Interpreting Solution set

All values of a variable that satisfies equality, inequality, system of equations, or system of inequalities can be termed as solution set. All the values that are in a set for whom any equation is satisfied can be interpreted with the help of any feasible solution may it be graphical or algebraic. Solution of any equation or in equation depends upon the domain of the subject and its availability.

For example if we assume a linear equation as 3x + 2y = 12, then we need to find all the values of x and y that satisfied the given equation. All the values of x and y that satisfies the above equation can be put under one definite set, such as one set would be (2,3), similar to this there can be many sets of solution available.

Let us take another example as 5x + 6y = 30. Then one of the solution set of an equation would be (6,0). Now it is true that there may be many solution-sets and when we integrate all of them together we call it as a set.

There are many different ways to solve an equation, out of which most popular is solving it by hit and trial method. The other most prominent is the matrix solution by substitution the coefficient values of the equation. Any solution set of an equation depends on the type of variable it is having.

While dealing with algebra we will come across many such equations having unique solution and having infinite solutions.

Moreover solution set examples could be found in the questions of probabilities and permutations and combinations. Algebra is that part of mathematics that deals more about numbers in the form of equations. Apart from the linear equations there more complex equations for which solution-set can be determined, more complex the equation is, and more complex is the solution-set. For linear equations there may be more than one variable, but most important is that the degree of the equation should be one.

Similarly we have quadratic equations whose degree is two, means we get two solutions for one variable each time we solve. More preciously the solution-set may consist of degenerated sets, which might be imaginary numbers, but still can provide the solution.
So hence all equations have some solutions unique, some or infinite, which together collected and can be names as solution set.

Real life problems as word problems

It is much similar to the word problems what we have seen in the mathematics. It is just about a word expression which we should have to convert it into the productive form or the algebra word problems or in algebraic expression. It is nothing but the conversion of the real word situation and translates it into the mathematical form or some expression that having a numerical type of approach. We are here to give you how to do algebra problems?

Say for example as x plus 5 means x+5, this is the conversion of the word expression into expression of mathematics. Algebra word problems help you will take from these categories more. Word problems consist of the sentences and we need to resolve into the problems very carefully and through our understanding and bit analytical skill.

As word problems are somewhat difficult so you have a good exposure of the problems before also. So practice more over these kinds of equations. Say for example as on my birthday, my weight is 45 kg. One year later I have put on x kg so which expression gives my weight correctly after one year. Sometime the equations like polynomial based problems are also there.

Now we will discuss more on algebra word problem help so that you can easily solve these kinds of problems. Say in probability you will learn about algebra 2 questions and answers type of complex problems. Firstly you have to understand what is important in the question and don’t jump to the solution, be cool and calm while solving and then form the equation after reading once then go through the needs and requirements is all there. If yes then proceed further by solving these problems with the formation of equations or out it into proper tabular format so that to avoid mistakes happened in the future.

Algebra is not about the plus and minus you should have to think how to frame the questions. Once you understand this concept it is very easier to solve the word problems. Sometimes it is long paragraph wise don’t hesitate to solve it is easier one many times, it just need only your attention and confidence to solve. Once you take your decision to solve proceed further step by step and find out the right solution. It is real life based problems we can say that because framing of the questions gives you the gist of the future or present and sometimes past also.

Rotational Symmetry

Guess an object rotated once, twice, and thrice or may be as many times, but still the observation angle and view remains same, those kinds of objects have rotational symmetry, means rotate them through their symmetrical line it will appear same.

An object might have more than one rotational-symmetry, for instance its reflections against any reflection line may also be considered as symmetrical line. The question often arrives in our mind that what is rotational symmetry. The answer is simple, any such object when rotated as many times but still appears same has rotational-symmetry.

As such talking more specifically in terms of science there are many theoretical explanations of the same, but it can be expressed as simple as stated above.

Many mathematicians have the perception that this could be the characteristics of the object, but preciously speaking this is not yet proved. Much rotational symmetry definition is under consideration, all ending with same conclusions.

Rotational symmetry of order n or n fold rotational-symmetry is known as discrete rotational form symmetry of the nth order.  For n = 1 means it is rotated 180 degrees, for n = 2 it means rotated at 120 degrees, and it goes on as the value of n changes.

We can say when a shape is rotated about any fixed point and if it comes to rest in such a position and looks exactly like the original. Some of the rotational symmetry examples are an equilateral triangle, or a tetrahedral structure, no matter how much we rotate the shape is always same, when we observe it from any angle.

Another good example is the pizza of round shape, all pieces of pizza are distributes in equal shapes and size. If the shape has rotational-symmetry then it must eighter have line symmetry or point symmetry. For example a pentagon star has five points and thus has five lines of symmetry but does not have points of symmetry.

Only an object which has point symmetry or line symmetry can have rotational form symmetry. Thus we can have lot more examples in real world. Let us take the alphabetical letters, let us talk about the letter “I”, it has two line of symmetry and one point of symmetry hence this would be considered as rotational-symmetry on the basis of its rotation. The geometry of rotational-symmetry is much better explained in rotational dynamics of physics.

