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.

Math bolg on grade VIII

Friends, today i am going to teach you one of the interesting topic of mathematics, formulas for measurements . In  grade  VII  we  study  about  some  geometrical  figures  like  -  line, Triangle, circle  etc ,  but  in  grade VIII of maharashtra board we  study,  what  kind  of  formulas  are  we  use  to  measure  these  geometrical  elements and  learn  formulas for measurements which is  a very important part of online tutoring . Here  measurement means finding  distance , area  and  volume  of  geometric figures  and  so  we  divide  measurement  of  geometrical  figures  in  three  units –

• measuring   distance 
• measuring   area  of   geometrical  figures
• measuring  volume of  geometrical  figures

So,  first  of  all  we  discuss  formula  of  measurement of  distance  :
 1 )   if  there is  two  points P( x₁ , y₁ )  and   Q(x_{2 ,}y₂)  in  a  plane ,   then  formula  to   measuring distance  between  these  two  points   PQ  is  –
    sqrt ((x₂ – x₁ )²  +  ( y₂ –  y₁ )² )
    here   sqrt  means  square  root
2)  if  there  is  two  points  P( x₁ , y₁ , z₁  )  and   Q(x_{2 ,}y₂ , z₂ )  in  a  space ,  then formula  to  measuring  distance  between  these  two  points  PQ  is –
    sqrt ((x₂ – x₁ )²  +  ( y₂ –  y₁ )²  + (z₂   -  z₁ )² )
    here   sqrt  means  square  root and to know more about measurements click here,
Then  we  further  proceed  to   discuss   measuring  area  of   geometrical  figures  :

1.  Area  of   triangle     =    1    *   b  *  h
                                          2

                       here   b  refers    base  length   of  triangle   and   h  refers  to  height  length  of  triangle
2.   Area  of   square       =     a²
               here  a  refers  to  one  side  of  square
3.    Area  of  rectangle   =      a* b
                here  a  refers  to  length  of  rectangle and  b  refers  to  width  of  rectangle
4.    Area   of   parallelogram   =   b  * h
                here  b  refers  to  one  side  length  and  h  refers  to  width  between  two  sides
5.    Area  of   circle   =   pi * r²
                here  pi = 3.14  and   r  refers  to  radius  of  circle
6.    Area  of  eclipse  =  pi * r₁  * r₂
                here  pi = 3.14  and  r₁  and  r₂ are  two side  radius  of  eclipse
7.     Area  of   sphere  =  4 * pi * r²
                here  pi = 3.14  and  r  refers  to  radius  of  sphere
These  are  standard  area  formulas  which  use  to  measure  area  of   geometric  figures  and    we  further  proceed  to  discuss  formula  of  measuring  volume  of  geometrical  figures –

1. Volume  of  cube  =  a³ 

         here  a  refers  to  each  side  of  cube
2.  Volume  of  Cuboid  = a * b * c
         here  a  shows  length , b  shows width  and  c  shows  height
3.  Volume  of  cylinder  =  pi *  r² * h
         here  pi = 3.14  , r  refers  to  radius  of  cylinder  and  h  refers  to  height  of  cylinder
4.  Volume  of  pyramid   =   1    *  b *  h                            
                                              3
         here  b  refers  to  width  of  pyramid  and  h  refers  to  height  of  pyramid
5.  Volume  of   sphere    =    4    *   pi  *  r² *  h
                                              3
         here  pi = 3.14  ,  r   refers  to  radius  of  sphere  and   h  refers  to  height  of  sphere


These  are   standard  formulas for measuring  volume  of   geometrical  figures and  You can also refer Grade IX blog for further reading on Basic constructions.Read more maths topics of different grades such as Range in Grade VIII  in the next session here. 

