Learn Divisor

Introduction to learn divisor:

In learning divisor many students undergo difficulties in it. They find difficulty in multi-digit divisor. To overcome that we should be careful in selecting the numerals, so it may help the students a lot  with the divisor learning–and also from the arithmetic operations used from the divisor. They have to group the numbers when they multiply the numerals with the quotient of the divisor. Please express your views of this topic What is a Divisor by commenting on blog.


Definition of Integers in learn divisor:


The divisor is the digit that the dividend is divided by (in long division). The dividend divided through divisor is known as the quotient (plus a remainder).

One of two or more integers that can be exactly divided into another integer, the number by which a dividend is divided. What does n|m mean? It explains n divides m. For an example 5|10, or 5 divides 10. Do you know when n divides m or when n is a correct divisor of m? I think you do. Understanding List of Composite Numbers is always challenging for me but thanks to all math help websites to help me out.


Let’s see an example on learn divisor:


1)10 divides 20 or 10|20, we know..

The value twenty is divided by ten is written as two x two x five.

The value ten can be written as two x five

20 = 2 X 2 X 5 = 2^2 X 5 and

10 = 2 X 5

2) But 15 does not divides 20,

The value twenty cannot be divide by 15

20 = 2^2 X 5 and

15 = 3 X 5

3) Then why does 10 divide 20 but not 15 ?

We already know, 20 = 2^2 X 5

15 does not divide 20 for the reason that, 15 contains a prime factor ( 3) which is not exist in 20.

Likewise, 14 does not divides 20 as it contains a prime factor (7) which is not exist in 20.

Note that, 14 = 2 X 7

4) Again, 8 does not divides 20. Why?

20 = 2^2 X 5

8 = 2^3

Do you get it? 8 do not include any prime factors which do not exist in 20, then why does 8 not divide 20?

Answer is very easy; just have a close look at the power of 2.

Learn About Algebraic Expressions

Introduction to learn about algebraic expressions:

The Expressions are a central concept in algebra. A variable can take various types of values. Its values are not fixed. Otherwise, a constant has a fixed value. We combine variables and constants with operations this forms the algebraic expressions. For this, we can use the operations of addition, subtraction, multiplication and division. We have already known some simple algebraic expressions like y + 3, p – 5, 4a + 5, 10y – 5a and so on. The above expressions were formed by combining variables with constants.


Learn about Terms of an Expression


When terms have the same algebraic factors, they are like terms. When terms have different algebraic factors, they are unlike terms. For example, in the expression 5xy – 8x + 7xy – 17.

Learn about Like terms:

5xy and 7xy because the factors of 5xy are 5 and x and y. and factors of 7xy are 7 and x and y both factors are same. So these are like terms.

Learn about Unlike terms:

The terms 5xy and –8x, have different algebraic factors. They are unlike terms. Similarly, the terms, 5xy and -17 are unlike terms. Also, the terms –8x and -17 are unlike terms.

Learn about Coefficient of the Term:

The numerical factor is said to be the numerical coefficient or simply the coefficient of the term. The above example the coefficient of x is -8.

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Learn about Kinds of Expressions


Monomial Expression:

An expression with one term is called a monomial. For example, 5pq, 10z,

Binomial Expression:

An expression, it contains two unlike terms is called a binomial. For example,

a + b, b – 5

Trinomial Expression:

An expression, it contains three terms is called a trinomial. For example, the expressions p + q + 17, a b + a +b

Polynomial Expression:

An expression with one or more terms is known as a polynomial. Thus, a monomial, a binomial and a trinomial are all the type of polynomials. For Example: 4x^3+x^2+7x+2

Factoring Quadratic Polynomials

Introduction:

In algebra, the sum of a countable number of monomials is referred as  a polynomial. The way of writing a polynomial as a product of two or more simpler polynomials is referred as factorization.
The process of factorization is also known as the resolution of factors. Factoring quadratic polynomials is one of the basic operations of polynomials. The process of factoring the quadratic expression ax^2 + bx +c is explained below: Looking out for more help on Factor a Polynomial in algebra by visiting listed websites.

