Wednesday, July 10

how to find the perimeter of a polygon

In geometry which is a branch of math, there are primarily two types of polygons.
(a) Regular polygons
(b) Irregular polygons

The phrase ‘perimeter of polygon’ refers to the sum of all the sides of a polygon. First let us try to understand how to find the perimeter of a polygon that is irregular.

Perimeter of irregular polygons:

An irregular polygon is the one in which the measure of all the sides of the polygon are unequal. To find the perimeter of such a polygon, there is no other way but to add the lengths of each of the sides. For that the measure of each of the sides has to be known. If the lengths of all the sides of an irregular polygon are not known, then one cannot find its perimeter.

Example 1: Find the perimeter of the following irregular polygon.


Solution:
In the above polygon all the sides of the polygon are given. Therefore the perimeter would be
= 5 + 4 + 3 + 2 + 6
= 20 inches

Example 2: Now consider the following irregular polygon.

Solution:
The perimeter of this polygon cannot be found as some of the sides are not known and there is now way by which we can calculate them as well.
Now let us learn to find the perimeter of a polygon that is regular.

Perimeter of regular polygons:

For regular polygons, the formula for finding the perimeter would be as follows:

P = n * a

Here, P = perimeter of the polygon, n = number of sides of the polygon and a = measure of the length of the side of the polygon.

In a regular polygon all the sides are of equal length. Thus if the polygon has n sides each of length a, then the perimeter would be

= a + a + a + a + …. n times
= a * n

Example : Find the perimeter of the polygon below


Solution:
We see that each of the sides of the given polygon ( an equilateral triangle in this case) is 2 inches. Therefore the perimeter would be,
P = 2 * 3 (because there are 3 sides in the given polygon)
P = 6 inches <- answer="" p="">
In general Area and Perimeter of Polygons have different methods of solution based on whether the polygon is regular or irregular. Usually there are set formulas for area and perimeter of regular polygons. However for irregular polygons there are no well defined formulas and the calculations have to be done using various methods on case to case bases.
In case of irregular polygons, all the sides of the polygon have to be given. If not, then the unknown sides have to be calculable using basic concepts of geometry. Once we find all the sides, then adding them up would give us the perimeter of the irregular polygon.
For finding the area of an irregular polygon, we divide the polygon to rectangles and triangles with known dimensions. Then add up the areas thus found.

Tuesday, July 2

Learning Linear Algebra

The study of linear sets of equations and the transformations they undergo together is called Linear Algebra. While Learning Linear Algebra we come across many topics like, Systems of Equations, Matrices, Determinants, Euclidean n-space, Vector Spaces and Eigen Values & Eigen Vectors. These topics can be learnt with ease using many courses offered as online course Linear Algebra.
Math Linear Algebra
In this article we shall learn in brief about system of equations and matrices. The general linear equation is given by a1x1+ a2x2+ a3x3+…..+ anxn = b; here there are n unknowns and the known numbers are x1, x2, x3…xn . The solution set of which would be given by the set of numbers s1, s2, s3….., sn such that if we equate x1= s1, x2=s2, x3=s3….., xn=sn then the equation is satisfied.

 This means that when the solution set values are substituted on the left hand side of the equation that would be equal to the value b on the right hand side.  Solving system of equations involves different methods in which use of matrices is one of them. Some of the methods are used in solving the systems of linear equations are Gaussian Elimination and Gauss-Jordan Elimination. A series of steps in this method help in solving the given equations.

Let us now take a quick look at Matrices, a matrix is a rectangular array of numbers in the form of rows and columns and each of these elements is called an ‘entry’. The size of any given matrix is denoted with n rows and m columns using nxm. For instance, 3x4 shows it is a matrix consisting of 3 rows and 4 columns.
The matrix that has only one column is called a column matrix (vector) and the one which has only one row is called a row matrix. A matrix is denoted using an upper case letter and the entries using lower case letters. The entry in the ith row and jth column is denoted as ‘aij’ when the matrix considered is ‘A’. A square matrix is the one which has equal number of rows and columns and is denotes as nxn.
In a square matrix the entries a11, a22…ann which form the numbers in a diagonal is called the main diagonal of the matrix. This article gives a very brief outline of Matrices. To learn and understand the various topics under linear algebra with ease and to get Linear Algebra homework help there are many online courses that are offered to one and all.