What is Inverse Variation

In the given two variables if the value of one variable increases and another one decreases   that is if the values change in opposite manner then it’s said to be inverse variation.
For example if the speed increases the travelling time decreases and if speed decreases the travelling time increases

Inverse variations  occurs  when two variables such as XY are always equation to some constant K ie, one value  decreases and the other value increases

No of workers Days to complete the  work
        15               15
        10               18
         5               20

As the   number of worker decreases the days to complete the work increases.  Here inverse variations occur.

Inverse Variation Formula

Inverse-variation between two variables say  y and x is given by the formula    y  =  k/x
Here the variable K is known as the constant of proportionality.  Y varies inversely with x.
Y varies inversely with x, if there is a nonzero constant k such that xy = k or ,  y  = k/x, where k ≠ 0.
In inverse-variation its seen that , when the value of one  variable increases the value of the other value  decreases in proportion so that the product remains the same always.

Inverse Variation Word Problems

While solving inverse-variation word problems we use the formula ‘y = k/x   or any other variables relevant to the problem. Find the value of ‘k’ which is the constant of proportionality.
Plug in the value of k in the formula y=k/x .  Use the information from the given problem and solve for the variable.  An example problem is solved below.

If a car travels at the rate of 20 miles per hour it takes 1.5 hours to reach its destination.  If the same car travels at the rate of  50 miles per hour  what will be the time taken to reach its destination.

Use  the formula     t  =  k/s  where  t = 1.5 and s = 20 .  Plug in the values of t and s and find k.
         
                                   1.5  = k/20    (multiply by 20 both sides)
                                   K  = 1.5 x 20
                                   K =  30

Substitute the value of ‘k in the  equation   t = k/s
Here   k = 30  and  t  =  50
                                    t = 30/50

                                    t =  0.6 hours
The car takes 0.6 hours if it travels at the speed of 50 miles per hour.    Here it is found that as the speed of the car increases the time taken reduces.  So   there is inverse-variation here.

Dot product

There are two types of multiplication of two vectors. That is, as two types of products. One of them gives the result as a vector quantity and the other results as a scalar quantity. They are termed as cross product and dot product respectively.  The names are given so because in the first case a ‘x’ is used to indicate the multiplication and in the second case a ‘•’ indicates the same operation. Because of the fact that a dot product results in scalar quantity, the dot-product is also called as scalar dot product.

If P and Q are two vectors, then the dot-product two vectors is denoted as P • Q and its magnitude is given by lPl*lQlcos θ, where θ is the angle between the vectors P and Q. Unlike vector products, scalar products of vectors are commutative.

A typical example of a scalar product is the calculation of work done W by a force F while displacing the object by a displacement D. As per physics concept, W = lFl*lDlcos θ, where θ is the angle between the force and the direction of displacement. Although, F and D are vector quantities, the work done W is a scalar quantity. The work done by a force has no direction. .

From the definition, P • Q = lPl*lQlcos θ, let us analyze when the dot product is zero. It is obvious that it happens when P or Q or both are 0. But there is one more possibility. The scalar product becomes 0 when the angle between the vectors is 90o, even though P and Q may have finite values, since cos 90o = 0. In other words, the scalar product of two vectors acting at right angle is 0. We can illustrate this fact with a practical example. We all know that the weight W of an object placed on a smooth horizontal surface is a vector acting vertically. Suppose the object is moved horizontally by a displacement D, the vectors W and D act at right angle. Since the scalar product is 0, the work done W is 0. That is, no work is needed to be done in moving an object at a horizontal level subject to the assumption that there is no friction between the object and the horizontal surface. That is why we find that only a little amount of effort is needed to push even a heavy object horizontally. It is not zero as per theory, because the assumption of ‘no friction’ is practically impossible and the little effort only accounts for work done against the frictional force.

Mathematical Induction Definition

It is a process that is used to trueness of a given statement for all positive integers. This method includes proving the first statement true firstly from the series of infinites statement and then it is proved that the any one of the other statements from that series is also true.

The Principle of Mathematical Induction includes following steps to prove that a statement is true for all natural numbers:

1)Basis step: the statement is proved true for the least natural number which is normally 0 or 1.
2)Inductive step: In this step it is assumed that the statement is true for some natural number p that lies between least possible value of n to any natural number n, and then it is proved that the statement is true for the next natural number (p+1) also.
The value of n for basis step is decided from 0 and 1 on the basis of given question or statement.

Proof of Mathematical Induction
Let statement S(n) is false for some values of n. Let no be the least value of n such that S(no) is false. no cannot be 0, because S(0) is true. Therefore no should be 1+n1. As n1
Let us take some Examples of Mathematical Induction to solve some problems:

Mathematical Induction Proofs Examples

Q.1) Show that the sum of first natural numbers from 0 to n is given by n(n+1)/2.
Sol.) The above statement can be written mathematically as:
0+1+2…..+n = n(n+1)/2

Let us use mathematical indction process now:
Basis step: in this step we will put least possible value of n in above equation, which is 0 for this case.
S(0) =>
 0 = 0(0+1)/2
In above equation LHS is 0 and RHS also solves to 0. Hence basis step hold true and thus statement is true for n=0.

