Mixed Fractions

Mixed Fraction: - The mixed numbers are like 2 1/2 which is two and one half or 35 3/32 which is equal to thirty five and three thirty two seconds.  To express the mixed fractions we have to keep exactly one blank between the whole numbers and the fractions. The mixed fraction are in the form of 3 2/3 ( three and two third) , 7 1/5 ( seven and one fifth) , 13 1/7 ( thirteen and one seventh ) , 113 7/100 ( one hundred thirteen and seven hundredth) and so on.

The rules for adding the mixed numbers: - To add the mixed number, convert them into the fraction. The algebraic formulae can be used for the addition of these, for example a/b + c/d = (ad + bc) / bd.

Example :- Let us add the two mixed fractions a b/c and d b/c

Solution :- The given fraction are a b/c and d b/c. The first step which is to be used is to convert the mixed fraction to the fraction. The fraction of the mixed term a b/c is equal to (ac + b)/c. The procedure is to convert the mixed fraction to the fraction is to multiply the whole number to the denominator of the fraction and then add the fraction. The total value is divided by the denominator of the fraction. The fraction of the mixed term d b/c is equal to (dc + b)/c. Now we have to add the two fractions as below.

(ac + b )/c + (dc + b)/c. As the denominators of both the terms are same, the numerators can be added directly as below
(ac + b + dc + b ) /c =   (ac +2 b + dc) /c

The rules for subtracting the mixed numbers: - To subtract the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the subtraction of the fractions, for example a/b - c/d = (ad - bc) / bd.

Example :- Let us subtract the two mixed fractions a b/c and d b/c
(ac + b )/c - (dc + b)/c

As the denominators of both the terms are same, the numerators can be subtracted directly as below
{(ac + b) – (dc + b)} /c = (ac + b – dc - b) /c = (ac – dc)/ c = (a – d) [by canceling the common term in the numerator and denominator]. 

The rules for multiplying the mixed numbers: - To multiply the mixed numbers or the fractions we have to convert them into the fractions. The algebraic formulae can be used for multiplication of the fractions, for example a/b *c/d = a c /bd

The rules for dividing the mixed numbers :- To divide the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the division of the fractions, for example a/b ÷ c/d = a/b ×d/c = ad/bc.

Fraction Calculator Online: - The fraction calculator online is a tool which can be used to add, subtract, to multiply and to divide the fractions. We have to enter the fractions to be calculated, enter the function is to be carried out (i.e add, subtract, multiply, divide). The output in the form of the fraction will be displayed in the output window.

Direct and Inversely Proportional

Proportionality:

A quantity is said to be proportional to another quantity if change of one of the quantities is always accompanied by the change of the other. This property is known as proportionality.

Proportionality is of two types:

(i)                  Direct proportionality

(ii)                Inverse proportionality

Direct Proportionality:

A quantity is said to be directly proportional to another quantity if the change in both of them is in the same direction. This means that if one of the quantities increases then the other also increases. If one of the quantities decreases then the other also decreases.

It is denotes by the symbol a. If ‘a’ is directly proportional to ‘b’ then:

We write as a a b ==> a = k * b where k is the proportionality constant.

i.e.  a / b = k = constant à a1 / b1 = a2 / b2

Graph:

Let us consider that x a y. If we plot the values of x and y on a graph sheet we obtain the graph showing the relation between these two quantities. Generally, the graph of directly proportional quantities is a straight line. Thus by seeing the graph we can conclude the relation and proportionality between two quantities.

Inversely proportionality:

A quantity is said to be inversely proportional to another quantity if the change in both of them is in the opposite direction. This means that if one of the quantities increases then the other quantity decreases. If one decreases then the other increases.

Inverse proportionality is also uses the symbol a but the reciprocal of the second quantity is written.

If ‘a’ is inversely proportional to b then:

We write as a a 1 / b

You can see that the reciprocal of b is written to indicate inverse proportionality. We can also say that ‘a’ is directly proportional to the reciprocal of ‘b’.

If a a 1 / b à a = k / b where k is the proportionality constant.

i.e. a * b = k = constant à a1 * b1 = a2 * b2

Graph:

Let us consider x is inversely proportional to y i.e. x a 1 / y

Now plot the values of x and y on the on a graph sheet we obtain the graph showing the relation between the two quantities. Generally the graph is not a straight line but a curve. On seeing the graph we can analyze the relation and proportionality between the two quantities.

Problem:

If the volume of a gas at a given temperature is 2 liters when its pressure is 1 bar, then what will its volume when the pressure increases to 3 bars? (Volume in inversely proportional to pressure)

Sol: Given, initial volume v1 = 2 l

Initial pressure p1 = 1 bar

Final pressure p2 = 3 bars

As volume is inversely proportional to pressure we have, p1 v1 = p2 v2

Now v2 = p1 v1 / p2 = (2 * 1) / 3 = 0.66 liters (approx.)

Thus final volume the gas is 0.66l

Properties and Area of a Rectangle

Rectangles
For a normal 4th grader, a rectangle would mean a plane figure that has four sides. However, more precisely in geometry, a branch of math, a rectangle is a special type of a quadrilateral that has 4 right angles. It would look as shown in the picture below:

Properties of a rectangle:
1. It has four sides.
2. It has four angle and all the angles are right angles.
3. It has four vertices.
4. Each pair of opposite sides are congruent.
5. Opposite sides are parallel.
Examples of rectangles:
1. Top of a book.
2. Face of a cuboid.
3. Top of a table.
4. Front of a cupboard.
5. Etc.

The Area of the Rectangle:
The Formula for the Area of a Rectangle is as follows:
A = l * w
Where,
A = area of the rectangle,
L = length of the rectangle
W = width of the rectangle.

