What is periodic Table?

The periodic table contains the table of chemical elements arranged in a particular order. It shows the symbols of the element, it gives the full name of the element; it also gives their atomic and mass number.  The elements are arranged in columns and rows based on their structure and properties. This arrangement in columns and rows are called groups and periods.  Since the horizontal rows are named as periods it is called “the periodic-table”.

Periodic table for kids can be explained in a simple manner so that it can be easily understood. The periodic table basics are listed below.
It is nothing but it is like a usual table containing rows and columns.  These rows and Columns give us the information about the various elements.  This is of immense help to chemists to help them understand how and why elements react.  There are more than 100 elements arranged in the periodic-table. The table is read just like any other information from left to right and top to bottom.  The elements are arranged in order of their atomic number in ascending manner.  Rows are called as periods and columns are called as groups.

Periodic table names:   It contains the names of more than 100 elements.  The first one found on the extreme upper left corner of the table is hydrogen. The atomic number of Hydrogen is 1. The element that is located at the extreme right top corner is Helium, which has an atomic number of 2. Elements are listed in this pattern.  To find the next element one has to look down one row to the left side of the table to find the next element. Which will be lithium, then comes Beryllium, Carbon, Boron, Nitrogen and so on.  

Periodic table charges refer to the ionic charge of the corresponding element.    If the number of protons and   number electrons is not same then the atom gains or loses   the charge. The charge of an electron is negative and the charge of a proton is positive.  If the atom is not reacting then the charge will be zero since the number of electrons and protons are equal.    The charge of the atom becomes negative if it gains an electron since electron has negative charge, this leads to the atom become an anion.   The charge of the atom becomes positive if there is a lose in the electron.  This leads to the atom becoming cation.

Basics of Simple Interest

The term interest refers to the cost of borrowing money. The interest calculation varies from plan to plan and is also based on the lenders and the time period of lending/depositing the amount. This interest is calculated in different ways such as interest only on principal, interest on principal and interest so far earned/incurred, monthly interest, cumulative interest etc.  The amount for which the interest is calculated can be a loan (amount borrowed for need) or a deposit (amount deposited as savings).

Definition
The interest which is calculated only on the principal amount borrowed or deposited is termed to be Simple Interest that is denoted as SI in short.  This type of interest does not include the interest so far incurred or earned on the principal amount.

When an amount is borrowed, the amount borrowed is called the Principal. The duration which the borrower takes to return the Principal is termed as the time period and is calculated in number of days/months/year.  The next and most important part is the rate of interest which states the interest percentage for the given principal amount.  All the three put together explains what is Simple Interest. The interest percentage is directly proportional to the lender and the time duration to repay the amount.   Also it depends upon whether it is a loan or a deposit.

Formula for SI Calculation
Formula 1: “R as number”
The Simple Interest Formula is given by S.I = (P*N*R)/100.

Formula 2: “R as percentage”
The Simple-Interest Formula is given by S.I = P*N*R.

This shows how to calculate Simple Interest using the interest for the given principal amount P, with rate of interest R/period of a given period of time P.

Example of SI Calculation
If a principal amount of Rupees 1000 is borrowed/deposited by a person for a period of 2 years with 3% rate of interest, then the simple-interest is given by

By applying the values of P, N and R given, we get

Formula 1: R as number

We know that the rate of interest is 3%. We take R = 3

S. I.  = (P * N* R)/100
S. I.  = (1000 * 2 * 3)/100 = 6000/100 = 60

Formula 2: R as percentage

We know that R = 3% = 3/100 = .03

S.I. = P * N* R
S.I. = 1000*2*.03 = 60.00 = 60

Applications of SI
In post offices, schemes such as MIS pay simple interest for the amount deposited for five years as recurring deposit.

Credit cards charging simple-interest for the amount to be paid is more beneficial.

Markup

This topic is related to business, fundamentally buying and selling. The total expense that you incur on an item before selling is the actual cost of that item. The price at which you sell the same item is your revenue. The difference of (price – cost) is the profitor the income you get out of the sale. If, for any reason, the difference is negative, then it is no more a profit but a loss.  Let us continue our discussions with only the situations of profit.