Method of factoring quadratic polynomials:


Let us consider the coefficients a, b and c as integers and a is not equal to 0. When the coefficients a, b and c satisfy certain conditions, the quadratic expression ax^2 + bx +c can be factorized.

First, we consider a simpler case with a = 1 and b and c as integers.Now, we have to factorize  x^2 + bx + c. We try to write the integer constant term c as a product of two integers p and q such that p + q = b. If we find solution in our attempt, then

x^2 + bx + c  =  x^2 + (p + q)x + pq

=  (x^2 + px) + (qx + pq)

=  x(x + p) + q(x + p)

=  (x + p) (x + q)

General Rule for factoring quadratic polynomials : If the constant term c of quadratic expression x^2 + bx + c can be expressed as a product of two integers p and q such that the sum p + q is the coefficient b of x, then x^2 + bx + c = (x + p)(x + q).

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Example for factoring quadratic polynomials


An example for factoring quadratic polynomials is given below:

1) Factorize x^2 + 5x + 6.

Step 1 : Possible factorization of 6 is    6 = 1 x 6

6 = 2 x 3

Step 2 : Sum of factors are                      1 + 6 = 7

2 + 3 = 5

Step 3 : Factorization:

Now we need to compare the coefficient of x and the sum of the factors. We find that the sum of the factors 2 and 3 is the coefficient of x. The factorization of quadratic expression is explained below:

x^2 + 5x + 6   =   x^2 + (2 + 3)x + 6

=   (x^2 + 2x) + (3x + 6)

=  x(x + 2) + 3(x + 2)

x^2 + 5x + 6  =  (x + 2) (x + 3)

Continuous Probability Learning

Introduction to continuous probability learning :

Definition: Learning to define continuous probability of an event occurs when the one or more events occurred. Consider any two probabilities A and B. when the event A occur and it depends on the other event which is already occurred then we can say the conditional probability of above two events are P(A | B). Learning  conditional probability of events using the following formula :

`P((A)/(B))=(P(A U B))/(P(A))`

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Learning to solve conditional probability problems 1:


Learning some problems to find the conditional probability.

Pro 1:Consider a population, the probability a men life at least 70 years is 0.70 and is 0.65.if he won’t live more than 80 years. If a man is 70 years old, find the conditional probability that he will survive on 80 years. If A subset of B then P (A U B) = P (A)

Solution:Let us take A is the event that he lives to 70 years and B is the event that he will live at least 80 years.

So given that P (B) = 0.55 and P (A) = 0.70

So Conditional probability P (`(B)/(A)) = ( P (A U B)) / ( P (A))`

The given condition is P (A and B) = P (B) = 0.65

Conditional probability P (`(A)/(B)` ) = `(0.65)/(0.70)`

P (`(A)/(B)` ) = 0.9286

Is this topic Introduction to Probability hard for you? Watch out for my coming posts.

Learning to solve conditional probability problems 2:

A box contains red and blue marbles. We are choosing two marbles without replacement. Probability of choosing red and blue marbles is 0.45 and choosing the red marbles on the first draw is 0.57. Find the probability if the second marble is blue if the first one is red?

Solution:Probability of choosing red and blue marbles is 0.45

Probability of choosing red in the first draw is 0.57

So probability of choosing second marble ids blue then the probability is

P (`(Blue)/(Red) = ( P ( Red and Blue))/ (P(Red))`

P (`(B)/(R)` ) = `(0.45)/(0.57)`

P (`(B)/(R)` ) = 0.79 = 79 %

How to Learn Geometry

Definition:

Geometry is the defined as the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. However, geometry is used daily by almost everyone. Geometry begins with undefined items, definitions, and assumptions; these lead to theorems and constructions. It is an abstract subject, but easy to visualize, and it has many concrete practical applications. Please express your views of this topic What is a Acute Angle by commenting on blog.


