Algebra Word Problems

In the field of mathematics where many forms are present algebra is one such form that deals with the basic concepts of Mathematics. We all know that Maths is a game of mathematical operations, so to deal with addition, subtraction, multiplication, division with the known and unknown values of the variables is known as algebra. In a simple way to say is that when we play computer games then we deal with jumping, running or finding secret things and in Maths we play with letters, numbers and symbols to extract the secret things. Now let’s see what is included in algebra.

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WHAT IS THERE IN ALGEBRA?
1. Algebraic terms – Algebraic terms are those terms which are infused with the values of variables. For example 3a.
In this example 3 is the numerical coefficient and a is the variable. The main work of numerical coefficient is also to deal with + 3 or – 3 of the variable.

2. Algebraic expressions – Algebraic expressions deal with a logical collection of numbers, variables, positive or negative numbers in the mathematical operations. For example we can write an expression – 3a + 2b.

3. Algebraic equations – Algebraic equations refer to the equivalency of expressions considering both the left hand side and the right hand side. For example an equation is x – 35 = 56k2 + 3. Over here both left hand side and right hand side are dealing with expression. If both the expressions are equal then that is said to be an equation.

ALGEBRA WORD PROBLEMS

The main part in Algebra is to solve the word problems, which we also did in Pre Algebra Word Problems.  It can be harder if you don’t think and do it but it can be easy if you think and do it. If we say in English how are you and the simple answer to the question is I am fine which can be simply understood but to convert the English language in to the language of maths requires certain techniques. You need to practice a lot with proper understanding of the translations. Let’s go step by step.

1. First of all read the entire word problem carefully. Make it a point that we should not start solving the problem by just reading half of a sentence. Read it properly so that you can gather the information that what you have and what you still need to have.

2.  Then, pick the information so that it can be changed into certain variables. Some might be known and some might be unknown. If required then draw pictures and symbols in the rough work so that you can deeply understand the problem without leaving any loopholes.

3. Note the key words as for different mathematical operations there are different key words –

a) Addition – increased by, total of , sum, more than, plus
b) Subtraction – decreased by, minus, less than, fewer, difference between
c) Multiplication – of, times. Multiplied by, product of, increased or decreased by the factor of
d) Division – per, out of, quotient
e) Equals -  is, gives, yields, will be


4. After going though these expressions carefully you can solve various word problems logically. Limit Problems are part of Calculus.

Rational Inequalities

Rational Inequalities : - The rational expressions are written in the P(x)/q(x) form where p(x) and q (x) are the polynomials i.e p(x), q(x)∈z(x), and q(x)≠0 are known as rational functions. the set of rational function is denoted by Q (x).

Therefore Q(x) = (p(x))/(q(x)) (where p(x), q(x) ∈z(x) and q(x) ≠0. for, if any a(x)∈z(x), then we can write a(x) as  ( a(x))/1, so that a(x)∈z(x)  and therefore,(a(x))/1  ∈Q(x);i.e  a(x)∈Q(x). The rational function example is given below.   

(2x+4)/(x²+5x+8)>0
(ax+b)/(cx+d)<0 p="">
Rational inequalities means the left hand side is not equal to the right hand side of the equation. The inequalities are associated with the linear programming.  The step by step procedure to solve the inequalities is given here. 

The very first step to solve the inequality problems is to write the equation in the correct form. There are two sides in the equation, left hand side and the right hand side. All the variables are written in the left hand side and zero on the right hand side. The second step is to find the critical values or key. To find this we have to keep the denominator and the numerator equal to zero.

The sign analysis chart is prepared by using the critical values. In this step the number line is divided into the sections. The sign analysis is carried out by assuming left hand side values of x and plugs them in the equation and marks the signs. Now using the sign analysis chart find the section which satisfies the inequality equations.

To write the answer the interval notation is used. Let us solve an example (x + 2) / (x² – 9) < 0. Now put the numerator equal to zero, we get x = -2. By putting the denominator equal to zero we get x = +3 and x = -3.  Mark these points on the number line. We get minus three, minus two and plus three on the number line.
Now try minus four, plug the value of x equal to minus four we get – 0.286.

Plug x equal to minus 2.5 we get + 0.1818. Now plug 2 in the equation we get minus zero point eight.
Now we try x equal to plus four which is at the right hand side of the number line. We get plus 0.86. The sign is minus, plus and plus sign. The equation is less than zero. To get less than zero we need all the negative values of the equation.

For x equals to plus three and minus three the equation is undefined. For x equals to minus two the equation is zero. The equation is satisfied by the value of x which is less than minus three. Therefore the answer which is the solution of the equation is minus infinite to minus three.

Rational Expressions Applications : - The equations which do not have equal sign are called the expression. The expressions are used to simplify the equations. The expressions are written to form the conditions of the problems.

According to the direction of the question the expressions are formed to solve a problem.

Set Operations

Set Operations: - The collection of the distinct objects is called the sets. For example the number 5, 7 and 11 are the distinct object when they are considered separately. Collectively they can form a set which can be written as [5, 7, 11]. The most fundamental concept of the mathematics is the sets.  The objects which are used to make the sets are called the elements. The elements of the sets may be anything, the letters of the alphabet, people or number. The set are denoted by the capital letters.  The two sets are equal when they have equal elements. The set can be represented by two methods, roster or tabular form and the sets builder form. In the roster form the elements are separated by commas. For example a sets of positive and odd integers less than or equal to seven is represented by {1, 3, 5, 7}.

The set of all the vowels of the alphabet is {a, e, I, o, and u}. The sets of the even numbers are {2, 4, 6,}. The dots indicate that the positive even integers are going to infinite. While writing the set in the roster form the element must be distinct. For example the set of all the letters in the word “SCHOOL” is {s, c, h, o, and l}. In the set builder form all the elements of the sets have the common property which is not possessed by any element which is outside of the set. For example vowels have the same property and can be written as {a, e, I, o, and u}. The set in the set builder form can be written as V = {x; x is a vowel in the English alphabet}.  A = {x; x is the natural number and 5 < x < 11}. C = {z; z is an odd natural number}.

Operations on Sets: - Like the mathematical operations, there are number of operations which are carried out by the sets. We will study them one by one. Union of sets is the union of the all the elements of a set and all the elements of the other  taking one element only once. Let A = {2, 3, 4, 5} and B = (4, 5, 6}. Therefore the union  of the two sets is written as A ∪B  = {2, 3, 4, 5, and 6}. Intersection of two sets A and B is the set of all elements which are common in both the sets. If the intersections of the two sets are equal to null then the set is called the disjoint set.

The difference of sets A and B can be defined as the A – B. It is equal to the elements which belong to set a and the element which does not belong to the set B. The operation on the sets such as union and intersection satisfy the various laws of the algebra such as the associative laws, the commutative laws, idempotent laws, identity laws, distributive laws, De Morgan’s laws, complement laws and involution law. These all the laws can be verified by the Venn diagram. 

Dispersion Statistics

Measures of Dispersion : - The dispersion is also known as the variability is the set of constant which would in a concise way explain variability or spread in a data. The four measures of dispersion or variability are the range, quartile deviations, average deviation and the standard deviation. The difference between two extreme observations in the given data is known as the range. It is denoted by R. In frequency distribution, R = (largest value –smallest value). It is used in statistical quality control studies rather widely. Median bisects the distribution. If we divide the distribution into four parts, we get what are called quartiles, Q(1 ),Q2 (median) and Q(3.)

The first quartile Q(1,) would have 25 % of the value below it and the rest above it; the third quartile would have 75% of values below it. Quartile deviation is defined as, Q. D.  = 1/2  ( Q3-Q1). If the average is chosen a, then the average deviation about A is defined as A.D. A.D. (A) = 1/n ∑|(xi- A)|  for discrete data. The Standard deviation is also called as the Root mean square deviation. The formula for the standard deviation is given as Standard deviation,σ=√(1/n ∑(xi- x ̅)^2 ) for discrete data.