Inductive step: let us assume that the equation is true for a number p that lies between 0 and n. Now we will show that it is true for p+1 also.
S(p+1)=>
0+1+2…..+p + (p+1)= ((p+1)((p+1)+1))/2
LHS can also be written as
(p(p+1))/2 + (p+1) , as we assumed that the statement is true for p natural number.
(p(p+1))/2 + (p+1) = (p(p+1)+ 2(p+1))/2
= (p^2+p+2p+2)/2
= ((p+1)(p+2))/2  
=((p+1)((p+1)+1))/2 = RHS.
Therefore S(p+1) is true. Hence from basis and inductive steps we come to the conclusion that S(n) is true for all natural numbers.

LCM and its methods

LCM of two numbers is the smallest number (non zero) that is multiple of both.

When we add or subtract any fraction, we make use of the least common denominator and least common denominator is nothing but the least common multiple of more than two numbers.

LCM - least Common Multiplies actually the least possible common multiple of two or more than two numbers. We can find the least common multiple by using two methods: -

Methods for finding LCM of numbers:-
1. LCM Using Prime Factorization– In this method we find the prime multiples of all the numbers for which we need to find the Least common multiple.
We find the common factors among them and the uncommon factors. Least Common Multiple is product of common factors and product of uncommon factors.

LCM Examples- Suppose we need to find the least common multiple of 15 and 25.
We see that 15 = 3 times 5 and 25 is 5 times 5.
Now we see common factor is 5 and uncommon factors are 3 and 5.
So the least common multiple will be 3 times 5 times 5 which equal 75. Hence the least common multiple of 15 and 25 is 75.

2. Common division method- The other method of finding the least common multiple is the common division method in which we arrange the numbers together separated by commas.
We start dividing with the smallest prime number and go on dividing, till none of the numbers can be divided any further.

For example:- If we need to find LCM of 20 and 30 by common division method, we first divide both of them by 2, we get 10, 15. Then we divide again by 2, we get 5, 15. Now we divide by 3, we get 5, 5 and lastly by 5, we get 1, 1.
Now we multiply all the prime numbers by which we divided. They are 2, 2, 3, 5 which is 2 times 2 times 3 times 5 and that gives 60. Hence the least common multiple of 20 and 30 is 60.

We can try different problems for LCM Practice.
We can find the least common multiple of more than two numbers also.
This is usually helpful when we are adding and subtracting the fractions.

Rounding Decimals

Teaching rounding decimals is very easy it helps us in day to day calculations as well. So let us learn how to round the decimals. A decimals rounding is nothing but using decimals and making them a whole round figure. Which makes math’s more easy and understandable. Let us use the example to understand rounding decimals. As we have already studied rounding decimals lesson in our school, for round to decimals let us Say, how to round 66.99 to the nearest tenth? So the number is 66.99 and we have to round to nearest “Tenth”. We will be doing by using four simple steps. So step 1.  Is to identify the digit to be rounded. In this case it is 66.99, and we are asked to round the nearest tenth. The tenth digit is 66.9 the very first 9 to the decimal, because it is one tenth of the whole. Step 2. Is look at the digit at the immediate right of the one to be rounded that it another 9 in this case. Step 3. Determine the digit if it is greater than 5 or less than 5. If greater than 5, we add 1 to the digit being rounded, and if it is less than 5, we leave it unchanged. The digit is greater than 9 here, and 9 is greater than 5 so we will add 1 to it at the tenth place. So 9+1 gives 10, 0 goes after the decimal and 1 gets carried over to 6 now add 66+1 is 67.09. Step 4. We replace all digits to the right one being rounded with zeros. Hence it will become 67.00 or 67 after rounding of decimals.

Example, we buy a bag full of taco chips costs $ 0.4582 per ounce. How much is this to the nearest cent? How do you Round a Decimal? Lets us solve it. Our question says a bag of taco chips is 0.4582 dollars; we need to round to the nearest cent. So a cent is 1/100 of a dollar. Step 1. This is to identify the digit to be rounded. It is the cent digit which is 5( in 0.4582). In case of 0.4572 the zero to be rounded. Here the digit to be rounded is 5. Step 2. Look at the digit to the right of it. Our immediate right of 5 digit is 8. Step 3. We look at digit 8 and determine that if it is less than 5 or greater than 5. In this case it is greater than 5,which implies that we add 1 to the original digit or the rounding digit. Which means 0.4572, as 7 is greater than 5, we ass 1 to 5 which is our original digit. As a result we get 0.4672. last step, replace all the digit to the right of the one being rounded by zeros. So, 0.4682 we replace all the digits which is after 6 with zeros. So we get 0.4600 or 0.46.

