Eight Grade Math

 In the previous post we have discussed about Math for 8th graders and In today's session we are going to discuss about Eight Grade Math.


-->

In Eight Grade Math, we study different topics related to numbers. We come across the chapter of Rational numbers. In rational numbers, we study about how rational numbers are formed, their standard form, the positive and the negative rational numbers and how to express the series of rational numbers on a number line.

Further we also learn in Eight Grade Math, about how to compare two rational numbers and how to perform the mathematical operations on rational expressions. We study different properties of rational numbers too.

Here are some of the properties of rational numbers:

1. Closure Property: By closure property we mean that if we have two rational numbers, then sum, difference, product and quotient of two rational numbers is also a rational number.

2. While talking about the Commutative property of rational numbers, we say that it holds true for addition and multiplication of rational numbers but not for the subtraction and the division operation, so if the two rational numbers are p1/ q1 and p2/q2, then we get :

(p1 / q1 + p2 / q2) = (p2 / q2 + p1 / q1)

(p1 / q1 * p2 / q2) = (p2 / q2 *  p1 / q1)

(p1/q1 - p2 / q2) < > (p2 / q2  - p1 / q1)

(p1/q1 divided by p2 / q2) < > (p2 / q2 divided by p1 / q1).

To know about the Lateral Area of a Cylinder, we can learn the formula from online math tutor help, which is available for free. We can also download the cbse syllabus for class 11, from Internet, which is available on website of cbse. 

Math for 8th graders

Here we are going to discuss about the topic rational number. We will study about the rational numbers in detail in 8th Grade Math.

We define the set of rational numbers as the numbers which we can write in the form of p / q, where p and q are any integers, such that q <> 0. Further exploring the definition of rational numbers we say that any number which can be either written in the form of terminating decimal number or a repeating decimal number, then we say that the given fraction number is a rational number. All the natural numbers, whole numbers, positive or negative integers and the fraction numbers are the part of the family of rational numbers.

Let us look at the properties of the rational numbers. They are as follows :

1. Closure Property: If we have a pair of rational numbers, then we say that the sum, product, difference and the quotient of the given two rational numbers is also a rational number. Thus we say it mathematically as follows : If p1/ q1, p2 /q2 are any two rational numbers, then we have

P1/ q1 + p2 / q2,

P1/ q1 - p2 / q2 ,

P1/ q1 * p2 / q2 .

P1/ q1 divided by p2 / q2 are also the rational numbers.

2. Associative property of rational numbers also holds true for the addition and multiplication, but not for the operation of division and subtraction. So we have

( P1/ q1 + p2 / q2 ) + p3 / q3 = P1/ q1 + (p2 / q2 + p3 / q3) ,

( P1/ q1 * p2 / q2 ) * p3 / q3 = P1/ q1 * (p2 / q2 * p3 / q3) .

Also we have the commutative property, Additive identity, Multiplicative identity, Additive and multiplicative inverse property which holds true for the rational numbers.

Questions based on Osmosis and Diffusion can be seen in the cbse latest sample papers.  

How to factoring quadratic equations

The factorization of a quadratic equation (mid - term splitting) leads to real roots of the equation. The quadratic equations when solved using the formula, it is the case where real and imaginary roots both are possible to occur. That is by using the direct formula also we are able to factorize the quadratic equation, but imaginary factors are possible. The direct formula for a quadratic equation wx² + vx + z is given as follows: x = [- v ± Sq root(v² – 4wz)] / 2w

The ±sign is put before the root sign because according to the property of root there are two possible roots when a root is solved. Thus we get either two real or two imaginary roots for our quadratic equation. In case of imaginary roots we can also say that the two roots are complex conjugate of each other. Let's us suppose an example of a quadratic equation to understand the factoring quadratic equations:

Suppose our quadratic equation is given as follows: 5x² + 8x = -6. First we have to arrange the given equation such that it is present in the standard form of a quadratic equation and then start solving it. In our equation we can - not factorize by using mid - term splitting and so we need to use the direct formula to solve for x as follows:

Substituting the values of w, v and z in the formula we get, 

x = [- 8 ± sq root(8² – 4 * 5 * (6))] / 2 * 5 ,

Thus we get x = -8/10 ±56i/10 .

According to the concept of Round to the Nearest Tenth, the values of the numbers are rounded to their nearest tenth number. For instance, 304 will be written as 300 and 256 is written as 260. These concepts are very important and have been explained in the icse sample papers 2013 in detail. (know more about How to factoring quadratic equations, here)

right triangles

In the previous post we have discussed about adjacent angles and In today's session we are going to discuss about right triangles. A right triangle in the mathematics can be defined as that triangle in which we have 1 angle equal to the right angle which means that 1 angle is equal to the 90 degree. The right triangles are also sometimes called as the right angled triangles. It should be known that the relationship among the different sides and the angles of any right angled triangle forms the basis for the trigonometry.

Now let us talk about the terminology in the right triangles. In any right angled triangle the side which is opposite to the right angle or we can say the 90 degree angle is known as the hypotenuse whereas the sides which are adjacent to the 90 degree angle are known as the legs of the right triangle. (know more about right triangles, here)

Also when the lengths of the 3 sides of the right angled triangle are the integers then that right triangle is known as the Pythagorean triangle whereas the lengths of the 3 sides are collectively called as the Pythagorean triple.

We all know that in any type of the triangle the area is calculated by multiplying 1 / 2 with the length of the base of that triangle and then multiplying that result with the height corresponding to that base of the triangle. But the case of a right angled triangle is very easy. In any right angled triangle, when 1 leg is considered as the base of the triangle then the other leg gives the height of the triangle. Thus the area in any right angled triangle can be calculated by just multiplying 1 / 2 with the product of the lengths of the 2 legs of the triangle.