Thursday, May 16

Sequence numbers


Informally, if we want to define sequence (seq) it will be something as an arrangement of events, elements, terms etc. It is a manner or discipline to keep the things in order. It may be a set of members. The length of the same is determined by the number of ordered elements. It is the arrangement of similar objects. The same elements can appear many times at different positions in the same arrangement.

Sequence definition:
A function f(x) which has domain and range, where x may be set of the natural numbers is called as its definition. The seq are of the following types.
( 1) Finite
(2) Infinite
Finite seq:- In finite form the number of elements is countable. For example
A= ( 1,3,5,7,……………111)
B =(2,4,6,8………………112)
Seq of any finite length ‘n is termed as an n-tuple and Finite seqs might also include empty form ( ) which will have no elements.
Infinite seq:- In the Infinite form the number of elements is not countable. For example
A =(……….. -3,-2,-1,0,1,2,3……………….)
Infinite seq is infinite in both directions. It has neither a first nor a final element is called a bi-infinite or two-way infinite. For example a function from all the integers included into a set, such that the seq of all even integers ( …-8,-6 -4, -2, 0, 2, 4, 6, 8,10,12… ), is found to be bi-infinite.
There are many important integer sequential forms and these are as follows.
(A) The even numbers which can be divided by 2.
(B) The odd numbers which cannot be divided by 2.
(C) The prime numbers that have no divisors except 1 and themselves.
The Fibonacci number Sequences:- It is nothing but in which elements are the sum of the previous two elements. The first two elements are either 0 or 1. This is (0,1,1,2,3,5,8,13,21,34,65,99...).
Formula for the Fibonacci:- It  can be defined using a recursive rule along with two initial elements.
 ,   with   a0 = 0  and  a1 = 1.
Where, 0 and 1 are initial elements of the Fibonacci sequence.

Special seq:- some of the special seq forms are given below.

(1) Arithmetic
(2) Geometric
(3) Square of numbers
(4) Triangular
(5) cube of numbers
(6) Roots of numbers
(7) cubic roots of numbers
(8) A set of vowels.
(9) indexing of the documents
Examples and notation
It is a list of elements with a particular order. These are useful for the study of the functions, spaces, and other mathematical structures by using the properties of convergence . The basis for series is sequences. These are used in differential equations and analysis. These are also used to find the patterns or to solve the puzzles and can be used in the study of prime numbers.

Tuesday, April 30

Solving simple equations


Simple equations are equations those have just one variable and which can be solved by using very algebraic operations. At times, you can even solve them mentally. These are also called one step equations because in one step. Since they involve variables and need to perform algebraic operations for solutions, they are also referred as algebra simple equations.

In many cases the one step equations are based on word problems and those are framed as per the statement of the word problem. Let us illustrate a few examples to understand how to solve simple equations.

Let x + 1 = 5. The solution is to find the value of x. We see on the left side a 1 is added to the variable. So to isolate that, we need to undo that addition by the inverse algebraic operation, which is subtraction. Since 1 is added to the variable, in the undoing process we need to subtract 1 on the left side. But since this is an equation doing any operation only on one side is prohibited since the balance of the equation is disturbed. Hence to maintain the balance a 1 must also be subtracted on the right side.

So the method of solution is (x + 1) – 1 = 5 – 1, or, x = 4, which is the solution. The solution can be checked by plugging that value in the original equation. That is, if x = 4, then the left side becomes as, x + 1 = 4 +1 = 5 = right side. Hence the solution is correct.

Let x – 3 = 4. Here the undoing operation is ‘add 3’ on both sides. So, (x – 3) +3 = 4 + 3, or, x = 7 which is the solution.
Next let us the equation 3x = 9. In this case, the variable is multiplied by 3 on the left side. In this case, the undoing operation is ‘divide by 3’ on both sides. So, (3x/3) = (9/3) or, x = 3, which is the solution. Similarly the undoing operation for a division is ‘multiply by the same number’ on both sides. For example, if, (x/2) = 4, then, (x/2)*2 = 4*2 or, x = 8.

