Study Online Factors

In online study on factors, it is essential to know the definition of factors.The set of numbers which produce the remainder zero by dividing a particular number. These sets of numbers are called as factors. We can express a number by the multiples of another two numbers called as factors.

For Example,

                   10= 2 x 5

                   32=4 x 8

     Here 2, 5 are the factors of 10 and 4, 8 are the factors of 32.

     Types of Factors and Sample problems:


For the online study on factors, knowing the types of factors is an essential one. The types of factors for online study are as follows:

There are two types of factors,

Prime Factors
Composite factors

Online study on Prime Factors:
     The numbers that can be expressed as the multiple of one and that number itself are called as prime numbers. The factors of prime numbers are called as prime factors.

Consider the following Examples,

3= 1 x 3 (1, 3 are the prime factors of 3)

5= 1 x 5 (1, 5 are the prime factors of 5)

7= 1 x 7 (1, 7 are the prime factors of 7)

Online study on Composite Factors:

     The factors that are not a prime numbers are called as composite factors.

Consider the following example,

24=4 x 6

Here the numbers 4, 6 are not a prime number. Therefore, 4, 6 are called as composite factors of 24.

Example:

Find the factors of 130.
Solution:

130= 1 x  130

130= 2 x  65

130= 5 x  26

130= 10 x 13

So the factors of 130 =1, 2,5,10,13,26,65,130.

Important note: We can express all the numbers except “1” as a product of prime numbers are called as prime factorization.

Consider the following Example,

24 = 4 x 6

     = (2 x 2) x (2 x 3)

24 = 23 x 3

Here 2, 3 are the prime numbers. Hence, we conclude that all the numbers can be expressed as a product of prime numbers.

Prime factorization tree:

     By using the prime factorization tree, we can get all the prime factors of the given number.

Example:

Find the prime Factors of 124
Solution:

                                       124

                                         /  \

                                     2     62

                                             /  \ 

                                          2   31

                            124 = 2 x 2 x 31


Practice Problems on factors:


Following are the practice problems given for online study on factors.

1. Find the factors of 524.

2. Find the Prime factors of 100

3. Find the composite factors of 1056.

Histogram Calculation

A histogram is a graphical representation of frequencies of a variable. The variable range is equally divided and is scaled on the x axis and is continuous. The frequencies of the variable in each range is plotted on the y-axis and a rectangle is drawn  for each interval with the height of the rectangle equal to the frequency of the range and the width representing the range. The width of the rectangles in each range will be equal since total range is equally divided into intervals. The walls of adjacent intervals share the same line.The height of the histogram make the interpretation of the frequencies of each interval easier compared to a tabular representation. The more the height of the rectangle in an interval the more is the frequency within the interval.

Parts of Histogram:

Title:

The title section in the histogram briefly explains the information about the variable that is used in the graph.

Bars:

The bars are explained by thier height and width. The height represents the frequency of observations falling into the interval. The width represents the interval.

Legend:

The legend provides extra information about the relation to the documents where the data came from and how the dimensions were gathered.

Horizontal X-Axis:

The horizontal X-axis provide the scale value, which represent the dimension that fit into the data. These dimensions were usually recognized to the periods. Plot the horizontal X-axis points with in the bar chart which respect the values of Vertical Y-axis.The dimension determine the interval to be fixed.

Vertical Y-Axis:

The vertical Y axis represent the scale of frequencies and is optimized to represent the data on the graph. The units are acquiesced and is linear starting from zero.

Histogram calculation - Conditions in Histograms:

The scale should be supposed to have all the data values. The scale divides the range into equal parts.


Steps to construct histograms:

• Classify the data of the given histogram
• Identify the time period for the data
• Tabulate the data for the given histogram.
• Establish the range of the given data
• come to a decision for the number of height and width of each bar
• Count the number of items in each bar.
• Make a bar chart using the data given.

Histogram calculation - Example Problems:

Histogram calculation - Problem 1:

Calculate the histogram for the given data.

5    10    15    20    25    30    35    40    45    50    55    60
2      4      7    10      5      7      7      4      4    10      7      9



Histogram calculation - Problem 2:

calculate the histogram for the following given data.

100-200    200-300    300-400    400-500    500-600    600-700
45             34             89              23            67               98



Histogram calculation - Problem 3:

calculate the histogram for the given data set.

1   2      3      4      5    6      7      8      9    10
7    3    32    56    15    4    78    34    45    56

Frequency Distribution Tables

Frequency distribution:

The frequency distribution represents the number of observations with in the interval These are  graphically or tabular format.But the intervals must be mutually exclusive and exhaustive,and these are used with in a statistical context.The frequency distribution is a table it contains classes and its representing frequency.The class means quantitative or qualitative type in which the placed of raw data.The data is two types .1)Primary data 2)Secondary data.Primary data is collected by who are the investigator,some times an investigator uses another investigator of primary data.This type of data is called secondary data.The secondary data grouping and presenting in form of table.These type of tables are called Frequency  distribution tables.The data form is in the table shows below.

