Largest Number

Let us study about the largest number. When we enter into the world of mathematics we can come to know that the numbers rule that world.
The various kinds of numbers with their different combinations form this math world. These numbers can be classified into two types basically as smallest and largest numbers.
The largest numbers are nothing but the largest values among the given ones. Some of the examples are given below.

Please express your views of this topic Factor Number by commenting on blog.

Largest number – example 1:

Arrange the following number series and find out the largest number: 524, 32, 43, 254, 234, 74, 930, 157, 338, 99, 86, 50, 290, 98 and 78.



Solution:

The given number series is as follows: 524, 32, 43, 254, 234, 74, 930, 157, 338, 99, 86, 50, 290, 98 and 78.
Do comparing each and every number of the series with the other numbers and arrange them from the largest numbers to the smallest numbers order as follows:
32 < 43 < 50 < 74 < 78 < 86 < 98 < 99 < 157 < 234 < 254 < 290 < 338 < 524 < 930
32, 43, 50, 74, 78, 86, 98, 99, 157, 234, 254, 290, 338, 524 and 930
Therefore the largest number in the series is found to be as ‘930’ among all other numbers.



Largest number – example 2:

By solving the following number series (4 * 4), (10 - 1), `(9/3)` , (16 + 7), (11 + 9), (12 – 8), (25 – 11) and find out the large number among them.



Solution:

The given number series is (4 * 4), (10 - 1), `(9/3)` , (16 + 7), (11 + 9), (12 – 8), (25 – 11)
The steps that are carried out to solve the given number series and to find the largest number among all others are as follows:

(4 * 4), (10 - 1), `(9/3)` , (16 + 7), (11 + 9), (12 – 8), (25 – 11)

(16), (9), (3), (23), (20), (4), (14)

Arrange all the numbers after solving in the series as from the largest numbers to the smallest numbers as follows:

23, 20, 16, 14, 9, 4, and 3

So by solving the given number series the largest number is found to be as ‘23’ among all other numbers.

I am planning to write more post on Rounding Calculator and Radical Calculator. Keep checking my blog.

Largest number – exercises:

Give the large number that gets divisible by ‘13’ between 100 and 200. (Answer: 195)
Give the large number that gets divisible by ‘21’ between 200 and 300. (Answer: 294)
Give the large prime number between 900 and 1000. (Answer: 997)

Axis of Symmetry Equation

The most interesting part of High-school mathematics is probably the visualization of the problem. The intrinsic part of learning mathematics is knowing the Function! Functions can be visualized by putting them in a graph in the form of equations. The axis of symmetry equation is such a graph in which there is a line which is the axis of symmetry of the given function. The line of symmetry acts as a mirror to one part of the graph. Thus we can call the axis of symmetry equation function, the mirror function!
Visualization of Axis of Symmetry Equation

Let us now visualize the simplest axis of symmetry equation: y = |x|;

For this function,

y = IxI

So, our graph would be: mod

So, we can see that the y-axis acts just like a mirror in this function!

Now, let us take the Quadratic equations. At a glance, this wouldn't have anything to do with the Absolute Value function but if we take a little deeper look, we will find that all quadratic equations can be expressed by the modulus function, i.e. it is again an axis of symmetry equation.

Let us take the example y = x2 (a parabola). This equation is the same as y = | x2 | because any squared number is always positive!

The graph of this equation would be: axis of symmetry

So, once again, we can see that the y-axis acts just like a mirror in this function. So the equation is an axis of symmetry equation!

Now, if the vertex is shifted to (2, 1) (say), the equation would be : y - 1 = (x - 2)2

The graph of this equation would be: axis of symmetry

Having dealt with the axis of symmetry equations where axis is either y-axis or a line parallel to y-axis, we can easily understand that the same thing will happen if the axis is x-axis or a line parallel to the x-axis if the form of the equation becomes : x = | f(y) |;

For example the parabola, y2 = x(i.e. x = |y2|) will have the x-axis as the "mirror"!

The graph of this equation would be:axis of symmetry

Here, we can see that the x-axis acting just like a mirror in this function. So the equation is an axis of symmetry equation!

There can be other axis of symmetry equations where the axis of symmetry can be inclined to the x and y axes at different angles. Let's take the case of the functions, y = ln x and y = ex. If we plot these curves, the line y = x acts as the axis of symmetry to these equations.

Here we see the axis of symmetry equationinclined axiswith inclined symmetry
Axis of Symmetry Equation-real Life Examples

There can be numerous examples of the axis of symmetry equations, in fact if you stand in front of the mirror and your shape can be defined by any function, your image and you, together can be described by the axis of symmetry equation or the modulus of that function!!!! If we look around, we'd just have to appreciate mathematics for giving us wonderful analogies of the real world! That is the beauty of Mathematics!!!

Solve Precalculus Inequalities

Precalculus inequalities are  statement that one algebraic expression is greater than (or less than) another algebraic expression is called as precalculus inequalities.

There are three rules for preparation algebraic inequality:

The same quantity can be added or subtracted from each side of inequalities.

Each side of inequalities can be multiplied or divided by the same positive quantity.

If each side of an inequality is multiplied or divided by the same, negative quantity is equivalent to the first.

Properties for Solve Precalculus Inequalities:

Trichotomy property for solve precalculus inequalities:

For any real number, a and b, exactly one of the following statements is true:

pq

The trichotomy property indicates that exactly one of the following statements is true about any two real numbers. Either

The first is less than the second

The first is equal to the second, or

The first is greater than the second one.

Transitive property for solve precalculus inequalities:

If p,q and s are real numbers with p
If p,q and s are real numbers with a>b and b>c, then a>c.

The first part of the transitive property indicates that:

If a first number is less than a second number and the second is less than a third, then the first number is less than the third.

The second part is similar, with the word “is greater than” substituted for “is less than”.

Addition property for solve precalculus inequalities:

Any real number can be added to (or subtracted from) mutually sides of an inequality to produce another inequality with the same direction.

To solve the addition property, we add 3 to both sides of the inequality 1<2 get="get" p="p" to="to">
3+1<3 p="p">
4<5 p="p">
We note that the < sign is unmovable (has the similar direction).

Subtracting 5 from both sides of 15<5 change="change" direction="direction" does="does" either.="either." inequality="inequality" not="not" of="of" p="p" the="the">
15-5<5-5 p="p">
10<0 p="p">
Algebra is widely used in day to day activities watch out for my forthcoming posts on Simplify Fractions and Volume of a Cube. I am sure they will be helpful.

Multiplication property for solve precalculus inequalities:

If both sides of an inequality are multiple (or divided) by a positive number, another inequality results wilt the same direction as the original inequality.

To solve the multiplication property, we multiply both sides of the inequality 3<5 2="2" by="by" get="get" p="p" to="to">
3(2)<5 p="p">
2<10 p="p">
The < symbol is unaffected.

Dividing property for solve precalculus inequalities:

Dividing both sides by 2 does not change the direction of the inequality either.