The profit definitely has a bearing on the decision of fixing your price. If the market situation is favorable you can fix the price at your discretion as long as it is more than the cost you have incurred plus the income you desire to have. But in the competitive world the situation is not so.

So you have to very prudently decide the amount to be added by which you are neither priced out nor suffer a loss. The amount such added is called as markup. So basically it is also refers to the profit but decided purely on the cost and not on the favorable market conditions. Hence, mark up is the minimal profit in business.

Now let us go to the mathematical part like markup percentage, calculating mark up or mark up calculation.  Suppose ‘x’ is the cost of purchase of an item plus the other incidental expenses you incurred. Let ‘y’ be the minimum amount that you feel the transaction is worth.

All said and done that is the amount of income for your survival which cannot be considered as a part of the actual cost. So the percentage of markup is [(y/x)*100]%. Generally a merchant uses this percentage as a key for similar items and for a brief period.

Now let us do an actual mark-up pricing example. Suppose you buy an item for $95 and incurred $5 subsequently to cover up all the incidentals and overheads. so the actual cost of the item is $100. Your experience prompts you that only with 20% mark-up on each sale you can sustain in business. So the sale you could decide is $120. But the merchant will not immediately decide on this.

If he senses that the particular item is in great demand or others are selling at higher price, he will sell only at the market price and hence in such case the actual markup is much higher than he envisaged. On the other hand, even with the minimal adding of 20% if he is priced out in the market he will decide to stop selling that particular item.

Using Elimination to Solve Linear Equations

There are different types of equations in mathematics. There can be different methods to solve these equations. One can use solve by elimination method to solve the equations. The equations can be linear and also quadratic in nature. Sometimes they are cubic in nature. This method is used to solve a set of linear equations. If the numbers of variables to be found out are ‘2’, then a set of equations must be given to find them.

A set of equations means two equations must be given. The number of equations can be deciding factor in finding the solution. In this method an unknown variable is eliminated first. Then the value of the second variable can be easily found out. Once the value of the first variable is found out then this value can be substituted in any one of the given equations and the second variable can be easily found out. This is one of the easiest techniques that can be used to find the unknowns in linear equations. For solving equations having three unknowns three equations must be given, otherwise it is not possible to find the unknowns. This is a prerequisite to find the unknowns.

The ‘solve by elimination technique is used when a set of linear equations are to be solved. This technique can be very useful and simple to use. There are various techniques available to solve linear equations. This is one of the simplest methods and can easily understood and used.

This is why it is most commonly used by the students. It is also easy to cross-check the answers. The solution obtained must satisfy the given equations. The process of solving by elimination can make the task of solving of linear equations very easy.

Once solve by the elimination method is learnt, the other methods to solve the linear equations can also be learnt. This is the basis for the other methods. So, one must thoroughly understand this method. Here elimination refers to the fact that a variable is first eliminated in this method. Once a variable is eliminated the second variable can be easily found out. Then this value helps in finding the other variable.

This can be done by using the principle of substitution. The value found out is substituted back into the equation and the value of the other variable is easily found out.

Diagonal Matrix Properties

A (mxm) matrix is a square array of numbers which has equal number of rows and columns here denoted by ‘m’. The elements of a square matrix that start from the left corner towards the right corner like a diagonal is called the Main Diagonal of a Matrix. For example, a 3x3 matrix given by
A=[-1  3  4]
 [2   1  0]
 [7  0   5]
The elements [ -1,1,5] form the elements of the main diagonal. Now consider A=(aij) to be a square matrix then the elements in aii is the diagonal of a matrix.  A square matrix in which all elements except the elements of the leading diagonal are zero is called a Matrix Diagonal. In this all the non diagonal elements are zero. For instance, a square matrix given by A=[aij]nxn to be a diagonal-matrix , aij is equal to zero wherever ‘i’ is not equal to ‘j’. Example of a diagonal-matrix can be given as follows, a 3x3 matrix given by
A=[2    0    0]
     [0   -1    0]
     [0    0    5]

