How To Do Distributive Property

Introduction Distributive Property :-

The distributive property or distributive law makes numbers easier to work with. In mathematics when we use the distributive property, we are actually mounting up the simplified ones.

The Distributive Property in Algebra:

Is left-distributive over + if, given any elements x, y, and z of S,
x · (y + z) = (x · y) + (x · z);

is right-distributive over + if, given any elements x, y, and z of S:
(y + z) · x = (y · x) + (z · x);

is distributive over + if it is both left- and right-distributive.
(Source:- Wikipedia)

The problems solved here helps you learn the distributive property easily. I like to share this The Definition of Distributive Property with you all through my article.


Solved problems Based on Distributive Property:-


Learning problem 1

13 ( 4 + 6 ) solve it by using 'Distributive Property.

Solution:-

= 13 ( 4+ 6 )

The Distributive Property states that

a(b + c) = ab + ac

here

a = 12

b = 4

c = 6

Multiply each number within the parentheses by the number outside the parentheses

13(4 + 6) = 13 • 4 + 13 • 6

= 52 + 78

= 130

Learning Problem 2

11 (5 + 6) solve it by using 'Distributive Property.

Solution:-

= 13 (4+ 6)

The Distributive Property states that

a (b + c) = ab + ac

Here

a = 11

b = 5

c = 6

Multiply each number within the parentheses by the number outside the parentheses

11(5 + 6) = 11 • 5 + 11 • 6

= 55 + 66

= 121

Learning Problem 3

10 (2 + 6) solve it by using 'Distributive Property.

Solution:-

= 10 (2+ 6)

The Distributive Property states that

a (b + c) = ab + ac

Here

a = 10

b = 2

c = 6

Multiply each number within the parentheses by the number outside the parentheses

10(2 + 6) = 10 • 2 + 10 • 6

= 20 + 60

= 80

Learning Problem 4

7 (2 + 6) solve it by using 'Distributive Property.

Solution:-

= 7 (2+ 6)

The Distributive Property states that

a (b + c) = ab + ac

Here

a = 7

b = 2

c = 6

Multiply each number within the parentheses by the number outside the parentheses

7(2 + 6) = 7 • 2 + 7 • 6

= 14 + 42

= 56

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Practice Problems Based on Distributive Property:-


1) 1 ( 2 + 6 ) solve it by using 'Distributive Property.

Answer:- 8

2) ( 4 + 2 ) 3 solve it by using 'Distributive Property.

Answer:- 18

3) 12 ( 4 + 2 )  solve it by using 'Distributive Property.

Answer:- 72

4) ( 4 + 5 ) 2 solve it by using 'Distributive Property.

Answer:- 18

5) ( 7 + 2 ) 5 solve it by using 'Distributive Property.

Answer:- 45

Radius Math Term

Introduction to radius math term:

In day to day life, we often came across some unique math terms. Radius is one of the special math terms that falls under this category.
Radius of a circle is nothing but the line segment from the center of the circle to its perimeter. In other terms, half the diameter is the radius.
In this article of radius math term, we are going to find the radius of the circle by several methods. I like to share this Partial Fraction Decomposition Calculator with you all through my article.

Formulas for math term radius:


Radius from circumference:

If the circumference of a circle ( C ) is given, the radius can be calculated by the following formula:

r  = C / 2`pi`

Radius  from diameter:

If the diameter (D) of the circle is given, the formula for finding the radius is

r  = `<< D/2>>`

Radius from area:

If the area of the circle is given, then the radius can be calculated using the formula

r =  `sqrt(A/pi)`

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Example problems for math term radius:


Example 1:

Find the radius, if the circumference of the circle is 90 cm

Solution:

Radius of a circle, r = C / 2`pi`

= 90 / (2 * 3.14)

= 90 / 6.28

= 14.33 cm

Example 2:

Find the radius of the circle, if its diameter is 62 cm.

Solution:

Radius of a circle, r = `<< D/2>>`

= `<< 62/2>>`

= 31 cm

Example 3:

Given the area of a circle is 94 m2. Find its radius.

Solution:

Radius of a circle  r =  `sqrt(A/pi)`

= `sqrt(94/3.14)`

=  5.5 cm


Practice problems for radius math term:


1) Find the radius, if the circumference of the circle is 110 cm.

Answer:  17.52 cm

2) Find the radius of the circle, if its diameter is 92 cm.

Answer: 46 cm

3) Given the area of a circle is 124 m2. Find its radius.

Answer: 6.28 m

Application of Matrices

Introduction to learn applications of matrices::
A Matrix in plural matrices or less commonly matrices, is a rectangular array of numbers. Matrices are a key tool in linear algebra. One use of matrices is representing linear transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a constant; matrix multiplication corresponds to composition of linear transformations.
(Source : wikipedia)

In this article, we shall learn the applications of matrices. I like to share this Augmented Matrix with you all through my article.