Let us see how one step equations are formed from word problems.
‘Ben is 5 years older than Joan and Joan is 3 year old. How old is Ben?’ Let ‘x’ be the age of Ben. Since he is 5 years older that Joan, the equation is x – 5 = 3 and the solution is x – 5 + 5 = 3 + 5 or x = 8. Hence Ben is 8 years old.

Tuesday, April 9

Templates for the Diagrams Representing Set Operations


The Venn diagram template can be associated with set theory in mathematics. Sets represent collection of objects which are similar in nature. So, the all real numbers together form a set. Even the integers together can form a set. These are similar numbers which is necessary to form a set. The triple Venn diagram template is used when three sets intersect.

There is a common area when the sets intersect. There are various operations that are to be learnt in the set theory. The operations like union or intersection can be used to solve various problems. The set theory is used to solve the problems in a simpler manner.

They give a clear understanding of the problems. The 3 circle Venn diagram template is used when three sets are intersecting. These templates can be very useful and helpful. They must be chosen carefully.

The Venn diagram templates are used to make these better understandable so that the problems can be better understood and can be solved easily. Even Free Venn diagram templates are available online and can be chosen at will to solve the problems. They can be really helpful. A wide variety of them are available and the right choice must be made. For this to happen one must be thorough with the concepts in set theory and these diagrams.

These diagrams are pictorial representations of the problems in the set theory. Once they are drawn one is able to understand the problem better and can solve it more easily and in a smaller amount of time. So, one must learn how to draw these diagrams.

The intersection of two sets in these diagrams denotes common area between the two sets. It contains the common elements which are present in both the sets. So, the common area in these diagrams is the intersection and denotes the elements which are present in both the sets.

The same can be true if there more than two sets. In case of three sets also this is the case. The common area denotes the elements which are present in all the three sets. The union of sets denotes all the elements present in the sets of which the union is found. There is also the concept of empty set. If there is no common area in the diagram, then it denotes an empty set. It basically means there are no common elements between the sets.

Wednesday, April 3

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.

Wednesday, March 27

Parallelogram Vector addition


One of the most common vector operations that is frequently encountered is addition operation. Addition of two or more vectors to arrive to a vector sum is called vector addition. Consider two vectors,  u=(u1, u2) and  v=( v1, v2), the sum of these vectors would be vector u + vector v= (u1+v1, u2+v2). This sum is called the resultant vector. There are various methods to find the resultant vector namely, parallelogram method, component method, graphical method, cosine method, polygon method etc.  

Parallelogram Vector addition:  In this method first two vectors are drawn such that their initial points coincide. Then the other two lines are drawn to form a parallelogram. The resultant would be the diagonal of the parallelogram drawn from the initial point to the opposite vertex of the parallelogram.

Vector addition component method is one way used in adding vectors. Component means ‘part ’and hence they can be considered as the coordinates of the point that is associated with the vector. In a Euclidean plane consider two vectors, u=(u1, u2) and v=( v1, v2), the resultant vector which is the sum of these vectors is given by, u+v = (u1+v1, u2+v2). In a three dimensional space, given vectors u=(u1, u2, u3) and v=( v1, v2,v3) the method would be similar to the method used in addition of vectors in a Euclidean plane. So, u+v = (u1+v1, u2+v2, u3+v3). We can finally conclude that vector-addition is just like the normal addition, component by component.

Let us now learn the vector addition graphical method, consider two vectors, u=(4,3) and v=(1,4) in the plane. Using the component method of vector-addition the sum can be given as, u+v = (4+1, 3+4) = (5,7). Using the graphical method we get the same resultant vector by taking one vector whose direction and magnitude is unchanged and placing its end at the unchanged vector’s tip, and joining the origin and the new location of the displaced vector using an arrow. This procedure in general works for addition of vectors. For any two given vectors u and v in the plane, the sum of the vectors in general can be graphically represented as the vector addition diagram given below

As the vectors in a two dimensional space lie in the same plane, any two vectors in a three dimensional space also lie in the same plane and hence graphical method works well for vector-addition in a 3-dimensional space.