Let us now take a look at the Diagonal Matrix Properties:
• If A and B are two diagonal matrices of the same order then, A+B and AB are also diagonal matrices.  Also AB=BA, that is they commute
• A square matrix is diagonal only if it is triangular and normal
• If A is a diagonal-matrix then the adjugate of A is also a diagonal matrix
• Any diagonal matrix is symmetric
• It is a scalar matrix when all the entries of the diagonal are equal
Determinant Diagonal Matrix
The determinant of a 2x2 square matrix A=[a    b] can be defined as |A|=ad-bc. It is denoted by det(A)
    [c    d]
Let A=[aij] be a nxn matrix where n is greater than or equal to two. We can define determinant,
|A|= a11C11 + a12C12+……..+a1nC1n which is equal to summation(i= 1 to n)[a1iC1i]. If A is a nxn diagonal matrix then |A| is the product of the elements in its main diagonal.
Let A = [a11       0       0……     0]
             [0         a22     0 ……   0]
     [0          0      a33……    0]
     [..         ..      ..     ..]
     [..         ..      ..     ..]
     [0          0        0…. .ann]

be a nxn diagonal matrix, then the determinant of this matrix can be given by a simple formula, det (A)=|A|= pi(i= 1 to n)[aii]
Consider the following example,
A=[2   0    0]
      [0   5    0]
      [0   0    1]

then det(A)=|A|= (2)(5)(1)=10
Note that if any of the elements of the main diagonal is zero then determinant would also be zero.

Process of Solving Inequalities

In mathematics both equalities and inequalities are present. The equalities give rise to equations. In equations the ‘equal to’ sign is used. There are various methods to solve the equations. The best and the easiest method must be used in order to arrive at the answer quickly and efficiently.

The concept of solving inequalities by addition and subtraction is very easy to understand and use. The concept can be very easy to understand if one is thorough with the concept of basic arithmetic operations like addition and subtraction.

The concept of inequality can be easily understood with the help of examples. The examples can help in explaining the concept very clearly.

In solving inequalities using addition or subtraction, the basic arithmetic operations of addition and subtraction are used. The inequality is called an inequality because the terms on the left side will not be equal to the terms on the right hand side. This is the basic reason for the term being called an inequality. This is the basic difference between the equality and inequality. In equality the terms on the left side will be equal to the terms on the right hand side.

The process of transposing will be used to solve the equations. Even in the case of inequalities the same process or principle is used. This is very simple principle. The terms on one side of the equation are moved to the others side; while doing so the sign of the term changes.

The positive term becomes the negative term and the negative term becomes the positive term. The multiplication becomes division and division becomes multiplication. So, one must be very careful while transposing the terms.

An inequality is formed when a term may be less on side or the other side. This is nothing but a side is surplus and the other side contains slack variable. So, the surplus has to be reduced to convert the inequality into equality.

But the same method can be used to solve both types of problems. Once the solutions are obtained for these types of problems it is necessary to check the feasibility of the solutions. All the solutions obtained might not satisfy the given equations. So, one has to be careful in selecting the final solutions. These solutions which are selected must satisfy the given equations. In case of inequality the solution must satisfy the inequality.

Plan your study schedule with your private tutor

It is proved that private tutoring helps students in excelling in exams. It nurtures students’ understanding skill and brush up their knowledge. Students can schedule their learning sessions with preferred private tutor and experience a better session.

Private tutors help students to perform well
Private tutoring has achieved a noticeable popularity among students of any grade. Class timings of school are limited and sometimes it is not adequate to learn a subject thoroughly. To handle this problem, private tutoring is taken as a useful solution by many students. This facility allow students to understand any topic at their pace. Most importantly, with this service, students get extra time to clear their doubts and understand each concept precisely. Private learning helps student in enhancing their performance in exam. Apart from assisting to complete the prescribed curricula, private tutors also help students in solving different worksheets that builds a good learning skill.