In order to get more help in understanding the topics: right triangles, Equation for Force and icse board papers 2013, you can visit our next article.

adjacent angles

We say that the angle is formed when we have two rays going in the different directions and they have a same starting point called the vertex. The measure between the two rays is done in the terms of degrees. So we say that the angles are measured in degrees.

 Let us now look at the adjacent angles. The pair of angles formed such that :

a)    The two angles have a common vertex.

b)    The two angles have one  common arm and two uncommon arms.

c)    The common arm exist in between the uncommon arms.

If all the  above three conditions are satisfied , then we say that the pair of angles is called the adjacent angles.

In case the pair of adjacent angles are supplementary, then we conclude that the two uncommon arms of the adjacent angles form a straight line. Thus if the sum of the two adjacent angles  is supplementary, then we say that the two uncommon arms is in fact the straight line.

The reverse of the above statement is also true, which says that if the two adjacent angles are formed on the straight line, then the pair of angles are supplementary.

 We also call these pair of angles as the linear pair of angles.

So by the term linear pair, we simply mean the pair of the adjacent angles which are formed on the straight line and so the sum of this adjacent pair of angles is automatically 180 degrees.

To learn about the  Kinematics Equations, we can take the help of online tutors. These online math tutor can be used any time on your pc without any cost.  icse books free download can also be done by internet when ever required. All you need to have is the P.C. and an internet connection.

Equation of Circle

In mathematical geometry, a round shape in which all the points on the boundary are at same distance from the center is known as a circle. The general Equation of Circle is given by: S 2 + T 2 = r2, where ‘r’ is the radius of a circle (Here S is along to the horizontal axis and ‘T’ is along the vertical axis). All of the points on the boundary of a circle are at fixed distance from the center is known as the radius of a circle. The general equation for circle is given by: (know more about Circle, here)

⇒ (x – s)2 + (y – t)2 = r2;

Now put the value of s, t, and r is 4, 5, 6 respectively then we get:

⇒ (x – 4)2 + (y – 5)2 = (6)2; on further solving the equation of a circle we get:

⇒ x2 + 16 – 8x + y2 + 25 – 10y = 36; So the equation of a circle is:

⇒ x2 + y2 – 8x – 10y + 41 = 36; we can also write it as:

⇒ x2 + y2 – 8x – 10y – 5 = 0; This is the required equation of a circle.

Here we can also write the general form of a circle using the constant value in place of numbers. So the equation of circle using constant is given by:

⇒ x2 + y2 + Sx + Ty + U = 0, here we will also see the equation of a unit circle. We know that ‘1’ is the radius of unit circle, if the radius of a unit circle is more or less then ‘1’ then the circle is not unit circle. The general equation of a unit circle is given by:

⇒ x2 + y2 = 1; let’s discuss how to graphing linear equations. In mathematics there are many methods through we can easily plot the graph of linear equation. To study more about the linear equations go through the online tutor of tamilnadu board of higher secondary education and In the next session we will discuss about adjacent angles. 

isosceles triangle

By the term triangle we mean that it is a closed polygon which if formed by joining 3 line segments. There are different types of triangles based on the lengths of the line segments.

They are : Equilateral Triangle,  isosceles triangle, and scalene triangles. By the term Isosceles triangle, we mean that the triangle which has a pair of the two sides equal. In such a triangle, if we have pair of the  sides equal, we say that the pair of the angles which are formed by the  pair of sides is also equal.  So we come to the conclusion that in the isosceles triangle we have two sides equal and two angles equal. (If you want to get more information about isosceles triangle, Refer this)

If we drop a perpendicular from the angle which is unequal in the isosceles triangle, we observe that this perpendicular is the perpendicular bisector to the line segment which is opposite to this angle. Thus we also called this line as the median of the isosceles triangle.

We further observe that this median, which is also the perpendicular bisector is the line of symmetry of the given isosceles triangle. Any isosceles triangle has only one line of symmetry and this line of symmetry divides the  given isosceles triangle into two equal halves. We also call it the mirror half of the triangle.  On the other hand an equilateral triangle has 3 lines of symmetry and a scalene triangle has no line of symmetry.

If we need to learn about  What are Perpendicular Lines, we can visit the online math tutorial and take the help of the modules based on the above topic in the tutorial. We also have cbse syllabus 2013 online which can be downloaded to understand the pattern of the question paper in the upcoming examination. It guides the students to  know about the important questions  for the exams and In the next session we will discuss about Equation of Circle.

supplementary angles

Angles are formed when we have two rays going in the different directions such that they have the same vertex. The pair of angles is called a supplementary angles, if we have the sum of the two angles equal to 180 degrees.  Thus simply if we say that the given angle is x degrees, then the supplementary of the given angle will be 180 – x degrees. (want to Learn more about supplementary angles, click here),

So if we have any angle say 110 degrees, then the supplement of the given angle 110 will be 180 – 110 = 70 degrees.

 Thus we say that the  supplementary of 50 degrees will be 180 – 50 = 130 degrees.

 Now if we want to know the measure of the angle which is equal to its own supplementary, then we will proceed as follows :

 Let the angle measure be x, then its supplementary angle will be 180 – x degrees.

 Now if we say that  both the angles are of the same measure, then we write it mathematically as follows :

X = 180 – x

On adding x on both the sides we will get :

X + x = 180 – x + x

Thus the above given equation becomes :

2 * x = 180 degrees

Now we will divide both the sides of the equation by 2 and we get :

2 * x / 2 = 180 / 2

Or x = 90 degrees.

