Solve Algebra Squared Fraction

Fractions:

In algebra, A certain part of the whole is called as fractions. The fractions can be denoted as a/b , Where a, b are integers. We can multiply two or more fractions. There are three types of fractions in math,

1) Proper fractions

2) Improper fractions

3) Mixed fractions

In this article we are going to see how to solve algebra squared fraction and some example solved problems on algebra squared fraction.

Solve Algebra Squared Fraction :

Steps to solve algebra squared fraction:

Step 1: We know that ( a/b)^n = a^n / b^n . So we can take square for the numerator and denominator.

Step 2: If it is possible we can simplify it.

Let us see the example problems.
Example Problems on Solve Algebra Squared Fraction :

Problem 1:

Solve (4/5)^2

Solution:

Given , (4/5)^2

We need to squared the given fraction .

(4/5)^2

We know that ( a/b)^n  = a^n / b^n .

We can apply the above rule,

(4/5)^2 = 4^2 / 5^2

= 16 / 25

Answer: (4/5)^2 = 16 / 25

Problem 2:

Solve (3x / 4xy) ^ 2

Solution:

Given, (3x / 4xy) ^ 2

We need to find the squared the given fraction,

We know that (a/b)^n = a^n / b^n .

(3x / 4xy) ^ 2 = ((3x) ^ 2) / ( (4xy)^2)

= (9x^2) / ( 16x^2y^2)

now we can simplify it,

Divide by x^2 on both numerator and denominator,

(9x^2) / ( 16x^2y^2) = (9x^2)/x^2 / ( 16x^2y^2) / x^2

= 9 / 16y^2

Answer: (3x / 4xy) ^ 2   = 9 / 16y^2

Problem 3:

Solve (( x^2 - 4 ) / ( x + 2) )^2

Solution :

Given , (( x^2 - 4 ) / ( x + 2) )^2

We need to find the squared the given fraction,

We know that ( a/b)^n = a^n / b^n .

(( x^2 - 4 ) / ( x + 2) )^2 = ( x^2 - 4 )^ 2 / ( x + 2)^2

Before that we can simplify the given fraction,

(( x^2 - 4 ) / ( x + 2) )^2 = (((x+2)(x-2))/ ( x + 2))^ 2

= ((x - 2)^2)

= x^2 - 2x + 4

Answer: (( x^2 - 4 ) / ( x + 2) )^2 = x^2 - 2x + 4

Results and Discussion Section

Here we are going to see about the result and discussion, whether the solution to the given equation is correct or not. By simplifying the equation we find the value for the particular variable. when we substitute the value in the equation the equation balances on both sides.

For example: x – 4 = 0.

Here    x – 4 = 0

Add 4 on both sides. We get

x = 4

The solution is x = 4

According to the topic we have to check whether x = 4 is a solution to the given equation or not. By substituting the value the in equation we can see that both sides have same value.
Example Problem for Result and Discussion Section:

Given equation is discussion section  x + 7x + 5x + 3x – 5 + x = 29

Solution:

Given question is x + 7x + 5x + 3x – 5 + x = 29

Step 1:

Arrange the given question as per the x term and the numbers

x+ x + 3x + 5x + 7x – 5 = 29

Step 2:

Add the x term first

2x + 3x + 5x + 7x – 5 = 29

Step 3:

Add all the  ' x ' term in the equation

17x – 5 = 29

Step 4:

Add both sides by 5.Then we get

17x = 29 + 5

Step 5:

17 x = 34

Divide by 17 on both sides.

`(17x)/17` = `34/17`

x = 2

Discussion section about the result:

The result is 2. Here we are going to discuss about the result for given equation, whether the solution x =2 is correct or not.

Given question is x + 7x + 5x + 3x – 5 + x = 29

The result is x =2

Here we need to substitute the x value in the equation if that both side value is equal means the solution is correct. Otherwise the solution is not correct.

x + 7x + 5x + 3x – 5 + x = 29

Substitute the value x = 2 in the given equation

2 + 7( 2) + 5( 2) + 3(2) – 5 + 2 = 29

2 + 14 + 10 + 6 – 5 + 2 = 29

34 – 5 = 29

29 = 29

Both the sides are equal, so given solution is correct for the equation.
One more Example Problem for Result and Discussion Section:

Solve discussion section  16x - 12 = 20

Solution:

Add 12 on both sides. we get

16x - 12 + 12 = 20 + 12

16x  = 32

Divide by 16 on both sides.