Flexibility makes private tutoring a one stop solution for students

Several positive features have made private learning service a useful learning option to students. It is flexible and beneficial as it provides extra time to students to get better understanding on each topic. Numbers of learning centres are available and students can choose any centre that comes near to their residence. They can choose online learning service as well. With this service, they can schedule their learning sessions from home. Moreover, experienced and well-trained tutors are the asset of this learning platform. They sincerely guide students and nurture their talent in a better way.

Different modes of tutoring
Online tutoring and live tutoring at centres are two types of private tutoring service where students can take additional learning help to complete their topic thoroughly. There are some differences between these two methods.
* Live tutors assist students in a group whereas online tutors provide one-on-one session to students. Hence, online assistance is more beneficial to students who feel hesitate to ask questions in class.
* Live tutors are only available in learning centres and students need to visit the centres to attend allotted classes whereas in online platform, students can schedule their own sessions from home at their convenient time. Hence, online learning help saves students time.

Online private tutors are more popular than live tutors
Several positive aspects have made online tutoring more popular than live private tutoring. Online learning mode is flexible, safe and fun-to-use. The expert tutors are available 24 hours and they not only help in understanding different topics but also assist in completing homework and assignments on time. Additionally, the usage of advanced and user-friendly tools like virtual whiteboard and attached chat option make each session interactive and productive. So students can take assistance from tutors to perform better in exams.

Solve Math Problems Better Under the Guidance of Expert Tutor

Learn simple steps and techniques for solving complex Math problems from highly experienced tutors. Getting help from qualified tutors will improve your Math problem-solving skills and also make you confident during examinations.

Math has a high importance as it is widely used in several fields like Banking, Finance, Engineering, Research, Science and Technology and many more. By utilizing mathematical skills, students can improve their problem solving capabilities. Needless to say, mathematical concepts are required
in every walk of life and it is best  to learn the subject in a detailed manner. Learning this subject from a well-qualified tutor will certainly upgrade students knowledge and make them better in dealing with tricky and difficult Math sums. Apart from this, Math also helps students to understand the world in a better way as mathematical principles are being used in day to day life of every individual.

It is good to learn Math from an experienced tutor who possess sound knowledge of the subject. Whether you opt for a live tutor or an online tutor, it is important to understand Math concept in a step-by-step manner. A live tutor will teach you the subject in a classroom or in a learning center but online tutor will give you the freedom to choose a preferred location and a convenient time before starting a session. Learning problem-solving techniques under the supervision of an Online Math tutor will be quite advantageous as it will makes you confident, save your time and cost.

Getting help from an expert tutor not only upgrade your knowledge but also assist you in completing homework and assignments in a flexible manner. Right from understanding the concept of College Algebra to Calculus, Geometry or Trigonometry, an online tutor will give instant solutions to all your Math problems. Online Math tutors generally make use of various teaching methods to make a learning session more interesting for students. A whiteboard is mainly used by the tutor to demonstrate  the steps involved in solving a math problem. Moreover, chat option is also being used by the tutor to comprehend student's queries.

Many students face trouble while doing Math problems and end up their day with lots of frustrations. Getting accurate Math answers in an elaborate way from an experienced tutor can minimize students worries, which they often face while doing math sums. A student can also opt for a free learning session with a preferred tutor so as to understand the importance of an online tutor as well as online tutoring. Experienced tutors not only work on your problems but also explain and teach you the methods and steps for solving any math question in an easy manner.

Linear Programming Simplex Method

In Linear Programming Simplex Method is a method used for problem solving. This method is used for a problem of the form:
Minimize or maximize : c.x
Subject to
Ax=b, xi ≥0  
X is the variables of the problem (x1,x2…) while c is coefficients of the objective function (c1,c2,…). A is a matrix and b are constants≥0.
To solve the linear programming problems we must follow standard form which is achieved as:
A new variable should be added for all those variables whose upper bound is other than zero such that the new variable is the difference between original variable and upper bound.
For example: if x≥3 then introduce a new variable y such that
Y=x-3 and x=y+3.
For rest of the inequality constraints, new variable known as slack variable is introduced to remove inequalities. Like:
X+2y≤3 and –x+3y≥2 is replaced as:
X+2y+z=3 and –x+3y-z=2 such that z≥0.
Let us solve some problems using The Simplex Method