`-4/2`<`8/2`

-2<4 p="p">Example for Solve Precalculus Inequalities:

To solve the  precalculus inequality 25≤12y-3<17 p="p">
Solution:

This inequality means that 12y-3 is between 25 and 17. We can solve it by isolating a between the inequality symbols.

25≤12y-3<17 p="p">
28≤12y<20 p="p">
14≤6y<10 p="p">
7≤3y<5 p="p">
The solution set is {y|7≤3y<5 5="5" in="in" interval="interval" notation="notation" or="or" p="p">

Types of Coordinate Systems

The coordinate systems use values or numbers for the location of a point or a shape in the system. And the values or the numbers are called as the coordinates of the point and are related between the various systems either in the two dimensions or three dimensions. In the following article we will see in detail about the topic types of coordinate systems.
More about Types of Coordinate Systems:

There are basically two types of systems like the two dimensional coordinate systems and the three dimensional coordinate systems. The two dimensional coordinate systems are,

1. Cartesian coordinate systems

2. Polar coordinate systems

Some of the three dimensional coordinate systems are,

1. Cartesian coordinate system.

2. Spherical coordinate system.

3. Cylindrical coordinate systems.

Cartesian coordinate system:

cartesian coordinate system

A point in the cartesian coordinate system has the coordinates (x,y,z). Other three dimensional coordinates in terms of Cartesian coordinates,

Spherical coordinates: `r = sqrt (x^2+y^2+z^2)` ; `theta = arccos (z/r)` ; `phi = atan2(y,x)`

Cylindrical coordinates: `rho = sqrt (x^2+y^2)` ; ` phi = arcsin (y/rho)` ; `z = z`

Spherical coordinate system:

Spherical coordinate system

A point in the spherical coordinate system has the coordinates (r, theta, phi). The other three dimensional coordinates in terms of the spherical coordinates,

Cartesian coordinates: `x = r *sin theta*cos phi` ; `y = r *sin theta*sin phi` ; `z = r cos theta`

Cylindrical coordinates: `rho = r*sin theta` ; `phi = phi` ; `z = r*cos theta`

Cylindrical coordinate system:

My forthcoming post is on solving algebraic proportions and how to solve proportion problems will give you more understanding about Algebra.

Points in the cylindrical coordinates have coordinate values (rho, phi, z). The relation between the cylindrical coordinate system and the other three dimensional coordinate systems,

Cartesian coordinates: `x = rho*cos phi` ; `y = rho*sin phi` ; `z = z`

Spherical coordinates: `r = sqrt (rho^2+z^2)` ; ` theta = arccos (z/r)` ; ` phi = phi`

Polar coordinate system:

polar coordinates

Points in the polar coordinate system has the coordinates (r,theta). The relation between the polar coordinates and the two dimensional Cartesian coordinates are,

Cartesian coordinates: `x = r* cos theta` ; `y = r* sin theta`
Example Problem on Types of Coordinate Systems:

1. Find the equivalent Cartesian coordinates for the point with the polar coordinates (6, 50).

Solution:

`x = r* cos theta`

`= 6*cos 50`

`= 6*0.64`

`= 3.8`

`y = r* sin theta`

`= 6*sin 50`

`= 4.6`

My previous blog post was on General Equation of an Ellipse please express your views on the post by commenting.

Practice problem on types of coordinate systems:

1. Find the equivalent spherical coordinates for the location of the point (7.8, 50, 5) in the cylindrical coordinates.

Answer: (9.3, 50, 50)

fundamental laws of algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics and number theory, algebra is one of the main branches of pure mathematics.Source: Wikipedia

I like to share this Limit Laws with you all through my article.

Fundamental Laws of Algebra:

We list the fundamental rules and properties for useful algebra and give examples on they may be used.

Suppose that a, b and c are variables or applied algebraic expressions.

Commutative laws for addition.

a + b = b + a

Example:

5+2=2+5

Commutative laws for multiplication.

a * b = b * a

Example:

6*5=5*6

Associative laws for addition.

(a + b) + c = a + (b + c)

Example:

(6+2)+5=6+(2+5)

Associative laws for multiplication.

(a * b) * c = a * (b * c)

Example:


(5*4)*2=5*(4*2)

Distributive laws of addition over multiplication.

a * (b + c) = a * b + a * c


And


(a + b) * c = a * c + b * c

Example:
2*(5+3)=2*3+2*5

Fundamental Laws of addition in algebra:

For the addition of positive and negative number, the follow rules, established in the First Course, apply make easy to solve algebra equation:

A number represented by a letter is called a literal number, and any number expression in which more than numbers of symbols are variable is called a literal expression.

Laws for fundamental addition in algebra:

1)  To add two numbers with similar signs, find the sum of their total values, and prefix the sign common to both.

2)  To add two numbers with different signs, find the diversity of their total values, and prefix the sign of the one with the larger total value.
Order of Operations for Fundamental Laws of Algebra:

In long math problems with +,-,x,%,(), and exponents in them, you have to know what to do first. Without follow the same rules, you may get unlike answers. You can easily keep in mind the silly sentence, Big Elephants Destroy Mice And Snails, you can commit to memory the order of operations, and you must follow.

Big               “B” means Brackets. You need to do operation in side parentheses first.

Elephants    “E” means an exponent, you must calculate exponents next.

Destroy        “D” means division. Start on the left of the equation and perform all multiplication and divisions in the order in which they appear.

Mice             “M” means multiply

And               “A” means addition.

Snails           “S” means subtract. Always on the left hand side and perform all additions and subtractions.


Example for fundamental laws of algebra:

Solve 12%2(10-6)-2+4 using order of operations.

Solution:

Big                 reduce brackets. 10-6=4                                                                          12%2*4-2+4

Elephants       reduce exponents. We do not have any exponents.                             12%2*4-2+4

Destroy           make divide operation in order from left to right     12%2=6                 6*4-2+4

Mice                carry out multiply operation in order from left to right 6*4=24                24-2+4

And                 do addition operation in order from left to right 24-2+4=24+2=26        26



Answer is 26.

Percent of a Number Calculator

The word per cent  means per hundred.  Its symbol is %.
90% means 90 parts out of 100 parts.  A fraction can be converted to a per cent.  For example 50/100 is a fraction.  It can be converted to percent as 50%. A decimal can be converted to per cent.  For example 0.33 is a decimal.  It can be written as
33%.
Percent of a number can be calculated by the following methods.
# Find 8% of the number 125
Solution 8% can be written as 8/100
so 8% of 125  is  8 x 125                1000
_______   =          _____     =    10

100                       100
Answer 8% of the number 125 is 10
#  What percent of the number 36 is 9?
Solution:  Let the number be x
Then x% of the number 36   =   9
We can write  x  times 36   =   9
100

36x   =  9
100
Transposing 36/100 to the other side, we get    x  =  9 times 100   =  900  =  25
36                  36
Hence the answer is 25% of a number 36 is  9
Fractions of Percent of a Number Calculator:
Fractions can be converted into percents.
# Convert the following fraction 9/20 into percent
To convert into percent we multiply the fraction by 100
We get  9  x   100   =    900  =   45     Solution  is  45
20                    20
# Find  6½% of 75 metres.