It is customary to denote the longer side of the rectangle as length and the shorter side as width. Another custom is to denote the horizontal sides as the length and the vertical sides as the width of the rectangle. It is shown in the picture below.

Let us now try to understand how to calculate the area of a rectangle with the help of a sample problem question.

Example 1: Find area rectangle from the figure shown below:

Solution:
From our formula for area of a rectangle we know that,
Area = A = L * W
For this problem,
L = length = 5 units and
W = width = 3 units
Therefore substituting these values of L and W into the above formula for area of the rectangle we have,
A = 5 * 3 = 15 sq units <- answer="" p="">
If instead of being given the measures of length and width, we are given the co ordinates of the vertices of the rectangle then its area can be found out as follows:

Consider a rectangle with the vertices at A (x1,y1),B  (x2,y2), C (x3,y3) and D (x4,y4) taken in clock wise direction. Therefore we know that if AB is the length of the rectangle then BC would be the width of the rectangle. The distance AB can be found using the distance formula as follows:

Length = L = AB = √[(x2-x1)^2 + (y2-y1)^2]

Similarly the distance BC can also be found using the distance formula as follows:

Width = W = BC = √[(x3-x2)^2 + (y3-y2)^2]

Both the above can be now used to find the area of the rectangle as follows:

A = L * W.

Sample problem:
Find the area of a rectangle having vertices at (3,7), (0,7), (3,-2) and (0,-2)

Solution:
First let us sketch a graph of the said rectangle.

From the picture we see that
Length = L = 3-0 = 3 and
Width = W = 7 – (-2) = 7+2 = 9

Therefore,
Area of rectangle = 3 * 9 = 27 sq units.

Define Absolute maximum

Optimization is one of the most vital applications of differential calculus, which guides the business and the industry to do something in the best way possible. Business enterprises ever need to maximize revenue and profit. Mathematical methods are employed to maximize or minimize quantities of interest. Absolute maximum value is when an object has a maximum value.

In mathematics, the maximum and minimum of a function, identified collectively as extrema , is the largest and smallest value that the function obtains at a point either within a given local or relative extremum (neighborhood) or on the function domain in its entirety.

A function f has an absolute maximum at point x1 , when f(x1) =  f(x) for all x. The number f(x1) is called the maximum value of ‘f on its domain. The maximum and minimum values of the function are called the extreme values of the function. If a function has an absolute maximum at x = a , then f (a) is the largest value that f that can be attained.

A function f has a local maximum at x = a if f (a) is the largest value that f can attain "near a ." Simultaneously, the local maxima and local minima are acknowledged as the local extrema. A local minimum or local maximum may also be termed as relative minimum or relative maximum.
Both the absolute and local (or relative) extrema have significant theorems linked with them Extreme Value Theorem is one of it.

To find global maxima and minima is an objective of mathematical optimization. If a function is found to be continuous on a given closed interval, then maxima and minima would exist by the extreme value theorem.
Moreover, a global maximum either have to be a local maximum within the domain interior or must lie on the domain boundary. So basically the method of finding a global maximum would be to look at all the local maxima in the interior, and also look at the maxima of the points on the boundary; and take the biggest one.
For any function that is defined piecewise, one finds a maximum by finding the maximum of each piece separately; and then seeing which one is biggest

In mathematics, the extreme value theorem signifies that if a real valued function f is continuous in the closed and bounded intermission [x,y], at that moment f should attain its maximum and minimum value, each of it at least once. That is, there prevail numbers a and b in [x,y] in such a way that:
F(a) = f(c) = f(b) for all c summation [x,y].
A related theorem is also known as the boundedness theorem which signifies that a continuous function f in the closed interval [x,y] is bordered on that interval. That is, there always exist real numbers m and M in such a way that:
m = f(c) = M for all c summation [x,y].
The extreme value theorem thus enhances the boundedness theorem by demonstrating that the function is not only bounded, but also accomplish its least upper bound as its maximum as well as its greatest lower bound as its minimum.

Algebra variable

Algebra is a subject which helps us to find unknown quantities with minimal information. But how do we carry all mathematical operations when something is not known? We assign letters from alphabet (mostly small case letters) for the unknown.

Such letters are only called as variables or changeable. Why the name is selected as such? Because the value assigned to these could be varied according to our wish but gives the correct information at a required condition. For example I earn $100 per day.

How do I calculate the total earnings after a certain number of days? Here the word certain is really uncertain! In other words it is an unknown. So we assign ‘x’ as the number of days and I can formulate that my total earnings in dollars will be 100x. Now my job is simple. I just to need to replace ‘x’ by the actual number of days I decide and I get the required information correctly.

Thus basically the changeable are employed as algebra variables or simply math variables. But why we had done away with the prefixes ‘algebra’ or ‘math’ and we simply refer as such?
Because, such uses of letters even in other subjects are subjected to mathematical or algebraic operations.
The types of variables are many. In most cases more than one forms are used. For example one may represent an input and another may be the corresponding output. Obviously the former can be assigned any value and hence it is described either as independent forms and the latter, because of the dependence on the input is referred as dependent forms.

In most of the cases they are represented by the letters ‘x’ and ‘y’ respectively. In such cases they are related by equations or functions.
Further, different types of variables are used depending on the context. For example when you study about distance versus time for a moving object, lettert is used for the unknown time and ‘s’ is used for the corresponding distance.

The same letter tis also used for denoting temperatures in normal scale. (whereas, T is used for temperature in absolute scale!). Letters p and v are used for denoting pressure and volume respectively.
We mostly see that the letters at the second half of English alphabet are mostly used for the unknown quantities. It is just a convention and the first of half is generally reserves for constants. However use of letters like ‘a’, ‘b’ ‘c’ is common in geometry.
In addition to English alphabetical letters Greek letters are also extensively used especially in the topics of trigonometry.