Linear programming

What is linear programming?

Linear programing is the process of finding a way to obtain the best result in a give mathematical model for some list of requirements represented as linear equations.

Example of linear programming:
Consider the function D = 5x + 4y.We will find the minimum value of this function in the region defined by the inequalities x ≥ 0, y ≥ 0, x + 2y ≤ 4, and x + y ≤ 3.The possible region determined by the given constraints is shown.

 The vertices are (0,0), (3, 0), (2, 1), and (0, 2). In order to find the minimum and maximum value of D let us evaluate the value of D = 5x + 4y at each of the four vertices and we find that the minimum value of the function subject to the   constraints is 0.The process adopted in the example above is   called Linear Programming.See the link on solve linear programming

Least Common Multiples of 12 and 18

Let us find the least common multiple of 12 and 16

Before we find the least common multiple of 12 and 16,see this link to understand what is meant by least common multiple.

Solution: The multiples of 12 are 24, 36, 48, 60, 84,....

              The multiples of 16 are 32, 48, 64, 80

According to the list of multiples, 48 is the smallest number that is found in both the lists.

Therefore, 48 is the least common multiple of 12 and 16.


I would also like to suggest you to see this link on how to find least common multiple.In this link you will be taught how to find the least common multiple of binomial and trinomial expresstions.

Types of Graph

Definition of graph: A graph is a representation of a set of objects which are connected by links.

There are different types of graphs:

1.Directed graph

2.Undirected graph

3.Mixed graph

4.Multigraph

5.Simple graph

6.Weighted graph

7.Half-edges, loose edges

Properties of graphs: 1.Two edges of a graph are said to be adjacent if they have a common vertex.

                                2. If a graph has only one vertex and there is no edges, the graph is called the trivial graph.

                                3. The graph having no vertices and edges is called the null graph.

                                4.The graph having vertices and no edge is called the edgeless graph.

Classes of graphs: Graphs are classified as : 1.regular graph

                                                                   2.complete graph

                                                                   3.Finite and infinite graph

See the given link of How to Graph Set of Linear Functions or Simultaneous Equations

Graphs are again classified in terms of connectedness. The details and explanation will be discussed later.
                                                                       

Mixed fractions

What is a mixed fraction?
A mixed fraction is a combination of a whole number and a proper fraction.
Example of mixed fraction:1 ¹/₃, 2 ¹/₄, 16 ²/<sub>5 are all mixed fraction.

How to add and subtract mixed fractions: Example1. - Add 1</sub> ¹/_{3 and} 4 ¹/_{4
Soln. 1} ¹/_{3 +} 4 ¹/<sub>4
= 4/3 + 17/4
= (16 + 51)/12
= 57/12
= 4 (9)/12

Example 2. - subtract</sub> 1 ¹/₃ and 2 ¹/₄
Soln. 2 ¹/₄ - 1 ¹/_{3
= (9/4) - (4/3)
= (27 - 16)/12
= 11/12}
I will also give you a link for you to learn how to add and subtract mixed fraction with a whole number.

<sub>Conversion of improper fraction to mixed fraction:Following are the steps to convert improper fraction to mixed fraction.

1.Multiply the whole number and the denominator of the fraction.
2.Add the product of the two with the numerator of the fraction.
3.write the result that you get on the numerator and denominator will be the same.</sub>

Percentage

We can calculate percentage of any quantity by dividing the number by 100.calculation of percentage is used in our daily life whether it be business, offices or institutions.

In schools and college, percentage is used in assessing the marks acquired by students.

Let us see some example on how to calculate percentage.
Find 25percent of 80
     It is calculated as the following:
     (25/100)x 80
     =200/10
     =20
Hence 25 percent of 80 is 20.           
                                      
You may also be asked to find the percentage of 100kg of rotten potatoes out of 500kg.
Here you need to find how many percentage is rotten from 500kg if 100kg of potatoes are rotten.

Let's see how to calculate:
 (100/500)x100
  =100/5
  =20
Hence 20 percent of potatoes are rotten. 

Addition of fractions

Let us see learn how to add fractions by the following examples:
Example1.1/2+4/6
         = (3+4)/6)
         = 7/6
       
Example2.3/2+6/2+4/8
        =(12+24+4)/8
        = 40/8
        = 10/2


You can visit some websites where you can find questions on adding  fraction examples.You can practice at home to solve the problems.The answer for the questions will be provided  online.The method for adding fractions are also explained.

You can see the following steps to add fraction.
There are two types of fraction for addition
Take L.c.M of denominator and multiply with numerator.
Add the numerators.You will get a new fraction.
Reduce the fraction to the lowest possible terms.

The above examples are solved by following the given steps.
You can also avail the link given here on How to add and subtract mixed fractions.
Next time we will learn how to add and subtract fractions with whole numbers.

Distance formula

Distance may be defined as the measure of length from one point to another.The formulae of distance d between the points (x1,y1) and (x2,y2) is given as : d = square root of ((x2-x1)^2 + (y2-y1)^2)

Example: Suppose (-1,-2) and (-3,5) are the two points,then the distance d between the two points can be calculated by using the above formulae.Then we get , d = square root of ((-3-(-1))^2 + (5-(-2))^2 = sq.root of ((-3+1)^2 + (5+2)^2) = sq.root of ((-2)^2 + 7^2) = sq.root of 4 +49 = sq.root of 53.
There distance d between the two points = sq.root of 53.

Distance mid-point formula: According to the distance formula, (x1, y1) and (x2, y2) are the end points of the distance d.

The mid point distance formula is written as: ((x1+x2)/2)((y1+y2)/2)units.

Example of finding midpoints between two points:
Suppose if we are to find the mid-point of the two points (6,3) and (9,1).By using the mid-point formula ((x1+x2)/2)((y1+y2)/2)units, we can write as: ((6+9)/2)((3+1)/2) units = (15/2)(4/2) = 7.5 x 2 = 15 units.