Solution  Step 1 convert  6½ into improper fraction.  We get 13/2
Step 2   (13/2) %  of  75  is   13 x  1    x  75   =   975  = 4.88 metres.
2    100                 200
Decimals can be converted to percent.
# Find 7.5% of 80 kgs.
Solution  : -  7.5% can be written as (75/10) times 100
7.5% of 80 kgs =  7.5  x  80  =   600  =  6 kgs.
100                 100

Word Problems on Percent of a Number Calculator:
Let us do some word problems on finding the percent of a number.
# A basket contains 350 eggs.  12% of the eggs are rotten.  Find the number of good eggs.
Solution. Total number of the eggs in the basket = 350
Percent of the rotten eggs = 12%
Number of rotten eggs =  12% of 350
=  12  x  350  = 4200  = 42
100                 100
Number of rotten eggs = 42
Number of good eggs  =    350 - 42  =  308
Answer 308 eggs are good.
#If 15% of the workers in a  factory are females, and the number of male workers  is 272, find the total number of workers
in the factory.
Solution:-  Let the total number of workers be x
Then percentage of females = 15%'
Percentage of male workers = 100 -15= 85%
Male workers are 272
Hence 85% of  x  = 272
That is 85x  = 272
100
Therefore   x  =  272 . 100  =  27200   =  320
85              85
Answer:  There are 320 workers in the factory.
Practice Problems:
1.Find the percent of 3/4 (answer 75%)
2.If x% of 75 is 9, find the value of x (Answer 12)
3.What % of 2/7 is 1/35 (Answer 10%
4.What % is $15 of $ 120(Answer 12.5%)
5. Find the number whose 13% is 65(Answer 500)

Algebraic Expressions Product

An expression is the important topic in algebra. Algebraic expression product  is the combination of variables and constants with basic arithmetic operators, they are add, subtract and then divide and multiplication.  For Example (p-9) is the algebraic expression.  Here p is the variable and 9 is the constant value and then ‘-‘is the subtraction operation.  In algebraic expressions product we can multiply the two algebraic expressions.
General Process of Algebraic Expressions Product

General process of algebraic expressions product: Now we have to assume the algebraic expressions (fx + gy) and (ax +by). This contains the below steps. They are,

Step 1: First we have to take multiplication of first term of the first expression with second expression. That is fx (ax+ by).

Step 2: Now we have to multiply the inner term values. That is fx(ax) +fx(by). It gives afx2 +bfxy

Step 3: Now we have to take multiplication of Second term of the first expression with second expression. That is gy (ax+ by) .

Step 4: Now we have to multiply the inner term values. That is gy(ax) +gy(by). It gives agxy +bgy2

Step 5: No we have to add the step 2 and step 4 values. That is afx2 +bfxy + agxy +bgy2

Step 6: Take the common terms we can get, afx2 + xy(bf + ag) +bgy2

This is the general rule of algebraic expressions product.
Problems Using the Algebraic Expressions Product

Problems 1 : Take algebraic expressions product of (j + 5) and (k− 6)

Solution: multiply the two given algebraic expressions

(j + 5)(k− 6)

= j(k − 6) + 5(k − 6)

= jk – 6j + 5k − 30

Problems 2 : Take algebraic expressions product of (2u +8)(u2− 2u − 10)

Solution: multiply the two given algebraic expressions

(2u +8)(u2− 2u − 10)

= (2u) (u2− 2u − 10) + (8) (u2− 2u − 10)

= (2u3 – 4u2 – 20u) + (8u2 – 16u − 80)

= 2u3 + 4u2 – 36u − 80

Solving Factoring Rules

Factoring is the method of finding out the multiples of an expression. The expression may be algebraic equation or a real numbers. It is like making an expression into simpler one by splitting them with multiplication. There are many factoring rules and also there are many formulas for factoring rules. Here the rules are divided into four types which are Greatest Common Factor (GCF), four terms, three terms and two term Expressions.
Solving Factoring Rules i and Ii

Solving Factoring Rules I: Greatest Common Factor (GCF)

GCF is the basic factoring rule in both integers and algebraic expressions. When factoring, always start with finding the largest expression and dividing them into simple term.

Ex: Factor the expression 14 x2 y3 + 21x

Sol:

Factor of 14 is 1, 2, 7, 14

Factor of 21 is 1, 3, 7, 21

Therefore GCF of 14 and 21 is 7.

Also GCF of  x2 y3 and x is x.

Therefore GCF of  14 x2 y3 + 21x is 7x.

First we have to factor the GCF. So the factors are,

7x (2 x y3 + 3)

Solving Factoring Rules II: 4 – Terms (Factor by Grouping)

Sometimes there are four or terms in an expression and also they wont have common factors. In this case, group up together terms to  get common factors.

Ex: Factor 2x3 – x2 + 18x – 9

Sol:

The given expression 2x3 – x2 + 18x – 9 does not have common factors for all of the terms. Here we have to group the terms.

= 2x3 – x2 + 18x – 9
= x2 (2x – 1) + 9 (2x – 1)
= (2x – 1) (x2 + 9)
Solving Factoring Rules Iii and Iv

Solving Factoring Rules III: 3 – Terms (Trinomials)

In order to factor trinomials, we have to begin with multiplying two binomials.

Ex: Factor 2x2 + 2x – 60

Sol:

= 2x2 + 2x – 60
= 2x2 + 12x – 10x – 60
= 2x (x + 6) – 10 (x + 6)
= (2x – 10) (x + 6)

Solving Factoring Rules IV: 2 – Terms

Difference of Squares       a2 – b2 = (a + b) (a – b)

Difference of Cubes          a3 – b3 = (a – b) (a2 + ab + b2)

Sum of Cubes                  a3 + b3 = (a + b) (a2 – ab + b2)

Decimal Equivalent for Fractions

Decimals are a kind of fractional number. The decimal 0.6 signifies the fraction 6/10. The decimal 0.75 signifies the fraction 5/100. Decimal fractions forever have a denominator based on a exponent of 10.We know that 5/15 is equal to 1/3 since 1/5 times 5/5 is 5/15.In this article we shall discuss the table of decimal equivalent for fractions.
Table of Decimal Equivalents for Fractions:

Example problems- table of decimal equivalents for fractions:

Example problem1:

Convert the decimal value into fraction, 0.5

Solution:

Here to convert 0.5 into fraction, multiply and divide by 10,

=   0.5*10*`1/10`

=   5*`1/10`

= `5/10`

= `1/2`

Equivalent fraction is `1/2.`

Example problem2: table of decimal equivalents for fractions:

Convert the decimal value into fraction, 0.75

Solution:

Here to convert 0.5 into fraction, multiply and divide by 100,

=   0.75*100*`1/100`

=   75*`1/100`

= `75/100`

= `15/20` =` 3/4`

Equivalent fraction is `3/4` .