Geometrical objects:

Point

A point is the most fundamental object in geometry. It is represented by a dot and named by a capital letter. A point represents position only; it has zero size

Line

A line can be thought of as connected set of infinitely many points. It extends infinitely far in two opposite directions. A line has infinite length, zero width, and zero height. Any two points on the line name it. The symbol ↔written on top of two letters is used to denote that line. A line may also be named by one small letter

Collinear points

Points that are lie on the same line are called collinear points. If there is no line on which all of the points lie, then they are non-collinear points.

Plane

A plane has an infinite set of points forming a connected flat surface extending infinitely far in all directions. A plane has infinite length, width, and zero height (or thickness). It is usually represented in drawings by a four-sided. A single capital letter is used to denote a plane. The word plane is written with the letter so as not to be confused with a point . Is this topic How to find Area of a Circle hard for you? Watch out for my coming posts.


Postulates:

Postulate 1: A line contains at least two points.

Postulate 2: A plane has at least three non-collinear points with it.

Postulate 3: If we cross through any two points, there is exactly one line.

Postulate 4: If we cross through any three non-collinear points, there is exactly one plane.

Postulate 5: If two points lie in a plane, then the line joining them lies in that plane.

Postulate 6: If two planes intersect with each other, then their intersection is a line.

Learn Matrices Online

Introduction about learn matrices online:

Matrix  is a rectangular array of numbers, arranged in rows and columns.

The number of rows (say m) and the number of columns (say n) find the order of the matrix.

It can be written as m x n (to be read as m by n).

Learn matrices online:

The plural  of matrix is matrices. Matrix is usually denoted by capital letters such as A, B, C,…

The Matrices can be defined as the set of m × n numbers arranged in a rectangular form. here m is row of the matrix and  n is column of the matrix.

for eg: A =  `[[a,b],[c,d]]` is a matrix of order 2 × 2.

Matrices Online is  used to verify the consistency of the linear equation system. Also, used to find the solution for the linear equation system with two and three variables by matrix inversion. Please express your views of this topic Matrices Calculator by commenting on blog.



Example Problems to learn solving online matrices:


Learn online solving  matrices Example 1:

If  `[[2,x],[x,1]]` = `[[5,2],[4,1]]`   ,then find x?

Solution:

Since the given matrices are equal,their corresponding elements must be equal.

Equating the corresponding elements, implies

2 - x2 = 5 - 8

x2 = 5 then

x = `+- sqrt(5)` .

Learn solving online matrices Example 2:

Evaluate determinant   = `[[1,2,2],[2,2,1],[1,2,1]]`

Solution:

= 1(2-2) - 2(2-1) + 2(4-2)

= 0 - 2(1) + 2(2)

= -2 + 4

= 2.

Learn solving online matrices Example 3:

If  `[[x+3,z+4,2y-7],[-6,a-1,0],[b-3,-21,0]]` = `[[0,6,3y-2],[-6,-3,2c+2],[2b+4,-21,0]]`   Find the values of a, b, c, x, y and z.

Solution:

Since the given matrices are equal,their corresponding elements must be equal.

Equating the corresponding elements, implies

x + 3 = 0,

z + 4 = 6,

2y – 7 = 3y – 2,

a – 1 = – 3,

0 = 2c + 2,

b – 3 = 2b + 4,

By Simplification,

Answers:

a = – 2,

b = – 7,

c = – 1,

x = – 3,

y = –5,

z = 2.

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Learn online solving Matrices Example 4:


Find the values of a, b, c, and d from the following matrix equation:

`[[2a+b,a-2d],[5c-d,4c+3d]]` =` [[4,-3],[11,24]]` .