Wednesday, March 20

Understanding the Concept of Linear Combination


In mathematics the equations are very important. The problems can be converted into equations and then the equations can be solved with the help of multiple methods. The solutions obtained must be checked for their feasibility. This is because all the solutions obtained will not be feasible. Only the solutions that are feasible must be selected, otherwise the answer might go wrong. One must be very careful in selecting the solutions of the equations.

In case of linearity the degree of the given equation is ‘1’. The linear combination can be formed with the help of an equation. There need to be constants for this process to be performed. It also involves the simple process of addition. Basically it is the formation of an expression. This expression can be formed with the help of constants and the simple addition process.

The term in the expression must be multiplied with a constant and the answers obtained must be added. This will give the required solution. The linear combinations can be very helpful and can have various applications in mathematics.

The process of solving linear combinations must be learnt thoroughly to appreciate the concept. The examples will help in better explaining the concept. This concept is related to the concept of linear algebra. It is a branch of algebra and has to be learnt to understand this concept. This concept is also very helpful in vector theory and is a part of vector theory.

Vectors are different from scalar quantities. Both are quantities which have magnitude. The difference between vector and scalar quantities is that the vector quantities have a direction attached to them. But the scalar quantities do not have a direction attached to them.

This is the basic difference between the two quantities. So, one has to be very careful in dealing with these types of quantities. The direction can play a very important role. If the direction is not properly denoted the answer can go wrong. So, in the case of vector quantities, the direction plays a very important role. But in case of scalar quantities the magnitude plays the most important role.

The change in magnitude can make the whole answer go wrong in the case of scalar quantities. But in the case of vector quantities both magnitude and direction are important. Even if one goes wrong the whole answer goes wrong.

Monday, February 25

Definition Subset


Sub-set means part of. If we say that America is part of world then America is said to be sub-set of world. So we Define Subsets as the set which is part of another set. For example in library are many book shelves according to the subjects.
Consider a book shelf containing English books. Then the English book shelf will be sub-set of all library shelves.
Consider students in your class which makes a set C and all students in your school which makes a set S. the set C will be sub-set of set S as it is a part of set S or we can say it is contained in set S.  :

Definition Subset / Subsets Definition – sub-sets are those sets whose all elements are contained in some another set.

Let us see properties of sbsets:
Symbol used for sub-set is:  or . Example if A is sub-set of B then we write it as: A B or A B.
A sub-set may be sub-set of more than one set. For example: consider set N = {11, 33, 77}. This set can be sbset of any set containing the elements 11, 33 and 77 like {1, 2, 11, 33, 77}, {a, e, 33, 11, 77} and many more.
All sets are sub-set of themselves. For example set S={English, math, science, GK}. Then the set S contains these elements. We can say that S is sub-set of S.

An empty set (φ) is sub-set of itself and also sub-set of all other sets.
A set is said to be proper sub-set of another set if another set contains the sub-set and also one or more extra elements with it. For example: set A = {c, v, r, e, m} is proper sub-set of B = {c, v, r, e, m ,s} as it contains extra elements s with the elements of A. A set can be proper sub-set of more than one set. For example above set A is also proper sub-set of C ={c, v, r, e, m, d, s, a}. Null set is proper sub-sets of all the existing sets.

If A is sub-set of set B then B is known as super set of B.

A superset can also be sub-set of some other set.

If A is sub-set of B and B is also sub-set of A, then A is equivalent set of B or A=B.

Possible Number Subsets in set A = 2n where n is number of elements in set A.
These are the properties of Subsets Math

Wednesday, February 20

Venn Diagram Math


The relationship between two or three sets can be shown diagrammatically in the form of circles called the Venn-diagram. They were first given by the mathematician John Venn and hence the name for the diagrams. As we know set is the collection of unique things which are called the elements of the set. Consider the following Venn Diagram examples, List of odd numbers less than 15 forms a set which can be denoted as set A={1,3,5,7,9,11,13} here each element is unique.

Let us now learn how to draw Venn-diagrams to show the relationship between two sets. Let set A={1,3,5,7,9,11}, a set of odd numbers and set B={1,2,3,5,7,11} set of prime numbers.
The Venn-diagram of these two sets would be as given below,

In the above diagram the two circles represent each set, set A and set B. Both the circles with all the numbers represent ‘Union of sets’.