Thus we say that the angle 90 is such that it is equal to its own supplement angle.

In order to learn about How to Find Slope of a Line, we can take the help of online math tutorials.  icse guess papers 2013 are also available online, which can guide the students to understand the  concept and the patterns of the question papers in the fore coming  board examinations and In the next session we will discuss about  isosceles triangle.

Parallel Lines

Before going to discuss anything about the parallel lines we should first have an understanding of the term parallelism. The term parallelism is generally used in the geometry which is referred as a property in the space of the Euclidean of 2 or more than 2 lines or the planes or any combination of lines and planes.

We can say that in a plane any 2 lines are said to be parallel if their intersection does not happen or if they do not touch each other at any point. This definition can also be frames in another way as follows. The lines are said to be parallel when their plane is the same and they exist at the same distance in the complete length of the lines. By this we mean that it does not matter at all that how long we are extending the lines, if they are parallel then they will not meet ever. (Know more about Parallel Lines in broad manner, here,)

Now let us discuss about how we show any two parallel lines. For showing that any 2 lines are parallel we make use of the parallel sign which is represented by ||. Thus if we write EF || GH then it means that the line EF is parallel to the line GH.

Let us now pay some attention on the construction of the parallel lines. Suppose we are given a line say x and a point p and we have to construct a line y parallel to the x through the point p. We can make this line y by considering the fact that it should have the same distance from the line x everywhere. There is another way to draw the line y that is we have to consider a random line passing through the point p and crossing the line x at point q. Then we have to take this point q to the infinity.

In order to get more help on the topics: Parallel Lines, College Algebra Problems, icse board papers 2013, you can visit our next article and In the next session we will discuss about supplementary angles.

What are Lines in Geometry

In the previous post we have discussed about Perpendicular Lines and In this blog we are going to discuss about Lines in mathematical world. As we know that slope of a line is generally the measure of an angle of a given line from the x-axis or Y-axis. Here we are going to deal with line in the coordinate plane geometry. In coordinate plane the two lines are present which are vertical and horizontal line. The line whose x- coordinate remains unchanged and y –coordinate changes according to the given values is known as vertical line. A line which move up and down and that line is also parallel to the y – axis of the coordinate plane is known as vertical line. There is no slope defined for vertical line. Equation of a vertical line is given by: x = s. (want to Learn more about Geometry, click here),

A line whose y- coordinate remains unchanged and x – coordinate changes according to the given coordinates is known as horizontal line. A line move straight left and right and also parallel to the x – axis of the coordinate plane is known as horizontal line. The slope of a horizontal line defined as zero. The equation of a horizontal line is given by: Equation of a line is y = t; here the value of 'y’ represents the coordinates of any point on the line and the values of ‘t’ represents the line which crosses the x – axis. Let’s see how to solve equation of Lines in the Coordinate Plane. In mathematical geometry math problem solving is a tool which is used to set a given procedure and see what and when these given procedure should apply. If we want to identify the procedure then it is necessary to know the situation of problem. Before entering in the examination hall please go through the iit sample papers, it helps to solve the problem very easily. This is all about lines and other related topics.

Perpendicular Lines

Hi friends, here in this blog we are going to understand an important topic that is Perpendicular Lines. If we have two lines and these lines make the angle of 90 degree with each then the lines are said to be Perpendicular Lines.

Let we have two lines then the slope of one line is negative to the other line. When the slope of one line is ‘s’ then the slope of other line is = -1 / s. let's see how we construct perpendicular lines in geometry.

To plot the perpendicular lines we have to follow some steps, we have to discuss each step one by one.

Step1: To graph perpendicular lines first of all we take a line of any length.

Step2: Then we have to put a point on the line and the point is named as ‘P’.

Step3: For the construction of perpendicular lines we need compass.

Step4: Then using the compass we make an arc on both side of point ‘P’ and both the arc named as ‘U’ and ‘V’.

Step5: Now we have to put the compass on point ‘U’ and measure the length of compass greater than the point ‘P’.

Step6: Then from point ‘U’ make an arc on the upper side of a line.

Step7: Then with the same length we have to make an another arc from point ‘V’ to the upper side of a line. Both arc cross each other.

Step8: Then using pencil we have to draw a line which meets at the cross point.

And we get another line which is perpendicular to the given line and that line makes exactly 90 degree angle to the other line.

We get this figure when we follow all the above steps:

For the pre algebra practice we have to study the different topics in algebra which are number, fractions, factoring, mixed etc. To get good result please focus on sample papers for cbse board and In the next session we will discuss about What are Lines in Geometry.

How to work out gcf calculator

In the previous post we have discussed about How to use Least Common Multiple Calculator and In today's session we are going to discuss about How to work out gcf calculator. Greatest Common Factor or Highest Common factor that is abbreviated as GCF, provides the largest numerical value that divides the other numbers with zero remainder .There are some methods of solving the GCF are as division method or factorization method .GCF calculator is an online tool that helps in calculating the factors accurately.  (know more about gcf calculator , here)
There are some examples for defining the GCF calculations are as follows:
Number 24 that is also denoted as 24 = 12 * 2 = 8 * 3 = 4 * 6, that denotes form of product of two integers. If we talk about the divisor of 24 then these are 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24 that divides the 24 and 1 , 2  , 3 , 4 , 6 , 9 , 12 are some divisor of the value 36. There are some common divisor are 1 , 2 , 3 ,6,12 and among these factors greatest common factor is 12 .So the GCF of the 24 and 36 is 12. Finding the gcf with the help of gcf calculator will be very easy process. For finding the gcf with the help of gcf calculator user should enters the list of integers for which they want to find the gcf that is separated by commas or spaces .It will be written as gcf( 36 , 24 ) = 12. For reduce the fraction to its lowest terms we can use gcf and if both numerator and denominator have the common gcf if 14 / 35 = 7 * 2 / 7 * 5 = 2 / 5 as an example of gcf( 14 , 35 ) = 7. In this example gcf calculator define the smaller value as the denominator and higher value as the numerator.
Organic chemistry help is the useful tutorial to understand the various concept of organic chemistry. Tamilnadu state board syllabus helps the students of the Tamilnadu state board to understand the related chapters of the subjects of their respective classes.