Learn Geometry and Transformation on a coordinate plane

Hello Friends, in today's class we all are going to discuss about one of the most interesting and a bit complex topic of mathematics, geometry. In VIII grade gujarat board students  are  familiar  with algebraic expressions  of  mathematics , but  geometry  is  new  subject  for  them  . First  of  all   one  question  arises  in  student’s  mind  what is  geometry ? Basically  when  we  study  about  shape like Triangle, size  and figure  of  space   than  this  study  is  called  as  a  geometry (Learn more about natural numbers here),

like when two points  meets  it  makes  line  and  four  lines  make  rectangle  , square  etc  .  For  understanding  the properties  of  geometry  what  grade VIII  student  supposed  to  do,  they  have  to  study  about  coordinate  system  and   Transformations on a coordinate plane . coordinate  system  is  basically  a  system   which tell   position  of   points  or  position  of  geometry element . We  can  understand coordinate  system  by  coordinate  plane  .  So,  first  of all  we  have  to  understand coordinate plane . coordinate  plane  is  basically   two  intersecting  lines (combination  of  horizontal  and  vertical  lines)  where  horizontal line  is  known  as  a  x  coordinate  and  vertical  line  is  known  as  a  y coordinate , for  example  we  have  a  line  on  a  plane   whose  one  point  coordinate  is  (0,1)  means  x  direction  coordinate  is  0  and  y  direction coordinate  is  1 .  But  when  we  want  to study  about  3D  coordinate  of  space,  we  have  to  include  z coordinate  on  plane  which  shown  outward  direction  form plane  and  in  3d ,  coordinate  plane  work  as  a  ( x, y, z )  coordinates .
                                       Now , when  we  move  some  point  with  same  distance  in  same  direction  on   plane ,  then  this  process  is  called  as  a  Transformations on a coordinate plane . We  can  understand  Transformations on a coordinate plane  with  an  example  like  when  we  move  table  from  one  place  to  another in  room ,  then  we  have   to   move  each  bottom legs  of  table    with  same  distance in  same  direction .  In  mathematical  point  of  view , when  we  move  one  object P (x , y)   with      a  units  in  horizontal  direction ( moves  right )  and  b  units  in  vertical   direction ( moves  upward)   on   coordinate  plane ,  then   after  this  Transformations on a  coordinate plane  process   coordinate  of  object  P( x +a , y +b) . Here  one  think  is  noted  that  in  horizontal  direction  moves  right  shows  positive  movement  and  moves  left  shows  negative  movement  of  object   and  in  vertical  direction   upward  movement  shows  positive  movement   and  downward  movement  shows  negative  movement  of  object .  So, we  take  some  example  to  understand  Transformations on a coordinate plane –
Let  one  point  A (2 , 3)  is  moves  right  with  2 unit  and  moves  up  with  5  unit  then  after  transformation  coordinate  of   A ( 2 + 2 ,  3 + 5 )  or   A ( 4 , 8 ) .
Similarly when  A( 2, 3 )  is  moves  left with  1 unit  and  move  up  with  3  unit  then  after   transformation   coordinate  of  A ( 2 – 1 ,  3 + 3)   or   A ( 1 ,  6)
So, I  hope  u  understand  about  geometry  portion  of Eighth Grade syllabus and Read more maths topics of different grades such as Basic constructions
in the next session here.

Factors and Number Sequences in Grade VIII

We have studied a lot about numbers till now. We have come across different types of numbers in various grades of different education board, which includes Natural Numbers (used for counting), whole numbers (used for measurements), Even and Odd Numbers. Now we will proceed and learn more about Math problems associated with it.
Let us first talk about Factors of a given number. Any number which exactly divides any number is called the factor of any number. Let us take 16.
 16 is exactly divisible by 1, 2, 4, 8, and 16 . So 1,2,4,8, and 16 itself are the factors of 16.
Also we know that 1 is the factor of all the numbers as any number multiplied by 1 gives the number itself.
Multiple of a number: A number is multiple of all its factors, which means if we multiply any factors we get the number. In the above mentioned example we find that 16 comes in the table of 1, 2, 4 ,8 and 16 , so 16 is the multiple of all its factors. To find the multiples of any number we simply write its table as
 multiples of 5 are 5, 10, 15, 20, 25, ------- and so on.
Primes : Any number is called a prime number if it has only 1 and itself as two of its factors. It means that a number is prime, if it is divisible by 1 and itself.
Composite Numbers: The number which has factors other than 1 and itself are called Composite Numbers. The smallest composite number is 4 as the factors of 4 are 1, 2 and 4 itself.
 Co- primes: If we have two numbers, not necessary primes, such that they do not have any common factor except 1 are called co- prime numbers. Ex 4 , 5.  Here common factors of 4 & 5 is only 1. So they are co- prime numbers.
Twin Prime Numbers: A pair of  numbers which have two consecutive odd numbers are called twin primes. Ex: 3, 5   and  11, 13 are consecutive odd as well as consecutive prime numbers.Know more about factors here,
We come across such number sequences in our day to day mathematical work.