The green shaded region represents the numbers in both the sets, this is called the ‘Intersection of sets’, the yellow region represents only A, that is the element present in only set A and the orange region represents the element present in only set B.

Thus one glance at the Venn-diagram gives all the possible information which gives a logical relationship between the sets. At times we see Venn diagram lines, these lines are provided to write the elements or at times lines are used to show the relationship between the sets using line shading.

Venn diagram can also be used to show the relationship between three sets which would be a three circle Venn diagram.

For example: In a group of 23 students, 12 read mathematics, 15 read statistics and 11 read Physics, 4 read Mathematics only, 7 read Statistics only, 3 read Physics only, 4 read all the three subjects, 1 reads only Math and Statistics not Physics, 3 read only Mathematics and Physics not Statistics and 3 read only Statistics and Physics not Mathematics. Let us now write down all the given data to draw the 3 circle Venn diagram.

The total number of students is denoted using the symbol ‘µ’ which is ‘mu’, n(MUPUS)= µ=23, Mathematics =n(M), Physics=n(P) and Statistics=n(S). Intersection is shown with a symbol ‘∩ ‘showing the elements common to both the sets.

Elements not present in a set is shown by compliment, not A is A’. n(M)=12, n(P)=11 and n(S)=15. n(M∩P∩S)=4 and hence, n(M∩P)=7, n(P∩S)=7 and n(M∩S)=5. In the Venn-diagram below each color represents a logical relationship,


Friday, February 15

Least Common Multiple of two Numbers


There are various mathematical concepts and terms that are to be learnt. The term LCM stands for least common multiple. This is an operation and can be carried out on two integers. The answer we obtain is the least common multiple. This denotes the least number that is fully divisible by both the integers.

Now how to find LCM is the question. It is a rather an easy task and can be performed if one is familiar with the basic arithmetic operations of mathematics and number theory. Division is the operation that is most frequently used. So, one must be confident in the division of two numbers.  So, to understand how to find the least common multiple one must be thorough be with the basic concepts in mathematics otherwise it would be really tough to proceed ahead.

So, what is LCM and how can be it found out. The numbers are selected first, for which it has to be found out. Then they are divided by a common number which should be able to divide all the numbers chosen completely. The number which is not divisible by the common number is written as it is for the next step to be performed. In the next again the numbers are divided by a number which is able to divide all the chosen numbers and this continues till ‘1’ is obtained at the end of all the steps of division. The numbers chosen are divided several times till ‘1’ is obtained in the end. Then the product of all the numbers which were used for division is obtained and this gives the least common multiple of the chosen numbers in the beginning.

So, what is the LCM of ‘3’ and ‘2’ is a question that must be answered and can explain the concept better. First a number which divides both ‘2’ and ‘3’ have to be found out. Since there is no common number that divides both ‘2’ and ‘3’ their product is found out. On multiplying ‘2’ and ‘3’, ‘6’ is obtained. So, ‘6’ is the least common multiple of ‘2’ and ‘3’. If two numbers are not divisible by a common number in the beginning itself, then their product itself is the least common multiple of the two. This is apt in the case of two prime numbers. In the example both ‘2’ and ‘3’ are prime numbers.

Tuesday, February 12

Multiplying Even and Odd Numbers

Even and odd numbers are two of the most important topics in basic arithmetic. All real numbers can be classified into even numbers and odd numbers. Even number is a number that can be evenly divided into two parts or can be exactly divided by 2. For example: I have to shed almost 12 kg post pregnancy weight . Here, 12 is an even number. On the other hand, odd number is a number that cannot be evenly divided into two parts or cannot be exactly divided by 2. For example: I have already shed 7 kg post pregnancy weight . Here, 7 is an odd number. There are some trend that can be seen while multiplying even and odd numbers. Let’s have a look at the multiplication rules for even and odd numbers in this post.

Multiplying Even Numbers
When two even numbers are multiplied, the result is always an even number. The formula is Even * Even = Even. For example: I need to list 6 different hobbies for kids for 2 magazines. 6 hobbies for kids * 2 magazines = 12. The result is an even number.