Solution:

since the given matrices are equal,their corresponding elements must be equal.

equating the corresponding elements, implies

2a + b = 4,

5c – d = 11,

a – 2b = – 3,

4c + 3d = 24,

By Simplification,

Answers:

a = 1,

b = 2,

c = 3 and

d = 4.

Learn online Solving matrices Example 5:

If A = `[[sqrt(3) ,1,-1],[2,3,0]]` and B = `[[2,sqrt(5),1],[-2,3,1/2]]` , find A + B.

Solution:

Since, the two matrices is of the same order.

Matrix addition is possible, implies

A+B = `[[sqrt(3)+2,1+sqrt(5),-1+1],[2-2,3+3,0+1/2]]`

= `[[2+sqrt(3),1+sqrt(5),0],[0,6,1/2]]` .

Math Superstars Answers

Introduction to math superstars answers:

The basic concepts of number, measurement, algebra, and geometry. In number basis of simple addition, subtraction, multiplication, and conservation. The Superstars program is to provide the additional or extra challenge that self- motivated to the students. Usually more than one way to solve the problems but student needs the opportunity to discover the shortcut methods. It is best way to learn the math’s and easily understand the concept. Understanding Math Answers Fast is always challenging for me but thanks to all math help websites to help me out.


Math superstars problems 1 to 5::


Math superstar problem 1:  If the 24th day of the month falls on Saturday, on what day did the 8th fall?

Solution:  Students can use a calendar or make a chart with “Su, M, T, W, Th, F, Sa” at the top and begin numbering      backward putting 24 under Saturday. They may also realize that the 17th and 10th fall on Saturdays and count back from the 10th.

Answer: Thursday

Math superstar problem 2:  Together, 12 tigers  and 6 horses weigh 1050 pounds. The horses all weigh the same -- x pounds. Each tiger weighs 55 pounds.

What is the  weigh of one horse?

Solution: Students will probably solve this by first finding the total weight of the 12 tigers: 12 × 55 = 660 pounds. Then   they will compute 1050 - 660 = 390 pounds, the weight of the 6 horses. Then 390 ÷ 6 = 65 pounds per horse.:

Answer: 65pounds

Math superstar problem 3: The sum of 3 consecutive numbers is 405. What are the numbers?

Solution:   Students may use the guess-check-revise method. Some students might know that the numbers they seek are about 1/3 of the total, and approximate the numbers by dividing 405 by 3. This gives 135, which is the middle number.

Answer: 134, 135, 136

Math superstar problem 4: . How many corners are on 3 squares?

Solution: 1 square = 4 corners

3 square = 4 × 3 = 12 corners

Answer: 12 corners

Math superstar problem 5:  How many sides are on 2 squares?

Solution: 1 Square = 4 sides

2 Square = 4 × 2 = 8 sides

Answer: 8 sides

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Math superatars problem 6 to 10::


Math superstar problem 6:  I saw 1 cat and 2 dogs outside. How many legs did i see?

Solution: 1 cat = 4 legs

2 dogs= 8 legs

8 + 4 = 12 legs

Answer: 12 legs

Math superstar problem 7:  I have 6 balls, half of them are red. How many are Red?

Solution: Total balls = 6

Half of ball = 6/2 = 3

Answer: 3 Red balls

Math superstar problem 8:  I have a nickel and 2 pennies, how much money do I have all together?

Solution: 1 nickel = 5 pennies

So, Total = 5 + 2 = 7 pennies

Answer: 7 pennies

Math superstar problem 9: There are 3 cars, 2 bicycles, and 4 tricycles in the neighbor's garage. How many wheels are there in all? Forget about any "spare tires"!

Solution:  There would be 12 wheels on the 3 cars, 4 on the 2 bicycles, and 12 on the 4 tricycles

Answer: 28wheels

Math superstar problem 10: .I saw three triangles, how many sides are there all together on two triangles?

