Algebra is widely used in day to day activities watch out for my forthcoming posts on taylor series cos and cbse previous year question papers class 12. I am sure they will be helpful.

Practice problem:

Problem1: find the value of 9²- 4² and prove it?

Answer is 65

Problem2: find the value of (2x + 1) and (x + 3)?

Answer is 2x2 + 6x + 1x + 3.

Solving Sums Integrals

Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In this case it is called an indefinite integral. Integral can be classified as definite and indefinite integral. (Source: Wikipedia)

I like to share this Table of Derivatives and Integrals with you all through my article.

Examples problems for solving sums integrals

Solving sums integral problem 1:

Integrate the given function ∫ (234x2 + 132x4 - 5x) dx.

Solution:

Given ∫ (234x2 + 132x4 - 5x) dx

Integrate the given function with respect to x, we get

∫ (234x2 + 132x4 - 5x) dx = ∫ 234x2 dx + ∫ 132x4 dx - ∫ 5x dx.

= 234 (x3 / 3) + 132 (x5 / 5) - 5 (x2 / 2) + c.

= 78x3 + `(132 / 5)` x5 - `(5 / 2)` x2 + c.

Answer:

The final solution is  78x3 + `(132 / 5)` x5 - `(5 / 2)` x2 + c.

Solving sums integral problem 2`:`

Find the value of the integration

`int_2^5(x^6)dx`

Solution:

Integrate the given function with respect to x, we get

`int_2^5(x^6)dx`  = `(x^7 / 7)`52

Substitute the lower and upper limits, we get

= `((5^7 / 7) - (2^7/ 7))`

= `((78125 / 7) - (128 / 7))`

= `(77997 / 7)`

Answer:

The final answer is `(77997 / 7)`

Solving sums integral problem 3:

Integration using algebraic rational function ∫ 7dx / (17x + 37)

Solution:

Using integrable function method,

Given function is ∫ `(7dx) / (17x + 37)`

Formula:

∫ [L / (ax + c)] dx = (L / a) log (ax + c)

From given, L = 7, a = 17, and c = 37

Integrate the given equation with respect to x, we get

= `(7 / 17)` log (17x + 37)

Answer:

The final answer is `(7 / 17)` log (17x + 37).

Practice problems for solving sums integrals

Solving sums integral problem 1:

Integrate the given function using integrable function ∫ `(14 / (11x + 12))` dx

Answer:

The final answer is `(14 / 11)` log (11x +12)

Solving sums integral problem 2:

Integrate the given function using integrable function ∫ `(15 / (21x + 92))` dx

Answer:

The final answer is `(15 / 21)` log (21x + 92)

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Solving sums integral problem 3:

Integrate the given function ∫ (7.9x2 - 12.9x) dx

Answer:

The final answer is `(7.9 / 3)` x3 - `(12.9 / 2)` x2

Difference of Two Means

Mean is defined as the average for the total number of values given in the data set. It is the sum made between the given data set and it is divided it by the total number of values given in the data set. Tis is called as the mean for the given data set. This can also be called as arithametic mean or sample mean. The difference between two mean is nothing but the taking difference for the two sets of data (mu_1 , mu_2 ) and calculating the difference for that mean.

Difference of two mean  = mu_1 - mu_2

Where as

mu_1 is the mean value for the first set of data.
mu_2 is the mean value for the second set of data.

Mean is calculated by the way,
mu = (Sum of all the values given) / (Total Number of values)


Steps for calculating the difference of two means:

Get the two sets of data for calculating the mean.
Measure the sum for the first set of data and divide it by the total number of data's given. This is the mean for the first data set
Measure the sum for the second set of data and divide it by the total number of data's given. This is the mean for the second data set.
Now measure the difference of the two means from the mean calculated.