The Process of Division of Decimal Numbers

The number systems are very interesting to study. Various arithmetic operations can be carried out on them. Multiplication is one of them. The reverse of this operation is division. Division is bit more different from addition and subtraction. The division of natural and whole numbers are almost similar. Any number divided by ‘0’ gives infinity. Infinity is a number which is very large and not defined. Zero on dividing by zero also gives infinity. This question was first raised by the great Indian mathematician Ramanujam.  Zero divided by any number gives zero. Any number divided by the number one gives the same number. This is the same with multiplication. Any number multiplied with one gives the same number. But any number multiplied by zero gives zero.

The process of dividing decimals with whole numbers is similar to the division of two whole numbers. The simple difference is in the placement of the decimal point. This is very important as the placing of decimal point can change the whole number. So, one has to be very careful about this. There can be dividing decimals problems in mathematics and have to be solved carefully. The decimals division is very much similar to the division of natural numbers or the whole numbers. The placement of the decimal point in the final answer is very crucial. Many dividing decimals examples can be used to explain the concept. The following steps to dividing decimals have to be followed to arrive at the final answer.

These are very easy steps.
If a decimal number is divided by a whole number, the division is carried out as usual. The digits present after the decimal point are noted and the decimal point is placed at the same point in the final answer. If the division is carried out between two decimal numbers then the decimal numbers are first converted into whole numbers by multiplying by 10, 100 or thousand and so on depending on how many digits are present after decimal point in the denominator. If the numbers to be divided are 8.68 and 1.2, then the numbers are multiplied by 10. The numbers now become 86.8 and 12. Now the division process is carried out. First ‘868’ is divided by 12. In the final answer decimal point is placed one point from the right. This is how the final answer is got.

Set Theory Rules are as follows

(A∪A) = A,  (A∩A) = A (idempotent law)
(A∪B)’ = A’∩B’, (A ∩ B) ′ = A ′ ∪ B ′ (De Morgan’s law)
A∪ B = B ∪ A, A∩B = B∩A (commutative law)
(A ∩ B) ∩ C = A ∩ (B ∩ C), (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative law)
A ∪ φ = A, A ∩ U = A, A ∪ U = U, A ∩ φ = φ (identity law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)   (distributive law)
(A’)’ = A (involution law)

Basic Set Theory Properties are:
A st is inherently unordered. This means that the order of elements in a st does not make a new st. Change in order of elements inside a st is feasible. For example if st A = {a, x, c} then A = {a, c, x} = {x, c, a} = {x, a, c} = {c, a, x} = {c, x, a}.
Each element in a st is distinct. Multiple repetition of an element makes no difference. For example if st A = {2, 3} then A = {2, 2, 3, 3} = {2, 3, 3} = {2, 2, 3} = {2, 2, 2, 3, 3, 3, 3} = {2, 3} and so on.
Above rules and properties of Set Theory Help you to solve Set Theory Questions easily.
Let us see some Set Theory Problems and Solutions now:

Q.1) prove that (A ′ ∩ B) ′ ∩ (A ∪ B) = A
Sol.) lets begin with LHS of the above problem:
LHS = ((A ′) ′ ∪ B ′) ∩ (A ∪ B) (by De Morgan’s law)
= (A∪ B’) ∩ (A ∪ B) (by involution law)
= A∪ (B∩ B’) (distributive law/property)
= A ∪ φ (complement law)
= A (identity law) = RHS
Hence, proved.

Q.2) in a community if 70 % people can speak English and 60% people can speak French then find the number of people who can speak both languages.
Sol.) Number of people who can speak English = n(E) = 70 %
Number of people who can speak French = n(F) = 60%
Number of people who can speak both languages will be n(E∩F) = ?
Total number of people in the community n (E∪F)= 100%
Now n (E∪F)= n(E) + n(F) - n(E∩F)
100=70 + 60 - n(E∩F)
100=130 - n(E∩F)
n(E∩F)=30%

Articles – a, an, the

An article is a major concept of language learning. In English an article is considered as an adjective that provides some valuable information about a noun in a sentence. In many other different languages an article is considered as a part of speech. An article can be classified based on definite and indefinite characteristics. These are called definite article and indefinite article.

Definite Article

A definite article is that which defines that a noun is particular and has something definitely addressed. Definite article is used for nouns that have been already used in a speech or for nouns that are uniquely specified. In the English language, there is only one definite article ‘the’. Definite article ‘the’ is used for both singulars and plural nouns. For example: Give me a Barbie doll. Give me the Barbie doll. In the second sentence, a specific Barbie doll is referred to while in the first sentence not any particular Barbie doll is referred.
1) Ram said to his wife, “Please pack the gifts for kids that I bought yesterday”. (Here Ram is speaking about those particular gifts for kids that he bought the previous day.)
2) The Taj Mahal is made by Mughal Emperor Shah Jahan. (Here, Taj Mahal is a uniquely specified noun.)

