Language and Literacy Essay Example | Topics and Well Written Essays - 3000 words

Language and Literacy - Essay Example There seems to be no agreement as to when language was first used by humans. Some estimates date as far back as two million years ago, during the time of Homo habilis, while others date as recent as forty thousand (40,000) years ago, during the time of Cro-Magnon man. What is unarguably clear, however, is that language development is a continuous process affected by several social factors and that most contemporary human languages are a blend of several primitive ones. One main feature of human language is arbitrariness of symbols and sounds. A symbol or sound only needs to be attached to a particular concept or meaning, or even applied to the rules of grammar and becomes a part of the language. For instance, while the word 'nada' is conceptualised to mean nothing in the Spanish language, for Croatian speakers, it means 'hope' (Hudson, 2000). Through the course of this essay, I shall attempt a discourse of the various social factors that come into play in language, within the context of literacy development. In this regard, three students currently undertaking a basic skills "Brush Up Your English" course at Halton College will be used as case studies. After a brief analysis of what has been said and researched on the impact of social factors on language development, I will give a brief account of the backgrounds of the three students in this group, before examining how the duo of region/geography and gender (two main social factors) have affected language development and literacy in these students. Language and Social Factors Sociolinguistics are social sciences that consider the interactions between languages and society as a whole. It is an established body of knowledge that studies language on a social basis. Thus, it involves an interest in interaction, variability and diversity in language (Deumert, 2005). Or as described by Trask (1999), it is "the study of variation in language, or more precisely, the variation within speech communities." (Trask, 1999, p.283). This field of interest only dates back to the 1950s, this perhaps explain why most of the social influences on language development are still not fully comprehended. Within the Sociolinguistics school, there are two broad approaches to language variation; prescriptivism and descriptivism. Prescriptivists tend to be found among the ranks of language educators and journalists, and not in the actual academic discipline of linguistics. They hold clear notions of what is right and wrong and tend to advocate what they consider as 'correct' use of language according to set rules (Hudson, 2000). Describing this school of thought, Thorne (1997) states that "it is associated with formal written and spoken language and is used in dictionaries, grammar books and language handbooks." (p.92). To further buttress this point, Thorne (1997) cited the example of the original version of the National Curriculum for schools' emphasis on Standard English (SE) being taught as "the language of wide social communication and was generally required in formal contexts" (p.138). Descriptivists, on the other hand, do not accept the prescriptivists' notion of "incorrect usage." They prefer to describe such variance as 'non-standard'. Thus, they see Standard English as "only one variety among manylinguistically speaking it can not legitimately be

Text Linguistics Essay Example for Free

Text Linguistics Essay Module I. Classificationally meaningful characteristics of the text as an integral and independent object of investigation. Lecture 6. Conceptual variability of linguistic interpretations of the text essence and status and their reflections in the models of the text descriptions. Problem for discussion Evolution of the text description approaches. Models of the text descriptions. Grounds for the chosen models and schemes of the text descriptions as a scientific object. It has already been mentioned that despite the fact that there are many publications devoted to problems of text linguistics. There does not exist an adequate definition of the text that would find satisfaction with all researchers. The difficulties that arise when trying to work out an universally acceptable definition of the text can be explained by the fact that scholars study the text in its various aspects : grammatical, stylistic, semantic, functional and so on. The text can be studied as a product ( text grammar) or as a process (theory of text). The text-as-a-product approach is focused on the text on the text cohesion, coherence, topical organization, illocutionary structure and communicative functions; the text-as-a-process perspective studies the text production, reception and interpretation. Text can be understood as an instance of (spoken or written) language use ( an act of parole) , a relatively self-contained unit of communication. As a â€Å"communicative occurrence† it meets seven criteria of textuality (the constitutive principles of textual communication): cohesion, coherence, intentioanality, acceptability, informativity, situationality, and intertextuality and three regulative principles of textual communication: efficiency, effectiveness and approapriateness.