When an even number is multiplied by an odd number, the result is always even. The formula is Even * Odd = Even. For example: I have to survey minimum of 10 baby care center across 5 cities. 10 baby care center * 6 cities = 60. The result is an even number

Multiplying Odd Numbers
When two odd numbers are multiplied, the result is always an odd number. The formula is Odd * Odd = Odd. For example: I have to buy 7 play toys for 7 kids. 7 * 7 = 49. The result is an odd number.
When an odd number is multiplied by an even number, the result is always even. The formula is Odd * Even = Even. For example: She has to buy 7 dresses for 4 kids. 7 * 4 = 32. The result is an even number.
These are the multiplication rules for even and odd numbers.

Tuesday, February 5

Outer product of vectors in R^3


Outer product of vectors in R^3
Cross product definition:
The outer product is also called the cross product of vectors. The formal definition is as follows:
If vector x = (x1,x2,x3) and vector y = (y1,y2,y3) are vectors in R^3, then their outer product is denoted by x X y and is defined as
x X y = (x1,x2,x3) X (y1,y2,y3)
         = (x2y3 – x3y2, -(x1y3 – x3y1), x1y2 – x2y1) <- cross="" is="" p="" product="" the="" this="" vector.="">Note that the cross product of two vectors is always another vector. Alternatively in matrix notation we can write the cross product identities as follows:
x X y = (|(x2&x3@y2&y3)|, - |(x1&x3@y1&73)|, |(x1&x2@y1&y2)|)

Properties of cross product:
(1) x X y = -y X x
That is because when we do y X x we interchange the rows of the cross product determinants. Interchange of rows results in negative value of the same determinant.

(2) x X x = 0
That is because if a determinant has two identical rows, then the value of the determinant is zero. When finding x X x, the two rows would be identical.

(3) x X ky = kx X y = k(x X y).
Here, k is a scalar. If one of the rows of a determinant has a common factor k, then the k can be taken out of the determinant and the resulting determinant value when multiplied by k gives us the value of the original determinant.

(4) x X (y + z) = x X y + x X z
This is just distributive property. The distributive property is applicable to determinants there for it also applies to the outer product.

Cross product rules:
The following rules apply to all outer products of vectors. The outer product is also called vector product of two vectors.

(1) The vector product of two vectors is always another vector.
(2) The vector product is not commutative as we saw in the first property above.
(3) The vector product is not defined for vectors in R or R^2. It is defined only for vectors in R^3 or higher order dimensions.

Sample problem:
Find the outer product of the vectors x = (1,2,3) and y = (-1,3,5).
Solution:
x X y = (|(2&3@3&5)|, - |(1&3@-1&5)|, |(1&2@-1&3)|) = (10-9, -(5+3), (3+2)) = (1,-8,5) <- answer.="" p="">

Monday, January 28

Cost Price

Price can be classified into cost price and selling price. Cost price is the value that is paid by someone to purchase goods or products. Cost price is popularly represented as C.P. Selling price on the other hand is the price that a store or product owner sells a product. Selling price is popularly represented as S.P. Below is two examples on cost price and selling price.
Example 1: Maria bought action toys for 80 rupees per piece and sold it out for 100 rupees per piece. Here, Rs.80 is the cost price of action toys and Rs.100 is the selling price of the same.
Example 2: Rohit bought building toys collection for Rs.2500 from online. He sold the same collection for Rs. 2000. Here, Rs. 2500 is the cost price and Rs. 2000 is the selling price of the building toys collection.
Cost price can be classified into four main forms – Actual cost, Last cost, Average cost and Net realizable value.
Actual Cost Price: Actual cost is the complete amount of a product including duty taxes, service taxes and so on. For example: Christy bought role play toys online for Rs.599 exclusive of shipping charges that is Rs.99. Here, actual cost of the role play toys is 599+99 = Rs.698.
Last Cost Price: Last cost is the value of an item on its last purchase. For example: I bought by mobile phone at Rs.13500 and now the same phone model’s price is Rs. 10000 only. Here, Rs.13500 is the last cost price of the mobile phone.
Average Cost Price: Average cost is the average of new and old stock values. For example: the new price of a product is Rs. 10 and the old price was Rs.8 and therefore, the average cost is Rs.9.
Net Realizable Value: Net realizable value is the average price of a product in the market.  For example: The average value of Barbie dolls in market is Rs.199 and it can also be referred as Net realizable value of Barbie dolls.
These are some basics on Cost Price.