Solution: 1 triangle has 3 sides

3 triangle has 3 × 3 = 9 sides

Answer: 9 sides

Algebra 2. Quadratic Functions

General Form of quadratic function:

In general the quadratic function is of the form f(x) = ax^2 + bx + c,

Where a, b, and c are numbers and a not equal to zero.

The graph of a quadratic function is a curve and it is called as a parabola. Parabolas may open upward or downward and different length in "width" or "steepness", but they all have the same basic "U" shape. For example, y = 2x^2is a quadratic function since we have the x-squared term. y = x^2+1/x would not be a quadratic function because the 1/x term is x-1 which does not valid the form.


Sketching Parabola


To sketch the graph of the quadratic equation, following steps are used  :

(i)Verify if a > 0 or a < 0 to decide if it is opened upwards or downwards respectively.

(ii)If a>0, the quadratic function has a minimum value, and if a<0 a="" br="" function="" has="" maximum="" quadratic="" the="" value.="">
(iii)The x-coordinate of the minimum or maximum point is given by

X=-b/2a

We need to substitute x-value into our quadratic function Then we will have the (x, y) coordinates of the minimum or maximum point. This is called the vertex of the parabola.

(iv)To find the he coordinates of the y-intercept, substitute x = 0. This is always simple to find!

(v)We substitute y = 0 to find the values of y-intercepts in the quadratic equation.


Example for Quadratic functions


Sketch the graph of the function y = 2x^2 ? 8x + 6

Solution:

We first identify that a = 2, b = -8 and c = 6.

Step (i) Since a = 2, a > 0, it is a parabola with a minimum point and opens upwards

Step (ii) The x co-ordinate of the minimum point is

X= -b /2a  = -8/  2(2)= 2

y value of the minimum point is

y = 2(2)2 - 8(2) + 6 = -2

Hence the minimum point is (2, -2)

Step (iii) the y-intercept is found by substituting x = 0

y = 2(0)2 - 8(0) + 6 = 6

Hence (0, 6) is the y-intercept.

Step (iv) The x-intercepts are found by setting y = 0

2(x^2 - 4x + 3) = 0

2(x - 1)(x - 3) = 0

Hence x = 1, or x = 3.

Precalculus with Limits

Introduction:

In general, the limit is an extension up to which something can go. In mathematics, a limit is an intended height of a function. In other words, a limit is defined as the boundary of a specific area. The function limit is most often used in calculus problems.

Generally it is written as

`lim_(x->c)`  f (x) = K

Where    c - real number (when f(x) is a real valid function).


Properties:

Let the given function be f(x) and x approaches to h.
State all the possibilities.
Simplify the given function to apply the limits and solve.
Replace x for h.
Simplify the function.
The limits of a function could be obtained.

Example problems:

Problem 1: Find the limit of the function f (x) = 3x as x approaches 6.

Solution: Given f (x) = 3x

Substituting the value of x the equation becomes

f (6) = 3 (6) = 18.

So, the limit of f (x) = 3x as x approaches 6 is 18.

Problem 2: Find the limit of the function 9x^2 + 2x – 5 as x approaches 2.

Solution:

Given 9x^2 + 2x – 5

Substituting the values of x

= (9) (2)2 + 2 (2) – 5

= (36 + 4 – 5)

= 35

Hence the limit of 9x^2 + 2x – 5 as x approaches 2 is 35.

Problem 3: Solve (|x^2 - 5x + 6| / (x – 2))  for x approaching to 1

Solution:

Given (|x^2 - 5x + 6| / (x – 2))

Factorize the numerator |x^2 - 5x + 6|

x^2 - 5x + 6 = 0

(x – 2) (x – 3) = 0

The roots are x = 2 and x = 3.

Substitute the roots in the given function

((x – 2) (x – 3) / (x – 2))

Applying the limit

= ((1 - 2) (1 – 3) / (1 – 2))

= (-1) (-2) / (-1)

= 2 / (-1)

= -2

Hence the solution of (|x^2 - 5x + 6| / (x – 2)) is -2

Understanding Derivative Functions is always challenging for me but thanks to all math help websites to help me out.