Difference of two means - Example Problems:

Difference of two means - Problem 1:

Find the difference of two means from the given two data set. 2, 3, 4, 5, 6, 7 and 1, 2, 3, 4, 5, 6

Solution:

Mean for the first set of the data given

mu_1 = (2+3+4+5+6+7) / 6

= 27 / 6

=  4.5

Mean for the second set of the data given

mu_2 = (1+2+3+4+5+6) / 6

=  21 / 6

mu_2    = 3.5

Difference of the two means is given by

Difference of two mean = mu_1 - mu_2

= 4.5 - 3.5

Difference of two mean = 1

Difference of two means - Problem 2:

Find the difference of two means from the given two data set. 24, 23, 24, 25, 26, 27 and 11, 13, 13, 14, 15, 16

Solution:

Mean for the first set of the data given

mu_1 = (24+23+24+25+26+27) / 6

= 149 / 6

=  24.8333333

Mean for the second set of the data given

mu_2 = (11+13+13+14+15+16) / 6

=  82 / 6

mu_2    = 13.6666667

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Difference of the two means is given by

Difference of two mean = mu_1 - mu_2

= 24.8333333 - 13.6666667

Difference of two mean  = 11.1666666

Is a Cube a Polygon

Cube is not a polygon,because cube is a three dimensional shaped figure .But polygon is a two dimensional object. Generally polygon must be flat, plane figure and it’s made up of line segment. Here we are going to study about cube and polygon shape and its example problems.


Shape of cube:

Cube is a regular solid three-dimensional figure it has six square faces .it has 12 edges of equal length and 8 vertices it is otherwise called as regular hexahedron.

Shape of polygon:

It will be triangle, quadrilateral, pentagon, hexagon, octagon, heptagon…etc. All are two dimensional shapes.

Example problems for cube:

Example: 1

Find the volume of the cube with side length 6 meter.

Solution:

We know that volume of the cube is a 3

Given a = 6

Therefore a3 = (6)3

= 6 * 6 * 6

= 216 meter cube

Example: 2

Find the surface area of the cube each side length of a cube is 16 feet.

Solution:

We know that surface area of cube is,

A = 6a2

Here the given  a = 16 feet

Substitute the a value in the above formula we get

A= 6*162

162 = 16*16 = 256

Therefore the area of the cube is

= 6*256

=1536 feet square

The surface area of the given cube is 1536 feet square.

Example problems for polygon:

Example: 1

Find the perimeter of the polygon which is hexagon shape with side length is 9 feet

Solution:

We know that perimeter of the hexagon is =6*a

a represents the side length

Therefore perimeter = 6 *9

= 54

Perimeter of the given hexagon is 54 feet

My forthcoming post is on syllabus of class x cbse and tamil nadu text book will give you more understanding about Algebra.

Example: 2

Find the perimeter of the regular pentagon with side length is 8 meter

Solution:

We know that pentagon is one of a polygon and formula for finding the perimeter is 5 *a

= 5 * 8

= 40

Perimeter of the pentagon is = 40 meter

Unit of Measurement for Volume

The volume is defined as the space occupied by any object in three dimensions. There are various units for the measurement of the volume of the objects. The most commonly used is the standard international unit of measurement, the cubic meter unit. But there are other units of volume measurement. In this article we will see them in detail.


Unit of measurement for volume:

The various units of volume measurement are related with the standard international unit of volume measurement the cubic meter. With the relations the various units can also be related to one other. The various units of volume measurement and their relations are,

1 cubic meter = 1000 L = 264.2 gallons

1 cubic meter = 35.31 ft3 = 1.308 yd3

1 gallon = 0.1337 ft3 = 3.785 L

1 cubic feet = 7.481 gallons = 0.0283 m3

1 cubic yard = 27 ft3 = 202 gallons = 0.7646 m3 = 764.6 L

1 imperial barrel = 163.7 L = 6.10 m3

The above relations can be used for the conversion between the various units above by relating with each other.