Question) Use Simplex Method to solve the following:
P=3x+4y subject to:
x+y≤4
2x+y≤5
x≥0,y≥0

Solution)  Since we have two constraints, we will introduce 2 slack variables p and q:
x+y+p=4
2x+y+q=5
We rewrite our objective function as −3x−4y+P=0 and thus system of equations become:
x+y+p=4
2x+y+q=5
−3x−4y+P=0
This gives initial simplex table:
X y p q P
1 1 1 0 0 4
2 1 0 1 0 5
-3 -4 0 0 1 0

Find the column with the most negative entry among x,y,p,q and P (here this is −4). Find pivot row by dividing each entry in the constant column by the entry in the corresponding in the pivot column. In this case, we get 4/1 as the ratio for the 1st row and 5/1 for the ratio in the 2nd row. The pivot row is the row corresponding to the smallest ratio which is 4 in this case. So our pivot element is in the 2nd column, 1st row =1.Now, perform the following row operations to convert the pivot column to a unit column:
R2→R2−R1
R3→R3+4R1
So, simplex table is changed to:
X Y p q P
1 1 1 0 0 4
1 0 -1 1 0 1
1 0 4 0 1 16
The variables are given the value in the constant column in the row where a value 1 is in the unit column. All variables above a non-unit column is given 0 value. So y=4, p=1, P=16, x=0, and q=0.
Thus, maximum occurs when x=0, y=4 and the maximum value is 16.
This is how Simplex Method Solver works.

Change of base formula

In logarithms, many times we encounter with the problem when there is a logarithm whose value is not known to us for that particular base, but we know its value for some other base.
For such cases we work by changing the base of logarithms by using Logarithmic Change of Base Formula which is easy to implement.

The Change of Base Formula for logarithm:
log a (x) = log b (x) / log b (a)
Here ‘a’ is base of logarithm of x before change and, b is base of logarithm of x after changing the base.
So above Log Change Base Formula can be used when you need to change base of logarithm from ‘a’ to b provided that the new base is useful to you, otherwise changing a base will not be useful.
Let us see proof of Logarithms Change of Base Formula and then we will go through some examples:

Proof:
We know that Raising ‘a’ (base of log a (x)) with the power of log a (x) gives x:
 x = a^(〖log〗_a  (x))………………………………………….(1)
Similarly, raising b with the power of log b (a) gives a:
a = b^(〖log〗_b  (a))        …………………………………………(2)
When we replace a in (1) with b^(〖log〗_b  (a)) given in equation (2), we get:
x = a^(〖log〗_(a ) (x)) =〖〖(b〗^(〖log〗_b  (a)))〗^(〖log〗_a  (x)) =  b^(〖log〗_b  (a)  〖* log〗_(a ) (x))     ………(3)
Applying log b() on both sides of equation (3) we get:
log b (x) = log b 〖(b〗^(〖log〗_b  (a)  〖* log〗_(a ) (x)))    
Using the log power rule we get:
log b (x) = log b (a)*log a(x)
Log a (x) = log b (x) / log b (a)
Now let us see some examples using the above formula:

Example 1) Change the base of following logarithm to 2: log12 64.
Solution) Using change of base formula this can be evaluated as:
log12 (64) = log2 (64)/log2 (12)

Example 2) Solve the following by changing base to e: log10ee
Solution) log10 ee = loge ee / loge 10
= e/ln 10 (loge 10 is written as ln 10)
= e/2.302   (as ln 10 =2.302; use calculator for this)

Example 3) Solve log9 27.
Solution) here it is difficult to find the value directly so we will change the base to 3 as 9 and 27 is square and cube numbers of 3 respectively.
Log 9 27 = log 3 27 / log3 9
= log3 33 / log3 32
= 3/2   (As loga bc = c(loga b) and also loga a = 1; these are other properties of logarithms)