Practice Problems:

Determine the limit of the function f(y) = -2y as y approaches to 0.5
Answer: -1
Find the limit of f(x) = sin x as x approaches to 0
Answer: 0
Find the limit: f(x) = x^2 - 5x + 6 when x tends to 2
Answer: 0
Solve the following: `lim_(x->3)` (x^2 + 8x + 3) / (x3 + 2x + 1)
Answer: 18 / 17

Language Proof and Logic

Introduction to Language Proof and Logic

Statement is logic. Logic is a set of sentences in a perfect language. Theorems are individual sentences. Equivalence is the two declarations are logically equivalent if they have the identical truth values for all grouping of truth value of their variables. Language proof done in logic. Now we will see the examples for language proof and logic. I like to share this Population Correlation Coefficient with you all through my article.

Basic Operations- Language Proof and Logic

Negation (~p)    : If the condition is true then change into false and vice versa.

Disjunction (pvq)  : All declarations answer will be true. Only the answer will be false if two declarations are false.

Conjunction (p^q)  : If two statements are true then the result will be true otherwise false.

Conditional (p?q) : Truth of the statement ‘p’ is sufficient to truth of statement ‘q’.

Bi-conditional (p?q):  p?q to be true, both p and q must have the equal accuracy values. Otherwise it is false.

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Examples-language Proof and Logic

Example 1

What is the conjunction form for the following statements?

P: Michel reading newspaper.

Q: Peter reading story books.

Solution:

Given statement is as follows,

P: Michel reading newspaper.

Q: Peter reading story books.

Conjunction is a ‘and’ declaration. The representation of ‘and’ condition is ^.

We can write the above sentence in the following way.

‘Michel reading newspaper and Peter reading story books’.

So logical form = P v Q.

Example 2

Solve the logical term for the following statement.

‘If you go to the theater early then you can buy the ticket easily’.

Solution:

Let P: If you go to the theater early.

Q: Then you can buy the ticket easily.

The given state is ‘if-then’. The sign of ‘or’ is ?.

So logical form=P?Q.

These example problems are used to understand about the language proof and logic.

Calculate Inches to Cubic Feet

Introduction to calculate inches to cubic feet:

The area measurement is done with various units of area measurements. There are various type of units for the measurement of the area like the metric system, SI units etc. The inches and feet are the basic units of the area measurement commonly applied for the small areas. In this article we will see more about the measurement of area using the inches and square feet.

More on Calculate Inches to Cubic Feet:

The area measurements of the small areas are measured with the basic units like the square inch and the square feet. The relation between the square inch and the square feet helps in the conversion of the area measured using the square inch to square feet. The relation between the square inches and the square feet is given by the relation,

1 square feet = 144 square inches

The relation can also be written as the form for the inches as,

1 square inch = `1/144` square feet

Example Problems on Calculate Inches to Cubic Feet

1. Convert the given area of 152 square inches into its equivalent square feet.

Solution:

1 square feet = 144 square inches

1 square inch = 1/144 square feet

152 square inch = `152/144` square feet

152 square inch = 1.05 square feet

2. Convert the area of 245 square inches into square feet.

Solution:

1 square feet = 144 square inches

1 square inch = 1/144 square feet

245 square inch = `245/144` square feet

245 square inch = 1.70 square feet

3. Convert the area of 683 square inches into its equivalent square feet.

Solution:

1 square feet = 144 square inches

1 square inch = 1/144 square feet

683 square inch =` 683 /144` square feet

683 square inch = 4.74 square feet

Practice problems on calculate inches to cubic feet

1. Convert the given area of 1231 square inches into its equivalent square feet.

Answer: 8.54 square feet.

2. Convert the area of 510 square inches into square feet.

Answer: 3.54 square feet.

3. Convert the given area of 195 square inches into square feet measurement.

Answer: 1.35 square feet.