Example problems on unit of measurement for volume:

1. A container volume is measured to be 15000 liters. Convert the volume into m3 and gallons.

Solution:

1 cubic meter = 1000 L

1 liter = `1/1000` m3

15000 liter = `15000*(1/1000)` m3

15000 liter = 15 m3

1000 liter = 264.2 gallons

15000 liters = 15*264.2 gallons

15000 liters = 3963 gallons

2. A water tank can store 20.5 m3 of water. Convert the volume into yd3 and gallons.

Solution:

1 cubic meter = 1.308 yd3

20.5 cubic meters = 20.5 * 1.308 yd3

20.5 cubic meters = 26.8 yd3

1 cubic meter = 264.2 gallons

20.5 cubic meter = 20.5*264.2 gallons

20.5 cubic meter = 5416 gallons

Algebra is widely used in day to day activities watch out for my forthcoming posts on Multiplying Mixed Number Fractions and polynomial function degree. I am sure they will be helpful.

3. Convert 1.35 Barrel into cubic meters.

Solution:

1 imperial barrel = 6.10 m3

1.35 imperial barrel = 1.35*6.1 m3

1.35 imperial barrel = 8.23 m3
Practice problems on unit of measurement for volume:

1. Convert the volume of 560 gallons into ft3 and liters.

Answer: 74.9 ft3 and 2119.6 L

2. Convert the volume of 2.5 yd3 into liters.

Answer: 1911.5

Median Frequency Table

Median is defined as one of the most important topic in mathematics. Mainly it is used to find the middle values. The values are given in the frequency table. By using the table we can find the median. Both the even numbers and the odd numbers, the median can be find. In this article, we are going to find the calculation of median from the frequency table.

I like to share this Median in Statistics with you all through my article.

Explanation to median frequency table

The explanation to median frequency table are given below the following,

Median for odd values:

Median = `(n + 1)/2`,      if the n value is odd.

Median for even values:

Median = `(n/2)+1` , if the n value is even.

Example problems to median frequency table

Problem 1: Find median for the following frequency table,
Values 2 3 4 5 6
Frequency 3
4 5
6
7

Solution:

Step 1: Given:

Values = 2, 3, 4, 5, 6

Frequency = 3, 4, 5, 6, 7

Step 2: Find:

Values = 2 + 3 + 4 + 5 + 6

= 20 ( Its a even function)

Step 3: Formula:

Median = `(n/2)+1` , if the n value is even.

Step 4: Solve:

Median = `(n/2)+1`

= `(20/2) + 1`

= 10 + 1

= 11

Therefore, the median is in the position of 11.

Step 5: To find position:

Add values and the frequencies, we get,
Values 2 3 4 5 6
Frequency 3 4 5 6 7
Position 2 + 3 = 5 5 + 4 = 9 5 + 9 = 14

Since the frequency is at 11 position, it will be between the 9 and the 14 position, So, 4 is the median value.

Result: Median = 4

Problem 2: Find median for the following frequency table,
Values 1 3 4 5 6
Frequency 2
4 6
8
10

Solution:

Step 1: Given:

Values = 1, 3, 4, 5, 6

Frequency = 2, 4, 6, 8, 10

Step 2: Find:

Values = 1 + 3 + 4 + 5 + 6

= 19 ( Its a odd function)

Step 3: Formula:

Median = `(n + 1)/2`,  if the n value is odd.

Step 4: Solve:

Median = `(n+1)/2`

= `(19 + 1)/2`

= `20/2`

= 10

Therefore, the median is in the position of 10.

Step 5: To find position:

Add values and the frequencies, we get,
Values 1 3 4 5 6
Frequency 2 4 6 8 10
Position 2 + 1 =3 4 + 3 =7 7 + 6 =13

Since the frequency is at 10 position, it will be between the 7 and the 13 position, So, 4 is the median value.

Result: Median = 4


My forthcoming post is on Divide Polynomials and Regular Convex Polygon will give you more understanding about Algebra.

Practice problems to median frequency table

Problem 1: Find median for the following frequency table,
Values 3
5 7 9 10
Frequency 5
6
8
8
9

Answer: 7

Problem 2: Find median for the following frequency table,
Values 3
6
9
12
15
Frequency 2
4
6
8
10

Answer: 9

Study Definition of Subset

Online gives the definition of subset as the elements of subset are contained by another set. By studying the definition of subset we can understand that subsets are part of another set which is used a symbol `sube`. Online gives a clear definition of subsets which helps to study problems easily.