Example problem 3: table of decimal equivalents for fractions:

Convert the decimal value into fraction, 0.25

Solution:

Here to convert 0.25 into fraction, multiply and divide by 100,

=   0.25*100*`1/100`

=   25*`1/100`

= `25/100`

= `1/4`

Equivalent fraction is `1/4` .

Example problem 4: table of decimal equivalents for fractions:

Convert the decimal value into fraction, 0.35

Solution:

Here to convert 0.35 into fraction, multiply and divide by 100,

=   0.35*100*`1/100`

=   35*`1/100`

= `35/100`

= `7/20`

Equivalent fraction is `7/20` .

Example problem 5: table of decimal equivalents for fractions:

Convert the decimal value into fraction, 0.45

Solution:

Here to convert 0.45 into fraction, multiply and divide by 100,

=   0.45*100*`1/100`

=   45*`1/100`

= `45/100`

= `9/20`

Equivalent fraction is `9/20` .

Is this topic physics problems hard for you? Watch out for my coming posts.

Example problem 6:

Convert the decimal value into fraction, 0.55

Solution:

Here to convert 0.55 into fraction, multiply and divide by 100,

=   0.55*100*`1/100`

=   55*`1/100`

= `55/100`

= `11/20`

Equivalent fraction is `11/20` .

Example problem 7:

Convert the decimal value into fraction, 0.3

Solution:

Here to convert 0.3 into fraction, multiply and divide by 10,

=   0.3*10*`1/10`

=   3*`1/10`

= `3/10`

Equivalent fraction is `3/10` .

The following table will explain you the decimal equivalent for the given fraction

Practice Problems - Table of Decimal Equivalents for Fractions:

Practice problem 1:

Convert the decimal value into fraction, 0.2

Result: `1/5` .

Practice problem 2:

Convert the decimal value into fraction, 0.8

Result: `4/5` .

Practice problem 3:

Convert the decimal value into fraction, 0.9

Result: `9/10` .

Congruence And Similarity

Robert finds two marbles that look exactly the same.
John has got a new baby. People visited him to congratulate John and say that the baby looks like him.
Note the two words exactly  and  looks like.
In mathematics these two words are defined as congruence and similarity.
Definition on Congruence and Similarity
Between congruence and similarity, let us see what is meant by similarity.
Two shapes are said to be similar if the shapes are same. The sizes may be different. The following is the example.

In the above diagram you find the two triangles look alike but the sizes are different. If the difference in size  is in a right proportion then the triangles are said to be similar.
That is, for the above triangles  to be similar, the ratio of the corresponding sides of the triangles must be same. Besides the measures of the angles must be equal.
Thus, the complete definition of similarity of two figures is both the figures must be of the same shape and the corresponding sides must be in the same ratio. Also the corresponding angles must be congruent.
The concept of similarity has many practical applications. You see a building. It is impossible to show the building to the same size on a drawing. What is normally done is to draw a smaller but the same shape of the building to a certain ratio. This ratio is called as the scale factor.
Definition on Congruence and Similarity
in congruence and similarity, we have seen what is meant by similarity of two figures.
If two figures are to be congruent, they must be of same shape and also of be same size.

In the above diagram you find the two triangles are of same shape and he measures of the corresponding sides and the measures of the corresponding angles are equal. Hence the two triangles can be said as congruent.
Thus, the complete definition of congrency of two figures is both the figures must be of the same shape and of the same size. Also the corresponding angles must be congruent.
Please note that the transformation of figures does not affect congruency.
The congruency of two shapes can be established if they fulfill certain geometric conditions. The concept of congruency helps in many geometric solutions.

Arc Of Circle

Arc of circle is the important concept in geometry.  Arc of circle is the one of the segment of the circumference of the circle.  Arc of the circle is the part of the circle.  In this topic we have to discuss about the area of the arc of circle with their example problems.
Brief Explanation of Area of Arc of Circle
Figure of the arc of the circle:

Area of the arc of the circle:
    Here area of the arc is also denoted as area of the sector.  It is defined with the help of the following formula,
A = `(theta)/(360)`  p r2
    We have substitute the central angle and radius value in the above formula. Here p is equal to 180 degree.  Now we can get the area of the arc of the circle.
Example Problems:
Example 1:
Find the area of the arc of a sector if the central angle is equal to 120 degree and radius is equal to 10 cm.
Solution:
Given central angle = 120 degree and radius r=10 cm
The formula for area of the arc of the circle is equal to
A = `(theta)/(360)`  p r2
Substitute the central angle and radius and p value we can get the value for the area of the arc of the circle
A=`(120)/(360)`x 180 x 102
Simplifying this we can get,
A=6000 Square centimeter
Therefore the area of the arc of the circle is 6000 Square centimeter

I am planning to write more post on solving equation, solving linear equations with two variables. Keep checking my blog.

Example 2:
Find the area of the arc of a sector if the central angle is equal to 270 degree and radius is equal to 22 inches.
Solution:
Given central angle = 270 degree and radius r=22 inches
The formula for area of the arc of the circle is equal to
A = `(theta)/(360)`   p r2
Substitute the central angle and radius and p value we can get the value for the area of the arc of the circle
A=`(270)/(360)`x 180 x 222
Simplifying this we can get,
A=65340 Square inches
Therefore the area of the arc of the circle is 65340 Square inches

Pre Algebra Two Step Equations

An equation is a mathematical statement that asserts the equality of two expressions. Equations consist of the expressions that to be equal on opposite sides of an equal sign. (Source: From Wikipedia).
The following rules are used to solve the equation and the equation does not change: Add or subtract any variable or number to the both sides of the equation. Multiply or divide any variable or number to the both sides of the equation.
2 step equations is the method of solving the equations by 2 steps.  Now, we are going to see some of the problems on pre algebra two step equations.

I am planning to write more post on  System of Equations. Keep checking my blog.

Problems on Pre Algebra Two Step Equations:

Example problem 1:
Find the value of x: -4 x + 6 = 30
Solution:
Step 1: Subtract 4 on both sides of the equation
-4x + 6 - 6 = 30 – 6
-4 x = 24
Step 2: Divide by -4 on both sides of the equation
`(-4x) / -4 = 24 / -4`
x = -6.
So, x = -6 is the solution of the given equation.
Example problem 2:
Find the value of m: 2m - 2 = 12
Solution:
Step 1: Add 2 on both sides of the equation
2m - 2 + 2 = 12 + 2
2m = 14
Step 2: Divide by 2 on both sides of the equation
`(2m) / 2 = 14 / 2`
m = 7
So, m = 7 is the solution of the given equation.

Few more Problems on Pre Algebra Two Step Equations:

Example problem 3:
Find the value of t: 12 – `(t / 2)` = 6
Solution:
12 –` (t / 2) ` = 6
Step 1: Subtract 12 on both sides of the equation
12 –` (t / 2)`  – 12 = 6 – 12
-`(t / 2)`  = -6
Step 2: multiply by -2 on both sides of the equation
-` (t / 2)` * -2 = -6 * -2
t = 12
So, t = 12 is the solution of the given equation.
Example problem 4:
Find the value of p: `(p / 2)` + 8 = 13
Solution:
Step 1: Subtract 8 on both sides of the equation
`(p / 2) ` + 8 - 8 = 13 – 8
`(p / 2) ` = 5
Step 2: multiply by 2 on both sides of the equation.
`(p / 2)` * 2 = 5 * 2
p = 10.
So, p = 10 is the solution of the given equation.