(cf. de Beaugrande and Dressler 1981, Maljaer 1991) 1. Regular Principles of Textual Communiction. The principle of efficiency requires that a text should be used with a minimum effort hence the use of plain ( stereotyped and unimaginative)  language. Which, however booring and unimpressive, is easy to produce and comprehend. In contrast, effectiveness presumes leaving a strong impression and the creation of favourable conditions for attaining a communicative goal; this presupposes the use of creative ( original, imaginative) language which, however effective, may lead to communicative breakdown. The principle of appropriateness attempts to balance off the two above mentioned principles by seeking an accord between the text setting and standards of textuality. Common text features. Some common text features found in books are Table of Contents, Glossary, Index, Bold Words, Headings and Titles, Maps, Diagrams, Illustrations and Charts. Why is Understanding Text Features an important reading strategy? Text Features help us to identify the big ideas and topics that the author is focusing on. Visual text features such as maps and charts help to support the information the author presents in the text. How do we use text features to help us understand what we are reading? Before reading, preview the kinds of text features throughout the book. Think about what the big ideas the author wants you to remember. Make sure to read captions Functional Classification  The functional classification identifies illocutionary text types according to the type of the dominating illocutionary act (see 10.2): representative or assertive type (e.g. research reports, public notices, administrative texts, weather forecasts, diaries, CVs, lectures), directive type (e.g. commands, orders, invitations, instructions, directions, giving advice), expressive type (e.g. apologies, thank-you notes, greeting, condolences, compliments, toasts, congratulations), commissive type ( e.g. promises , pledges, swears, offers, vows, contracts, bets), declarative or performative type (e.g. , nominations, appointments, dismissals, accussations: I find you guilty as charged, marriage ceremonies, testaments, certificates). Texts viewed from this perspective satisfy diverse communicative needs of the society members. Situational Classification  The situational classification sorts out texts according to the sphere of activity (e.g., private, official or public, such as a private letter, a  letter addressed to an institution) and form of communication (dialogical and monological, spoken and written texts). Strategic Classification  The strategic classification deals primarily with the topic and the ways of its expansion 9 the term slohove postupy is used in Slovak stylistics to denote macrocompositional principles, cf. Mistrik 1997): narrative, descriptive, and argumentative.

Plyometric Exercises and Their Benefits for Football Players

Plyometric Exercises and Their Benefits for Football Players The benefits of polymetric exercises for soccer players and the  importance of when in a training session these type of exercises should  be implemented. Introduction The subject of exercise and physiology is a broad topic. Researchers have known for many years that exercise benefits the body in various ways and there is not a practicing physician anywhere that does not recommend it to his/her patients. But the actual effects of exercise on a person’s ability to perform an activity are not well understood. Whether the increase in ability to perform is due to what is known as â€Å"muscle memory†, increased muscle mass, or simply based on repetitive motion is hard to determine in a quantitative sense. In the next few pages a case will be made for what are known as polymetric exercises specifically for increased soccer playing ability. In order to explore the subjects of exercise and kinesiology it is necessary to have an adequate amount of background information beforehand therefore before delving directly into the core subject matter, a few general principles will be discussed to provide a framework of thinking for the remainder of the paper. A few principles that are crucial to the understanding of the effects of polymetric exercise are muscle physiology, various exercise regimes, and finally the basics of polymetrics and why they work. Muscle Physiology The human body is made up of hundreds of muscles from the most minute (i.e. the muscles that close your eyelid) to very large conspicuous muscles (quadriceps, gluteus maximus, etc.). The large muscles are generally the ones that human beings â€Å"train† to become stronger and more adept, however all muscles have the same basic structure. Each muscle is made up of a series of strands of tissue known as muscle fibers. These fibers work together as one large unit to form what we know as muscle groups. The major muscle groups that most people know about and understand the location of are the bicep brachi(biceps),pectoralis major (pecs), quadriceps femoris (quadriceps), and rectus abdominas (abs). These groups are the major muscle groups that people tend to â€Å"workout or train† in order to lose weight or tone up and are the muscles with which the majority of the population is most familiar with. While a working knowledge of the major muscle groups is sufficient for the everyday layperson going to the gym, to really understand what is involved in the build up of muscle mass and the accumulation of what is known as â€Å"muscle memory† it is necessary to understand what is going on at the cellular level. Muscle cells work together with the nerves to perform actions. The body’s nerves create an electrical impulse triggered by a potassium gradient which then stimulates the muscle cells into action. Even though this entire process takes much less than a second, there are still ways of maximizing the efficiency and output of the muscle cells. Some of the muscle cells are part of what are known as fast twitch muscle fibers and others are part of slow twitch muscle fibers. These concepts will be discussed more in the next section. Fast twitch versus slow twitch muscles Experts usually split muscles into two general groupings or types. The first type is called fast twitch muscle fiber. Fast twitch