`(16x )/ 16` = `32 / 16`

x = 2

whe we substitute x =2 in the equation 16x - 12 = 20. we get

16(2) - 12 = 20

32 - 12 = 20

20 = 20.

Thus the equation satisfies.

Raw Score Formula

Raw score is defined as the unique, unchanged transformation or calculation. When we bring together the information, in which the numbers we get hold of are known as raw score or raw data. Raw score can able to find the precise location of each data in a given distribution.

percentage of sampling error = (value of statistic-value of parameter)*100.

In this article we are going to see about raw score formula.
Raw Score Formula:

The raw score formula for variance can be given as follows,

Raw score formula

Where `sigma` and X indicates mean.
Raw Score Formula Related to Z Score Formula:

Z-score formula can be given as follows

Z = (x) - (X) / (S.D)

where x indicates a data value

X indicates mean (average of the given values)
S.D indicates standard deviation.

Using the formula of Z score we can find the raw score formula

But to find the raw score formula we have to solve this formula for x

hence the raw score formula for x can be given as follows


Z* (S.D) + X= x

Where x indicates a data value or raw score

X indicates mean (average of the given values)
S.D indicates standard deviation.

Example problem for raw score formula:

Let us take an example where the mean value X has to be 280 and the standard deviation is let to be 10.The data values are given as follows x1 = 300 and x2 = 285.Find the raw score value using raw score formula.

Solution:

Z = [(300) - (280)] / (10)


Z1= 2

Z= [(285) - (280)]/ (10)



Z2 = .5
Finding Raw Score using raw score formula.

Z = 2.2 hence


2.2 = (x-280)/10

(2.2) * 10 + 280 = x

x = 302

Z = 1.1

1.1 = (x- 280) /10.

(1.1) * 10 + 280 = x

x = 291

Solve a Parabola with Fraction

Parabola is a half circle and a curve formed by an intersection of right circular cone. A set of all points are same distance from a fixed line is called as directrix and also fixed point called as focus not in the directrix. The midpoint between the directrix and focus of the parabola is called as vertex and the line passing through the vertex and focus is called as axis. Let us see about how to get the value of vertex, latus rectum, focus and axis of symmetry in parabola equation. Let us see about solve a parabola with fraction in this article

The general form the parabolic curve is y = ax^(2) + bx + c or  y^(2) = 4ax . Substitute the above formula to find the vertices, latus rectum, focus and axis of symmetry.

Example 1 for Solve a Parabola with Fraction – Vertex:

Find the vertex of a parabola where y = x^ (2) + 7/2 x + 3

Solution:

Given parabola equation is y = x^ (2) + 7/2 x + 3

To find the vertices of a given parabola, we have to plug y = 0 in the above equation, we get,

0 = x^ (2) + 7/2 x + 3

Now we have to factor the above equation, we get,

So x^ (2) + 3/2x + 2x + 3 = 0

x(x + 3/2) + 2 (x + 3/2) = 0

(x + 3/2) (x + 2) = 0

From this x + 3/2 = 0 and x + 2 = 0

Then x = - 3/2 and x = - 2

So, the vertices of given parabola equation is (-3/2, 0) and (-2, 0) .

Example 2 for Solve a Parabola with Fraction – Focus:

What is the focus of the following parabola equation where y^ (2) = 10x

Solution:

Given parabola equation is y^ (2) = 10x is of the form y^ (2) = 4ax

To find the focus of a parabola use the formula to find the value of focus value.

We know that the formula for focus, p = 1 / (4a)

Now compare the given equation y^(2) = 10x with the general equation y^ (2) = 4ax . So, that 4a = 10

From this p = 1 / (4a) = 1 / 10

So, the focus of a parabola equation is (0, 1/10) .

Other Example Problems to Solve a Parabola Fraction

Example 3 for Solve a Parabola with Fraction – Axis of Symmetry:

What is the axis of symmetry of the parabola where y = 4x^ (2) + 8x + 14 ?
Solution:

Given parabolic curve equation is y = 4x^ (2) + 8x + 14

From the above equation, a = 4 and b = 8

So the axis of the symmetry of the given parabola is -b/ (2a) = -8/ (2 xx 4) = -8/8 = - 1

Therefore, the axis of symmetry for a given parabolic curve equation is -1 .