Learn Application of matrices to check properties:


(1) Matrix addition is commutative:

If A and B are any two matrices of the same order then A + B = B + A. This property is known as commutative property of matrix addition.

(2) Matrix addition is associative:

I.e. If A, B and C are any three matrices of the same order.

Then A+ (B + C) = (A+B) +C.  This property is known as associative property of matrix addition.

(3) Additive identity:

Let A be any matrix then A + O = O + A = A. This property is known as identity property of matrix addition. The zero matrix O is known as the identity element with respect to matrix addition.

(4) Additive inverse:

Let A be any matrix then its matrix is –A then the property is A + (- A) = (- A) + A = O. This property is known as inverse property with respect to matrix addition. The negative of matrix A  i.e. - A is the inverse of A with respect to matrix addition. Please express your views of this topic answers for algebra 2 problems by commenting on blog.


More Applications to learn matrices:


(1) In general, matrix multiplication is not commutative i.e. AB ? BA

(1) In general, matrix multiplication is not commutative i.e. AB ? BA

(2) Matrix multiplication is always associative.

I.e. A (BC) = (AB) C

(3) Matrix multiplication is always distributive over addition.

I.e. (i) A (B + C) = AB + AC

(ii) (A + B) C = AC + BC

(4) AI = IA = A where I is the unit matrix or identity matrix. This is known as Identity property of matrix multiplication.

This is how, we can learn the applications of matrices.

Learn Online Inequalities

Definition of learn online inequalities:

Inequality is described as two real numbers or two algebraic expressions are communicated with functioning a symbol as ‘<’ (less than), ‘>’ (greater than), ‘≤’ (less than or equal) and ≥ (greater than or equal). Learn online inequalities are the method of learning the inequalities by online. The various inequalities are learned online as follows,

Numerical inequalities
Literal inequalities
Double inequalities
Strict inequalities
Slack inequalities
Linear inequalities

Types :


From learning online inequalities, they can be explained as follows,

1) Numerical inequalities:

Inequalities which enclose arithmetical lone without any variables is known as numerical inequalities

Eg: 5 < 8; 5 > 4

2) Literal inequalities:

Inequalities which have one or more variables are labeled as literal inequalities.

Eg: p< 5; q > 2; m ≥ 4; n ≤ 6

3) Double inequalities:

An inequality which contains two sign (< or > or ≤ or ≥) is named as double inequality.

Eg: 2 < b < 8; 3 ≥ t ≥ 8

4) Strict inequalities:

If an inequality holds a symbol < or >, then it is learned as strict inequalities.

Eg: Ax + B < 0; Ax2 + Bx + C > 0

5) Slack inequalities:

If an inequality involves a sign ≤ or ≥, then it is called slack inequalities.

Eg: Ax + By ≤ C; Ax + By ≥ C

6) Linear inequalities:

An inequality may have one variable with linear is called linear inequality through one variable; If it contains two variables, then it is called linear inequality through two variables.

Eg: Ax + By < C; Mx + C > Y


Rules & Example for learn online inequalities :


RULES :    Inequality contains the following rules for learning online,

Rule 1: All sides of an inequality can be added or subtracted by means of equal numbers without affecting the symbol.

Rule 2: Same numbers may be multiplied or divided as of both sides of an inequality.

Ex :  Solve online 40 u < 200 when

(i) ‘u’ is a natural number,

(ii) ‘u’ is an integer.

Sol :      Given 40 u < 200

From online,

40u / 40 < 200 / 40 (Rule 2)

u < 5.

(i) When ‘u’ is a natural number, then the statement gives,

1, 2, 3, 4

The solution set is {1, 2, 3, and 4}.

(ii) When ‘u’ is an integer, then the solution is given as,

..., – 3, –2, –1, 0, 1, 2, 3, 4

The solution set is {...,–3, –2,–1, 0, 1, 2, 3, 4}

Sat Math Practice Problems

Introduction:

Scholastic Assessment Test or Sat reasoning test is a test which is used to get admissions in US universities and colleges. Sat test deals with quantitative questions and English skills. Sat test is conducted for 3 hrs and 45 minutes. It consists of three parts; they are i) Critical reasoning ii) Math aptitude and iii) Writing.  Math section of Sat questions deal with quantitative questions and logical reasoning questions. Math questions are given with multiple choices. Students should Practice math problems to get good scores in Sat test. I like to share this Define Geometry with you all through my article.


Sat Math Problems:


Example 1:

There are 240 balls, 4 times as many are brown, and the rest is black. How many are green and how many are black colors?

Solution:

There are 4 times as many brown balls as there is black color,

Hence, there must be 4 brown for each 1 of black color.

Given that 4 + 1 = 5,

Therefore,

240/5 = 48

On solving this we get, 48 sets of balls with 4 greens and one of black color in each set.

The total is then,

48 * 4 = 192 green and 48 of black color

Answer Check:

Number of green balls = 192 balls

Remaining black balls = 48

Total balls = 192 + 48

= 240 balls.