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Practice Problems on Pre Algebra Two Step Equations:

1)     Find the value of p: (p / 3) + 6 = 7 (Answer: p = 3).
2)     Find the value of q: 4q + 2.5 = 6.5 (Answer: q = 1).
 3)   Find the value of t: 2 t + 11 = 2 (Answer: t = -4.5)

Measures of Central Tendency Simplified

The Mean, Median and Mode are the measures of central tendency. Mean is the average of the values in the given data. Median is the middle term or value in the given data. Mode is the value that occurs most often or most number of times in the given data.

Definition of Mean Median and Mode
Arithmetic Mean can be defined as the sum of the values given in the data divided by the number of values.
Median can be defined as the middle value in a list of data arranged from the smallest to greatest.
Mode can be defined as the value which occurs most often in a given list.
Let us consider the data 6,7,9,6,8,4,3. The Mean of the given data would be (6+7+9+6+8+4+3)/7= 43/7 = 6.14. To find the median we need to arrange the given data in the ascending order and the middle value would be the median. 3,4,6,6,7,8,9. The middle value in the arranged list is 6 and hence the median is 6. Mode is the value that occurs the most number of times, 6 is such value and hence the mode.

To understand the Mean, Median and Mode of the given data let us consider a simple example problem. The responses of nine people as to how many times they visit a grocery store a month were, 8, 10, 9, 5, 8, 4,11, 8, 9.  Here let us first arrange the given data in the ascending order. 4, 5,8,8,8,9,9,10,11. Mean is the average of all the data values given, that would be[sum of the data values/total number of data values]. The sum of the data values is (4+5+8+8+8+9+9+10+11=72) the total number of data values is 9. So, the Mean = 72/9 = 8.  The middle value of an arranged data list is the Median. Here we have odd number of values and hence the middle value in the list, 8 is the median.
Mode is the value that occurs the most number of times in a list, 8 is the value which is occurring most number of times and hence the mode

What Does Mean Median and Mode Mean
Mean is sum of the data values divided by the number of data values. It is most useful when the data set has no outliers. Median is the middle value in a sorted list (If even numbered, 2 middle values-average them). It is most useful when the data set has outliers and there are no big gaps in the middle of the data. Mode is the data value(s) that occurs most often in a set of data. It is most useful when the data set has many identical values.

Percent Deviation

Percent deviation problems bring the fact that the average of a set of measurements, such as weight, does not necessarily reflect the fact none of the data in the sample will be at the calculated average.
The percent deviation gives a number that indicates where the majority of measurements are. Percent deviation is most useful  in statistics and chemistry problems. It finds how far a measurement, on average, will deviate from the  mean value. This problem gives a set of measured  numerical data.
Deviation or Error Percent.

Percent deviation :
In order to find percent deviation first of all you need to find the Average deviation and Mean. After finding the two things you divide the average deviation into the mean then multiply by 100% . To Find the average deviation you need to subtract the mean from a measured value.  Mean value can be measured as sum the data values and then divide that number by the number of data values.
percent deviation = (Average Deviation/Mean) x 100

The following steps are required to find the percent deviation:
ex:  4, 5,9
Step 1: find the mean value of the measurements. Sum the data values and then divide that number by the number of data values
Mean =  (4+5+9)/3  = 18/3  = 6;
Step 2: Find the Average Deviation. Find the deviation of each  Data value, sum them and then divide by the number of measurements. The Deviation is calculated like as absolute values of subtraction of the each value from the mean.
Deviation = ( |4-6|+|5-6|+|9-6|);
= ( 2+1+3)
= 6
Average Deviation = 6/3 = 2  
Average Deviation = 2
Step 3: By using the Formula of percent deviation substitute the Average deviation and Mean then we get Percent deviation
Percent Deviation =  (Average Deviation/Mean) x 100
=  (2/6) x 100
= 33.33%

Exercise on Parent Deviation
1) A student  measured the length of his  desk as  81cm.Later he found  that it was actually 75cm long. What was his percent deviation ?
2) A student the volume of  a piece of rock to be 25 grams and the accepted value is 19 grams. What is the deviation in percentage?

Introduction to solve by elimination method

Introduction to solve by elimination method

This is one of the methods used to calculate the unknowns involved in different simultaneous equations. Here we need to have equations equal to the number of unknowns. Now, let us discuss the elimination method by solving few simultaneous equations of two unknowns.
Example to Solve by Elimination Method

Ex 1: Solve the following simultaneous equations using elimination method

          3x + 2y = 18

          4x – y = 2

Sol:  Here let us call the equations as 1 and 2.

         3x + 2y = 18 ---------`|->` (1)

         4x – y = 2------------`|->` (2)

(2) × 2  `rArr` 8x – 2y = 4

(+) (1)  `rArr` 3x + 2y = 18

         `rArr` 11x      = 22

          `rArr`   x = `(22)/(11)` = 2

                   x = 2

Now plug in x = 2 in the equation number (1) as follows:

3x + 2y = 10   `rArr` 3(2) + 2y = 18

                      `rArr` 2y = 18 - 6 = 12

                      `rArr` y = `(12)/(2)` = 6.

                      `rArr` y = 6

Therefore, the solution of the given equations are x = 2, y = 6.
Example to Solve by Elimination Method

Ex 2: Solve the following simultaneous equations using elimination method

          3x + 2y = 23

           x – y = 1

Sol:  Here let us call the equations as 1 and 2.

         3x + 2y = 23 ---------à (1)

          x – y = 1------------à (2)

(2) × 2  `rArr` 2x – 2y = 2

(+) (1)  `rArr` 3x + 2y = 23

         `rArr` 5x   = 25

         `rArr` x = `(25)/(5)` = 5

                x = 5

Now plug in x = 5 in the equation number (1) as follows:

3x + 2y = 23 `rArr` 3(5) + 2y = 23

                   `rArr` 2y = 23 -15 = 8

                   `rArr`  y = `(8)/(2)` = 4

                   `rArr` y = 4.

Therefore, the solution of the given equations are x = 5, y = 4.
Practice Problems to Solve by Elimination Method

Solve the following simultaneous equations using elimination method:

(i) 4x + 3y = 10

        x + y = 3

Answer: x = 1, y = 2.

(i) 5x + 2y = 16

    3x + 4y = 18

Answer: x = 2, y = 3.

Introduction of passport to algebra and geometry

Introduction of passport to algebra and geometry:

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.

Concepts of Passport to Algebra and Geometry:

Algebra: The following are some of the concepts of passport to algebra.

Number theory : The number theory has a natural numbers and whole numbers which can show below. The counting of numbers 1, 2, 3, 4, 5…is known as natural numbers and the whole numbers have o along with the natural numbers.