Elementary row operations

A matrix is an arrangement of expressions, defined in general terms. The items that are arranged are called as elements. We repeat that it is only an arrangement; thereby a matrix does not suggest any algebraic operation between the elements. Due to this fact a matrix can undergo certain operations with its rows, called as matrix elementary row operations or simply as row operations. The row operations are also called as row transformations. Such transformations can be done on columns also.  Let us study the elementary row operations one by one.
1) In a matrix, a row or a column can be interchanged. For example,

a   b   c     can be interchanged as, d e f  or,  as,  a c b                                                                                                              d   e   f                                   

a   b   c             

d   f   e                                                                                                                                                                                 g   h   i

i    g   h            

g   i    h 

In the above example, the row interchange isdenoted as R2 <->R1 and the column interchange is denoted as C2 <-> C3.

2) A row or column can be modified by multiplying by a non- zero real number.For example,

a   b   c     can be modified as,   k[a]  k[b]   k[c]    or,  as,  k[a b c]                                                                                                             d   e   f                               d       e     f            k[d  e   f]                                                                                                                                                                                g   h   i                               g        h     i            k[g  h  I ]

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> kR1 and as C1 -> kC1

3) A row or column can be modified by multiplying by a non- zero real number. For example,

a   b   c     can be modified as,   a + kd    b + ke   c + kf      or,  as,   a + kb   b   c                                                                                                              d   e   f                                            d            e          f                        d + ke    e   f                                                                                                                                                                                 g   h   i                                            g             h          i                        g + kh    h   i

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> R1 + kR2 and as C1 -> C1 + kC2

These elementary transformations are extremely useful in further topics of matrices, like finding inverse of a matrix.For example, if A is an invertible matrix and B is its inverse, the formulas are,

A = I A and I = BA where I is the identity matrix of the same order.

Start with the equation A = IA.

Plug in the given matrix for A only on the left side and write only the identity matrix times A on the right. That is let the symbol A or the right remain as symbol A.

By repeated elementary row transformations, try to reach in the equation form I = BA. Then the matrix represented by B is the inverse of A.

Understanding of end behavior of functions

End behavior definition:
With respect to polynomial functions, the end behavior of the function refers to the direction and position of the two ends of the function. For example the graph of the function y = x^2, would be as follows:

Note that both the ends of the graph are going upwards. However if we see the graph of the function y = x^3, it would look as follows:

In the above graph, both the ends of the graph point in different directions.

End behavior of polynomial functions:
Polynomial functions have primarily 4 types of end-behaviors. They are as follows:
(a) Both ends going up.

(b) Both ends going down.

(c) Left end going up and right end going down.

(d) Right end going up and left end going down.

It is possible to know the end behavior of a polynomial function without actually graphing it. We use the end behavior model for that. The following are the end behavior rules that help us model the end behavior of polynomial function.

(1) If the degree of the polynomial function is even, then both the ends would point the same direction. Therefore for an even degree polynomial function, both ends would either go up or both the ends would go down.

(2) If the degree of the polynomial function is odd, then both the ends would point in opposite direction. Therefore for an odd degree polynomial function there are two possibilities, either left ends goes up and right end goes down or left end goes down and right end goes up.

(3) If the coefficient of the leading term is positive, then the right end of the polynomial function would always point up. Therefore if the polynomial is of even degree with positive leading coefficient, then both ends would point up, and if the polynomial is odd with positive leading coefficient, then as right end points upward, the left end would point down wards.

(4) If the coefficient of the leading term is negative, then the right end of the polynomial function would always go down. Therefore if it is an even degree polynomial with negative leading coefficient, then both the ends would go down whereas, if it is an odd degree polynomial with negative leading coefficient, then as right end goes down, left end would go up.

End behavior asymptotes:
Polynomial functions do not have asymptotes. However, if it is a rational function, then it would have asymptotes.