The Square of the standard deviation is known as the variance. It is denoted by the square of sigma. Out of these measures, the last σ is widely used as a companion to x ̅ on who is based, when dealing with dispersion or scatter. Measure of dispersion is calculated for the data scattering. Deviation means how a value is deviated from it mean or average value. The mean of the two groups of the data may be same but their deviation may be high.

Central Tendency Measures : - The central tendency measures are also called the statistics central tendency. The clustering of data about some central value is known as the frequency distribution. The measure of central tendency is the averages or mean. The commonly used measures of central values are mean, Mode and median. The mean is the most important for it can be computed easily. The median, though more easily calculated, cannot be applied with case to theoretical analysis. Median is of advantage when there are exceptionally large and small values at the end of the distribution. The mode though easily calculated, has the least significance. It is particularly misleading in distributions which are small in numbers or highly unsymmetrical. In symmetrical distribution, the mean, median and mode coincide.

For other distributions, they are different and are known to be connected by empirical relationship. Mean – Mode = 3 (mean – median). The sum of the values of all the observations divided by the total number of observations is called the mean or average of a number of observations.  The value of the middle most observations is called the median. Therefore to calculate the median of the data, it is arranged in ascending (or descending) order. The observation which is found most frequently is known as mode.

The central tendency measures and the variability or dispersion are used in the statistical analysis of the data.

Interpolation

Interpolation is a concept that is used in numerical analysis. It means finding an intermediate value from the given data. That is, for a set of function values, sometimes a situation arises to know what the value of the function is, for some intermediate value of the variable.

Let us explain the concept with a simple example. Let the ordered pairs of the data of a function be (0, -1), (1, 7), (2, 22), (3, 40), (4, 69), (5, 98) and we need to find the function value of 3.8 of the variable. The attempt to find that value is called interpolation and there are several methods. Let us describe those one by one.

The first method is to round the given value of the variable to the nearest value in the data and take the corresponding value of the function. So for the given data, the value 3.8 of the variable can be rounded to 4 and assume the value of the function approximately as 40. But as one can easily see it is a crude method and far from accurate. This method can only be used just for guidance.

The next method is called as linear interpolation. That is, the function is assumed to be linear in the interval that contains the required value of the variable and accordingly the value of the function is determined. In the given example 3.8 falls in the interval [3, 4]. Considering the function to be linear in this interval, the slope of the function in this interval is (69 – 40)/(1) = 29. So the value of the function at x = 3.8 is, 29(0.8) + 40 = 63.2. Though this method also is not very accurate still the accuracy is much better than that in method 1. This method, even at the cost of sacrificing some accuracy, is preferred because it is easy to work with. In general the linear interpolation y at a point x  is given by the formula y = [(y_b – y_a)/(x_b – x_a)](x – x_a) + y_a, where [a, b] is the interval in which the required point occurs. This is the algorithm used in any linear interpolation calculator.

It is always possible a curve rather than a straight line could better cover the points plotted from a data. In other words a polynomial function can give a better interpolation. Finding a suitable polynomial function for a given data is called as polynomial regression.  Suppose we consider the same data, the three degree polynomial function f(x) = 3x² + 5x – 1 will be very close to the desired results. In such a case evaluating the function for x = 3.8, the value of the function is will be a very accurate interpolation. The accuracy can further be improved when a polynomial function of degree same as the number of data points is determined.

A high level interpolation polynomial can be derived by extensive methods and such an interpolation is called as Lagrange interpolation introduces by the famous Italian mathematician.

Analytical Geometry

Analytical Geometry:

Another term for analytical geometry is Cartesian geometry or co ordinate geometry. It refers to the study of relationships between points, lines, planes etc against a back drop of the co ordinate system, be it in three dimensions or two dimensions. In our lower grade normal geometry, we knew of geometric shapes such as triangles, angles, squares, rectangles etc. The same shapes in co ordinate geometry are described by the co ordinates of their vertices, or by the equations and slopes of the lines joining these vertices.

The basic of analytical geometry deals with the concept of lines. The topics covered are parallel lines, perpendicular lines and inclination of lines with respect to the coordinate systems or with respect to each other. A line is described by its slope or gradient. This slope can be found using the co ordinates of two points that lie on the line. If two lines have the same gradient then they are said to be parallel to each other. If the product of the slopes of two lines is -1, then the two lines are said to be perpendicular to each other. The slope can also be defined by the tangent of the angle that the line makes with the positive x axis direction. Thus if we know the slope we can use the arctan function to find the inclination of the line with respect to the positive x direction.

All the other geometric shapes are studies with the help of this concept of straight lines. For example consider a quadrilateral. It is a closed shape formed by four line segments that are parts of four lines. If both the pairs of opposite sides are parallel and congruent to each other, (that means if both the pairs of opposite sides have the same slope and same length) then such a quadrilateral is called a parallelogram. In a parallelogram if all the adjacent sides are perpendicular to each other, (in other words, if the product of the slopes of adjacent sides is -1), then the parallelogram becomes a rectangle. Similarly other geometric shapes can also be studied in this way.

Analytical geometry is most useful in studying three dimensional objects. This is sometimes also called analytical solid geometry. In three dimnesional space, there are three coordinate systems that can be used. Besides the Cartesian co ordinate system, there can also be the cylindrical coordinate system and the spherical co ordinate system. All these three systems can be interchangeably used to study various types of curves, surfaces or solids in space. Just like how a point in the Cartesian co ordinate system is defined by its x y and z co ordinate, in the cylindrical co ordinate system it is defined by three parameters. They are, (a) its distance from the origin (r), (b) the angle of the line joining the point and the origin with the positive x axis (θ) and (c) its perpendicular distance from the x-y plane (z co ordinate).

Also check out the video streaming of Analytical Geometry

Variables

                                                                                                               
What are Variables?

A variable is something which always varies. That is it does not have a fixed value. In any situation if we are not sure of a value for a quantity then we represent such things using alphabets like x, y, z, a, b, c etc known as variables.

Let’s assume that Peter is working in a hotel and earns $12 per hour and on top of his hourly wage he also get tips for each hour. This expression $(12 + x) would give how much peter earn in a given hour. Here x denotes the tips earned. Now, you might also realize that number of tips or the amount of tips Peter make per might change dramatically from hour to hour. It varies consistently. Since, we are not sure of the tips Peter make each hour exactly, this value is called a variable.

Random Variables

In this section i will introduce you to the concept of a random variable. For me this is something I had lot of trouble for getting my head around. That’s really because it called variable. Generally, variable are unknowns used in algebraic equations/ expressions.

For example: x + 2 = 15. We can find the value of x, by subtracting 2 from both sides.
Or if we have an equation in two unknowns that is y = 2x + 5, here x is an independent quantity and y is a dependent quantity. So, we can assume any x value and find respective value for y. In this case we will have infinite combinations of x and y satisfying the equation.

A random variable is kind of same thing that it can take multiple values but it is not something which you really ever solve for. A random variable is usually denoted by a capital letters say X. It can take bunch of different values but we are never solving for it. In fact it a function that maps from you from the world of random processes to the actual numbers. Let us see one example.

Let the random process be it is going to rain tomorrow or not. Now, how are we going to quantify this, let say X = 1 if it rains tomorrow and X = 0 if does not rain. It is not compulsory that we have to use 1 and 0. Here we can assign to any number. So, we need to keep in mind is a random variable is not a traditional variable.

It is defined as a numerical value to each outcome of a particular experiment. For every element of an experiment’s sample space, it can take only one value. When a random variable can take finite (countable) finite set of outcomes then it is known as discrete random variable. Individual outcomes for a random variable are denoted by lower case letters. A continuous random variable would take on any of the countless number of values in the given line interval.

Using Matrices

In this article we shall study about one of the methods used to solve system of linear equations using matrices. Before we study about the method let us first see few definitions

Consider the following system of simultaneous non – homogeneous linear equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂

Expressing the above equations in matrices, we get

These equations can be represented as a matrix equation as AX = D, where







Here A is called the coefficient matrix.X is called the variable matrix.
D is called the constant matrix.