Indefinite Article

An indefinite article is used for a noun that is not particular and uniquely specified. It can be a noun that the speaker is mentioning for the first time. For example: Give me a Chota Bheem DVD. Give me the Chota Bheem DVD. Here, in the first sentence the speaker is talking about no particular Chota Bheem DVD while in the second sentence the reference is definite. In the English language, there are two indefinite articles ‘a’ and ‘an’. The indefinite article ‘an’ is used before words that begin with a vowel sound. At times even a word that starts with a consonant but sounds like a vowel. For example: an hour, an MBA, etc. Also the indefinite article ‘a’ is used before words that begin with a consonant sound even if the word begins with a vowel. For example: a European, etc. These were the basics about article in English language.

Introduction for Solving Number Patterns

Solving number patterns is the main topic under the solving number systems. In this lesson we have to study about the concepts of divisors, factors, multiples and prime numbers. Also in this topic we have to discuss about properties of number patterns. Here we have to discuss about Divisors, Factors ,Multiples,Even,Odd numbers,Prime numbers and Composite numbers. These are the main topics in the solving number patterns.

Solving Divisors, Factors and Multiples Number Patterns:

Divisors:

Step 1: Consider the number 96. Divide 96 by 4. When we divide 96 by 4, the remainder is 0.

Step 2: 4 is the divisor of 96. Similarly if we divide 96 by 2, we will get 0 as the remainder.

Step 3: Again 2 is the divisor 96. By applying the same process, 1,2,3,4,6,8,12,16,24,32,48,96 are also divisors.

Step 4: Thus the divisors of 96 are 1,2,3,4,6,8,12,16,24,32,48,96.

Note:

The divisor in the division algorithm is different from this divisor. In division algorithm the divisor indicates the denominator only.

Example:

Find the divisors of (a) 28, (b) 35

Solution:

(a) Solving The divisors of 28 are 1, 2, 4,7,14, and 28

(b) Solving The divisors of 35 are 1, 5, 7 and 35

Solving even and Odd Number Patterns and Prime and Composite Patterns:

The numbers that are divisible by 2 is called as even numbers.
If we divide the even number by 2, the remainder is always zero.

The numbers that are not divisible by 2 is called as odd number.
If we divide an odd number by 2, the remainder is always 1

Now observe the following:

(a) 2 + 2 = 4 ; 2 + 6 = 8 ; 4 + 8 = 12 ; 10 + 20 = 30

(b) 2 × 2 = 4 ; 2 × 6 = 12 ; 4 × 8 = 32 ; 10 × 20 = 200

From the above examples the following are true.

Even number + even number = even number
Even number × even number = even number
Odd number + odd number = even number
Odd number × odd number = odd number
These are the main rules used for solving number patterns.

Prime numbers and Composite number patterns

From the above, find the numbers that have only 2 divisors?

2, 3, 5 and 7 have only two divisors.

What are the numbers which have more than 2 divisors?

4, 6, 8, 9 and 10 have more than 2 divisors.

The natural numbers that have only two divisors are known as the prime numbers.
The natural numbers that have more than two divisors are known as the composite numbers.

Example:

Solving prime numbers between 20 and 30.

Solution:

The numbers between 20 and 30 are

21 22 23 24 25 26 27 28 29

∴ the prime numbers are 23 and 29

Definition of parabola

Definition of parabola:
A curve in a plane having two ends and one vertex is called a parabola. It is a conic section. That means it is a section of a double cone intercepted by a plane. When a double cone is cut by a plane such that the plane is parallel to any one slant length of the cone, the cross section thus obtained is called as such.

However in co-ordinate geometry we do not study parabola problems as sections of double cone. We use the characteristic property of these to define it and use that to find its equation.

It so happens that in the plane of every parabola there is a line (called the directrix) and a point (called the focus) such that for every point on this the distance from the directrix and the distance from the focus are found to be equal. This is the definition most commonly used in co-ordinate geometry to define these and to derive its equation.

Types of parabola:
Primarily these are of two types and they are horizontal and vertical. In simple terms, the horizontal form of this is x as a function of y. Therefore the equation of a horizontal parabola would be x = f(y) type. On the other hand in a vertical form of this the equation is y as a function of x. Therefore it would look like this: y = f(x).

Another way of classifying these would be: positive and negative parabola. For the horizontal form, if it opens towards the positive side of the x axis, it is called the positive para-bola; where as if it opens on the negative side of the x axis, it is called the negative para-bola. Similarly, for the vertical form, if it opens towards the positive y axis it is called a positive para-bola and if it opens towards the negative side of the y axis, it is called a negative form of same. Below are the pictures of the various types of these:

1. Horizontal positive form of the same:

As you can see in the above picture, this one opens to the right.
X = f(y) = p(y-k)^2 + h
2. Horizontal negative form:

The above form opens to the left.
X = f(y) = -p(y-k)^2 + h

3. Vertical positive form of the same:

A vertical positive form opens up (as can be seen in the picture above).
Y = f(x) = a(x-h)^2 + k

4. Vertical negative form:

Lastly the vertical negative form would open down.
Y = f(x) = -a(x-h)^2 + k

How to calculate Area of Ellipse?