Data Range:

It is Highest value-lowest value

Class width

Range/ desired number of classes

Upper/lower class limit – upper/lower class Limit of next class

Upper class boundary – lower class Boundary

Class midpoint

Xm   = ( Lower bound + Upper bound)/2

Or

Xm = (Lower Limit + Upper Limit) /2


Frequency Distribution table:


.We can represents The frequency distribution table in number of ways.

1)Group frequency table or graph(polygon,chart)

2)Regular frequency table or Bar graph(Histogram)

When the data is from nominal or ordinary scale then we use Bar graph.

When the data measurements consists more categories than the listed in regular table then we use Group frequency distribution.


Important Notes:


1)The grouped frequency distribution table must use when the range of scores is large, causing a regular frequency table to have too many entries in the score categories (X column). The guidelines for a group frequency table include   approximately 10 rows in the table

2)  The Interval width of 2,5,10,20,50,100 should be used  it depends on the number of rows chosen

3)  Each interval of the  first (lowest) value of should be a multiple of the interval width

4) Note all intervals without missing any, The top interval must contain the highest observed X value and the bottom interval should contain the lowest observed X value.

Example:

For  the given data below into a frequency distribution table and show percentage of each category.
8, 9, 8, 7, 10, 9, 6, 4, 9, 8,
7, 8, 10, 9, 8, 6, 9, 7, 8, 8



x    f    ρ
10    2    0.1
9    5    0.25
8    7    0.35
7    3    0.15
6    2    0.10
5    0    0
4    1    0.05

Solving Irrational

solving irrational numbers:

Irrational numbers in difference to rational numbers aren’t presented a fraction of the shape: m / n,  where m and n are integers. There are numbers of a new type, which are calculated with any accuracy, but can’t be changed by a rational number. They can appear as outcome of geometrical measurements,

for example:

- a ratio of a square diagonal duration to its side length is equal to √2.

- a ratio of a circumference duration to its diameter length is an irrational number  Pi



Where can you find special irrational numbers?(solving irrational)


The answer to this depends on what you consider 'special'.  Mathematicians have proved that definite special numbers are irrational, for example Pi and e.  The number e is the base of natural logarithms.   It is irrational, just like Pi, and has the value 2.718281828459045235306....

It's not easy to just "come up" with such special numbers.  But you can easily discover more irrational numbers after you've found that most square roots are irrational.  For example, what do you think of √2 + 1?  Is the result of that addition a rational or an irrational number?  How can you know?  What about other sums where you add one irrational number and one rational number, for example √5 + 1/4?

You can also add two irrational numbers, and the amount will be many times irrational.  Not always though; for example, e + (-e) = 0, and 0 is rational even though both e and -e are irrational.  Or, take 1 + √3 and 1 - √3 and add these two irrational numbers - what do you obtain?

Or, increase/separate an irrational number by a rational number, and you get an irrational number.  For example, √7/10000 is an irrational number that is moderately close to zero.  Yet another possibility to discover irrational numbers is to multiply square roots or other irrational numbers.  Sometimes that outcome in a rational number though (when?).  Mathematicians have also calculated what happens if you raise an irrational number to a rational or irrational power.

Yet more irrational numbers begin when you take logarithms, or calculate sines, cosines, and tangents.  They don't have any particular names, but are just called "sine of 70 degrees" or "base 10 logarithm of 5" etc.  Your calculator will give you decimal approximation to these.

Examples of Irrational Number


`Pi`,  √10 Are few examples of irrational numbers.

Solved Example on Irrational Number

Identify the irrational number.

Choices:
A. 0.3
B. 3 / 5
C. √6
D. √25
Correct Answer: C

Solution:
Step 1: The value of  is 2.4494897427831780981972840747059...
Step 2: The value of  is 5.
Step 3: The value of  is 0.6.
Step 4: Among the choices given, only  is an irrational number

FRACTIONS UNLIKE DENOMINATORS

A fraction  is a number that can represent part of a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

How to Add Fractions with Unlike Denominators:

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction.

Consider adding the following two quantities:

3/4 + 2/3

First, convert  3/4  into twelfths by multiplying both the numerator and denominator by three3/4 x 3/3 =9/12

Note that  3/3 is equivalent to 1. which shows that  ¾ is equivalent to the resulting  9/12.