We should remember the following facts related to numbers:
a. 1 (one) is a number which is neither prime, nor composite.
b. 2 is the only even number which is prime. All other prime numbers are odd numbers.
c. 2 is the smallest Prime number.
 Prime factorization: To find all the factors of any number which are all primes they are called prime factorization. Suppose we need to find the factors of 36.
 36 can be written as 36 = 2 * 18, but 18 is not a prime,
 further we write 18 as 2 * 9, but 9 is again not prime,
 again we write 9 as 3 * 3, which are primes,
so we can write 36 = 2 * 2 * 3 * 3, where we get all the numbers as prime factors.

We observe that factors can be written in different order, but the product remains the same. This is called unique property of factorization.  Factorization also helps in finding the L.C.M. and H.C.F. of any given numbers. Also various numerical sequences are formed using the factors.

So children, this is a brief article about prime numbers, factors, multiples, and number sequences and If you want to know about Variables in VIII Grade and also about Congruence in Grade VII then there are many websites available on the internet where you can get detailed knowledge about all the topics.

Probability in Grade VIII

When we talk about the possibility of occurrence of any event, it is called probability and its an important part of cbse syllabus.
Any occurrence of the experiment is called an Event. Events can be dependent events or independent events.
Children if we talk about a independent event, in any experiment if the event occurs independently then it is called independent event. For example, if we throw a dice, the possibility is of  getting any numbers 1, 2, 3, 4, 5 and 6. But we find the possibilities are independent that is it does not depend on anything else, Two events are called to occur independently, if they to not depend on one another. Suppose 2 coins are thrown, we find that on both the coins we have the possibility of getting either a head or a tail. The possible outcomes are ( HH, HT, TH, TT ) but all these 4 outcomes are not dependent on one another, so they are  called Independent events.

Now let's talk about dependent events. When we conduct any experiment such that the observation of 2nd depends upon the observation of first experiment, then the event is called dependent event. You will study this type of math questions in higher grades.
Conditional Probability of any event is the probability condition when we have two events A and B associated with the same random experiment. Then probability of occurrence of  of event A under the condition that event b had already taken place is called conditional event.
In Grade VIII we will learn about basic experiments of probability.To know more about it refer this,
Let us take an experiment of throwing a coin. The possible outcomes for this event will be Head and Tail. Now a question arises that

•     what is the probability of getting a head in this event?

        We observe that the event has one probability out of two for getting a head, so
        P (Getting head ) = 1/2

•  what is the probability of getting a tail in this event?

      We observe that the event has one probability out of two for getting a tail, so

      P (Getting tail ) = 1/2

   Similarly if we study the pack of cards, we know that there are 52 cards in a pack. out of which 13 are ♣ ,13 are ♥, 13 are ♠ , 13 are ♦
   So probability of getting a ♣, if one card is drawn = 13/52  = 1/4
        probability of getting a ♥, if one card is drawn = 13/52  = 1/4

       probability of getting a ♠, if one card is drawn = 13/52  = 1/4

       probability of getting a ♦ , if one card is drawn = 13/52  = 1/4

     

      Further we elaborate and see, what is the probability of getting a red card, if a card out of a pack of 52 cards is drawn?
             P ( getting a red card ) = 26/ 52
                                               = 1/2
    what is the probability of getting a black card, if a card out of a pack of 52 cards is drawn?

             P( getting a black card ) = 26/ 52

                                               = 1/2

   In this way we solve various other problems related to it. To get information about Percent and Rates in IX Grade  and Mean in Grade VIII you can refer Internet.