Tuesday, January 22

Rational expression



A rational expression (in abbreviation as RE) is one of the forms of expressions in algebra. The rational expression definition is an expression which is in fractional form of two expressions. A rational-expression is generally denoted as p/q, q =! 0.. If ‘p’ and ‘q’ are numbers, then it is a fractional number. If ‘p’ is an expression with variable/s and ‘q’ is just a number, then the RE becomes as a simple expression with each coefficients of p is divided by ‘q’.

Rational-expressions have certain constraints which a normal expression may not. As the first one, we said the denominator expression cannot be zero. Thus, a rational expression can have domain and range restrictions. That is, a RE is not defined for the real zeroes of the denominator part. Thus, the domain has to be excluded for such values of the variable/s, and correspondingly the range is affected.

As a convention, the denominators of REs cannot be left with radicals, negative exponents or complex numbers. Putting back them in proper way is called as solving rational expressions. Let us discuss how to solve rational expressions in such cases.

In case of a RE with a radical term in the form (a + sqrt b) in the denominator, multiply both numerator and denominator by the conjugate of (a + sqrt b), which is (a - sqrt b). Now as per the ‘sum and difference product formula’, the denominator becomes as a2 – b, free from radical terms.

In case of complex numbers, the method is exactly same. This process is also called as ‘rationalizing the denominator’.
REs have more prominent place in rational functions. A rational function f(x) is normally expressed in the form f(x) = [g(x)/h(x)]. The domain, range, continuity, asymptotes are all dependent on the nature of g(x) and h(x). A rational function will have vertical asymptotes at the zeroes of h(x). Determining the horizontal asymptotes (HA) of a rational function is a bit lengthy but can me summarized as below. If the degree of g(x) is ‘m’ and that of h(x) is ‘n’, then,

If, (m – n) < 1, then the HA is y = 0, that is, the x-axis.
If, (m – n) = 0, then the HA is y = b, where ‘b’ is the ratio of leading coefficients.
If, (m – n) = 1, then there is no HA but the slant asymptote, is y = mx + b, where ‘mx + b’ is the quotient part of the long division of g(x)/h(x).
If,(m – n) > 1, neither HA or slant asymptote.

Wednesday, January 16

Relatively prime numbers

Mathematics is the study of numbers. In branch of mathematics, we deal with different types of numbers which are grouped together.
 For example: - Odd numbers, even numbers, prime numbers, composite numbers, etc.

A prime number is one which has factors one and itself.
For example: - 2 is a prime number as it has two factors only that is 1 and itself. Similarly numbers like 3, 5, 7, and 11 are also its examples.

Now we talk about Relatively Prime Numbers. What is Relatively Prime - Relatively Prime Definition says they are those numbers which have the greatest common divisor as one only.

They are also called co prime-numbers. They are those numbers which share no common factors except 1. Let us look at some examples to understand this concept better.

How about number 7 and 15? If we make factors of 7, we see that number 7 is divisible by 1 and 7 only. For number 15, we see that it is divisible by 1, 3, 5 and 15. If we look at the factors of both the numbers, we see that the common factors between them are 1 only.

Hence they can be termed as relatively-prime-numbers. Similarly if we have numbers 8 and 22, we see that number 8 has factors 1, 2, 4 and 8 and number 22 has factors 1, 2 and 11.

If we look at the factors of both the numbers, we see that apart from number 1, 8 and 22 has a common factor 2 as well. Therefore we cannot call them as relatively-prime-numbers.

If we look at the Relatively Prime Numbers List from 1 to 10, we can make groups like 2 and 3, 2 and 5, 2 and 7, 2 and 9, 3 and 4, 3 and 5, 3 and 7, 3 and 8, 3 and 10, 4 and 5, 4 and 7, 4 and 9, 5 and 6, 5 and 7, 5 and 8, 5 and 9, 6 and 7, 7 and 8, 7 and 9, 7 and 10, 8 and 9 and 9 and 10.
They are forms a group of Relatively Prime nos. till 10.