An ellipse is nothing but the planar curve which is resulted by the intersection by a plane of a cone in a way such that it will produce a closed curve. Circles can be said as a special case of ellipse which can be obtained by cutting in orthogonal plane of the cone’s axis. Ellipse can also be said as the locus of all the points in the plane in which the sum of the distances of two fixed points will be a constant. Before going into the formula for area of an ellipse, it is essential to know the elements of ellipse which forms the structure of it. An ellipse is said to be a smooth and closed curve symmetric about its vertical and horizontal axis. The longest diameter of an ellipse is called as major axis and the shortest diameter of an ellipse is called as the minor axis. These both axes are the lines that pass through the centre of an ellipse.

With the measurement of both these axis only, the area of a ellipse will vary. Each of the above said axes bisects perpendicularly with the other. Also, the sum of the distance from the two focus of the ellipse to any point P on the ellipse will be equal to the major axis. It can also be said that each axis cut the other equally into two parts and they cross at right angles to each other. Also, if these both axes are equal in length, then it is called as circle.

Area of an Ellipse Formula
The standard equation for representing the ellipse equation is given as,
X^2 / a^2 + Y^2 /b^2 = 1.

This equation is a standard equation in which when the ellipse is centred with origin.
But algebraically the ellipse area formula is given in other terms with the help of semi major axis (half of major axis) and semi minor axis (half of minor axis). Thus the formula for area of ellipse is given as Pi*a*b, where ‘a’ is the semi major axis and ‘b’ is the semi minor axis. This formula is actually arrived from the formula of circle Pi multiplied by radius square. Here the radius is split into semi major and semi minor axis.
Also at some special cases, when an ellipse is given by the implicit equation which is given as,
AX2 + Bxy + CY2 = 1, then the area of the ellipse would be 2*Pi whole divided by square root of 4AC subtracted with B square.

Understanding Fractions Better

Numbers have always fascinated humankind. Indians are credited with the invention of zero. Indians have always been good with numbers and the world looks at us with awe. We have great strides and come a long way. From natural numbers, whole numbers, integers, real numbers and complex numbers. We have also studied fractions. Let us what are fractions to understand the concepts of denominator numerator. A number can written in the form of A/B where A is an integer and B is an integer. A can be called the numerator and B can be called the denominator. So we see numerator v/s denominator to extend the concept further. The numerator is written on top of the horizontal line and the denominator is written below the horizontal line. So we understand what is a numerator and denominator if we didn’t know it earlier.

We must understand numerators and denominators to understand fractions better. Operations of addition, subtraction, multiplication and division between the fractions can be performed. Here both numerator and also the denominator come into picture. If the denominators are dissimilar they must be made similar to proceed further with the operations. For this we need to take the LCM of the denominators. Taking the LCM is a simple operation. Unless the denominators are equal we cannot proceed further. Once the denominators are equal, we can just add the numerators if the operation to be performed is addition. If subtraction is to be performed we just need to subtract the numerators after making the denominators equal. So we must know the definition of numerator and denominator to perform these operations. Only when we know it, we can do something.

After making the denominators equal if they are not, we proceed to perform addition, subtraction or any operation with fractions. Sometimes the fractions can be in the form of mixed fractions. We need to be careful in such cases. We must first convert them into improper fractions without proceeding. They cannot be directly used. For the process of finding LCM we need improper fractions in our hand. Once we get the improper fractions we first check whether their denominators are same or not. If they are same, our problem becomes very simple and easy. The numerators can be just added or subtracted as the case may be. If they are not equal only then the process of taking LCM is to be performed and addition or subtraction is to be done.

Distributive Property of Addition

The distributive property is expressed in the mathematics expressions as following equation: a (b + c) = ab + ac. You can understand this as the sum of a (b + c) is corresponding to the sum of a times b and a times c. Distributive property of addition would be inaccurate to multiply ab and just add c, or to multiply ac and add b.

Order of Operations:

The distributive property of addition that remind that all contained by the parenthesis needs to be multiplied by the outside number. Distributive property, when they are knowledge the order of operations.

Concept of that the problems anywhere present are different mathematical operations, such as multiple, addition, subtraction, parenthesis, you have to work in a certain order to get the right answer. This arrange is the parenthesis, exponent, multiplication,division , addition and subtraction, that may be abbreviate to the  PEMDAS.

Example Problems (distributive Property of Addition):

Example problems:

When you include a mathematics problem that use parenthesis you need to solve what’s in the parenthesis first, by you can move about on to solve further problems. If the mathematics problem largely has known numbers, it is somewhat easy to solve. 2(10+5) becomes 2(15) or is also equal under the distributive property of addition is 2(10) + 2(5). What obtain additional difficult is when you are functioning with variables (such as a, b, x, y, and so on) in algebra, and when these variables cannot be joint together.

