Fraction Variable Solver

A fraction (from the Latin fractus, broken) is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (½, ?, ¾, etc.) and which consist of a numerator and a denominator.
Source – Wikipedia.


Fraction variable solver for Addition:

Variable Solver:

A variable solver is the way of thinker that refers to give proper solutions or values for the given fractions problems.

The following are some of the examples for Fraction variable solver in addition.

Simplify the fraction: `(3x)/7+(5x)/6`

Solution:

=` (3x)/7+(5x)/6`

The LCD for the denominators 7 and 6 is 42.

= `(6*3x)/42+(7*5x)/42`

We can express the denominators in the LCD form as

= `(18x)/42+(35x)/42`

We can simplify the numerators.

= `(18x+35x)/42`

Then by adding the numerators.

= `(53x)/42`

Simplify: `(3x)/5+(5x)/3`

Solution:

= `(3x)/5+(5x)/3`

The LCD for the denominators 5 and 3 is 15.

= `(3*3x)/15+(5*5x)/15`

We can express the denominators in the LCD form as

= `(9x)/15+(25x)/15`

We can simplify the numerators.

= `(9x+25x)/15`

Then by adding the numerators.

= `(34x)/15`

Simplify: `(2x)/5+(7x)/6`

Solution:

= `(2x)/5+(7x)/6`

The LCD for the denominators 5 and 6 is 30.

= `(6*2x)/30+(5*7x)/30`

We can express the denominators in the LCD form as

= `(12x)/30+(35x)/30`

We can simplify the numerators.

= `(12x+35x)/30`

Then by adding the numerators.

= `(47x)/30`

Fraction variable solver for Subtraction:

The following are some of the examples for Fraction variable solver in subtraction.

Simplify the fraction: `(6x+3)/12-(4x+5)/8`

Solution:

= `(6x+3)/12-(4x+5)/8`

The LCD for the denominators 12 and 8 is 24.

= `(2(6x+3))/24-(3(4x+5))/24`

We can express the denominators in the LCD form as

= `(12x+6-12x-15)/24`

We can simplify the numerators.

= `(x-9)/24`

Simplify: `(2x+3)/6-(6x+5)/5`

Solution:

= `(2x+3)/6-(6x+5)/5`

The LCD for the denominators 5 and 6 is 30.

= `(5(2x+3))/30-(6(6x+5))/30`

We can express the denominators in the LCD form as

= `(10x+15-36x-30)/30`

We can simplify the numerators.

= `(-26x-15)/30`

Simplify: `(8x+3)/12-(3x+5)/8`

Solution:

= `(8x+3)/12-(3x+5)/8`


The LCD for the denominators 12 and 8 is 24.

= `(2(8x+3)) /24-(3(3x+5)) /24`

We can express the denominators in the LCD form as

= ` (16x+6-9x-15)/24`

We can simplify the numerators.

= `(7x-9)/24`

Discrete Data

The nature of data is discrete if there are only a limited number of values probable or if there is a space on the number line between each 2 possible values.  Example, A 5 query quiz is specified in a Math class. The number of exact answers on a student's quiz is an example of discrete data.

In discrete data examples the number of exact answers would have to be one of the following: 0, 1, 2, 3, 4, or 5. there are not an endless number of values, therefore this data is discrete. Also, if we were to illustrate a number line and position each probable value on it, we would see a gap between each pair of values.

Definition of Discrete Data examples:

The discrete data is set of data having limited number of values or data points are called Discrete Data. The differing of the discrete data i.e. sets of infinite number of values is called constant Data. For example Data that can only take certain values. The number of apple in a box (you cannot have the half a student).

Examples: Number of hours prakash time spent for playing in a week this is an example of discrete data as playing period (finite) of ram will be specific.

Solved Examples on Discrete Data:

The table illustrates the set of books by Ravi. What type of graph is suitable to display the data in the table?

We use any type of graph is used to display the data
a. Bar graph
b. Histogram
c. Line graph
d. Stem-and-leaf plot

Solution:
Step 1: The data given in the table is discrete.
Step 2: Bar graphs are used to represent discrete data.
Step 3: So, a bar graph is proper to display the data shown in the table.