muscles are the muscles that react quickly to stressors (ie sprinting, jumping, or punching). The fast twitch fibers are usually what are termed the â€Å"shorter† muscle fibers. Even though the actual physical length of the fibers are no different from the â€Å"longer† or slow twitch muscle fibers, the term short refers to the type of energy they use and the types of activities that these muscle fibers are suited best for. Fast twitch muscle fibers are better suited for activities that require quick movements for short amounts of time. For instance, sprinters build up fast twitch muscle fibers, so do boxers, and to an extent soccer players do as well (Vannatta 2002). These muscle fibers are powered by â€Å"quick or fast† energy which is provided by ATPs produced by carbohydrates. In order to build up these fast twitch muscles specif ic types of exercises work best. General exercises (jogging, weight training, calisthenics) will build the muscle fibers up to a certain point, especially if this is the first activity in a while or ever for that matter. However, once a reasonable level of fitness is achieved it begins to get harder to realize increased fast twitch muscle enhancement. It is due to this new plateau of difficulty that researchers have developed different types of exercises to specifically address the fast twitch muscle fibers (Pollock et al 1998). There are several different types of exercises that can be used to overcome fast twitch muscle fiber plateaus. These exercises employ techniques that capitalize on movements that require quick employment of energy. A few examples of exercises would be anything that requires jumping, dexterity drills, or footwork drills. All of these exercises increase the muscle memory of the fast twitch muscle fibers. Muscle memory is defined as the tendency for muscles to â€Å"remember† or acquire a propensity for the motions of a specific action. This is due to a muscle’s direct feedback mechanism connected with growth. Once a muscle is stressed and the stressor is removed, the muscle takes time to recover. Depending on the amount of stress, the muscle can take varying amounts of time to recover. Also, once the muscle is stressed it will not only recover to the original state of the muscle but it will increase its resistance to stressors to the point of the initial stress event (Rhe a et al 2003). In other words, once the muscle fiber is torn by work (ie exercise) it will repair itself to a point where it can resist the same type of exercise stress again later and in the process builds up an increased muscle mass. It’s the accompaniment of muscle memory exercises and the increased resistance to stressors that leads to the abolishment of a fast twitch muscle fiber performance plateau. While fast twitch muscle fibers rely on quick use of energy, slow twitch or long muscle fibers require the use of sustainable energy found in slower burning sources such as protein and fats. These muscle fibers are better suited for types of activities that require muscle endurance. Slow twitch muscle fibers are responsible for activities such as long distance running, rowing, and cycling. Long muscle fibers must be equipped with the stamina needed for endurance events, as well as be able to work in conjunction with the fast twitch muscle fibers for quick bursts of speed. No one has solely fast twitch or solely slow twitch muscles. Each person has a specific ratio of slow to fast twitch muscle fibers that determines what kind of sports or activities that they are better suited to (ie endurance vs. sprints). In other words, marathon runners have more slow twitch muscle fibers than do sprinters and boxers have more fast twitch muscle fibers in comparison to cyclists. Although the ratio of muscle fiber types plays a role in determining the types of sports and activities that a person participates in, it is not a completely â€Å"firm† method of deciding on the type of performance expected from an individual. Some sports, like soccer, are activities that require a fair amount of both slow twitch and fast twitch muscle fibers in order to perform at the highest level, and this holds true for many sports (Ekblom 1986). Therefore the question is â€Å"how can athletes enhance the performance of both their slow twitch muscle fibers and fast twitch muscle fibers with one series of exercises?†. In the past there has not been a good answer to this question. Coaches and athletes have simply trained one set of muscle fibers on one day and then train the other muscle fibers on the next days. In this way all of the muscle fiber types were being trained, but not in a single training session, and thus the simulation of a game or competitive event was not accurately depicted. However, recently kinesiology has taken an interesting turn with regard to training multiple muscle groups at the same time. The next section will discuss a few of these techniques at length. Exercises to train multiple muscle groups Researchers have determined that there are sets or groupings of exercises that can effectively train both slow and fast twitch muscle fibers, as well as training multiple muscle groups, concurrently. This group of exercises is collectively known as polymetric exercises. The meaning of the word polymetric is just as its roots imply. These are exercises that employ techniques to train multiple (poly-) muscles at once using different movements (-metrics). There are several different types of exercises that are all considered polymetric exercises. These exercises include all non-isolationistic movement, or in other words, any exercise or movement that trains more than one muscle or muscle group at a time. These exercises can include plyometric exercises, isotonic exercises, polykinetic exercises , polytonic exercises or compound exercises. We will look at each type briefly as a preliminary procedure. Plyometric Exercise Plyometric exercises are a group of exercises that many organized sports teams and athletes are familiar with. Plyometrics are usually implemented in what are also known as â€Å"drills†. These can include such practices as box jumping, jump roping, line hops, etc. Basically, plyometrics serve to recreate certain situations that the athlete may encounter during a competitive event. This could be anything from jumping over a would be tackler, making a quick turn to avoid an opponent, or jumping over a hurdle. There are many plyometric drills that are employed by various coaches and one only needs to decide on the specific movements that are used in the activity that they are involved with in order for new drills to be designed. Plyometrics are very good at training the fast twitch muscle fibers to react with greater efficiency and at a higher rate than the original state of the muscle. Even though most polymetric muscles are good at training both types of muscle fibers, plyomet ric exercises in general do not do a good job of training the slow twitch muscle fibers and thus are considered an earlier stage of exercise development than other more advanced polymetric exercises. Isotonic Exercise Isotonic exercises are a group of exercises that stresses a constant load of resistance against the opposing muscle. These are most easily generalized as the weight lifting exercises that people perform in a gym such as: bicep curls, bench press, and standing barbell rows. Most experts agree that the use of free weights for these exercises is essential because free weights tend to employ more muscle groups at the same time in order to balance the weight. It is due to the act of balancing muscle groups that more muscle mass may be gained by using free weights instead of machines. Its important to realize that not all isotonic exercises can be called polymetric exercises. In most cases the use of isotonic exercises necessitates the employment of more than one muscle group or type of muscle fiber at a time and therefore may be classified as a polymetric exercise, however in some cases isotonic exercises may isolate a single muscle or muscle group in which case they can no longer be cons idered a polymetric exercise. Polykinetic Exercise Polykinetic exercise literally means multiple motion or multiple movement. Dancers and tennis players perform these exercises most frequently in order to increase their â€Å"spring† or vertical jumping height. Polykinetic exercises sometimes are misclassified as other polymetric exercises such as polymetrics. Polykinetics use multiple motion exercises to employ as many muscles as possible in one specific exercises. Again, polykinetics much like plyometrics are used mostly to train fast twitch muscle fibers, however if the exercises are performed for a longer period of time then these could also be used to train the slow twitch muscle fibers as well. Polytonic Exercise Polytonic exercise is not actually a separate division of polymetric exercise but is simply used interchangeably with polymetric and plyometric when discussing various sports training activities. Literally polytonic means â€Å"multiple tones† and was originally applied to Greek orthography. Compound Exercise Compound exercise is also simply another way to express the idea of an exercise which involves multiple muscle employment. However, most researchers acknowledge compound exercises as those that involve a great deal of balance as well as the actual movement of the exercise. There are now specific tools that are used in conjunction with compound exercises which include such items as balance boards, balance balls, posture correcting exercise balls, etc. All of these items are designed to force the person exercising to not only employ their muscles during the exercise, but also use them to balance themselves and the weight at the same time. Why does polymetric training work? The idea and principle behind polymetric training is to â€Å"get the most bang for your buck†. Exercise kinesiologists have developed methods to get the most muscle fatigue and consequent recovery in the least amount of time and energy expenditure. This age of technology is the fastest paced since the industrial revolution and does not appear to be slowing down any. Therefore it is imperative for today’s athletes to be able to train as many muscle groups as possible in one session, while still maintaining a high level of performance. Polymetric exercises allow this to happen and in some cases have been shown to provide the best possible training for the competitive event for which they are intended (Noda et al. 1998). The Ins and Outs of Polymetrics Polymetric exercises have been shown to improve the ability of many sports teams and athletes to perform at amazingly high levels when employed correctly. The types of activities that are performed are very important and should be specific to that particular sport. For example, it has been shown that soccer players need the recruitment of both fast twitch and slow twitch muscle fibers. Soccer is a sport that requires not only stamina to last the entirety of the game while running almost constantly, but also demands small bursts of speed and energy in order to outdistance your opponent or to save the ball (Reilly 2005). In the case of soccer then, it is absolutely essential to not just train for endurance or for speed, but for a combination of the two. By utilizing polymetrics it is possible to not only train both slow and fast twitch muscle groups, but is possible to do this at the exact same time. The optimal conditioning program is the implementation of a holistic