Rational numbers : The rational number can be defined as the integer of values is not equal to 0 which is in the form m / n, where m, n is the integers and n ≠ 0. The rational number is said to be in a proper form.

Linear equation: It is an algebraic equation which has the term of constant and product of a constant in single variable. The values are not fixed it can be represented as p, q and r.

Geometry: The following are some of the concepts of passport to geometry.

Radius: A radius is the distance from center of a circle to any point on that circle's circumference.

Perimeter :  The perimeter is a path which can be surrounds an area and also used for path or length.

Circumference: A circumference is the distance around a circle.

Example of Passport to Algebra and Geometry Answers:

Ex 1: We can be obtained the natural number by subtracting of 1 from in the given numbers as shown.

Predecessor of 13 = 13 – 1 = 12

Predecessor of 43 = 43 – 1 = 42

Sol:  We can be obtained the natural number by adding of 1 from in the given numbers as shown.

Successor of 97 = 97 + 1 = 99

Successor of 18 = 18 + 1 = 19

EX 2:   Determine rational number represented as 65.0.

Sol :   Let x = 65.0. Then x = 0.656565…

 100 x = 65.656565…

 100 x − x = (65.656565…) − (0.656565…)

                = 65.0000…

 99x = 65

 x = 65 / 99

    = 13 / 33

Ex 3:  Solve the linear equation for a variable:      x + 7 = 19

Sol :  Take away 7 from the both sides using the property 1

x + 7 - 7 = 19 - 7

x = 12

Ex 4: Solve: y - 3 = 16

Sol :  Add 3 to both sides using property 2

y - 3 + 3 = 16 + 3

y = 19

Q 5 : Find the area and perimeter of rectangle with length 10cm, width 2 cm.

Sol :  Area of rectangle = Length x width

                            = 10 * 2

                            = 20cm^2

Perimeter of rectangle = 2 (Length + width)

                                   = 2 (10 + 2)

                      Answer = 24cm

Q 6:  Find the area and circumference of the circle when the radius is 5cm.
 

Sol : Circle Area= (r= 5) (Pi=3.14 constant)

                                  = 3.14 * 5 * 5

                                  = 78.5cm^2

       Circumference = 2* pi * r

                                 = 2 * 3.14 * 5

                  Answer = 31.4 cm

Permutations and combinations probability

Math Permutations and Combinations We have learnt some techniques of counting the objects so that we need not count them one by one. Such techniques have been of interest since thousands of years. We have learnt to solve several counting problems through the fundamental principles of counting and the concepts of permutations and combinations.

Permutation is used when we are interested in different arrangements of the given objects while combination is used when we are interested only in selection of the given objects and it matters little which object is selected first. Note that ab and ba are two different permutations but the same combination. Let us now define permutations and combinations.

Permutation: The word permutation stands for arrangement. An arrangement that can be made with a given number of distinct objects by taking some or all of them is called a permutation.
The notation nPr or P(n, r) stands for permutation of n objects taking r at a time, i.e., arrangements of r objects out of n objects. nPr makes sense only when n > 0, r = 0 and n = r.

Combination: Sometimes we are not interested in arrangement butt only in selection. For example, out of the three persons a, b, c two can be presented in 6 ways; ab, ba, ac, ca, bc, cb. However there will be only 3 combinations: ab, ac, bc. Each of the different groups or selections which can be made by taking some or all of a number of things (irrespective of order) is called a combination.

Calculating permutations and combinations: We will use nPr = n!/(n-r)! for calculating permutations and we use nCr = nPr / r! for calculating combinations . Let us take an example for calculating permutations.
Example: Evaluate 7P3.7P3 = 7!/(7 – 3)! = 7!/4! = (7 × 6 × 5 × 4!)/4! = 210.and if we will Evaluate 5C2  to solve  5C2 = 5! / (2! (5 – 2)!) = 5! / (2! 3!) = (5 × 4 × 3!) / (2 × 3!) = 5 × 2 = 10.

Let us take some examples to understand permutation and combination probability.
Example: One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn.
Find the probability that the card drawn is an ace. Here we will use combination as out of 52 cards, one card can be drawn in 52 C 1 ways.

So, total number of elementary events = 52 C 1 = 52.
There are four aces in a pack of 52 cards, out of which one ace can be drawn in 4 C 1 ways.
So, favourable number of elementary events = 4 C 1 = 4.
So, required probability = 4/52 = 1/13.

Statistics Problems with solutions

Following are some of the few Statistics Problems from the wide ranging problems in statistics
The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does it signify?

Number of wickets 20-60 60-100  100-150    150-250   250-350     350-450
Number of bowlers    7              5             16                 12             2                3

Solution: the class size varies and the x(i)’s are large. We shall apply the step deviation method with a=200 and h=20. Let us tabulate the values

No. of wickets Number of x(i) d(i)=x(i)-200 u(i)=d(i)/20 u(i)f(i)
   Taken bowlers
   20-60 7 40         -160     -8    -56
   60-100 5 80       -120     -6    -30
   100-150       16         125        -75             -3.75             -60
   150-250       12         200 0       0        0
   250-350        2 300 100               5      10
   350-450 3 400 200      10      30
   ------------------------------------------------------------------------------
   Total       45    -106

So, u(bar) = -106/45 and hence, x(bar) = 200 + 20[-106/45]= 200 – 47.11 = 152.89
This tells us that, on an average, the number of wickets taken by these 45 bowlers in one-day cricket is 152.89

Let us solve Statistics problems involving confidence interval; A sample of 16 students is taken.  The average age in the sample was 22 years with the standard deviation of 6 years. Construct a 95% confidence interval for the average age of the population

Solution: The formula to construct a 95% confidence interval for the mean population can be given as,
  C.I. = [{Xn(bar)+z(alpha/2) sigma/sqrt(n)} , { Xn(bar)-z(alpha/2) sigma/sqrt(n)}]
Where Xn(bar)=22 is the sample mean, sigma=6 is the standard deviation given, n=16 is the sample size, z(alpha/2) is the cutoff point to the standard normal deviation. Since we want a 95% confidence interval , we consider alpha =0.05 and with the help of the normal distribution table we can arrive to z(alpha/2) = 1.96. Plugging in all the data we have in the formula, we get
        C.I = {[22+1.96(6/sqrt(16)], [22-1.96(6/sqrt(16)]= (24.94, 19.06) is the required confidence interval

Statistic problem to find the variance and variation for the data given, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
In this problem, the mean is calculated by step-deviation method taking 14 as the assumed mean. The number of observations n=10. Here the x(i) values are 6,8,10,12,14,16,18,20,22,24; d(i) values we get are, -4, -3,-2,-1, 0, 1,2,3,4,5 and summation of d(i) is 5. The deviation of mean [x(i)- x(bar)] values are -9, - 7,-5,-3, -1, 1, 3,5, 7, 9 and finally the [x(i)- x(bar)] are 81, 49, 25, 9, 1, 1, 9, 25, 49,81, summation of which is 330. Mean is calculated using the formula,
Mean x(bar) = assumed mean + [summation(i=1ton) d(i) *h]/n
       = 14 + (5/10)x 2 = 15
                      Variance (sigma^2) = 1/n summation (i=1to10)[x(i)-x(bar)]^2 = (1/10) x 330 = 33
Standard deviation = sqrt(sigma) = sqrt(33) = 5.74

Introduction toVector Calculus

The vector calculuswas developed by Willard Gibbs and Oliver Heaviside from quaternion analysis. In 1901, most of the notation and terminology used in multivariable calculus was established by Gibbs and Edwin Bidwell Wilson in book called vector analysis.