Augmented Matrices

The coefficient matrix augmented with constant column matrix, is called the augmented matrix, generally denoted by [AD]. Hence the augmented matrix of the above system of simultaneous linear equations is 

Sub Matrices

A matrix obtained by deleting some rows or columns (or both) of a matrix is called a sub matrix.

Definition (Rank of a matrix)

Let A be a non – zero matrices. The rank of A is defined as the maximum of the orders of the non – singular square matrices of A. The rank of a null matrix is defined as zero. The rank of A is denoted by rank (A).

It is to be noted that :-

If A is a non zero matrix of order 3 then rank of A is
(i)    1 if every 2 x 2 sub matrices is singular
(ii)    2 if A is singular and atleast one of its 2 x 2 sub matrices is non – singular.
(iii)    3 if A is non – singular.

Consistent and Inconsistent systems

A system of linear equations is said to be
(i)    Consistent if it has a solution
(ii)    Inconsistent if it has no solution.

A system of three simultaneous equations in three unknowns whose matrix form is AX = D has
(i)    A unique solution if rank (A) = rank ([AD]) = 3
(ii)    Infinitely many solutions if rank (A) = rank([AD]) < 3
(iii)    No solution if rank (A) is not equal to rank ([AD])

It is to be noted that the system is consistent if and only if rank (A) = rank ([AD])

The different ways of solving non homogenous systems of equations are
(i)    Cramer’s Rule
(ii)    Matrix inversion method
(iii)    Gauss – Jordan method

In Gauss – Jordan method we try to transform the augmented matrix by using elementary row transformations. So that the solution is completely visible that is x = α, y = β and z = γ. We may get infinitely many solutions or no solution also.

For solving a system of three equations in three unknowns by Gauss – Jordan method, elementary row operations are performed on the augmented matrix as indicated below.

(i)    Transform the first element of 1^st row and 1^st column to 1 and transform the other non zero elements if any in of 1^st row and 1^st column to zero.
(ii)    Transform the second element of 2^nd row and 2^nd column to 1 and transform the other non zero elements if any in of 2^nd row and 2^nd column to zero.
(iii)    Transform the third element of 3^rd row and 3^rd column to 1 and transform the other non zero elements if any in of 3^rd row and 3^rd column to zero. 

Matrices Calculator
We shall study about Matrices Calculator in some other article.

Right Circular Cylinder

Right Circular Cylinder :- If the number of circular sheets is arranged in a stack, if the stack is kept vertically up then it is called the right circular cyl. If the base is circular and the other sheets are kept at ninety degrees at the base, then it is called the right circular cyl.  The right circular cyl can be made by a rectangular sheet by folding it in round shape. The area of the sheet gives us the curved surface area of the cyl because the length of the sheet is equal to the circumference of the circular base which is equal to 2 π r . Therefore the curved surface area of the cyl is equal to the area of the rectangular sheet. The area of the rectangular sheet is equal to the product of the length and the breadth. The curved surface area of the cylinders is also equal to the product of the perimeter of the base and the height of the cylinders.  If the top and bottom of the cylinders is also consider because it is needed to make the circular tin, If the area of the bottom and top of the cylinders is included in the curved surface area we get the total surface area. It means total surface area of the right circular cylinder is equal to the area of the base plus the area of the top of the cylinder plus the curved surface area of the cylinder. The cylinder has two circular base at the bottom and at the top.The diameter of the base of the cylinder can be measured directly with the help of the scale. 

What is the Formula for Volume of a Cylinder :- As we know that the volume of cuboids is equal to the product of the length, breadth and the height. The volume is three dimensional. The cuboids are built up with the rectangle of the same size sheets. In the same way the right circular cylinders can be made by using the same principle. So, by using the same argument as for cuboids, we can see that the volume of a cylinders can be obtained by the product of the base and height of the cylinders.  Therefore, the volume of a cylinder = the base area of the cylinder X height of the cylinder. As we know that the base of the cylinder is the circle. The area of the circle is equal toπ r². The height of the cylinder can be assumed as h. Therefore the volume of the cylinder = π r²h, where r is the base radius and h is the height of the cylinder.

Example :- Let us find the volume of the cylinder which has the radius of the base is twenty one centimeter and its height is thirty five centimeter.

Solution :- The formula for the volume of the cylinder, V = π r²h
Radius r = 21 centimeter, height h = 35 centimeter
Therefore the volume of the cylinder V = π ×21² ×35 = 48490.48 cubic centimeters.

Trend Line

The trend lines are drawn for the prediction of the data. These lines are associated with the data series. The trend lines are used for any type of improvements from the available data. The trends lines can be drawn to follow the equation of a line, i.e y = mx + c, where m is the slope of a lines. The value of c is constant which decides the equation of a lines. The trends line is used for exponential or logarithmic formulae. We can choose the right  type of it for our data as per our requirements. Linear trend lines, logarithmic trend lines, Polynomial trend lines, power trend lines, exponential and moving average trend lines can be drawn as per our requirements. We can add, remove or modify a trend lines as required.

Now a days, online tools or calculators are available to draw a trend line. We can display the equation of any trend line on the chart. The R squared value can be displayed. We can format the present structure to draw a fresher one. The online tools are available to draw any type or desired line quickly to take the faster decisions. Prediction of whether forecasting, draught, can be made by the data available. These  are the key of the success of our business, trade, education and research.

Equation of a Line Calculator: - Line calculators are the online tool which can be used to find the slope of a lines. Slope of a line is the tangent of the line. Slope is the ratio of the rise to run.  A line can be written in the standard form, slope intercept form, intercept form, general form, point slope form, two point form.
The equation of line calculators can also be used to find the equations of the lines when the coordinates are given. This calculator has four inputs and one output. The input to the calculators is the value or the coordinates of two points. The output terminal shows the value of the slope which is ratio of the difference between the y coordinates to the difference of x coordinates. The four inputs to the line calculators are x- coordinate one, x_ coordinate two, y- coordinate one and y- coordinate two. After giving all the inputs we have to press the output button. In the output, we can get a fraction or the degree of the tangent. If we want to check the equation of a line then press the output button for equation of the line. The slope of the lines can be used to find the height and the base when we making the bridge or when a road is planned in the hill area. The tangent angle is required to hit a moving or stationary target.

In the hanging bridge technology the line calculators are used to find the slope of the tension wire to hold the bridges. All the computers which are used to find the moving target and to fire guided missiles use the line calculators to calculate the tangent angles.

Mixed Fractions

Mixed Fraction: - The mixed numbers are like 2 1/2 which is two and one half or 35 3/32 which is equal to thirty five and three thirty two seconds.  To express the mixed fractions we have to keep exactly one blank between the whole numbers and the fractions. The mixed fraction are in the form of 3 2/3 ( three and two third) , 7 1/5 ( seven and one fifth) , 13 1/7 ( thirteen and one seventh ) , 113 7/100 ( one hundred thirteen and seven hundredth) and so on.

The rules for adding the mixed numbers: - To add the mixed number, convert them into the fraction. The algebraic formulae can be used for the addition of these, for example a/b + c/d = (ad + bc) / bd.

Example :- Let us add the two mixed fractions a b/c and d b/c

Solution :- The given fraction are a b/c and d b/c. The first step which is to be used is to convert the mixed fraction to the fraction. The fraction of the mixed term a b/c is equal to (ac + b)/c. The procedure is to convert the mixed fraction to the fraction is to multiply the whole number to the denominator of the fraction and then add the fraction. The total value is divided by the denominator of the fraction. The fraction of the mixed term d b/c is equal to (dc + b)/c. Now we have to add the two fractions as below.

(ac + b )/c + (dc + b)/c. As the denominators of both the terms are same, the numerators can be added directly as below
(ac + b + dc + b ) /c =   (ac +2 b + dc) /c

The rules for subtracting the mixed numbers: - To subtract the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the subtraction of the fractions, for example a/b - c/d = (ad - bc) / bd.