Explanation to study definition of subset:

The definition of subset is as follows.

A set X is called as a subset of a set Y if some of or all the elements of X are existing in the set Y which can be denoted as X `sube` Y. We can also write the set as the set y is a superset of set X and denoted as Y `supe` X. A empty set is also taken as a subset for any kind of set.

Representation to study definition of subset:

The elements in X are existing in the set Y called X is the subset of Y.

Example problems to study definition of subset:

Example: 1

Write the subset relation for the following sets.

A = {m, n, o} , B = {o, p} and C = {m, n, o, p, q, r}

Solution:

Given sets are,

A = {m, n, o}

B = {o, p}

C = {m, n, o, p, q, r}

A and B:

The elements in A is not in B as well as the elements in B are not in A.

A and C:

The elements in A are in C. So, A is called as subset for the set C = {m, n, o, p, q, r} which can be represented as A `sube` C.

B and C:

The elements in B are in C. So, B is called as subset for the set C = {m, n, o, p, q, r} which can be represented as B `sube` C.

Example: 2

Say whether the set P = {12, 14, 15} is a subset for a set Q = {11, 12, 13, 14, 15}.

Solution:

Given sets are,

P = {12, 14, 15}

Q = {11, 12, 13, 14, 15}

The elements in P are in Q. So, P is called as subset for the set Q = {11, 12, 13, 14, 15} which can be represented as P` sube ` Q.

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Practice problems to study definition of subset:

Problem: 1

Write the subset relation for the sets C = {9, 7} and D = {9, 8, 7, 6}

Answer: C `sube` D

Problem: 2

Write the subset relation for the sets S = { } and P = {4, 5, 10}

Answer: S `sube` P

Preparation for Subset

In mathematics subsets are the terms used in set theory. The preparation depends on the elements in the sets. For the preparation for subset we have the derive a set by having elements of another set. The subset preparation uses the symbol `sube` .

For example, C `sube` A denotes C is subset of the set A and A `supe` C denotes A is a superset of C.

Understanding subset of a set is always challenging for me but thanks to all math help websites to help me out. 

Explanation to preparation for subset:

The preparation for subset is as follows.
Every set has a subset which is derived from the set. This subset may have all elements from the given set and it may be an empty set.

For example, set V = {l, p m , n}. Some of the possible subset for the given set V are { } , {l, p, m , n} , {l}, {l , p} , { m, n} , {p, m, n} etc. In this the set V is called as superset.

Example problems to preparation for subset:

Example: 1

Prepare the subsets of a set C = {2, 3}

Solution:
Given: C = {2, 3}

Subsets has the elements from the given set C = {2, 3} such as { }, {2}, {3}, {2, 3}

Example: 2

Which of the following is true for the sets A = {1, 2, 3, 4} B = {3, 4}?

a) B `supe` A

b) A `sube` B

c) A = B

d) B` sube` A

Solution:
Given A = {1, 2, 3, 4} B = {3, 4}
        B has the elements 3, 4 which is in A and has the elements 1, 2, 3, 4 where 1, 2 are not in B.So, B is a subset of A and A is a super set of B. (B `sube` A)
Answer: d

Algebra is widely used in day to day activities watch out for my forthcoming posts on Sum of Exterior Angles Formula and Multiplication Fractions. I am sure they will be helpful.

Practice problems to preparation fo subset:

Problem: 1

Which of the following is true for the sets X = {a, c, b, m, l, i} Y = {a, b, l}?

a) Y `supe` X

b) Y `sube` X

c) Y = X

d) X `sube` Y

Answer: b
Example: 2

Which of the following is true for the sets P = {p, m} Q = {p, q, r, s}?

a) P `supe` Q

b) P `sube` Q

c) P = Q

d) Q `sube` P

Answer: b