Ratios

Ratio is one of the most important concepts in mathematics. Ratio is used to make comparison between two values in mathematics. Ratio in math is represented by colon sign (:). Ratio in words is represented with “is to”. For example: 6 pillows for kids : 3 kids. This expressional can be written or said in word as 6 pillows for kids are to 3 kids.  Ratio can be written in several ways. Let’s have a look at the same in this post along with relevant examples. .

Ratio as a fraction: Ratio can be used as a fraction. For example:
7 apples / 2 kids
13 baby seating chair / 4 kids
30 bed / 2 rooms
Ratio using the word to: Ratio as mentioned above cam be used with the word “is to” or “to”.

For example:
8 baby seating chair is to 2 kids. Therefore each kid will get 4 baby seating chair .
Mohan will buy 3 baby beds online for his 3 nephews. Here, 3 baby beds online is to 3 nephews.
Shreya will buy 5 baby beds online and 2 baby beds from market. Therefore, 5 are to 2.
Ratio using colon: Ratio using colon is the most commonly used form of ratio. It is used to separate two or more values. For example:
8 apples: 4 kids
32: 16: 8: 4: 2
16 boys: 8 girls in the class.
If we multiply each value in a ratio with a non-zero number will result an equal ratio. For example: 2: 4 = 1: 2; 16: 8 = 2: 1; 45: 9 = 5: 1 and so on. Here, 2: 4 and 1: 2 can be referred as equal ratios.
These are some basics about ratio and its usage in mathematics.

Fraction and its types

Fraction is one of the most important concepts in mathematics. When an object is divided into two parts, then each part is called a fraction. For example: Two thirds of the class wanted phone for kidsas gifts. This can be written as 2/3 of the class wanted phone for kids as gifts. Let’s have a closer look at fraction and its types in this post.

Numerator and Denominator :

Every fraction has a numerator and a denominator. Numerator represents the number of parts in the fraction and denominator represents the number of parts in the whole object. For example: 5/6 of toys for infants are made considering hygiene and safety. Here, 5 is the numerator that represents the part of fraction and 6 is the denominator that represents the whole object, that is the number of Baby toys.

Fractions can be classified into three types – proper fraction, improper fraction and mixed fraction.
Proper fraction: When the numerator is smaller than the denominator, it is called a proper fraction. For example: 4/5 has spoken in favor of rechargeable flashlight . Here, 4 less than 5 and therefore, it is a proper fraction.

Improper fraction: When the denominator is smaller than the numerator, it is called an improper fraction. For example: 8/7 has spoken in favor of rechargeable flashlight . Here, 8>7 and therefore, it is an improper fraction.

Mixed fraction: Mixed fractions are formed with a whole number and a fraction. For example:1 2/4. Here, 2 is the numerator, 4 is the denominator and 1is the whole number.
these are some basics on fraction. Let’s do some exercises.

Find the type of fraction from the following
• 2/3:  Proper fraction
• 4/5:  Proper fraction
• 11/2: Improper fraction
• 17/6:  Improper fraction
• 2 4.5: Mixed fraction
These are the common facts about fraction in mathematics.

All about circles

General equation of a circle:
The standard equation of a circle is an equation that when plotted on the co-ordinate plane results in a circular curve. The equation is as follows:
(x-h)^2 + (y-k)^2 = r^2
Where, (h,k) is the centre of the circle and r is the radius of the circle. For example if the centre of the circle is (2,-3) and radius is 4. Then the equation of the circle would be:
(x-2)^2 + (y+3)^2 = 16
The graph of the above circle would look like this:

Standard equation of circle:
A circle with centre a origin and radius r is called a standard circle. Therefore the equation of such a circle would be:
X^2 + y^2 = r^2
Where r = radius of the circle.
For example the standard equation of a circle with radius 3 would be:
X^2 +y^2 = 9
The graph of such a circle would look as follows:

Parametric equation of a circle:
If the parameter is an angle of measure t, then the parametric equation of a circle is given by:
X = r cos(t) and y = r Sin(t), where r = radius of the circle and the angle t ranges from 0 to 2 pi. For example the parametric equations of a circle with radius = 2 would be:
X = 2 cos(t) and y = 2 sin(t).
To be able to find the equation of a circle we need to know two things primarily, the centre of the circle and the radius. However for the standard equation only the radius need be known.