How to use Least Common Multiple Calculator

In the previous post we have discussed about How to use Geometric Mean Calculator and In today's session we are going to discuss about How to use Least Common Multiple Calculator. To understand about the least common multiple calculator, we must first understand the meaning of the multiples. If we write the multiples of a number 5, we say that the numbers 5, 10, 15, 20, 25, 30, 35 . . . . are all the multiples of 5. It simply shows that the numbers which are completely divisible by the given number is it’s multiple.  So we say that the number 10, 15, 20 . . .  are completely divisible by 5, so they are the multiples of 5.  Now to learn about the least common multiples of the given two or more numbers, we say that the common multiples of the factors of the given numbers can be picked and then they are multiplied to get the desired number. Here if we need to find the least common multiple of the given numbers says 25 and 45, we will first write the prime factors of the two numbers and then we will pick the common factors and find their product to get the least common multiple of the given numbers. So we proceed by writing the factors of  25 and 45 as follows :
25 = 5 * 5 * 1
45 = 3 * 3 * 5 * 1
 Here we find the common factors of the two numbers as 5 and 1, so we get the least common multiple of 25 and 45 as 5 * 1 = 5 Ans. (know more about Least common multiple, here)
To understand more about least common multiple calculator, the online help is always available to practice on the required worksheets.
  We can learn about How to Find Circumference of the circle by the online math tutor, which is always available on the internet.  To know about the West Bengal Board Syllabus, we can visit its website and know the contents of the curriculum that are being covered in the syllabus.

How to use Geometric Mean Calculator

In the previous post we have discussed about Operations With Integers and In today's session we are going to discuss about How to use Geometric Mean Calculator. In mathematics whenever we study about series then geometric progression is one of the most complicated series. If a, b, c are in geometric progression then b/a = c/a =r here  r is called a common difference and its value will be same for all the elements of geometric progression.
If the value of a =2 , b=4 ,c=8 ,then according to geometric progression b/a = 4/2 , c/a = 8/4
now we get common difference r = 2 . If we have to extend the terms in geometric progression then we have to multiply last term by common difference. As in the above example 8 is last term and if we multiply it by common difference two then we will get 16 that will be the next term of the series.
If we have two numbers a and b and we are asked to calculate the geometric mean then it will be
G= √ab here G is called geometric mean for all the elements. Now we  will see the How we calculate the geometric mean. Suppose we have a two numbers first number a= 2 and another number is b= 8 and these numbers are in G.P then geometric mean will be G=√2*8 = √16 = 4 , in this way we get the geometric mean G= 4. If we talk about geometric mean calculator then it always ask for two values, the values should be in geometric progression, firstly it will multiply two value then it will calculate square root. In this way we can calculate the geometric mean of any two numbers with the help of geometric mean calculator.
Boolean algebra is one of the most important topics in CBSE syllabus so If we want score good marks you should go through Boolean algebra and would read the cbse class 12 sample papers.

Operations With Integers

Integers are endless series of numbers which start from minus infinite and goes up to plus infinite, such that every number in the series has its successor and each number has its predecessor. We will learn that how to perform   Operations with Integers. In order to perform the operations with integers, we say that all the mathematical and the logical operators can be performed on the integers. By logical operations, we mean that we can compare the two integers and arrange the series of integers in ascending or descending order. We know that all the positive integers are greater than 0 and all the negative integers are less than zero. So we come to the conclusion that  if we draw a number line , zero lies in the middle and the more we move towards right the positive number goes on increasing and the more we move towards left, the negative numbers goes on decreasing. So to compare the two numbers we say that the number on the right of the number line is always greater than the number on the left side.  Now we look at the mathematical operations on integers. We can add, subtract, multiply and divide the integers. We say that the integers satisfy the closure property for the addition, subtraction and multiplication operation. But the closure property does not hold true for the division operation. It means that when two integers are added the result is an integer, if two integers are subtracted or multiplied the result is an integer but division of two integers is not an integer every time.
  We can take the guidelines from Free Online Tutoring to understand the topics which we find difficult to solve otherwise. We also have CBSE Sample Papers online to learn about the  concepts of the question papers which have come in the past years for different subjects and In the next session we will discuss about How to use Geometric Mean Calculator.

How to Find The Area of a Circle

In today session we are going to discuss how to find the area of circle. The area of circle can be obtained by the following ways.
1: If we know the radius of the circle.
If the radius r is given then the area of the circle can be calculated by the using the method
   Area of circle = pi*r²
Where r is the radius of the circle and (pi) is the constant having fixed value 3.142 or 22/7
We can also understand how to find the area of circle in the CBSE Books for Class 8.
 2: If we know the diameter d of the circle
If diameter d is given then the area of the circle can be calculated by the using the method
     Area of the circle = pi * d²/4
Where d is the diameter of the circle and (pi) is the constant having fixed value 3.142 or 22/7
This topic is also important to find the confidence interval.
3: if we know the circumference of the circle
If circumference c is given then the area of the circle can be calculated by the using the method
             Area of the circle = c²/4*pi
Where c is the circumference of the circle and (pi) is the constant having value 3.142 or 22/7.
At the end of the above method to determine the area of the circle .there is an another way to find the area of the circle
By using the method of area of sector of the circle = (pi*R²*angle) / 360
Where r is the radius of the circle and (pi) is the constant having fixed value 3.142 or 22/7
For the area of whole circle angle must be equal to 360 degree. Therefore the area of the circle will be = (pi*R²*360) / 360 it implies that
Area of the circle = pi*R²   

Volume of a Sphere

Previously we have discussed about add radicals calculator and In today's session we are going to discuss about Volume of a sphere, It is a three dimensional surface in which every point on the given surface is equidistance from a point.