Example :- Let us subtract the two mixed fractions a b/c and d b/c
(ac + b )/c - (dc + b)/c

As the denominators of both the terms are same, the numerators can be subtracted directly as below
{(ac + b) – (dc + b)} /c = (ac + b – dc - b) /c = (ac – dc)/ c = (a – d) [by canceling the common term in the numerator and denominator]. 

The rules for multiplying the mixed numbers: - To multiply the mixed numbers or the fractions we have to convert them into the fractions. The algebraic formulae can be used for multiplication of the fractions, for example a/b *c/d = a c /bd

The rules for dividing the mixed numbers :- To divide the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the division of the fractions, for example a/b ÷ c/d = a/b ×d/c = ad/bc.

Fraction Calculator Online: - The fraction calculator online is a tool which can be used to add, subtract, to multiply and to divide the fractions. We have to enter the fractions to be calculated, enter the function is to be carried out (i.e add, subtract, multiply, divide). The output in the form of the fraction will be displayed in the output window.

Direct and Inversely Proportional

Proportionality:

A quantity is said to be proportional to another quantity if change of one of the quantities is always accompanied by the change of the other. This property is known as proportionality.

Proportionality is of two types:

(i)                  Direct proportionality

(ii)                Inverse proportionality

Direct Proportionality:

A quantity is said to be directly proportional to another quantity if the change in both of them is in the same direction. This means that if one of the quantities increases then the other also increases. If one of the quantities decreases then the other also decreases.

It is denotes by the symbol a. If ‘a’ is directly proportional to ‘b’ then:

We write as a a b ==> a = k * b where k is the proportionality constant.

i.e.  a / b = k = constant à a1 / b1 = a2 / b2

Graph:

Let us consider that x a y. If we plot the values of x and y on a graph sheet we obtain the graph showing the relation between these two quantities. Generally, the graph of directly proportional quantities is a straight line. Thus by seeing the graph we can conclude the relation and proportionality between two quantities.

Inversely proportionality:

A quantity is said to be inversely proportional to another quantity if the change in both of them is in the opposite direction. This means that if one of the quantities increases then the other quantity decreases. If one decreases then the other increases.

Inverse proportionality is also uses the symbol a but the reciprocal of the second quantity is written.

If ‘a’ is inversely proportional to b then:

We write as a a 1 / b

You can see that the reciprocal of b is written to indicate inverse proportionality. We can also say that ‘a’ is directly proportional to the reciprocal of ‘b’.

If a a 1 / b à a = k / b where k is the proportionality constant.

i.e. a * b = k = constant à a1 * b1 = a2 * b2

Graph:

Let us consider x is inversely proportional to y i.e. x a 1 / y

Now plot the values of x and y on the on a graph sheet we obtain the graph showing the relation between the two quantities. Generally the graph is not a straight line but a curve. On seeing the graph we can analyze the relation and proportionality between the two quantities.

Problem:

If the volume of a gas at a given temperature is 2 liters when its pressure is 1 bar, then what will its volume when the pressure increases to 3 bars? (Volume in inversely proportional to pressure)

Sol: Given, initial volume v1 = 2 l

Initial pressure p1 = 1 bar

Final pressure p2 = 3 bars

As volume is inversely proportional to pressure we have, p1 v1 = p2 v2

Now v2 = p1 v1 / p2 = (2 * 1) / 3 = 0.66 liters (approx.)

Thus final volume the gas is 0.66l

Properties and Area of a Rectangle

Rectangles
For a normal 4th grader, a rectangle would mean a plane figure that has four sides. However, more precisely in geometry, a branch of math, a rectangle is a special type of a quadrilateral that has 4 right angles. It would look as shown in the picture below:

Properties of a rectangle:
1. It has four sides.
2. It has four angle and all the angles are right angles.
3. It has four vertices.
4. Each pair of opposite sides are congruent.
5. Opposite sides are parallel.
Examples of rectangles:
1. Top of a book.
2. Face of a cuboid.
3. Top of a table.
4. Front of a cupboard.
5. Etc.

The Area of the Rectangle:
The Formula for the Area of a Rectangle is as follows:
A = l * w
Where,
A = area of the rectangle,
L = length of the rectangle
W = width of the rectangle.

It is customary to denote the longer side of the rectangle as length and the shorter side as width. Another custom is to denote the horizontal sides as the length and the vertical sides as the width of the rectangle. It is shown in the picture below.

Let us now try to understand how to calculate the area of a rectangle with the help of a sample problem question.

Example 1: Find area rectangle from the figure shown below:

Solution:
From our formula for area of a rectangle we know that,
Area = A = L * W
For this problem,
L = length = 5 units and
W = width = 3 units
Therefore substituting these values of L and W into the above formula for area of the rectangle we have,
A = 5 * 3 = 15 sq units <- answer="" p="">
If instead of being given the measures of length and width, we are given the co ordinates of the vertices of the rectangle then its area can be found out as follows:

Consider a rectangle with the vertices at A (x1,y1),B  (x2,y2), C (x3,y3) and D (x4,y4) taken in clock wise direction. Therefore we know that if AB is the length of the rectangle then BC would be the width of the rectangle. The distance AB can be found using the distance formula as follows:

Length = L = AB = √[(x2-x1)^2 + (y2-y1)^2]

Similarly the distance BC can also be found using the distance formula as follows:

Width = W = BC = √[(x3-x2)^2 + (y3-y2)^2]

Both the above can be now used to find the area of the rectangle as follows:

A = L * W.

Sample problem:
Find the area of a rectangle having vertices at (3,7), (0,7), (3,-2) and (0,-2)

Solution:
First let us sketch a graph of the said rectangle.

From the picture we see that
Length = L = 3-0 = 3 and
Width = W = 7 – (-2) = 7+2 = 9

Therefore,
Area of rectangle = 3 * 9 = 27 sq units.

Define Absolute maximum

Optimization is one of the most vital applications of differential calculus, which guides the business and the industry to do something in the best way possible. Business enterprises ever need to maximize revenue and profit. Mathematical methods are employed to maximize or minimize quantities of interest. Absolute maximum value is when an object has a maximum value.

In mathematics, the maximum and minimum of a function, identified collectively as extrema , is the largest and smallest value that the function obtains at a point either within a given local or relative extremum (neighborhood) or on the function domain in its entirety.

A function f has an absolute maximum at point x1 , when f(x1) =  f(x) for all x. The number f(x1) is called the maximum value of ‘f on its domain. The maximum and minimum values of the function are called the extreme values of the function. If a function has an absolute maximum at x = a , then f (a) is the largest value that f that can be attained.

A function f has a local maximum at x = a if f (a) is the largest value that f can attain "near a ." Simultaneously, the local maxima and local minima are acknowledged as the local extrema. A local minimum or local maximum may also be termed as relative minimum or relative maximum.
Both the absolute and local (or relative) extrema have significant theorems linked with them Extreme Value Theorem is one of it.

To find global maxima and minima is an objective of mathematical optimization. If a function is found to be continuous on a given closed interval, then maxima and minima would exist by the extreme value theorem.
Moreover, a global maximum either have to be a local maximum within the domain interior or must lie on the domain boundary. So basically the method of finding a global maximum would be to look at all the local maxima in the interior, and also look at the maxima of the points on the boundary; and take the biggest one.
For any function that is defined piecewise, one finds a maximum by finding the maximum of each piece separately; and then seeing which one is biggest

In mathematics, the extreme value theorem signifies that if a real valued function f is continuous in the closed and bounded intermission [x,y], at that moment f should attain its maximum and minimum value, each of it at least once. That is, there prevail numbers a and b in [x,y] in such a way that:
F(a) = f(c) = f(b) for all c summation [x,y].
A related theorem is also known as the boundedness theorem which signifies that a continuous function f in the closed interval [x,y] is bordered on that interval. That is, there always exist real numbers m and M in such a way that:
m = f(c) = M for all c summation [x,y].
The extreme value theorem thus enhances the boundedness theorem by demonstrating that the function is not only bounded, but also accomplish its least upper bound as its maximum as well as its greatest lower bound as its minimum.