Vector Calculus
Vector or multivariable calculus is primarily expressed in three dimensional Euclidean space R^3, and is the branch of mathematics that is concerned with differentiation and integration of vector field. The word multivariate calculus is used as a synonym for the broader subject of multivariable calculus. This multivariate calculus includes multivariable calculus as well as partial differentiation and multiple integration.
In differential geometry and in the study of partial differentiation equation, multivariable calculus plays a very important role. It is also used in the description of the electromagnetic field, fluid flow and gravitation field, extensively used in physics and engineering.
In vector or multivariate calculus, the basic objects are scalar field and vector field.

Algebraic operation
In multivariable calculus, the basic algebraic operations are referred to as vector algebraic being defined for a vector space and the globules applied to a vector field, and consist of scalar multiplication, vector addition, dot product, cross product, scalar triple product and vector triple product.

What is Vector Calculus?
Vector calculus is used to measure all the three dimensions of the space. For example to measure the variation of temperature, fluid velocity, fluid force, magnetic flux, etc., multivariable calculus gives the necessary mathematical notation and technique.

Vector fields
A vector field is a vector valued function. Vector field ?(x, y, z) is defined by a function that takes a vector and returns a vector. Example for vector field is the special variation of fluid is the spatial variation of fluid velocity ?(x, y, z) in a steady flow.

Vector field is an assignment of a vector to each point in a subset of Euclidean space. Vector field in a plane can be visualized as a collation of arrows of varying length or thickness with a given magnitude, and direction and each arrow attached to a point in the plane. Often, vector fields are used to model, for example, the direction and speed of a moving fluid through out space, or the direction and strength of some fields, like magnetic or gravitational force, as it change from point to point. Vector fields are generates the flow and vice versa.

Vector Calculus Solutions

More about Limit of a Sequence

A Sequence is an ordered list of numbers. Each number in a sequence is called a term.  For instance, in the sequence 1, 3, 6,9,…., 1 is the first term, 3 is the second term and so on. The notation a1,a2, a3…an is used to denote the different terms in the sequence. A Limit of a Sequence can be defined as, if the terms of a sequence {an} approaches a number L, as n increases, then lim-(n->infinity) a(n)-> = L. Sequence that have limit converge and the sequence that do not have limits diverge. Let us find the limit of the sequence converges or diverges, given  an= (3n-2)/n-1. Let us take f(x) = 3x-2/x-1. In finding the limit of a sequence, we write it as lim(x->infinity) [(3x-2)/(x-1)] which gives us 3, and hence lim (n->infinity)an=3 and hence the sequence converges.

So, the limit of a sequence (an) is L given any e >0, there is an N>0 such that modulus of (an-L) is less than e for all n>N. Using this definition finding limits of sequences is simple. Let us consider Limits of Sequences Examples for a better understanding of the given definition.

Example: Find the limit of the Sequence given by a(n)= 1/n^2. Let  X(n)=1/n. Then a(n)= X(n). X(n). we know lim(n->infinity) X(n)=0, we get lim-(n->infinity) a(n)  = lim-(n->infinity) X(n) .lim-(n->infinity) X(n) =0.0 =0. By induction we can generalize and show that lim(n->infinity) 1/np=0 for any natural number p>0

Example: Find the limit of the Sequence X(n) = (n^2+3n)/(3+n^2)
Solution: X(n) = [1+3n/n^2]/[(3/n^2)+(n^2/n^2) = [1+3/n]/[(3/n^2) +1]

Now applying the limits, we get, lim(n->8) [1+(3/n)]= lim(n->8)1 + lim(n->8)(3/n) = 1+0=1
Also, lim(n->8)[(3/n^2)+1] = lim(n->8)(3/n^2) + lim(n->8)1 = 0+1 = 1
Hence lim(n->8) X(n) = lim(n->8)[1+3/n]/ lim(n->8)[(3/n^2) +1] = 1/1 = 1

Consider the function,  f(x)= x^2 sin(1/x). When x gets closer to zero, the function g(x) = sin(1/x) has no limits. But this function g(x) = sin(1/x) is bounded shown as, -1 = sin(1/x) = 1 for any real number x.  Since x^2=0, we get –x^2= x^2 sin(1/x) = x^2. Hence when x get closer to 0, x^2 and –x^2 become very small in magnitude. We have, lim(n->0) x^2sin(1/x)=0. This is an example for the Pinching Theorem  which is also called the Sandwich Theorem or the Squeeze Theorem, is given as: Let h(x)=f(x) =g(x) for any x is an interval around the point a. If lim(x->a) h(x) = L and lim(x->a) g(x) = L, then lim(x->a) f(x) = L. Basically , the Pinching Theorem says that if a function is trapped between two functions, both of which are approaching a particular value, then the function trapped in between also approaches that same value.

Natural Logarithms Properties

Introduction to natural logarithm:The most important function-inverse pair in mathematics and science is the pair consisting of the natural logarithmfunction ln x and the exponential function ex. The key to understanding ex is ln x, so we introduce ln x first. The importance of logarithms came at first from the improvement they brought to arithmetic. The revolutionary properties of logarithms made possible the calculations of the great seventeenth-century advances in offshore navigation and celestial mechanics. Nowadays we do complicated arithmetic with calculators, but the properties of logarithms remain as important as ever.
natural Logarithm Function

The natural logarithm of a positive number x, written as ln x, is the value of an integral.Natural logarithm definition: The natural logarithmfunction

If x > 1, then ln x is the area under the curve y = 1/t from t = 1 to t = x. For 0 < x < 1, ln x gives the negative of the area under the curve from x to 1. The function is not defined for x = 0. We also have

Using x for everything would have us writing

with x meaning two different things. So, we change the variable of integration to t.
The graph of y = ln x and its relation to the function y = 1/x, x > 0. The graph of the logarithm rises above the x-axis as x moves from 1 to the right, and it falls below the axis as x moves from 1 to the left.

Properties of natural logarithms
The properties that made logarithms the single most important improvement in arithmetic before the advent of modern computers are listed below. The properties that made it possible to replace multiplication of positive numbers by addition and division of positive numbers by subtraction. They also made it possible to replace exponentiation by multiplication.
For any numbers a > 0 and x > 0
• The first property of Natural Logarithmsis Product rule , product of logarithm of x and y is equal to the sum of logarithm x and logarithm y: lnxy = ln x + ln y
• The second property of natural logarithmis quoitent rule, logarithm of division of x and y is equal to the difference of logarithm x and logarithm y:ln x/y = lnx – lny
• The third property of natural logarithm is Reciprocal rule:ln1/x = - ln x
• The forth property of natural logarithm is power rule: lnxn = n ln x.
Solving Natural Logarithms:  Let us understand the concept of solving natural logarithm with natural logarithm properties . Suppose we have to solve the natural logarithmwithout calculator ln e4. Now using the power property of natural logarithms, we have, 4 ln e. now simplifying using loge e = 1, we have the solution 4 . 1 = 4.