Now we will see the formula for finding the volume of sphere, the formula for finding the volume of a sphere is:

        Volume = 4 ⊼r3, Where r represents a radius of a sphere.

                        3

Some condition of sphere is also realized that the volume of a sphere is exactly two thirds of the volume of its circumscribed cylinder, which is known as smallest cylinder which can contain the sphere. If we want to find the radius of a sphere using the above formula:

           r = 3√3v, where v denotes the volume of a sphere.

                     4⊼

And the value of ⊼ is 3.14;

If we want to find the volume of a sphere using diameter then we use the formula

        Volume = 4 ⊼ (d/2)3,

                        3  

Sphere is an object in the three dimensional space. Its shape is just like as a round ball.  And the maximum distance through the sphere is known as diameter of the sphere. Diameter is twice the radius of the sphere.

Now we see how to find the volume of the sphere using diameter, with the help of example.

Example: - Find the Volume of a Sphere where its diameter is 30 inches.

Solution: we know that the volume of a sphere is

                Volume = 4 ⊼r3

                                3

Here we have to find the radius of a sphere with the help of diameter.

We know that the radius is half of the diameter.

        r = diameter

                  2

So putting the diameter value in formula

        r = 30

               2

        r = 15;

And the value of ⊼ is 3.14

So putting all the values in the formula

Volume = 4 ⊼r3

                3

             = 4 * 3.14 * (15)3

                 3

            = 4 * 3.14 * 3375

                3

           = 42390

                  3

             = 14130 inch3

We get the volume of a sphere using the diameter is 14130 inch3.

Multiplication Worksheets is very important concept of mathematics, it is also available in cbse class 8 books.

add radicals calculator

In our last post we talked about online math tutor, in this post we will focus on add radicals calculator. We have Add Radicals Calculator, which will help us to understand and learn about the operations on radicals. Radicals mean the numbers with the powers in the form of fractions. These fractions can be ½, 3/2, 2/5 etc. Let us consider the following addition of radicals.
Root ( 3 ) + root (3 ) = 2 root ( 3)
 We say that only similar radicals are added. It means that the radials with the same radicands are added.  So if we have the problem: 3* root (2) - 2* root ( 3 )  + 5 * root ( 3 )  - root ( 2) + 5
Here we will join together the radicands with root 2 one side and the radicands with root 3 on another side of the expression. Thus we get the following result:
= 2 * root ( 2) +  3 * root ( 3)  + 5
 Here we observe that the whole number 5 is not the part of any radical so it will neither be added to root ( 2) nor to root ( 3). So we get 5 as the separate number and it is not summed up to any of the radicals. On another hand the radicals which are similar, their coefficients are added up or their difference is calculated to get the result.
 Let us take another example:  root ( 18 ) + root ( 8 )
 = root ( 3 * 3 * 2 ) + root ( 2 * 2 * 2 )
   = 3 * root ( 2) + 2 * root ( 2)
 =  5 * root ( 2) Ans.

  We can always use Algebra Help, online to learn about the problems related to Algebra. Tamilnadu Board Statistics Sample Papers are also available to get time to time help on the curriculum.

online math tutor

Online math tutor can help us to learn about the integers and its properties.  If we talk about the integers, we say that the integers are the series of the positive and negative numbers which also has a number zero as the middle number.  Now we also observe that the online math tutor can be used to learn about the properties of the integers. All the integers satisfy the following properties:
Closure property:   Closure property holds true for the addition, subtraction and multiplication, but does not holds true for the division. It means that if two integer numbers are added then the resultant number is also an integer. Similarly by closure property of subtraction, we mean that if  we have two numbers, then the difference between the two numbers is also an integer and the product of the two integers is also an integer but the quotient of two integers is not necessary an integer every time.
Commutative property of integers also holds true for addition and multiplication, it means that if we have any two integers a and b, then by commutative property of addition and multiplication, we mean:  a + b = b + a
Similarly we have a * b = b * a
Associative property of integers also holds true for the addition and multiplication  :  It means that if a, b, and c are any integers, then  by associative property of integers, we mean that if the order of addition or multiplication is changed the result remains same . Mathematically we mean:
( a + b )  + c = a + ( b + c )
( a * b )  * c = a * ( b * c )

 We can use online help to understand about Area of a Regular Polygon, which is the part of syllabus of school of secondary education Andhra Pradesh. In next post we will talk about add radicals calculator

Representations of data

Previously we have discussed about differential equations formulas and In today's session we are going to discuss about Representations of data which comes under cbse 12th syllabus, It means presenting the data in form of graphs, diagram, maps or chart besides the tabular form. In this type of representation of data, visualization is provided in forms of graphs and charts. There are some basic needs for the representation of data:

. Sometimes it is difficult to understand the descriptive data as it is not easy to draw the results.