fitness approach as stated by Reilly (2005). While the types of exercises are extremely important, another aspect of training which is often overlooked is that of the time of training. This is not referring to the time of day, but rather the time in the training regime. It is believed that the sequence in the exercise routine is linked to the overall performance of the participant later in a competitive event. Using the concepts of strength and endurance as potential results, it is possible to make a few generalizations concerning training. First, an athlete will have the most energy during the beginning phase of an exercise or workout routine and will be able to produce the most power. We can say that this is when his/her strength is at its highest level throughout the entire routine. Secondly, if an athlete wanted to increase his/her strength to the optimal level he/she should train their bodies at the point when their strength is the highest initially so that the amount of stress on the muscles is the greatest, leading to the most growth. It would seem reasonable then to assume that in order to gain the most strength gains an athlete would do some sort of polymetric training early on in their workout routine. Since polymetrics would work multiple groups and the athlete is working these muscles early, the greatest increase in athletic ability would be in strength instead of endurance. However, if we assume that at the end of a workout period an athlete’s muscles are at their weakest point or are the most tired/stressed, then it is reasonable to say that the opposite is true if the polymetric exercises are performed at the end of a workout session. If performed at the end of an exercise period, polymetric drills will increase the endurance of an individual since the muscles will not be exerting the most force (ie strength) that they are capable of, but will instead be utilizing the longterm energy sources in the body. Conclusion Polymetric exercise encompass a wide range of workouts, drills, and exercise regimes which all help to stress the body’s muscles in many different ways. In the case of soccer players it is of the utmost importance that these techniques be employed. According to Reilly (2005) today’s soccer players are enjoying an increased physical ability and game tempo compared to decades in the past. This is due not only to better medicine and technology, but also to the increase of high end research performed in the disciplines of physiology and kinesiology. It is an obvious assumption that the temporal aspect of exercise is extremely important in determining what muscles are stressed and how. This paper has shown that in order to increase strength, athletes should perform polymetric drills at the beginning of an exercise routine and to increase stamina or endurance one should perform polymetric drills at the end of a practice or exercise period. This claim is significant in that it implies that polymetrics may be more important to athletes as they increase their basal fitness level and approach loftier performance levels in an effort to perform at their absolute best. It is necessary for research to continue to be done in this area so that athletes may continue to increase their performance levels naturally and without pharmaceutical enhancement.

The Myth of the American Dream :: ESL Essays

  Ã‚  Ã‚  Ã‚   Striving for success nobody thinks that he follows somebody’s well planned way. A single person or a small group does not create the notion of success, but it is created by our whole society. The myth of instant wealth is one of the most popular myths society uses. In fact society uses the hope of instant wealth to make people work harder. The fact that they do not have a real chance of obtaining that wealth by competing in the economic system stays invisible to the most of people.     Ã‚  Ã‚  Ã‚   When we imagine a successful person, we see a person, who is working on some company and is busy working all the time. This person has a nice car and beautiful apartment or house, where he does not spend much time because he is so busy. We get this impression since we were born. Movies, magazines, and news – everything supports this notion of a successful person.     Ã‚  Ã‚  Ã‚   Interesting thing is that notion of success did not change very much since the beginning of last century. There happened some variations but the idea stayed the same: working hard will bring you to the top of the society circle. This idea became very popular in the end of eighteen hundreds thanks to the stories, written by Horatio Alger. In spite of similarity of all his books, his works had an edition of hundred thousand copies. Simple idea of getting into upper class circles starting from the very down, was accepted by society as a model of success achievement. People have believed that if they will work hard than they can achieve success.     Ã‚  Ã‚  Ã‚   Richard Hunter, main character of the book â€Å"Ragged Dick† has been a typical example of American notion of success. According to this book everybody can became well recognized and financially prosperous if they would work hard and show their merit. Dick, â€Å"a young gentleman on the way to fame and fortune,† as his friend Fosdick from the story â€Å"Ragged Dick† describes him in the end of the story, climbs on the social ladder, starting from the very bottom. Being absolutely illiterate and having no money in the beginning, Dick gets into business circle of people, by working hard and showing his merit. Why did this story become so popular in the end of eighteenth century? People always need hope and this story gave hope to everyone. If person from the lowest class of society could get into the high class then everyone else was able to do the same.