Example 4 for Solve a Parabola with Fraction – Latus Rectum:

Find the latus rectum of the given parabola equation y^ (2) = 6x

Solution:

The given parabola equation is y^ (2) = 6x

To find the latus rectum, we have to find the value of p.

The parabola equation is of the form y^ (2) = 4a

Here 4a = 6

So, p = 1/ (4a) = 1/6

The formula for latus rectum is 4p .

From this, the latus rectum of the parabola is = 4p = 4 (1/6) = 4/6 = 2/3

Therefore, the latus rectum for the parabola equation is 2/3 .

Short Definition of Geometry

Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metria "measurement") "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. 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.( Source- Wikipedia)

Short Basic Geometry Definitions:
Short Definition of Geometry for Square:

In mathematical geometry, a square consists of 4 equal angles every side which includes 90° and 4 equal sides. Square area is calculated by-product of two opposite sides.
Short Definition of Geometry for Rectangle:

In mathematical geometry, a rectangle is equal to the square, it consists of 4 equal angles with 90° and 4 equal sides. Rectangle area is calculated by using the product of length and width.
Short Definition of Geometry for Triangle:

In mathematical geometry, triangles consist of 3 straight lines that are equal length and it is called as equilateral triangle. Triangle area is calculated by half of the product of bottom and height.
Example for Short Definition of Geometry:
To solve the Rectangle problems:

Area of rectangle= Length x width.

Perimeter of rectangle= 2 (Length + Width)

Example 1:

Find the area of rectangle that sides measure 6 centimeters and 4 centimeters respectively.

Solution:

Given that

Length – 6 centimeter.

Breadth – 4 centimeter.

The formula used to find out the area of rectangle is = length * breadth.

Area = length * breadth.

Substitute the given value in the formula

Area   = 6 * 4

The result of 6 and 4 = 24

Hence the area of the rectangle given is 24 square centimeter.

Example 2:

Find out the volume of the cube that sides measure 6 centimeter.

Solution:

Given that

Measure of cube length = 6.

The formula used to find out the volume of the cube is = a3.

While we insert the value in the formula we get the result is

Volume = 63.

= 216

Hence the volume of the given cube is 216 cubic centimeters.

To solve Square problems:

Perimeter of square= 2 * side

Area of square= side * side

Problem 3:

To find out the area and perimeter of square while side length is 4cm?

Solution:

Area of square = (side*side)

=4 * 4

=>16 cm2

Perimeter of square = 4 * side

= 4 * 4

= 16 cm

The Multiplication Rules

In algebra fundamental arithmetic procedure (addition, subtraction, multiplication, and division) in general used in day-to-day life. The multiplication defined as product of numbers. Multiplication is usually expressed in a*b or ab form. The multiplying numbers are having two terms i) Multiplicand and ii) Multiplier. In this article, we are going to discuss about the multiplication rules

The Multiplication Rules – Rules and Example Problems:

The multiplication rules - Rules:

Rule 1: The multiplication of two integers by the connected signs resolve be positive sign

a) Positive `xx` positive = positive

b) Negative `xx` negative = positive

Rule 2:  The multiplication of two integers by the different signs will be negative

a) Positive `xx` negative = negative

b) Negative `xx` positive = negative.

The multiplication rules – Example problems:

Example 1:

Evaluate 6802 `xx` 322

Solution:

6802 `xx` 322

------------------

13604

13604

20406

--------------

2190244

--------------

Therefore, the final answer is 2190244.

Example 2:

Find the solution for given problem

(- 9667) `xx` (- 26)

Solution:

(- 9667) `xx` (- 26)

---------------------------

58002

19334

-------------------

251342

-----------------

Therefore, the final answer is 251342.

Example 3:

Find the solution for given problem

8655 `xx` (- 124)

Solution:

8654 `xx` (- 124)

-----------------

34616

17308

8654

----------------

- 1073096

-----------------

So, the final answer is (- 1073096)
The Multiplication Rules – Practice Problems:

Practice problem 1:

Find the solution for given problem

(- 7897) `xx` (- 53)

Answer: So, the final answer is 418541

Practice problem 2:

Find the solution for given problem

(- 7790) `xx` (- 48)

Answer: So, the final answer is 373920

Practice problem 3:

There are 280 students in every class at the performing arts school. If there are 62 complete classrooms, how many students attend this school?