Algebra variable

Algebra is a subject which helps us to find unknown quantities with minimal information. But how do we carry all mathematical operations when something is not known? We assign letters from alphabet (mostly small case letters) for the unknown.

Such letters are only called as variables or changeable. Why the name is selected as such? Because the value assigned to these could be varied according to our wish but gives the correct information at a required condition. For example I earn $100 per day.

How do I calculate the total earnings after a certain number of days? Here the word certain is really uncertain! In other words it is an unknown. So we assign ‘x’ as the number of days and I can formulate that my total earnings in dollars will be 100x. Now my job is simple. I just to need to replace ‘x’ by the actual number of days I decide and I get the required information correctly.

Thus basically the changeable are employed as algebra variables or simply math variables. But why we had done away with the prefixes ‘algebra’ or ‘math’ and we simply refer as such?
Because, such uses of letters even in other subjects are subjected to mathematical or algebraic operations.
The types of variables are many. In most cases more than one forms are used. For example one may represent an input and another may be the corresponding output. Obviously the former can be assigned any value and hence it is described either as independent forms and the latter, because of the dependence on the input is referred as dependent forms.

In most of the cases they are represented by the letters ‘x’ and ‘y’ respectively. In such cases they are related by equations or functions.
Further, different types of variables are used depending on the context. For example when you study about distance versus time for a moving object, lettert is used for the unknown time and ‘s’ is used for the corresponding distance.

The same letter tis also used for denoting temperatures in normal scale. (whereas, T is used for temperature in absolute scale!). Letters p and v are used for denoting pressure and volume respectively.
We mostly see that the letters at the second half of English alphabet are mostly used for the unknown quantities. It is just a convention and the first of half is generally reserves for constants. However use of letters like ‘a’, ‘b’ ‘c’ is common in geometry.
In addition to English alphabetical letters Greek letters are also extensively used especially in the topics of trigonometry.

Linear Transformation

Let us try to give a simple introduction and explanation about linear transformations. Let us not scare the readers with hi-fi terms like ‘vector spaces’ ‘matrices’ and symbols like ‘e’ ‘Rn’ etc. Yes, let us involve those at a higher level after getting acquainted with what basically a linear-transformation means.

In a data different scores, that is, items described by numbers are exhibited. There is a possibility to express the items of the data in the form of a pattern. If such a pattern is in the linear form that the transformation of the data set to a pattern is called as linear transformation. It may be noted that such a transformation can be fairly accurate for a limited interval meaning limited number of items in the data set.

This type of transformation is obviously results as a linear function in the form a + bx (similar to mx + b in analytical geometry). Since linear functions are always ‘one to one’, this type of transformation is also referred as one to one linear transformation. Recording the scores of student in a class can be cited as one of the linear transformation examples.

Let us discuss about the basic concept of this type of transformation. Suppose Xi represents the item in general, of the given data, and if X’I is the same after the transformation of the data, then the linear relation is X’ = a + b Xi, where a and b are constants for the particular transformation. The letter a is called additive component and b is the multiplicative component of transformation.

These are analogous to y-intercept and slope of linear algebraic functions. One must know what should be mean and standard deviation of the transformed data and accordingly the values of constants a and b are determined. Because the condition of a linear-transformation is X’m =   a + b Xm and X’s = b Xs, where the subscripts m and s refer the respective mean and standard deviation.

Let us illustrate a linear transformation example. Suppose a data is describes the scores as 13, 16, 21, 21, 24. This has to be linearly transformed with a mean of 95 and standard deviation of 15. What is the formula of transformation?

The mean and the standard deviation of the given data are Xm = 19 and Xs = 4.42, rounded to nearest hundredth. The set of desired figures in the transformation is X’m = 95 and X’s = 15. Since, X’s = b Xs,
b = 3.39 and a = 95 - 3.39*19 = 30.59, rounded to two decimal place.
Thus the transform relation is X’ = 30.59 + 3.39X.

Linear Programming with the Help Of Simplex Algorithm

The concept of programming is very important. It is being used in the field of mathematics as well. Linear programming is a very important concept and is now very widely used in the field of mathematics. The Simplex method tutorial is a part of the linear programming model. This method is also called an algorithm. This algorithm is used as part of linear programming. This is used in finding a optimal solution.

The Simplex method examples can be very helpful in understanding and knowing more about this algorithm. For understanding this method a geometric figure called the polytope has to be studied. Basically a polygon is a geometric figure which has many sides. So, hexagon is a geometric figure which has six sides.

A pentagon is a geometric figure which has five sides. Similarly there are other geometric figures which have different number of figures and they are given various names. In Simplex methods the polytope plays a very important role as this gives the area which is under consideration for finding the optimal solution. So, this concept has to be learnt properly.

There is different number of vertices present in a polytope. To find the optimal solution, the process begins from any one of the vertices of the polytope and moves towards the vertex which shows the optimal solution. This can be represented in a standard form.

Another form can be used in this case, namely the canonical form. There are two methods that can be used. The two methods are called the M-method and the other one is called two-phase method. As the name suggests in the two-phase method there are two phases that are to be considered to arrive at the final solution.

The final solution is nothing but the optimal solution. The ultimate purpose is to arrive at the optimal solution. An example can be used to explain the concept. An equation will be given for simplification. There will also be some constraints given. The simplification has to be done keeping these constraints in mind.

The constraints can also be in the form of equations. These equations must be taken into account while performing the simplification procedure. Then they can be represented in the canonical form and a feasible solution is found for the variables present in the equation, keeping in mind the constraints given. Once this is done the optimal solution is found out.

Dot product

A dot product is an operation that which involves multiplication of two vectors to arrive to a scalar product.  Given two vectors, v=ai + bj and u=ci + dj, v.u read as ‘v dot u’ would be equal to a scalar product, ac + bd. So, basically the product would be a number and not a vector.
The dot product of two vectors would be a scalar even in a three dimensional space, R3.  So, in a three dimensional space given vectors v=ai +bj+ck and u=xi+yj+zk, the dot-product is given by v.w=ax + by +cz. The definition of dot product can be given as the dot product equation of vectors a’ and b’ such that a.b= ax. bx + ay.by = |a||b|cos(theta) .
Here |a| and |b| are the magnitudes of the vectors and theta is the angle between the vectors. It is read as modulus of vector a multiplied with the modulus of vector b, multiplied by the cosine of the angle between the two vectors a’ and b’.
Following are some of the important points to be remembered while finding the scalar product, i.i=1, j.j=1, k.k=1, i.j=0, j.k=0 and k.i=0, this shows that the scalar product of vectors which are perpendicular to each other is zero.
Some of the properties of dot-product are as given below,
• Commutative property: u.v = v.u
• Distributive property: u.(v+w) = (u.v) + (u.w)
• Associative property: (cv). u = v.(cu)= c(u. v)
• 0. u = u.0 = 0
• v.v =|v|2
• If v. v = 0 then v = 0
Let us now take a look at the dot product proof of distributive property given by u. (v+w)=(u.v)+(u.w)
Let the vectors to be, u=(u_1,u_2,u_3...,u_n ); v =(v_1,v_2,v_3...,v_n) and w=(w_1,w_2,w_3...,w_n). On the left hand side we have, u.(v+w) = (u_1,u_2,u_3...,u_n ).[(v_1,v_2,v_3...,v_n)+ (w_1,w_2,w_3...,w_n)]
          =  (u_1,u_2,u_3...,u_n ).[(v_1+w_1), (v_2+w_2), (v_3+w_3)…, (v_n+w_n)] on regrouping we get,
          = [u_1((v_1+w_1), u_2(v_2+w_2), u_3(v_3+w_3),…,u_n  (v_n+w_n)]
Applying the distributive property we get,
= [u_1v_1+u_1w_1, u_2v_2+ u_2w_2, u_3v_3+ u_3w_3….., u_n v_n+ u_n w_n]
Which can be written as, [u_1v_1, u_2v_2, u_3v_3…, u_n v_n] + [u_1w_1, u_2w_2, u_3w_3…, u_n w_n]
On re-writing the above expression we get, [(u_1,u_2,u_3...,u_n ). (v_1,v_2,v_3...,v_n)]+[ (u_1,u_2,u_3...,u_n ). (w_1,w_2,w_3...,w_n)]  which would be the expression on the left hand side, [u.v+u.w] and hence proved!Thus we can prove all the properties using the above computational method.

 u_n it vectors are the vectors with length of one u_n it.  For u_n it vectors u and v, the dot product of u_n it vectors is given by, u.v=cos(theta) where (theta) is the angle between the two u_n it vectors.