Problem for discrete data Examples:

The frequency tables situate for the number of worker and their salaries. Find out the distinction between the numbers of worker receiving a salary of 5000and1500.

Solution:

Step1:  The frequency column in the table specifies the number of worker who gets the particular salary.

Step2: From the table, the number of worker who gets a salary $1500 is 8.

Step3: The number of worker who get a salary $5000is 25.

Step4: The difference between the number of worker who get 5000 and those who get 1500 = 25 - 8 = 17

Step5: So, 17 more workers get a salary 4200 than those who get 3900.

Special Functions in Math

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics or other applications. In math, several functions are important enough to deserve their own names. Functions like algebraic and transcendental functions are elementary functions in math. The list of special functions in math is given below.

(Source: Wikipedia)

Understanding Absolute value functions is always challenging for me but thanks to all math help websites to help me out.

List of special functions in math:

Basic special functions
Number theoretic functions
Hypergeometric and related functions
Riemann Zeta and related functions
Antiderivatives of elementary functions
Elliptic and related functions
Iterated exponential and related functions
Gamma and related functions
Bessel and related functions
Other standard special functions
Miscellaneous functions

The example problems of trigonometric integral and logarithmic integral function are given below. Both trigonometric integral and logarithmic integral function are antiderivatives of elementary functions.

Example problems of special function in math:

Example 1:

Evaluate the integral int log (1 + x2) dx

Solution:

By integrating by parts method, taking log (1 + x2) as the first function and 1 as the second function, we get

 int log (1 + x2) dx  = int (log (1 + x2) . 1) dx

The above equation can be written as

 int log (1 + x2) dx  = log (1 + x2) int 1 dx - int [d/dx {log (1 + x2)} * int  1 dx] dx

= log (1 + x2) * x - int  (2x)/(1+x2) x dx

= xlog (1 + x2) - 2int (1 - 1/(1+x2) dx

= xlog(1 + x2) - 2int dx + 2int (dx)/(1+x2)

= xlog(1 + x2) - 2x +2 tan-1x + C

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Example 2:

Integrate the trigonometric function sin-1(cos x).

Solution:

Step 1: Given function

sin-1 (cos x)

Step 2: Integrate the given function sin-1 (cos x) with respect to ' x ',

int sin-1 (cos x)dx = int sin-1{sin pi/2 - x} dx

= int (pi/2 - x)dx

= pi/2 int dx - int x dx

= (pix)/2 - x^2/2 + C

Identifying Functions in Math

The mathematical concept of a function expresses the intuitive idea that one quantity (input) completely determines another quantity (output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the co domain of the function. (Source : WIKIPEDIA)


For example, f(x) = 3x2 .Here f(x) is a function that express three time the square of the number. In this article we are going to discuss about how to identifying the function.

Representation of functions in math:

The functions in math can be represented in four ways.

Verbal
Table of values
Numerical
Graphical

The functions in math can be identified by using the table of values,graphical and verbal statements.

Example for representation of functions:

1.Verbal representation :

The cool drinks that john get from the vending machine depends upon the code that he enter when provoked.

2.Table of values:

x             0             1             2               3                    4

y             3             7            11            15                 19

3. Numerical

f(x) = 4x+3

4.Graphical

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Fig(i) Graph of function

In the following section,  we are going to see how for identifying the function in math when we given the table of values.

Problems on identifying the function in math:

Problem 1:

Identifying the function for the following math table of values

x         -2           -1          0             1          2

y          8           -1          0             1          8

Solution:

Given   x         -2           -1        0             1          2

y          8            1          0             1          8

When we give x = -2 we get f(x) = y = 8 = x3

x = - 1 we get f(x) = y = -1 = x3

x = 0   we get f(x) = y = 0 = x3

x = 1   we get f(x) = y = 1 = x3

x = 2   we get f(x) = y = 8 = x3

So the function for given table of values id f(x) = y = x3

Answer: y = x3


Problem 2:

Identifying the function from the following verbal statements.

The cost for activate a device is 5000 per week It also costs 6 for each unit produced in the device. Identifying the function for daily cost x of run the device as a function of the number m of units produced.

Solution:

The   cost y equals the cost of $5,000 plus the cost of producing x units.