Answer: 17360 students attend this school

Practice problem 4:

Jesse wants his giant chocolate chip cookies to have 28 chips each. Today he is make 355 cookies. How many chips does he need?

Answer: 9940 chips does he need.

Variance Covariance Formula

Variance: Variance is used in the statistical analysis to find the the extent to which a single variable is varying from its mean value given a set of values.

Variance of a random variable X is often denoted as VAR ( X). It is also denoted by the symbol `sigma` 2X

Covariance: Co-variance is used in statistical analysis to find the extent to which 2 variable are varying together given a set of values for both these variables.

Covariance of two random variable X and Y is denoted as COV ( X,Y). Unlike variance which has the mathematical symbol sigma ( `sigma` ), Covariance doesn't have any kind of symbol to depict it.
Formulae for Variance and Covariance.

Lets consider 2 sets of data namely X and Y such that the values in X are x1 , x2 , x3 , x4 , ......xn and similarly the values in Y are y1 , y2 , y3 , y4 , .......yn

Now lets denote the mean of the X set of variables to be    Xm

Mean Xm = X1 + X2 + X3 +........ + Xn / n

Similarly lets denote the mean of the Y set of variables to be Ym

Mean Ym = Y1 + Y2 + Y3 +.......+ Yn / n

Formulae for Variance of X = ( x1 - xm)2 + (x2 - xm)2 + ( x3 - xm)2 + .......+ (xn - xm)2 / n

VAR ( X ) = (1/n) `sum` (xi - xm)2

Formulae for Variance of Y = ( y1 - ym)2 + (y2 - ym)2 + ( y3 - ym)2 + .......+ (yn - ym)2 / n

VAR ( Y ) = (1/n) `sum` (yi - ym)2

Formulae for Covariance of (X,Y) = ( x1 - xm) ( y1 - ym) + (x2 - xm)(y2 - ym) + ( x3 - xm)( y3 - ym) + .......+ (xn - xm)(yn - ym) / n

COV ( X , Y ) = (1/n) `sum` (xi - xm)(yi - ym)
Example Problem 1 on Variance and co Variance

Lets assume the below set of marks received by Bill and Bob in Maths, Physics, Chemistry, English and Biology for the purpose of solving Variance and Co Variance

table

Based on the Formulae given in the previous paragraph, Lets calculate the Mean Marks for Bill and Bob

Mean Marks for Bill = 16+12+14+18+20 /5 = 16.00

Mean Marks for Bob = 14+18+15+18+20 /5 = 17.00

Now lets apply the formulae of variance and find out the Variance of Bill and Bob

Variance of Bill = (16-16)2 + (12-16)2 + (14-16)2 + (18-16)2 + (20 - 16)2 / 5 = 40 / 5 = 8

Variance of Bob = (14-17)2 + ( 18-17)2 + (15-17)2 + (18-17)2 + (20-17)2 / 5 = 24 / 5 = 4.8

Co variance of ( Bill , Bob ) = (16-16)*(14-17) + (12-16)*(18-17) + (14-16)*(15-17) + (18-16)(18-17) + (20-16)*(20-17) / 5

Co Variance of ( Bill , Bob ) = 21/5 = 4.2

I am planning to write more post on What are Line Segments and Perpendicular and Parallel Lines. Keep checking my blog.

Example Problem 2 on Variance and co Variance

Lets assume a class of 4 student who have been asked to rate their liking toward 2 musical instruments Guitar and Piano on a scale of 10

Based on the above data, lets calculate the Mean of the people liking Guitar and Mean of people liking Piano.

Mean(Guitar) = ( 5+3+7+9)/4 = 24/4 = 6

Mean(Piano) = (5+9+9+5)/4 = 28/4 = 7

Var(Guitar) = ((5-6)2 + (3-6)2 + (7-6)2 + (9-7)2 )/ 4 = (1+9+1+4)/4 = 15/4 = 3.75

Var(Piano) = ((5-7)2 + (9-7)2 + (9-7)2 + (5-7)2 ) / 4 = (4+4+4+4)/4 = 16/4 = 4

Cov(Guitar,Piano) = [(5-6)(5-7) + (3-6)(9-7) + (7-6)(9-7) + (9-7)(5-7) ] / 4 = (2+(-6)+2+(-4))/4 = -6/4 = -1.5