Work and Time Calculation

Work and time are two of inter-related concepts in mathematics and science. Work and time related calculations are most often asked in almost all competitive exams. Taught in middle school classes, work and time calculation problems are worked out in SAT, MAT exams as well. The trick is to solve the problems within seconds. Let’s have a look at some of the facts related to work and time calculation in this post.

1. If a person can complete a work in n days, then the person can complete 1/n part of the work in one day. For example: She completed the process of researching, ordering and buying the Fisher Price toys for infants’ collection for her shop in 6 days. Therefore, she will complete 1/6 part of the work of researching, ordering and buying the Fisher Price toys for infants’ collection for her shop.

2. If the number of person to complete a particular work is increased, the time to complete the same work decreases. For example: 100 employees build about 1000 toy action figures in 10 days. If the number of employees is increased to 150, then they will build 1000 toy action figures in less than 10 days because the work is distributed among more workers.

3. If worker A has the capability of working twice as worker B, then A will take ½ of the time that B took to complete a work. For example: B designed the outlook of cot mobile for baby girls in 2 hours. A works twice as B and therefore, A designed the outlook of cot mobile for baby girls in ½ x 2 hours = 1 hour.

These are some of the most important facts to be known while working out work and time calculation in mathematics.  However, the list if not the ultimate one, there are many other such work and time related facts.

Absolute Error

When we do any calculations there are always chances of making mistakes, either we do addition, subtraction or anything, similarly when we measure height, distance or anything with the help of any measuring device there are chances of making a mistake so if we measure the same thing twice we may get different answers and this is due to the error in measuring. Error is not the mistake we have made because it does not give you the wrong answer. The uncertainty in measurement is termed as the error. There are many types of errors which occur in experimental studies.

1. Greatest possible error – This is the error we make when we do the approximation or rounding off to tenth, hundredth place.

2. Absolute Error– This is the error which occurs due to the inaccuracy in the measurement we do. Experimental scientists come across usually with this type of error. This is the amount of physical error we make in the process of measurement. Absolute Error Formula– It is usually denoted by delta x and is equal to difference between the calculated value and the actual value. Now How to Calculate Absolute Error or How to Find Absolute Error– We can find the absolute-error by finding the difference between the inferred value and the calculated value of the measurement. It usually signifies the uncertainty in the measurement process. For example: - If we find the length of stick as 1.09 centimeter though its actual length is 1 centimeter. Then the absolute-error that is delta x = Calculated value – Actual value which is 1.09 – 1 and that is equal to 0.09. Hence we can say that absolute-error is equal to 0.09. Absolute-error is always positive. Therefore we can call it as the absolute value of the difference of the two values which are the calculated value and the actual value.

3. Relative error – This type of error tells you about how good a measurement is relative to size of the thing which is measured. It expresses the ratio of absolute-error to the measurement that is accepted. This actually shows the relative size of the error of the measurement in relation to the measurement itself. The formula for calculating relative error is Relative error = Absolute error over accepted measurement.

Online Tutoring - A Real Time Learning

Online tutoring is emerging day by day due to its personalized learning sessions. These tutoring sessions are completely student driven, secured, flexible and affordable. Students can schedule a session on any subject with their tutor from the comfort of home.

A Beginning of Online Tutoring

Online tutoring came into existence with technological advancements. Learning new topics in a technology-oriented set up is quite fascinating for every student. It is the most comfortable form of learning a subject from any location. In this one-on-one learning program, students get maximum attention and ample time to clear their doubts from a preferred tutor. Online learning not only satiates student’s educational need but also make them confident during examination time. Apart from regular sessions, students also get homework and assignment assistance from an online tutor. This flexible learning program is specifically designed for K-12 grades. Moreover, a tutor covers all the topics that are being taught in the classroom session.

Why to Choose Online Learning Program

• Learning a subject from different locations and at convenient time is one of the notable features of virtual tutoring.
• All learning sessions are managed by qualified and experienced tutors.
• Every tutorial package is designed by keeping in mind the educational requirement and budget constraint of students.
• In a virtual classroom, every tutoring session is scheduled as per student’s availability.
• All queries of students are explained with the help of a whiteboard or through chat
• Regular assessment is done to improve students’ performance.
• Personalized attention, instant connection with tutors and curriculum based guidance is what an online learning program offers.

Make Learning More Interesting with Fascinating Features

An online learning service provides several interesting features, which keep students involved. Students can take unlimited tutoring sessions in safe and fun way. A whiteboard on a computer screen allow students to write their questions and get instant answers in a step-by-step manner. Apart from a whiteboard, a chat option and a real time audio also helps students to communicate with their tutor and get their doubts cleared in a better way. This personalized tutoring session improves students’ knowledge and also make them aware of new learning methods. Further, every learning session can be saved, replayed for revision purpose in a virtual classroom.

Instant Connection with a Tutor

Students can get an instant connection with an online tutor right from home. By using a broadband connection and a personal computer, a student can take a session on any topic from his or her favorite tutor. Along with this, a tutor also provide proper guidance to students during exams and while doing homework and assignment. Students who like to study alone and at their preferred time can opt for online learning program. This dynamic tutoring program is gaining importance worldwide due to its exciting learning tools and computer-integrated unlimited sessions.

Revise and prepare well before your exam with online tutors

Online tutoring – an innovative learning method

The demand for online tutoring has rapidly gone up in these highly competitive times. The advancement in new technologies makes this learning method more useful to students of different grades. Allied tools like the virtual whiteboard and an attached chat box allow students to communicate with their online tutors in a smart way. Due to these fascinating tools, online learning sessions can give the effect of face-to face sessions. Additionally  it is done in a safe web environment helping students concentrate on the subject. According to current research, online learning methods offer a modernized learning platform where students get better results by putting less effort. It has been observed that any student who gets individualized instruction performs better than a student who studies in a classroom environment.

The role of a online tutor

Online tutoring carries several positive aspects and most importantly, it constantly strengthens the students' learning skills and increases their self confidence. It helps students develop a positive attitude towards any subject. Apart from providing knowledge on different subjects, it helps students to improve their self esteem. All these are made possible due the remarkable assistance of online tutors who guide students as per their requirements. These well trained online tutors are available 24 x7. Due to these one-on-one learning sessions with expert tutors , students can tackle any learning problem smartly. These tutors provide a thorough understanding of any topic and also give comprehensible guidelines that help students score well in exams. They also assist in completing home work and assignments, on time.

The features of online learning sessions:

Few positive aspects are mentioned below and these explain why students should prefer to choose online  learning sessions to get good scores in exams:

(i) Broad Subject coverage is one of the main reasons behind the success of online assistance. With this service, students can opt for educational help on any topic of any grade.

(ii)  Online help is affordable compared to other learning methods. Students can select the topic as per their need and they are required to pay for the services they choose.

(iii) Experienced tutors and their 24 hours availability helps students achieve their goals.