For 1 unit the cost is $6 .So for x units 6x

So the total cost y,

y = 5000 + 6 x

Answer: y  = 5000 + 6 x

Pre Calculus Limits

Precalculus is the advanced form of secondary school algebra, is a foundational mathematical discipline. Pre calculus does not prepare students for calculus as the pre-algebra prepares students for Algebra .The concept of a "limit" is used to describe the values that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to the pre calculus (and mathematical analysis in general) and are used to define continuity, derivatives and integrals.(Source: From Wikipedia)

Fundamental results on pre calculus limits

(1) If f(x) = k for all x, then lim_(x->c) f(x) = k.

(2) If f(x) = x for all x, thenlim_(x->c) f(x) = c.

(3) If f and g are two functions possessing limits and k is a constant then

(i)lim_(x->c) k f(x) = k lim_(x->c) f(x)

(ii)lim_(x->c) [f(x) + g(x)] =lim_(x->c) f(x) +lim_(x->c)g(x)

(iii)lim_(x->c)[f(x) − g(x)] =lim_(x->c) f(x) −lim_(x->c) g(x)

(iv)lim_(x->c) [f(x) . g(x)] =lim_(x->c)f(x) .lim_(x->c) g(x)

(v)lim_(x->c)f(x)/g(x) =lim_(x->c) f(x) / lim_(x->c)g(x), g(x) ≠ 0

(vi) If f(x) ≤ g(x) thenlim_(x->c) f(x) ≤lim_(x->c) g(x).

Examples for pre calculus limits:

Example 1.

lim_(x->2)   3=0

Example 2.

lim_(x->2) 6 / (x+4)  =1.

Example 3.

lim_(x->8) x2-64 / x+8 = 64– 64 / 16 = 0

Example 4.

Evaluate lim_(x->3) x2 + 7x + 11/x2 − 9

Solution:

Let f(x) =x2 + 7x + 11/x2 − 9

. This is of the form f(x) =g(x)h(x) , where g(x) = x2 + 7x + 11 and h(x) = x2 − 9. Clearly g(3) = 41 ≠ 0 and h(3) = 0.

Therefore f(3) =g(3)h(3) =410 .

Hence lim_(x->3)x2 + 7x + 11 / x2 − 9 does not exist

Example 5.

A function f is defined on by f(x) = −x2 if x≤0

= 5x − 4 if 0 < x ≤ 1

Examine f for continuity at x = 0, 1, 2.

Solution.

(i)lim_(x->0) − f(x) =lim_(x->0)− ( − x2) = 0

lim_(x->0)+ f(x) =lim_(x->0) + (5x − 4) = (5.0 − 4) = − 4

Sincelim_(x->0) − f(x) ≠lim_(x->0)+ f(x), f(x) is discontinuous at x = 0

(ii)lim_(x->1) −f(x) =lim_(x->1) −(5x − 4) = 5 × 1 − 4= 1.

lim_(x->1) +f(x) =lim_(x->1) +(4x2 − 3x) = 4 × 12 − 3 × 1 = 1

Also f(1) = 5 × 1 − 4 = 5 − 4 = 1

Sincelim_(x->1) − f(x) =lim x → 1 +

f(x) = f(1), f(x) is continuous at x = 1 .

Scatter Plot Regression Line

Scatter plot

Scatter plots show the relationship among two variables by displaying data points on a two-dimensional graph. The variable that might be measured an explanatory variable is plotted on the x axis, and the reply variable is plotted on the y axis.

Regression line:

From a given scatter plots we can draw the best fit line that is known as regression line. The regression line is can be calculated by using least squares method because the regression line drawn by eyes will causes wide variations and it is not accurate one.


Least Squares method to get regression line

This method is based on the sum of the squares of the deviations of all coordinates from the regression line is at a minimum. In a graph, variable y is dependent one because the values of y are calculated from the independent variable x. The most suitable regression line contains the best fit of y on x.  The regression of x on y gives a different regression line which is not a correct one. The best fit line is described by

yn = c + dx

Here yn is the dependent variable and dx is the independent variable. The values of yn  is calculated based on the value of dx, c is the y intercept .

The value of c is calculated by

c = yn- dx

The value of d is the slope of the best fit line and it is calculated by

d =`"(n(Sigmaxy)- (Sigmax)(Sigmay))/(n(Sigmax)-(Sigmax)^2) `

Here n is the number of pair of the data.