Consider the equation 5(12a + 2), but we don’t know what the variable a stand for, we can’t add 12a + 2, but using the distributive property still allow us to just this expression because we identify this equation is equal to 5(12a) + 5(2). In order to simply the expression we can take each part separately and multiply it to 5, and we get 60a + 10.

Addition, Subtraction, Multiplication and Division

As the learning session for numbers is completed, the next session starts with addition, subtraction, multiplication and division. Understanding these calculations is the basics of learning mathematics. It is quite challenging to make kids’ understand the concepts of addition, subtraction, multiplication and division followed by order of operations, simplification and so on. In that case providing fun examples related to kids can be helpful. Let’s give it a try.

Addition: Addition denoted by the plus sign, “+”, is a mathematical operation that combines two or more collection forming larger collections.

For example: It is Tina’s birthday and she has received many beautiful dresses for girls – one from Annie aunty, another from grandpa and another one from her mother. Now adding all three we get, 1+1+1 = 3 dresses for girls. This mathematical operation is addition.

Subtraction: Subtraction denoted by the minus sign, “-”, is a mathematical operation that is inverse of addition. In this operation, a part is taken away from a collection and thus forming a smaller collection.

For example: Father brought 50 return gifts for kids on Tina’s birthday. Only 45 of her friends came for the birthday party. Now after subtracting the two, we can find, 50-45 = 5 return gifts for kids remaining. This mathematical operation is subtraction.

Multiplication: Multiplication denoted by the sign “x” is a mathematical operation that scales one number by another.

For example: Mother is planning to organize a fancy dress costumes competition on next Sunday. There are 5 groups of two members and therefore each group requires two costumes. Now by multiplying, we get 5 X 2 = 10 fancy dress costumes. This mathematical operation is multiplication.

Division: Division denoted by the sign “/” is a mathematical operation that includes a dividend, quotient, divisor and remainder.

For example: Tina has 5 best friends and she has 20 chocolates. She wanted to distribute the chocolates among them. Now diving the numbers we get, 20/5 = 4. Each of her friends will get 4 chocolates. This mathematical operation is division.

These are the basic operations of mathematics that form the first steps in learning mathematics for kids.

Problems related to consecutive integers

Many a times in arithmetic we face problems related to finding consecutive integers the sum of which is given to us. The said integers may be odd or even or neither. In this article we’ll try to understand how to solve such type of problems.

Formula for sum of consecutive integers:
As such there is no specific formula for finding sum of consecutive integers. Such problems are solved using algebra and arithmetic approach. The methods may however, vary from problem to problem. Let us look at the following example to understand this better.

Example 1: Three consecutive integers are there whose sum is 66. Find consecutive integers.
Solution: Suppose we assume that the middle or the second integer is n, then the first integer would be n-1 and the third integer would be n+1, because the integers are said to be consecutive, and we know that consecutive integers differ by 1. The three integers are therefore now: n-1, n and n+1. The sum of these integers is given to be 66. Therefore,
n-1 + n + n+1 = 66
3n = 66
n = 66/3
n = 22.
Since n was our middle integer, the other two would be n-1 = 21 and n+1 = 23. So the three consecutive integers are 21, 22 and 23.
Odd consecutive integers:
Odd consecutive integers are such that each of the integer in the list is an odd number. It may be negative or positive, but has to be an odd number. For example, …. -11, -9, -7, -5, -3, -1,  1, 3,  5,  7,  9…These are odd consecutive integers. They go up to infinity in both directions.

Example 2: The sum of three consecutive odd integers is -39. Find the integers.
Solution: Here again, assume that the middle integer is n. Then the odd integer before n would be n-2 and the odd integer after n would be n+2. So the sum
= n-2 + n + n+2
= 3n = -39
n = -39/3
n = -13.
Therefore the other two odd integers would be -15 and -11. So solution: -15, -13, -11 are the required three odd integers.
Even consecutive integers:
The method of solving problems of even consecutive integers is exactly same as that for odd integers.

Example 3: Find the three even integers whose sum is 0.
Solution: If the middle integer is n, then the other two would be n-2 and n+2. Therefore the sum
= n-2 + n + n+2
= 3n = 0
n = 0/3
n = 0.
So the integers would be -2, 0, and 2

Integers: Positive and Negative

Integers are the numbers consisting of positive numbers, zero and also the negative numbers.  The positive numbers and negative numbers are called as positive negative integers. Positive integers are the integers with positive sign for example, 2, 3, 1, 5 etc. and negative integers are the integers with negative sign. Examples of Negative Integers are -21, -3, -1, -5 etc. On a number line the integers are shown as given below

The following are the Positive and Negative Integers Rules:
Addition rules of Integers: 
When two positive integers are added the sum would be a positive integer; (3+5=8)
When two negative integers are added the sum would be a negative integer; [(-3) + (-5)= -(3+5)= -8]
When a positive integer and a negative integer are added, the numbers are subtracted and given the sign of the larger number;[ (3)+(-5)= -2]