. Easily draw the visual impression of the data

. Makes comparison easy

. Characteristics can be represented in a simplified way

. Some kind of patterns are made easy as population growth or distribution and density or age – sex composition , etc .

There are some rules that must be followed before designing the graphs,charts or diagrams

1. first of all a suitable graphical method is selected .

    
        Then after selecting the graphical method a scale is selected that is suitable to the data
        On the basis of Title,Index and Direction design must be followed.

There are several diagrams that are used for representing the data that can be categorized

in the following types :

. Some examples like Line graph, polygraph, histogram, pyramid and bar diagram etc. examples of One dimensional diagram.

. There are rectangular diagram and pie diagram examples of Two dimensional diagrams.

. Diagram as cube and spherical diagram examples of Three dimensional diagram.

Representation of data are done in many ways that are flow chart,line graphs,bar diagrams,pie diagrams , wind rose and star diagram etc .These are some examples of popular diagrams that are mostly used in representing the data.

Rainfall,Population growth,Temperature are examples of series of time, For representing the series of time e and series like birth rates and death rates etc are represented by drawing the Line graphs .

Two or more variables on a same diagram is represented by a Polygraph that is also a line graph ,that are shown by different line as example for showing the growth rate of different crops and These diagrams are also used for show the birth and death rates .

So there are different types of graphs and diagrams that are used for show the different type of representing data .In the next session we are going to discuss Sampling techniques.

In the next session we will discuss about online math tutor and You can visit our website for online tutoring for free.

Simple random sampling

Previously we have discussed about application of differential calculus  and In today's session we are going to discuss about Simple random sampling  which is fronm cbse previous year question papers class 12.

Some points to understand Simple random sampling:-

• The set of total observations that can be calculated in the experiment is called population.

• Sample can be defined as a set of observations drawn from the population.

• Statistics is related to sample which is a measurable characteristic of the samples like standard deviation.

• Sampling method is the procedure of selecting sample elements from the population that were made.

• Random number is a matter of chance having no relationship with the occurrence of other number.

• Simple random sampling related to sample method having following properties

1.  In Simple random sampling population can have n number of objects.

2.  Simple random sample that is chosen can also have N number of objects, here n ≠ N.

3.  All possible samples of N objects have same probability to occur.

Samples chosen with the help of simple random sampling are refers to the population this is the measure profit of choosing simple random sampling, it means that the conclusion will be valid.

There are several methods to get simple random sample for example lottery ticket is the example of simple random sampling in which a unique number is assigned to each member of N population. These numbers are placed in a bowl and then thoroughly mixed then a blind folded researcher selects n numbers from the bowl. The selected n numbers are called the samples taken from the population.

There are two other terms too; sampling with replacements and sampling without replacements.

In the above example of lottery ticket when a number is picked from the bowl; if it putted aside by the blind folded researcher then the probability of occurrence of this number is only once and if this number is putted back in the bowl then the number can be selected further too.

This is sampling with or without replacements.

If a population element has the chance to be selected more than one time then it is called sampling with replacements.

If a population element only can be selected one time then this sampling is called sampling without replacements.

In the next session we will discuss about Representations of data and You can visit our website for getting help from online tutors.

Sampling techniques

Sampling techniques:

Previously we have discussed about definite integral examples and In today's session we are going to discuss about Sampling techniques which comes under cbse books for class 11,  Sampling is required when information is being processed for the transmission from one place to another place. The information is divided into samples and then the information is transmitted. It’s easy to remove noise or other factors from the samples in spite of the whole information directly.

If there are a lot items in a population set then the analysis process would be too costly and time consuming for that population. Like if the customer base id too large then  it would be too costly to determine the satisfaction level of each customer. The sampling process defines the same thing in short.

Sampling is the risk that it's not representative of the population from which it is made . Basically sampling  is the main step in analyzing any analytical process after that its not actually possible to remove errors.

The main processes for the sampling techniques are

·         Determine objectives and population then

·         Determining the sample size that would be created

·         Selection of the sampling method

·         Then the last step is to analyze the sampling errors regarding the projection or other

Sample size can be defined as

                                Sample size = reliability factor/Precision

There are several advantages of the sampling

·         The actual air sample can be collected without any breakthrough

·         No degradation problem of trapping material

·         Moisture has no effect on sampling

·         Duplicate analysis of the sample can be performed.

In mathematics it can be defined as to take a function f and recreate it with the help of only certain values.

Sampling techniques can be understood in probability or non probability preference. In probability method each member of the population has a non zero probability of being selected. This includes random, stratified and systemic sampling. In non probability method members are selected from the population in a random manner. It includes convent sampling, quota sampling, snowball sampling etc., techniques of sampling or methods for sampling are described below.

In Simple random sampling each member has an equal chance for being selected. It’s the purest method of sampling.

In systemic sampling every nth record is selected from the population.

Stratified sampling reduces the number of errors and used when one or more stratums have a low incidence relative to other.

Convenience sampling is used when inexpensive approximation of truth is required.

Judgment sampling is an extension of convenience sampling as the name indicates samples are made on the judgment basis.

Snowball sampling is used when desired sample characteristic is rare. It’s a difficult method and cost prohibitive too.

In the next session we are going to discuss Simple random sampling and You can visit our website for getting math help for free.