Circumference of Circle Equation:
Suppose there is a circular path around a lake. If we start moving from one point on the path in one direction, then since the path is circular we shall come back to our starting point. The distance thus covered by us in making this trip is called the circumference of the circle. There is no accurate measure of centre of a circle. If we wish to measure the circumference of the base of a cylinder, we can wind a thread around it, and then measure the length of the wound portion of the thread. However this practical method is not too useful for theoretical calculations. Moreover if we wish to find the circumference of very large cylindrical silos then this method is not even practical. The empirical formula derived by mathematicians for circumference of a circle is as follows:
C = 2pir, where r = radius of the circle.

Relations and functions

In Math Relations and Functions play an important role. A set of an ordered pair is termed as the relation. So a relation just signifies the relation between the x and y value in a function.

We can say that relation is just the set of ordered pairs. A function on the other hand is the set of ordered pair in which for every value of x, there is one value of y. How to do Relations and Functions–To do the relations and functions, we express them in brackets just like {x, y}.

The first element of the bracket is x and the x values forms the domain of any function or a relation. The second element of the brackets is y and the y values forms the range of any relation or the function. Math Functions and Relations go hand in hand.

The relations are not functions but a function shows the relation between x and y values. Examples of Relations and Functions are given below: - (1,3) , (2,3) , (5,6) , (7,9). This is a relation consists of ordered pairs where domain is the set {1, 2, 5, 7} and range is {3, 6, 9}.

This signifies a relation and not a function as for both the values 1 and 2, the range is 3. But in a function, there should be just one value of y for every x value.
Graphing Relations and Functions – We can graph the functions and relations, by simply plotting the x values on the x axis and the y values on the y axis. The only difference between a relation and a function is that x value in a function has one and only one y value.

To check whether the given set is a function or not, we can do the vertical line test, in which we check that vertical line drawn through the graph will intersect the function at one point only. Hence if there is a vertical line for each pair then that means it is a function and if not that means it is a relation.

So we can conclude that a function looks like a relation only but the only difference is that it is loyal to its partner that means for every x value in the pair there will be just one y value.

Step Functions

In our day to day life we always experience step functions but we do not notice it mostly. If you have noticed about the calling rates of cell phone, as it is charged per minute. You generally pay a certain amount of money for the first minute. And as soon as we run over to the second minute you start paying. So let it be say, you have 1 min 2 sec. you pay the same. For example, if we understand by using an example, say a cell phone call rate is charged 40 cents per min. as soon as we go over 1 minute by 1 or 2 seconds we are charged double for the another minute as well that is another 40 cents been charged likewise we have to pay in all together 80 cents up to 2 minutes of a phone call. This was the brief example to understand the step function definition. When we graph it, it looks like steps. We notice on the graph, first minute of 0 to 1, 40 cents. And then as soon we hit 2, one second after 1 minute. We are charged 80 cents.

The step function graph looks like a staircase. What is the integer graph and how does it affects the function graph. F(x)=[(x)], it is a greatest integer function. It says nothing but give the greatest integer or less than or equal number in it. Let us say for example, [(3.7)] so as we know integers are all sort of whole numbers negative or positive. Here we are looking for the greatest or less than the or equal to 3.7. So 3.7 is not an integer by its own. So we have to go to the next smallest one, which is down to 3. So, [(3.7)] =3. Now what if the [(8)], here 8 is already an integer, so the greatest integer, less than or equal to 8 is 8. So [(8)]= 8.  Now what if the number is negative, say [(- 2.5)] here if we think about how we do it on a graph using step function on a number line. Step functions algebra 2 , we had studied that say like on a number line 3.7 is on the right side, we go down to 3. Then 10.2 on the number line , but we are going down to 10.but on number line – 2.5,we will not change the direction towards right side as we did in positive integers. But we end up going on -3.as we see on number line -3 is smaller than the -2.5.