Types of slope of regression line:

Types of slope of regression line:

Regression line through positive slope

Regression line through negative slope

Regression line through zero slopes.

Condition for positive slope:

The rate of y increases as the rate of x increases.


Condition for negative slope:

The rate of y decreases as the rate of x increases.



Condition for zero slopes:

No relation along with the variables x and y.

Prepare for Quadrants

Four Quadrants

The word Quadrant comes form quad meaning four. For example, four babies born at one birth are called quadruplets, and a four-legged animal is a quadruped.

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are (+,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant.


As we know there are six trignometric ratios there signs depends on this quadrants there are four quadrants in total namely first quadrant, second quadrant,third quadrant, fourth quadrant. so here we will see what is first quadrant.

The range of angles varies from 00 to 3600 .In a plane with a system of cartesian coordinates, the sign of  x  and y coordinate decides the type of  quadrant.

The sign of this trignometric ratios depends on the quadrant in which the terminal side of the angle lies.

quadrant

In first quadrant,

x>0,y>0

Range of angle is- 00 to 900

So, all  trignometric ratios are positive.

In Second quadrant,

x<0,y>0

Range of angle is -  900 to 1800

So, sin and cosec are positive and the rest four cos , tan, sec, cot are negative.

In third quadrant,

x<0,y<0

Range of angle is - 1800 to 2700

So, tan and cot are positive and the rest others are negative.

In fourth quadrant,

x>0,y<0

Range of angle is -  2700 to 3600

So, cos and sec are positive and the rest others are negative.

Tabular representation of quadrants

Quadrant X
(horizontal) Y
(vertical) Example
I Positive Positive (3,2)
II Negative Positive (-3,2)
III Negative Negative (-2,-1)
IV Positive Negative (2,-1)


Application of quadrants:

As we know that the value of trignometric ratios depends upon the sign of x and y  coordinates which is different for different quadrants, so if we know the quadrant in which the x and y lies we can compute the sign of trignometric ratio accordingly. And we can solve various trignometric equations too.

Proportion Confidence Interval

In statistics hypothesis testing are used to compute the probability for a particular hypothesis to be right. Hypothesis is specified as declaration which may or may not be precise. In statistics two hypothesis testing are consumed. They are null hypothesis and alternative hypothesis. The null and alternative hypotheses testing are opposed to every other. Various tests are used in statistics. The test of population proportionality is one of the most important in hypothesis testing. Confidence interval means we can specify the range. Let us see about the proportion confidence interval  .

Proportion confidence interval

The central confidence interval used for which the probability of being mistaken is separated uniformly into a range of proportions beneath the interval and another range exceeding the interval.

Confidence interval for proportion include the following steps

Confidence level

Sample size

Observed learn result

Assumptions

Proportion confidence intervals

When a 95% confidence interval is formed, each value in the interval is considered probable values for the parameter being expected. Values outer the interval are rejected as comparatively improbable.

If the value of the parameter illustrated through the null hypothesis is integrated in the 95% interval after that the null hypothesis cannot be rejected on the 0.05 level.

If the value specified through the null hypothesis is not contained through the interval afterward the null hypothesis is able to be rejected on the 0.05 level.

If a 99% confidence interval is formed, subsequently value outer the interval is rejected on the 0.01 level.

The formula used for 98% confidence limits for the proportion =`p+- 2.33sqrt((pq)/n)`

Examples for proportion confidence interval

A random sample of 600 oranges were obtained from a large consignment and 60 were of them are bad get the 98% confidence limits for the percentage number as bad oranges in the consignment.

Solution

98% confidence limits for the proportion `p+- 2.33sqrt((pq)/n)`

Calculation

Given n = 600, x = 60

p = `x/n = 60/600`

p = 0.1

p+q = 1

q = 1- 0.1

q = 0.9


By confidence limits for the proportion

`p+- 2.33sqrt((0.1xx0.9)/600)`

= 0.1 ± 0.0122

= 0.1122, 0.0878

Result

98% confidence limits for the percentage number as bad oranges in the consignment = [0.0878 x100, 0.1122 x 100]

= [8.78, 11.22]