Range in Grade VIII

Range in Class VIII.
Hello children! Previously we have discussed about column multiplication. Now we will study about the topic Range which comes under gujarat education board.
In statistics, range means to find the difference between the largest and the lowest frequency of the given data.
So, for finding range we first need to arrange the given data in ascending or descending order in order to get the accurate information.
The formula for range is:
Range = highest value - lowest value.
It actually gives the length of the smallest interval which contains all the information of the given data. It gives the variation of the spread of the data.
 Let us understand it more clearly through some examples.
The ages of the ten teachers in the school, they are 35, 28, 36, 26, 45, 30, 29, 35, 26, and 30. Now for finding range of age of teachers in a school, we first arrange the given data in ascending order. We get: 26, 26, 28, 29, 30, 30, 35, 35, 36, and 45.
We observe that the teacher with least age is 26 and the teacher with largest age is 45.
So Range = Maximum age - minimum age
                  = 45 - 26 = 19 years.
Which means that range of age of teachers in the school is 19 years.

In the next topic we are going to discuss Sampling techniques and You can visit our website for getting information about physics problem solver.

Mode in Grade VIII

Previously we have discussed about calculus problem solver and In today's session we are going to discuss about Mode which comes under gujarat board textbooks online, Its mathematical term that comes in statistics. It can be defined as particular value that occurs most number of the times in a list. In simple words mode is a most frequent value in a data set or most common value in a group. Mode or statistical mode is same thing. Let us understand mode with an easy example:-

We have a list of numbers :- 3, 5, 7, 2, 8, 9, 2, 4, 2

The first step to find mode is to arrange the list in ascending order than we will get.

2, 2, 2, 3, 4, 5, 7, 8, 9

The next step to find out particular number that has occurred most number of times.

Here the particular number is 2.

So the mode will be 2.

Mode is used to collect information about non-deterministic numbers, also called random numbers in a single quantity.

Now let us discuss how calculate mode when list is fractional with an example:-

Now we are talking a example of fraction list :- 3.2, 3.7, 4.1, 3.2, 5.6, 2.4, 3.2, 2.4

First of all arrange the list in an ascending order:- 2.4, 2.4, 3.2, 3.2, 3.2, 3.7, 4.1, 5.6

3.2 is a number that has occurred most number of times in the above list.

So the mode is 3.2

Advantage of mode

-provide support to mean,median to solve statistical problems.

-to identify wether an event has occurred more than one time or not.

-used for counting the number of times

Note:- A mode will not exist if there is no repetition of any number in a list. A list contain may be more than one mode if two numbers sharing same occurrence of time

The above described mode information will be truly helpful for grade VIII students.

In the next topic we are going to discuss Range in Grade VIII and You can visit our website for getting information about free online math help.

Math Blog on Median

In reference with data handling to be studied in Grade VIII  of CBSE math Syllabus today we are going to learn about median.
We have already learned about math questions on mean, which help us to find the average of the given collection of data.
As you have studied earlier that the raw data collected is to be arranged in the ascending or descending order in order to retrieve some information from it.
In this Blog we will discuss about median in math. Median is the mid value of the collected data. Suppose we have collected the age of 5 teachers in the school.
They are 35, 24, 45, 22 and 36.(Know more about Median in broad manner here,)
 To find the median, we will first arrange this data in ascending order:
We get:
  22, 24, 35, 36 45
We observe that 35 is the median.
Now we look at the data when the number of data is even, in such cases we select middle two terms of the data.
Then we find the average of these two values.
To sum up, we produce the formula for the same:
To find the median, the following steps are to be followed:
1. Arrange the data in ascending order; let the number of entries be n
2. Then we will find if the number of entries is even or odd
3 If the number of entries is odd then
    Median = (n+1) / 2 th term
And if the number of terms is even then
    Median = [ (n/2) th term + (n/2) +1th term ] /2
 So children we first check the number of entries in the given set of raw data and then accordingly proceed to find the median.

In the next blog we are going to discuss Mode in Grade VIII and if anyone want to know about How to calculate Median in grade 9th then they can refer to Internet and text books for understanding it more precisely and also know about some interesting questions like is square root of 7 a rational number.

Mean in Grade VIII

Previously we have discussed about properties of numbers worksheets and Today we will discuss about mean in math which you need to study in grade VIII of indian certificate of secondary education board which can provide you huge help with math. Mean is the average value of any given series. Mean is widely used in statistics. You will be using application of mean in higher class. Mean is very useful whenever you are asked to find the average of any given series. Let's see some examples to have a good idea about mean.(want to Learn more about Mean ,click here),

Example1 find the mean of the given series?

4,5,6,7,8,9

solution : This series contains 6 elements, so number of element = 6

mean= sum of all terms of the series/number of elements

mean = 4+5+6+7+8+9/6

mean=39/6

mean=6.5

So 6.5 is the required mean for this expression.

Mean is very useful and whenever you are asked to calculate the average of any quantity, it will simply give you the average of that particular quantity. We will see an example:

Example 2: find the average marks of Sachin in grade VIII. His marks are as fallows

hindi -67, math-80, english-76, science-76,computer-60.

Solution:

there are 5 subjects, so the series contains 5 elements,

now as we know mean of the series is = sum of all terms of the series/number of elements

mean=67+80+76+76+60/5

mean=359/5

mean=71.8

mean of the all the subjects is 71.8.

As you are seeing that mean is 71.8, so we can say that mean can be an integer or can be a real number

Example 3:Find the mean of the series given below

2,5,7,9,11

Mean of the series will be sum of all the values divide by number of values

There are 5 elements in the series.

Now as we know mean of the series is = sum of all terms of the series/number of element

2+5+7+9+11/5

34/5

6.8 is the mean for given series.

This is all about the Mean of any series. In this article we have dealt briefly about mean. If you still feel any problem with this and other topics like How to tackle eighth standard Algebra you can visit different websites to solve your problems.