Example 2:

Mani drives from his house to hills 150 miles away, and at the end of the day drives home. If Mani drives at a standard speed of 50 miles per hour, how long does Mani takes to drive the round trip?

Solution:

Here Mani takes 150 miles to reach the destination,

The total distance covered by Mani during the round trip = 150 + 150

= 300 miles

Mani drives at an average of 50 miles per hour.

Using the formula

Distance = speed x time

300 = 50 x X

Divide 50 on both sides,

300/50 = 50X/50

6 = X

Mani takes 6 hrs to complete the round trip.

Answer is 6 hrs.


Sat Math Practice Problems:

Practice Problems1:

In a class of 78 students 41 are taking Tamil, 22 are taking English and 9 students are taking both Tamil and English. How many students are not enrolled in any of the course?

A) 10

B) 15

C) 24

D) 34

Answer: C


Practice Problems 2:

Six years ago rani was X times as old as raji was. If rani is now 17 years old, how old is raji now in terms of X?

A) 12/X + 6

B) 11/X + 6

C) 17X

D) 18/X

Answer: B

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Practice Problems 3:

Which of the following numbers can be used to demonstrate that all prime numbers are not odd?

A) 2

B) 5

C) 11

D) 13.

Answer: A

Direction Vectors

Introduction:

In mathematics a direction vector that describes a line segment D is any vector of the direction vector,

AB?

Where, A and B are two distinct points on the line D. If v is a direction vector for the D, so is kv for any nonzero scalar k; and these are in fact all of the direction vectors for the line D. Under the some definitions, the direction vector is required to be a unit vector, in which case each line has exactly two direction vectors, which are negatives of each other equal in magnitude, opposite in direction. (Source:Wikipedia)

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Direction vector for a line in two-dimensional:


Any other line in two-dimensional Euclidean space can be described as the set of solutions to an equation of the form

ax + by + c = 0

Where a, b, c are real numbers. Then one direction vector of (D) is (- b,a). Any multiple of (- b, a) is also a direction vector.


Basic properties:


The following section uses the Cartesian coordinate system with basis vectors

a=a1e1+a2e2+a3e3 and
b=b1e1+b2e2+b3e3
Are equal if

a1=b1, a2=b2, a3=b3.


Magnitude and Direction of a Vector:


Let v can be a vectors given in component form by

                        v = < a,b >

The magnitude of the || v || of vector v is given by

                 || v || = sort (a 2 + b 2)

and the direction of the vector v is angle t in standard position and in counterclockwise direction is such that

                  tan(t) = v / u

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Vector methods:


Vectors and vector addition
Unit vectors
Base vectors and vector components
Rectangular components in 2-D
Rectangular coordinates in 3-D
Direction cosines
A vector connecting two points
Dot product
Rectangular coordinates
Projection of a vector onto a line
The cross product
The triple product
Rectangular coordinates
Triple vector product

Example:

The equation of a line is 2x - 3y + 15 = 0.

2x-3y=-15

2(-3)-3(3)=-15

-6-9=-15

So (-3, 3) is direction vectors for this line

Geometry in Our Daily Life

Introduction to Geometry in our daily life:

Geometry is a part of mathematics which is concerned with questions of shape, size, the relative position of figures and the properties of space. In earlier, geometry was a collection of empirically discovered principles which concerns about lengths, areas, angles and volumes which were developed to meet some practical need in surveying, construction, and various crafts. There are various fields of life where geometry is considered as an important field of study. Please express your views of this topic Similar Polygon by commenting on blog.


Geometry in Daily Life

The general applications of geometry in our daily life are as follows:

Geometry is the very useful to define relative physical locations.
Geometry can be used to analyze or design physical structures and objects and to determine the various location and elevation of physical features on the Earth's surface.
Major examples of the geometry in real life:  The shape and volume of food containers, the design of buildings, and the layout of streets and roadways. I have recently faced lot of problem while learning Finding Surface Area of a Cylinder, But thank to online resources of math which helped me to learn myself easily on net.

Major Applications of Geometry in Daily Life

Major Applications of Geometry in Daily Life are as follows:

In real life, Geometry is particularly useful in home building and improvement projects. If we want to find the area of the floor of a house, we can use geometry.
In turn, this information is very useful for laying carpet or tiles and also for telling an estate agent regarding the information that how big our house is when we want to put it on the market. Geometry involved in many our daily life problem.
If we need to reupholster a piece of furniture, it is necessary to estimate the amount of fabric that we need by calculating the furniture’s surface area.
Geometry also finds its application in hobbies. In order for the fish to thrive, the water in a goldfish tank needs to have a certain volume as well as surface area. We can calculate it easily using geometry.
Pastimes like quilting and some other design projects will use geometry extensively. How the shapes of a quilt block fit together can be understand which is basically dependent on geometry; so that is determining the amount of fabric we need. Geometry is used for find the angle of the object.