Online tutoring – a helpful way to revise any topic before exams

Students feel anxious before their exams and also they need a quick revision to test their expertise in a particular subject. In that respect, online assistance is quite beneficial and effective as students can schedule their sessions at a convenient time. They can also clear their doubts step-by-step before exams. Online learning help gives students the confidence to handle exam hassles in a smart way.

Online math tutoring - Learn math with live examples

Learn Math in a virtual classroom from the best online tutors. With the help of a whiteboard and animated live examples, students can easily understand every difficult Math problem. Moreover, online math help provides free math worksheets in order improve students problem solving skills.

Online Math tutoring is a smart and comfortable way of learning the subject from any location. Many students struggle a lot in Math subject and score low marks. Online Math help is the ideal option to overcome the anxiety and stress, which students often face while solving Math problems. As we know, Math subject has high importance in varied field like Engineering, Science & Technology, Banking, etc. therefore it is important for every student to learn the subject thoroughly. Online tutoring gives enormous benefits and ample time to students to master the subject . This cutting-edge mode of learning has gained immense appreciation across the globe due to personalized and interactive learning sessions.

Online Math tutoring sessions come with a wide variety of teaching and learning tools like whiteboard, attached chat options, recorder, dashboard, etc. With the help of these tools, a student can select a topic, start a session with preferred tutor at convenient time. Along with this, students can communicate through chat option and clear their doubts from an online tutors who are available round the clock in a virtual classroom. Moreover, every tutoring session can be recorded and replayed by the students to revise the topic as many times as need and want.

Many websites make Math learning more informative for students with live examples. Every topic is well explained with the help of graphs or animations, which keep students involved more in a online learning session. Right from understanding the basic concept of Math to Algebra, Calculus,Geometry and Trigonometry, each topic is illustrated with animated examples to make the session more effective and influential for students. Online Math help not only enhance your problem solving skills but also keep a tab on your performance. Regular feedback from an online tutor can actually help a student to do better in the subject.

Learning Math with live example is quite enjoyable and beneficial for those students who face difficulty in understanding the subject. Every tricky sum is being solved by highly experienced online Math tutor so as to give a thorough understanding of the subject. In addition to this, some websites provide free worksheets and math quiz to make the Math subject more interesting for students. Online Math tutoring provides different ways of learning with the help of live examples, which helps students to understand the logic behind every math problem. It is a great assistance  for students who find Math subject boring and difficult.

System of Linear Equations

System of Linear Equations is a collection of linear equation  Systems of Linear Equations that involve two equations in two variables are simplest to deal.
Suppose there are two linear equation in x and y, then each equation will represent a line in x-y plane. A solution to these equations will be the point where these lines intersect. Thus the solution will be unique value of x and y. If the equations represent parallel lines then there will be no solutions to this system. If the Linear System of Equations contains same coinciding lines then the solutions will be infinite in number.

Any system of linear equation can have following conclusions: no solution, unique solution or infinitely many solutions. A linear system is consistent if it has at least 1 solution and is said to be inconsistent if it has no solution.
Suppose a linear equation is 2x+y=0. Then there will be infinite points satisfying this equation. Like (x, y) ={(1,-2),(0,0),(2,-4)… and many more}. Now suppose there is another line x-y=0, then solutions to this will be (x, y)={(0,0),(1,1),(2,2) and so on}. A common solution to these equations is x=0, y=0. This is hence solution of this system of equations.
We can search solution of more than two equations also by drawing graph of the equations also.

Method of Solving Systems of Linear Equations:
By substitution:
Let two equations are a_1x+b_1y=c_1 and a_2x+b_2y=c_2. Solve first equation for x:
 a_1x+b_1y=c_1  
a_1x=c_1–b_1y or x=(c_1–b_1y)a_1
Substitute this value of x in second equation to get:
a_2((c_1–b_1y)a_1)+b_2y= c_2
Now you get an equation in y. Solve for y. now put the value of y in any of the two equations to solve for x.
You can substitute value of y also from an equation and then substitute it in other equation.

Systems of Linear Equations Word Problems
Q.1) Cost of 2 chairs and 1 table is 1000 while cost of 1 chair and 3 tables is 1500. Find cost of each.
Solution) let cost of chair=x and of table=y.
2x+y=1000………(1)
x+3y=1500………(2)
From second equation: x=1500-3y
Substituting x in first equation = 2(1500-3y)+y=1000
3000-6y+y=1000
3000-1000-5y=0
2000=5y
y=(2000/5)=400
Putting y=400 in first equation we get: 2x+400=1000
2x=(1000-400)=600
x=600/2=300
Linear equation can also be solved by equating coefficients:
Equate coefficients of x by multiplying equation (2) by 2:
2x+2(3y)=2(1500)
Subtract this equation from (1):
   2x+y=1000
-(2x+6y=3000)
     0-5y= -2000
5y=2000
Y=400
Now put value of y in any equation.
You can equate coefficients of y also and then subtract the two equations to get value of x first.

Divison of two numbers is given by the following relation

Divison of two numbers is given by the following relation:
m/n=q
Here m is being divided by n and the result of the divison is q which is known as quotient. Let see how to do Division by 2 Digit Numbers which means dividing any dividend by 2 Digit Divisor

Step 1) Put the two digit divisor before the divison braces and put the dividend no. below the divison bar.
Step 2) Check the first digit of the dividend. If it is smaller than the divisor then take the first two digits of the dividend. Now determine how many times of divisor produces those two digits of dividend or produces a number which is just less than the dividend digits. Let x times of divisor give above result.
Step 3) Now multiply the no. x by divisor, let the result is y. Put y under the first two digits of dividend.
Step 4) Subtract no. ‘y’ from first two digits of dividend. Let the result is z. number z will be less than the divisor. So, bring down the third digit of dividend beside z. Now again follow the same steps from step 2.
Step 5) Continue following these steps till no more digits are left in the dividend and you get a remainder which is less than divisor.

Let’s use method of Long Division Two Digit Divisors through some examples:
Example 1) Divide number 7139 by 16.

Step 1) Firstly, check the first digit of the dividend. It is 7 and is smaller than divisor 16. So we will consider two digits of dividend i.e. 71. Now the largest multiple of 16 which is smaller than 71 is 64. As 16*4 = 64 so, write 4 (quotient) on right hand side of dividends and 64 below 71. Now, subtract 64 from 71.  
16)7139(4                                                                
       64                                                                                                                                                                                                                                   .       7
Step 2) Now consider number 7. As this is smaller than divisor 16, so we will bring digit 3 down with number 7. Now we will repeat the above step again. Find the largest multiple of 16 which is smaller than 73. This number is 64 which is equal to 16*4 . Write 4 at the place of quotients and subtract 64 from 73.
 16)7139(44                                                                
      64
----------
      73
-     64
----------
      99
-     96
----------
-      3
Step 3) Repeat the above steps again. This is done below:
16)7139(446
     64
--------
     73
-    64
--------
     99
-    96
--------
      3

3 is remainder.
This is how a Division 2 Digit Divisor is done. You can practice Divisibility by 2 digit divisors: 234/34, 5678/89, 7553/123etc.

What is Ratio

When we say one banana for every three apples, the relationship between the banana and the apple is shown by a term called Ratio. It is used in comparing and showing the relationship between two entities. It is denoted using the symbol colon (:) between the two values.

In the above example the proportion between banana to apple would be banana: apple read as ‘banana to apple’ the value of which would be 1:3 read as ‘one is to three’.

Hence we can say ratios tell the relationship between two values that is how one number is related to the other. It may be denoted as a fraction also, for instance the two values which are to be compared are X and Y then the proportion between them can be shown either as X:Y or X/Y or just X to Y. In the above example the proportion shows that apples are three times bananas.

One important point to remember while writing the balance is that the order should not be changed that is the respective numbers should not be interchanged.