We cannot include 2 and 4 in that group as they have two common factors. Similarly number 6 and 10 are also not in that group as they are divisible by 1 and 2.

Wednesday, January 9

Dividing Decimal


Steps for dividing decimals. Let us say we have 31.773 ÷ 5.1. Let us set this up by using long division. We have 31.773 on the inside, being divided by 5.1. One this we have to remember that we have to get rid of the decimal in the outside number. Here we will be dividing decimals with decimals So, in math dividing decimals, we have outside number of 5.1, here we need to get rid of this decimal in the outside number.

Basically we have to push this decimal point as to the right as possible. In this particular example we can only push it over one space to the right. And whatever we do to the outside number same is applicable to the inside number as well. Let us learn how to divide decimals with decimals. So since we moved our outside decimal number on space to the right, we have to do the same thing to the inside decimal. Our 5.1 becomes 51 and our 31.773 becomes 317.73. So 31.773 divided by 5.1 is exactly the same as 317.73 divided by 51.

Now because we have got the rid of the decimal in the outside number, we can just divide this using normal long division. 51 goes into 3 in 0 times, as 51 is bigger than 3, moving to the next digit. 51 goes into 31, in zeros times, again since 51 is bigger. Moving on to the next digit, 51 go into 317 how many times? Well we know that 50 goes in 300, so we can say that 51 goes into 317 about 6 times. Multiplying 6 times 1 gives 6, and 6 times 5 gives is 30. Now subtracting the two numbers we get 11, now bringing down the 7. So it is 117 now, 51 goes into 117 about 2 times. Two times one is 2, and 2 times 5 is 10. So we get 102.

Subtracting 102 from 117 remains with 15. Bringing down 3 gives us 153. Before we go further to solve this problem, we would like to place the decimal point. The decimal point moves straight up, that goes directly between the two numbers that is, 6.2. so now how many time does 51 go into 153? So we know that 50 go 3 times to get 150, so 51 may also similar ways. So 3 times 1 is 3 and 3 times 5 is 15, So 51 times 3 gives us 153.and subtracting 153 from 153 , our remainder becomes 0.  This means that we can stop, so we did took the help with dividing decimals by shifting the decimal point and made division simple and easy.

Wednesday, January 2

Rules of Narration

Narration is one of the most important concepts in English grammar. While doing narration, certain rules needs to be followed which are termed as the rules of narration. Let’s have a look at some of the most important rules of narration.
Changing from Direct Speech to Indirect Speech:
When the exact meaning of a speech that is direct speech is conveyed in reported or indirect speech, no inverted commas should be used.
Direct Speech: Hary said, “I always buy Newborn Baby Essentials from online baby store as it is fast and easy.”
InDirect Speech: Hary said that he always buy newborn baby essentials from online baby store as that is fast and easy.

Changing Tenses in Narration:


1. If the reporting verb is in Present tense or future tense, the tense of the verb is not changed.
Direct Speech: The girl says, “Barbie doll is my favorite toy.”
InDirect Speech: The girl says that Barbie doll is her favorite toy.
2. If the reporting verb is in simple present tense, the tense of the verb is changed into simple past tense in the indirect speech.
Direct Speech: Rima said, “I buy Johnson & Johnson products for my baby.”
InDirect Speech: Rima said that she bought Johnson & Johnson products for her baby.
3. If the reporting verb is in present continuous tense the tense of the verb is changed into past continuous tense in the indirect speech.
Direct Speech: Hary said, “Geeta is dancing a folk dance.”
InDirect Speech: Hary said that Geeta was dancing a folk dance.
4. If the reporting verb is in present perfect tense, the tense of the verb is changed into past perfect tense in the indirect speech.
Direct Speech: Maria said, “I have a car.”
InDirect Speech: Maria said that she had a car.
5. If the reporting verb is in simple past, the tense of the verb is changed into past perfect tense in the indirect speech.
Direct Speech: The lady said, “Many people bought baby products from online stores.”
InDirect Speech: The lady said that many people had bought baby products from online stores.
These are some of the most important rules of narration.