Subtracting Positive and Negative Integers:
A positive integer subtracted from a negative integer would give a negative integer; (-7)-(4)=-7+4= -3
A negative integer subtracted from a positive integer would give a positive integer, here the numbers are added and given the positive sign; (5)- (-7)=(5+7) = 12
When two negative numbers are subtracted it gives a negative and a positive integer. The numbers are subtracted and given the sign of the larger number; (-5) – (-7)=-5+7= 2; -7 – (-5)= -7+5= -2

Multiplication Rules of Integers:
A  Positive Integer multiplied with a positive integer, the product would be a positive integer
3x5=15
A positive integer multiplied with a negative integer, the product would be a negative integer
3x (-5)=-15
A negative integer multiplied with a positive integer, the product would be a negative integer
(-7)x3=-21
A negative integer multiplied with a negative integer, the product would be a positive integer
(-7)x(-5)= 35

Dividing Positive and Negative Integers:
A  Positive Integer divided by a positive integer gives a positive integer, 15/5=3
A positive integer divided by a negative integer gives a negative integer, 15/(-3) = -5
A negative integer divided by a positive integer gives a negative integer, -21/3 = -7
A negative integer divided by a negative integer gives a positive integer, (-21)/(-7) = 3

Positive and Negative Integers Word Problems
The temperatures recorded in the Sahara Desert and Thar Desert are 138 degrees Fahrenheit and -55 degrees Fahrenheit respectively. Calculate the difference in the temperatures.
Sahara Desert Temperature= 138 degrees Fahrenheit
Thar Desert Temperature= (-55) degrees Fahrenheit
Difference in temperatures = Temperature in Sahara Desert – Temperature in Thar Desert
      = 138 – (-55) = 138 +55 = 193 degrees Fahrenheit (Answer)

Standard Deviation

Standard deviation is denoted by a symbol of Greek letter (s) and is shows how much variation exists from the mean or average or expected value. If the standard deviation is low, that will indicate that the point tents to very close to the mean, whereas standard deviation is high, the data points are spread out over a large range of values. Standard deviation has very useful important properties such as unlike variance and is expressed in the same units as the data. Mean standard deviation is mainly used in the statistic conclusion to measure confidence. Also used to find how set of data spread out.
Standard deviation (s) is the square root of its variance (s2), which is the average of the squared difference from the average of the mean. Mean (µ) is simple average value of given set of data.
For example just take different height of dogs, find out mean, variance, and standard deviation.
Height of dogs = 700mm, 570mm, 180mm, 530mm, 400mm
To find the mean
Mean = 700+570+180+530+400/5
= 2380/5 = 476
The mean or average height of the dog is 476mm.
Then to calculate the variance first subtracts the value of mean from the every height of the dog and then square the resulting values. Add the sum of squared values and divide with total number of dogs, resulting value gives the variance.
Variance = (700-476)2 + (570-476)2+ (180-476)2 + (530-476)2 + (400-476)2 / 5
  = 50176+8836+87616+2916+5776 / 5
= 155320/5 = 31064
The square root of the variance gives the standard deviation for the height of the dogs.
Standard deviation (s) = vvariance
= v31064
= 176.24 mm


Finding the standard deviation for population and sample, following formulas are used,
Standard deviation for population (s) = v(1/n ?_(i=1)^n¦(xi- µ)2)
Standard deviation for sample (s) = v(1/(n-1) ?_(i=1)^n¦(xi- " " )2)
Where,
µ,   - mean
When we have n value of data, if we are calculating variance, we should divide by n for the population and divided by n-1 for a sample. Standard deviation is used to measure the investment volatility, in finance. It is also called as historical volatility.

Mean and Standard Deviation
Mean and standard deviation is mainly used to find the center of the data set. Mean is defined as; it is the simple average value of the given data set and is represented by a symbol of Greek letter (µ).
Mean for population (µ) = 1/n ?_(i=0)^(n-1)¦xi
Mean for sample ( ) = 1/n ?_(i=1)^n¦x

Where,
n - Size of the sample or number of item in the sample
x, xi - Observed value or set of value

Finding Standard Deviation
Finding standard deviation the following steps should be followed.
First calculate the mean of given set of data by sum of given data divided by total number of data.
Then subtract the mean from each observed value.
Square the each difference and then add all the squared values to get their total sum. The resulting value divided by one less then the number of data in the data set.
The resulting value gives the variance.
Finally standard deviation can get from square root of the variance.

Find Standard Deviation
Find standard deviation for the list of numbers, 1, 3, 4, 6, 9, 8
Mean (µ) = 1+3+4+6+9+8/6
= 5.16
Variance (s2) = (1-5.16)2 + (3-5.16)2 + (4-5.16)2 + (6-5.16)2 + (9-5.16)2 + (8-5.16)2  / ( 6 – 1)
= 17.31+4.66+1.35+0.71+8.07+14.75 / 5
= 46.85 / 5 = 9.37
Variance (s2) is 9.37
We know that standard deviation is the square root of the variance
So the standard deviation (s) = vs2
= v9.37
s = 7.81