Graph and Slope of Lines in Grade VIII

Hello friends, Previously we have discussed about probability examples and today we are going to learn about Graph and Math problems related to it., slope of lines for grade VIII of icse board. The slope of a line is generally represented by m. Simply slope is the rate at which the path of a line rises or decreases or the slope is a number that tells how steep the line goes up and down or slope is a ratio of vertical to horizontal distances. From the equation of a straight line (want to Learn more about Slope ,click here),
y = mx + b
Here the slope of line m is multiplied by x and b is the y-intercept where line crosses the y-axis. This is the equation of line and sensibly named as slope-intercept form. The graphical form of this equation can be quite straightforward, particularly if the values of m and b are relatively simple numbers. In the slope of line if the value of y changes then the value of x also changes. For example in the line y = (2/5) x – 3, here the slope is m = 2/5. This means that, starting at any point on this line, we can get to another point on the line by going up 2 units and then going to right 5 units. In other words, for every unit that x moves to the right, y goes up by two-fifths of unit. Now the formula of the slope of the line, if two points (x_1,y₁) and (x₂,y₂) are given then the slope m of the line is
m=y₂-y₁/x₂-x₁, (x₁≠x₂)
For example the slope of the line segment joining the points (1, - 6) and (- 6, 2). Here x₁ =1, x₂=-6, y₁=-6 and y₂=2, now using above formula
Slope= m=y₂-y₁/x₂-x₁
          2+6/-6-1=- 8/7
So, m=- 8/7
The graphical form of slope is shown in the figure.

Now I am going to tell you the different types of slopes and their definitions. The different types of slopes are Slope of Parallel Lines, Slope of Perpendicular Lines and Negative Slope. Firstly the Parallel Lines, if the slopes of two lines are equal then these lines are parallel. The two parallel lines never intersect. If each line will cut the x axis at the same angle then slopes are
m₁=tan x and m₂=tan x
 m₁= m₂
This shows that if two lines are parallel then their slope must be same.
The next is Slope of Perpendicular Lines. Perpendicular lines are lines which makes right angle at their intersection. Slope of Perpendicular is defined as if there is change in y co-ordinate then there is also change in x co-ordinate. On other hand, when two lines are perpendicular then the slope of one is the negative reciprocal of the other one. That is if the slope of one line is m then slope of the other is -1/m. -1/m is the slope of the other line is the negative reciprocal of m and is called negative slope. For example if the slope m=0.342 then the negative reciprocal of 0.342 is -1/0.342.
From the above discussion I hope that it would help you to understand the slope of lines and if anyone want to know about Permutations and combinations then they can refer to internet and text books for understanding it more precisely. Read more maths topics of different grades such as Factors and Number Sequences in Grade VIII in the next session here.

Real Numbers in Grade VIII

In earlier sections we have discussed and practiced on rational numbers worksheet and Today we will discuss about the real numbers and  properties of real number for grade VIII of Maharashtra Board Syllabus.
So before starting, we will first try to find that what the real numbers actually are.
The real numbers can be defined as the set of numbers that consists of all the rational numbers(maximum used in rational expressions) together with all the irrational numbers. In general language we can say that all integers, small or large, whole number, decimal numbers are all real numbers
Except the imaginary numbers (the numbers which have negative terms under the roots are called imaginary numbers) all numbers are known as real numbers
For Example:

Given set A =  0, 2.9, -5, 4, -7, p , is a set which  consists of  natural numbers,  whole numbers,  integers,  rational numbers,  irrational numbers but all element of  set A represent the real number.

Now we will learn about some properties of real numbers which apply on all real number. These properties will be very helpful to solve algebraic problems. We will discuss each property in detail and will try to explain with some examples. Let's talk about  Commutative properties of real numbers
For addition:            a+b = b+a
For example: 1:     3+5 = 5+3 =8
     
                     2:      4 + 5 = 5 + 4 =9
   
For multiplication:       a*b =b*a
For example: 1:              3*7 =7*3 = 21
                    2:               5 × 3 = 3 × 5 =15
But this property does not satisfy for subtraction and division
                b-a ≠ a-b and   a/b ≠ b/a
  for example:    4 – 5 ≠ 5 – 4    and          4 ÷ 5 ≠ 5 ÷ 4
    Now let's move on other property that is Associative property of real numbers.
For addition : a+(b+c) =   (a+b)+c
For example:   (4x + 2x) + 7x = 4x + (2x + 7x)
                       (4 + 5) + 6 = 5 + (4 + 6)

For multiplication: a*(b*c) = (a*b)*c
For example:1:   (3x*4x)*3y = 3x * (4x*3y)
                   2:    (4 × 5) × 6 = 5 × (4 × 6)
The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.(want to Learn more about Real Numbers ,click here),
Associative property for real numbers does not satisfy in subtraction and division
                          (a-b)-c  ≠  a-(b-c)
For example:    (4-5)-6  ≠  4-(5-6)
                (a/b)/c ≠  a/(b/c)
For example (4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6)
Next is Distributive property of real numbers. In this we see that
a*(b+c) = a*b + a*c
(a+b)*(c+d) = a*c + a*d + b*c + b*d

for example 1:  4(x + 5) = 4x + 20
                   2: 3(4 – x) =12 -3x
                   3:  (a – 3) (b + 4) = ab + 4a – 3b – 12
Now comes to Identity property of real numbers
For addition:              a+0 = a
For example:           5y + 0 = 5y

The identity property for addition tells us that zero added to any number is the number itself.
Zero is called the "additive identity."

For multiplication:     a*1 =a
For example:  2c × 1 = 2c
The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself as a result. The number 1 is called the "multiplicative identity."

So this is all about real numbers. If you want to more details on the topics like range and Probability in Grade VIII then you can visit various websites on the internet.