Having problem with algebra keep reading my articles, i will try to help you.

Median in Mathematics

Median in mathematics is the middle value of the data, which separates the data into two equal halves; which means fifty percent of the numbers are above the median and fifty percent of the numbers are below the median. So, the Definition of Median can be written as a mathematical result that indicates that one half of the data or group is higher and one half lower. For instance, if there are 5 children in Smiths family of ages 5, 12, 7,9, 3, 15; how do we find the age of the middle child? We need to find the middle child first, so we need to put the ages in an increasing order. The order would be 3, 5, 7,9,12,15. From this ordered list, we can easily make out that 7 is the middle value and that is the Median of the data. So, we can conclude that the age of the middle child is 7 years.

If there is ‘n’ number of values in a given data; then the Median Formula is:
Median = (number of values in the data)/2= (n+1)/2
  Median Formula =(n+1)/2
While Calculating Median or Finding the Median of a given data, the steps are:
Step1. Arrange the list in the ascending order
Step2. Count the number of values in the list (whether even or odd)
Step3. If the number of values in the list is odd; then median is the value exactly in the middle of the list and if the number of values in the list is even; then the mean of the two middle values is the median

Let us Find the Median or Find Median of the given data.
 A marathon race was completed by 5 participants. How do you find the median of the race, given the data 3.6hrs 4.2hrs 3.2hrs 5.4hrs 4.8hrs?
Let us first arrange the data in the ascending order, 3.2, 3.6, 4.2, 4.8, 5.4
The number of values in the data, n=5. So, the third value 4.2hrs which is exactly in the middle is the median of the race.

The Jones Family drove through 6 states on their summer vacation. Find the median of gasoline price if the gasoline prices varied from state to state.  $3.29, $3.34, $3.27, $4.02, $3.60, $3.86
To find the median we need to arrange the data in increasing order.
$3.27,  $3.29,  $3.34,  $3.60,  $3.86,  $4.02
Here,  n=6 ; data has even number of values. So, median would be the mean of the two middle values. The two middle values are 3.34 and 3.60.
Median=(3.34+3.60)/2 = 3.47
The median of the gasoline rate is $3.47

Exponential and Logarithmic Functions

Exponential function: if a positive real number  other than 1  , then the function f(x) defined by f(x) = ax for all x belong to R is called as exponential function.the domain of this exponential function is R .It is evident from its graph that it is everywhere continuos.
Graph of exponential function

Graph of y= ax , where 0

Graph of y= ax, where a>1

The shape of the graph y= ax for any a>1 is essentially the same as that of y = 2x.The graph of y = ax for a typical base b>1 and is rising (when viewed from left  to right ).for a base b where 0

Example of exponential function: y= 2x , y=3x+3
Logarithm function:


Si nce the exponential function y= ax for a >0 , a not equal to1  is montonic  (its graph is either  always rising for all x always falling ) , it must have an inverse that is itself monotonic .the inverse function  is called the logarithm of x to the base a .
Definition of logarithm function : If a > 0 and a  not equal to 1, the logarithm x to the base b is the function y=   that satisfy ay=x , that is y=   mean ay=x. Graph of logarithm function 

 Here green line represents graph of y=ax and blue line represent y=   .
Because y = bx is a continuos  , increasing function that satisfies ax >0 for all x ,   must also be continuos and  increasing  , and its graph must lies entirely  to the right of y-axis.

Introduction to Polynomials

Monomial: Monomial is a constant or variable or product of variables or variables with exponents 0, 1, 2, 3….
Example: 
What is a Polynomial?  
Polynomial is an expression which includes constants, variables. There may be more than one variable and variables can have any real number as coefficients but variable exponents should be a whole number, that is, 0, 1, 2, 3…
Clearly, polynomial is an extension of monomial. Polynomial is an expression which involves one or more than one monomial which is separated by addition or subtraction. Each monomial in a polynomial is also called as a term.
Example: 

Polynomial is a general term it can use for all expressions which satisfies above conditions. More specifically, a polynomial with one term is called as a MONOMIAL, two terms is called as a BINOMIAL and three terms is called TRINOMIAL.

Adding and Subtracting Polynomials:
How To Add Polynomials:
Adding polynomials is similar to adding variables or constants, to add two or more polynomials we have to add like terms (or monomials) in those polynomials.
Example:

Subtracting polynomials:
Subtracting polynomials is similar to subtracting variables or constants, to subtract two or more polynomials we have to subtract like terms (or monomials) in those polynomials.
Example:

How to multiply polynomials?
Multiplying Polynomials is different from adding and subtracting polynomials, here each term in one polynomial should multiply with each term in another polynomial it has nothing to do with whether they are like terms or not.
Example:

Derivatives of Functions in Calculus

What are Derivatives?
Derivative is the measure of the rate of change at any given point on a curve. The rate of change of a function is the slope of a line.
The derivative of f(x) when the limit extends from h to 0 is given by the formula:

Derivative of Acceleration
Derivative of acceleration is the rate of change of acceleration with respect to time. It is referred to as ‘jerk’ in technical terms.
How to Find the Derivative?
We have general Derivative Formulas which help to find the derivative of the given function.
Let us consider the following examples:
Derivative Examples:

Prime Numbers

Prime numbers are positive whole numbers which are divisible by number 1 and itself.  Prime numbers have only two factors. The smallest prime number is 2. Now the question arises is 1 a prime number? No, 1 is not a prime number. So our next question will be why 1 is not a prime number? As per the definition of prime numbers, a prime number must have two factors but 1 only has one factor. It is the only positive integer with exactly one positive divisor.  Therefore it is not considered as a prime number.

Is 2 a prime number? 2 is a prime number as it is divisible by 1 and itself. It has two factors and as per the definition of a prime number, prime number must have two factors. Therefore, 2 is considered as a prime number and it is the only even number which is considered as prime. The other prime numbers are divisible by 2 also so they do not come under a category of a prime number.

Euclid proved that there are infinite prime numbers and there is always a prime greater than the largest known prime number. Many mathematicians tried finding out the largest prime number. The largest prime number found in 2008 was 2^43112609-1

Let us learn about relatively prime numbers. Two numbers that have only 1 as a common factor are called relatively prime numbers. Thus, any two prime numbers are relatively prime numbers. They are also called co prime numbers. However, composite numbers may also be co - prime to each other.

For example: - Numbers 3375 and 2744 are considered as relatively prime because:-
3375 = 1 X 3 X 3 X 3 X 5 X 5 X5
2744 = 1 X 2 X 2 X 2 X 7 X 7 X 7
3375 and 2744 have just 1 as a common factor so they are considered as relatively prime numbers.