If for instance there are 3 pencils for every 5 pens, the balance when considered as pencils to pens should also be written in the same order pencils:pens, 3:5 and not 5:3 which would mean pens to pencils

Let us now determine the value of Y, if X=6 and the balance of X to Y is 3:4. To find the value of Y first we need to determine how many times X is divisible by the corresponding part of the balance (3:4) which can be calculated by dividing 6 with 3 which gives 2.

Now we just need to multiply this 3 with the corresponding balance part of Y which gives 2x4=8. When the proportion is 3:4 and the value of X=6 then the value of Y=8.  Ratio definition can be given as comparison between two things which tells the relationship between the two. Let us now take a glance at the various ratio problems which help to understand the concept.

There are 8 children, 3 are boys and 5 girls. What is the ratio of boys to girls, girls to boys, the total children to boys and total children to girls? Given the total number of children=8, boys=3 and girls=5. So, the proportion of boys to girls is 3:5; the proportion of girls to boys is 5:3; the proportion of total number of children to boys is 8:3 and the proportion of total number of children to girls is 8:5.

Change of Base Formula for Logarithms

Logarithm is a means of expressing a number using exponents. Example log101000 is equal to 3 as 1000 is a cube of ten and can be written as log_10 10^3. Hence the value is 3.
The common base for logarithms is base ten and the other base is the natural logarithm base –e. At times while calculating logarithms we come across base other than 10 and the base e, in such cases the base change can be done using a special formula.

Logarithm change of base formula can be given as, log x to base a = log x to base b/log a to base b.  To understand how to arrive to this base change formula let us go through the following steps:
Consider y=log_a x, we get x = a^y
Taking log_b on both sides would result in log_b x = log_b a^y
Applying the power rule to the above equation gives, log_b x = y log_b a
Now dividing on both sides with log_b a gives, log_b x/ log_b a= y log_b a/ log_b a
So, we get, y = log_b x/ log_b a

Let us now consider a simple example, the value of log 27. This can be written as log_10 7/log_10 2

The value can be calculated as log 7=0.845 and log2= 0.3010. When these values are divided the final answer would be 2.80730…; thus using loga x= log_bx/log_b a, the change of base formula logarithms value of the given logarithmic expression can be found easily. Using Log base change formula it becomes easy to evaluate logarithms with different base. Here the logarithm is written as a fraction with the logarithm of the number as the numerator and the logarithm of the base as the denominator, such as log_a x = log x/log a.

Then each of the logarithms is evaluated using the log table or a scientific calculator, the final value is got by dividing these values. The evaluation of other logarithms with base different from natural logarithm base or the common logarithm base can be done using the base change formula, log_a x = log_b x/log_b a. Let us now evaluate the logarithm log_5 9. This problem can be solved by either using natural logarithm or the common logarithm. Using the natural logarithm that is base-e it would be, log_5 9 = ln9/ln5 = 2.1972/1.6094 which would be approximately equal to 1.3652… Now using the common logarithm that is the base ten it would be, log_5 9 = log 9/log5= 0.9542/0.6989 = 1.3652… Using either of the logarithms we arrive at the same result.

Simple interest

Definition:
Consider a house that one would have rented. The tenant has to pay some amount of money to the owner of the house as rent for using the property. Similarly if a person borrows money from another person, he has to pay some amount of money as rent for using the borrowed money. This charge paid for use of funds is called interest. Therefore the amount charged on a fixed amount of principal, that is lent by a lender for a specific period of time is called simple interest. In simple interest the principle amount over the period of loan remains constant and is not reduced or increased.
Formula for simple interest:
Some important terms related to simple interest:
(1) Principal (P): The money borrowed or lent.
(2) Interest (I): The additional amount paid to the lender, for the use of the money borrowed.
(3) Rate( R ): Interest for one year per 100 units of currency.
(4) Time (T): The time period for which the money is borrowed.
(5) Simple interest or (S.I.): When the interest is paid to the lender regularly every year or every half year, we call the interest simple interest.
(6) Amount (A): Principal + Interest = amount at the end of the term of T years.

Formula used for calculating simple interest is like this:
S.I. = P x R x T
100
A = P + S.I.

When we calculate simple interest, the following points need to be noted:
(1) Rate of 4% per annum means $ 4 for every $ 100 per year. Similarly a rate of 1.5% per month means $ 1.5 for every $ 100 per month = $ 1.5 * 12 = $ 18 for every $ 100 per  year = 18% per annum.
(2) When time is given in days, we convert it to years by dividing by 365. When time is given in  months, we convert it to years by dividing by 12. When dates are given, the day on which the sum is borrowed is not included but the day on which the money is returned is included, while counting the number of days.

Decimal to Hexadecimal

We are now going to look at Decimal to Hexadecimal converter. So let us understand what exactly a hexadecimal number and what its digits mean. So we are going to look at three digits of hexadecimal number the first unit represents units, which is 16 to the power of zero that is one. That represents units. The second digit represents tenths, which is 16 to the power of one. And the third digit represents hundredths, which is 16 to the power of two. That is nothing but 256. So important thing to do when one is working on how to convert a decimal to hexadecimal, is the start of working out how many hexadecimal number is going to have?

Let us understand it with an example, convert decimal to hexadecimal. Say number 74, here we need to decide, what we are going to and how many digits this hexadecimal number is going to have. Now because 256 is less than 74, there is any going to be two digits. So we now going to see, that in 16 to the power of one column, here we divide 74 by 16 and the result is 4. This means 4 times 16 is 64 and we have the remainder as 10. Now 10 is a single digit in a hexadecimal, simply represents a A , that tells us 74  = 4 A.

 Let us understand with a complicated example. This time it represents 680 as a hexadecimal number. Here that we see 680 is greater than 256 so we are going to have three digits in a hexadecimal number. What we going to do first is divide 680 by 256. And the result of that is 2. Two times 256 is 512, so our remainder is 168. Next we go back as what we did in our first example, we are going to divide 168 by 16. The result of this is 10. 10 times 16 equals to 160, as we are left with the remainder 8. Now we have three digits in hexadecimal number, thus we notice that we have 10, which is represented by ( A ) . This tells us that 680 when written as a hexadecimal number as 2A8. That is, 680 = 2A8. This is how we do a decimal to hexadecimal conversion. The method to do this is to keep on dividing the decimal number by 16 till it gets the most significant remainder.

Rules of Narration for Different Types of Sentences

Narration is one of the most important concepts in English grammar. While changing narration, it is very important to follow certain rules. These rules at times differ according to the types of sentences. Let’s have a look at the rules of narration for different types of sentences in this post.
Rules of Narration for Assertive Sentences:

Rule 1: If there is no object after reporting verb, then it should not be changed. For example:
• Direct Speech: He said, “I bought a play gun from Nerf India collection for my nephew.”
Indirect Speech: He said that he bought a play gun from Nerf India collection for his nephew.
Rule 2: If there is any object after the reporting verb, then say is changed to tell, ‘says’ to ‘tells’ and ‘said’ to ‘told’. For example:
• Direct Speech: She said to me, “Pre Nan Nestle Baby is healthy and nutritious for babies.”
Indirect Speech: She told me that Pre Nan Nestle is healthy and nutritious for babies.
Rule 3: ‘said’ can be replaced by replied, stated, and added and more as per the context of the assertive sentence. For example:
• Direct Speech: She said to him, “I am going to school today.”
Indirect Speech: She replied to him that she is going to school that day.
Rules of Narration for Interrogative Sentences:
Rule 1: In interrogative sentences, ‘said’ is changed to ‘asked’ while changing from direct to indirect speech. At times, ‘said’ is also changed to ‘enquired’ or related terms as per the context.
Rule 2: If the question is formed with is/are/am/was/were etc. then it is replaced by ‘if’ or ‘whether’.
Rule 3: While changing from direct to indirect speech, the question mark is removed as the reported speech is an indirect statement and not a direct question.
For example:
• Direct Speech: She said to him, “Have you bought anything from Philips Avent India brand?”
Indirect Speech: She asked to him whether he has bought anything from Philips Avent India brand.
These are some of the rules of narration that is defined as per different types of sentences.