 Secondly, 2/3convert  into twelfths by multiplying both the numerator and denominator by four:2/3 x 4/4 = 8/12. Note that 4/4  is equivalent to 1, which shows that  2/3  is equivalent to the resulting 8/12 .

9/12 + 8/12 = 17/12.


Simplify the Fraction

Example: Find the Sum of 2/9 and 3/12

Determine the Greatest Common Factor of 9 and 12 which is 3

Either multiply the denominators and divide by the GCF (9x12=108, 108/3=36)

OR - Divide one of the denominators by the GCF and multiply the answer by the other denominator (9/3=3, 3x12=36)

Rename the fractions to use the Least Common  Denominator(2/9=8/36, 3/12=9/36)

The result is 8/36 + 9/36

Add the numerators and put the sum over the LCD = 17/36

Simplify the fraction if possible. In this case it is not possible.

Example:

Find. 3/4  - 2/3

Solution:

Since the fractions have different denominators, they cannot be subtracted until they have the same denominators.

We can express both fractions as twelfths.

3/4 = 4/12 and 2/3 = 8/12

9/12 - 8/12 = 1/12.

Example:

Find

 5/6 + 2/3

Solution:

Since the fractions have different denominators, they cannot be added until they have the same denominators.

We can express both fractions as sixths.

2/3 = 4/6

5/6 + 3/6 = 8/6

Graphing Inequalities

In this article we are going to discuss about graphing inequalities concept.A coordinate graph is called as the Cartesian coordinate plane. The graph contains a couple of the vertical lines are called coordinate axes. The vertical axis of the y axis value and the horizontal axis value is the x axis value. The points of the intersection of those two axes values are called the origin of coordinate graphing pictures. In this graph linear equation is in the form of y = mx + c. I

Graph of inequalities

Below you can see some examples on graphing inequalities -

Problem 1 :

Solving the system of inequalities for two variables equation use graphing method 7x – 6y < 5 and draw the graph for a given inequalities.

Solution:

We find the plotting points of the given inequality equation. The given equation is

            7x - 6y < 5

We are going to find out the plotting points for a given equation. In the first step we are going to change equation in the form of y = mx + c, we get the following term

                7x - 6y < 5                  

                  7x < 6y + 5

In the next step we are multiply the value -1 to equation (1), we get

            -7x > - 6y - 5

               6y > 7x – 5

In the above equation we put x = -1 we get

            6y > -7 - 5

            y > -2

In the above equation we put x = 2 we get

            6y > 14 – 5

             y > 1.5

We draw a graph for given equation. We get





Solving the system of inequalities for two variables equation use graphing method 4x – 3y > 2 and draw the graph for a given inequalities.

Solution:

We find the plotting points of the given inequality equation. The given equation is

            4x - 3y > 2

We are going to find out the plotting points for a given equation. In the first step we are going to change the above equation in the form of y = mx + c, we get

      4x - 3y > 2

            4x > 3y + 2                 

In the next step we are multiply the value -1 to a above equation, we get

               -4x < -3y - 2

                   3y < 4x - 2

In the above equation we put x = -1 we get

            3y < -4 - 2

             y = -2

In the above equation we put x = 2 we get

            3y < 8 - 2

             y < 2

We draw a graph for given equation. We get

The graphical concept of x intercept and y intercepts is Very simple. The x intercepts are the points where the graph crosses the x-axis, and the y-intercepts are the points where the graph crosses the y-axis. The problems start when we deal with intercepts algebraically.

The X and Y intercept are drawn in the graph to find the slope of the graph from the given equation.

Then, algebraically,

an x intercept is a point on the graph where the point y =  zero, and
A y intercept is a point on the graph where the point x = zero.

Procedure for finding X and Y intercept

x Intercept

Step1: For finding the X intercept we need to arrange the given equation in the slope intercept form.

Step2: The slope intercept form is    y = mx + c

Step3: After this we substitute y=0.

Step4: The equation becomes mx + c = 0, now we need to add/subtract constant C on both sides.

Step5:  Then divide the constant by the slope.  x = c/m.

y Intercept

Step1: For finding the y intercept we need to arrange the equation in slope intercept form.

Step2: After this, substitute x = 0

The equation becomes y= c, this is the Y intercept.


y and x intercept for 2x+y=8.


2x+y=8

The given equation is in the form of slope intercept form i.e. Y = mX + C

To find X Intercept, put y = 0

0 = -2x + 8

Subtract 8 on both sides,

0 – 8 = -2x + 8 – 8

-2x = -8

Divided 2 on both sides

-2x/2 = `-8/2`

-x = -4

Here the x intercept is 4.

To find the Y Intercept, put x = 0

y = 0 + 8

y = 8

Here the y Intercept is 8.

Using this we can plot the graph.