Hair Oil Marketing Essay

Hair oil is a hair care product specifically intended to keep the moisture balance of the hair, as moisture is lost due to strong shampoos and harsh chemicals in water. It can also be considered as conditioner to make the hair soft and pliable. Hair oil can come from natural products such as coconut oil, fruit extracts, milk, lemon oil, rosemary oil and others. Modern hair oils contain fragrances from different natural sources of plants. Musk Hair oil is available with rich coconut and almond mixture of hair oil products with soothing male and famine cent. Indian Market in Hair Oil Industry: Market Trend: Light Hair Oil Break Up: Packaging: 5 C ‘s Of Marketing 1. The study of the 5 Cs of marketing arises is called situation Analysis. 2. Situation analysis is study of the current market or industry in which company wants to launch a new market. 3. In order to launch a new product, a company first needs to study market condition. 4. The conditions are about the number of competitors and their market share, the cost of producing the product, the profit ration etc. 5. So a quick SWOT analysis will reveal where does the company stand in the market and what strategy it should adopt in order to grab a MUSK’s share in the market. The Five C’s are 1. Customer Needs 2. Company Skills 3. Competitors 4. Collaborators 5. Climate or context Customer Needs/ Company Skills/ Competitors through SWOT Analysis: Musk Parent Company Red Cherry Multi Commodity Pvt Ltd Category Personal Care brands – Hair care Sector FMCG Tagline/ Slogan â€Å"Oil of Pride† USP New Product Launch of hair oil brand in India STP Segment Hair oil segment with natural ingredients (Coconut and Almond) Target Group Youth and middle aged and Old men and women in urban and rural area. Positioning 1. A hair oil which nourishes your hair and maintains style at the same time 2. Positioned on the platform of purity and originality of coconut with best quality and resulting in shiny hair and a clear complexion SWOT Analysis Strength 1. Newly Established product with male soothing fragrance and famine scents. 2. Provides shine and softness and makes hair healthy from inside†¨ 3. Contains trusted natural ingredients like Coconut and Almond 4. Strong distribution network across the country 5.Introduction of oil in the market with better fragrances, ingredients and innovation yet using traditional method to extract oil from raw Almond and Coconut. 6. Celebrity/ film star brand ambassadors Weakness 1. New Brand Launched recently 2. Will be Preferred by loyal customers, but youth find other brands attractive†¨ such as Hair Gel and other hair care products 3.Sticky and oily, stains the pillow when used overnight Opportunity 1.Expansion in foreign markets†¨2.Export potential†¨3.Innovation in other hair care products Threats 1.Aggressive competitors†¨2.Threat from new entrants or local players selling oil with natural ingredients†¨3. Well established Brand Like parachutes and Bajaj Almond. Competition Competitors 1.Marico’s parachute 2.Bajaj Almond 3. Dabur 4.Emami Customers 1. Market size and Growth: Total Market Size – 63% of the total Indian Hair Oil Market and growth is increasing 10% annually. 2. Market segments –Urban and Rural Men, Women with Young Age, Middle age and Older age. 3. Retail Channel – where does consumer actually purchase the product?: Product should reach every single possible household, Retail outlet. Kirana shop, Online and Purchase options on social networking sites with discounts. 4. Consumer Information source – where does the consumer obtain information about the  product? Social media penetration, Road Shows, displays, Exhibition and discount offering strategies online, at retail outlets, Kirana shops and at road shows. 5. Trends: how consumers’ needs and preference change over time? Consumer behavior understanding from time to time by getting feedback and surveys at retail out lets, Online, social networking sites, dedicated consumer retention team through consumer forum specially designed for consumer for MUSK Hair Oil. Company: MUSK HAIR OIL 1. Product line I. Coconut based hair oil a. Soothing Male scent b. Soothing female scent II. Almond based hair oil a. Soothing male scent b. Soothing female scent 2. Image in the market MUSK Hair Oil was established on 24th September 2013 in a typical for niche and rural market, Red Cherry Multi Commodity Pvt Ltd envisioned offering trusted quality products made from 100% Coconut and Almond raw material for hair, skin and hair care. The company presently launching their flagship brand â€Å"MUSK Hair Oil† encased in an attractive cylindrical cardboard label (Blue for Male Customer and Pink for Female Customer) gives product a brand new look in keeping with high quality of the oil it encases and hope to successfully cater to diverse competitive markets in India and the suburbs. 3. Technology and Experience Age-old traditional methods and processes are utilized in producing this oil. The almond oil is lightweight, golden-brown oil, which is extracted from sweet almond nuts. The oil from bitter almond nuts on the other hand, is  extremely poisonous and should be avoided. Half of the weight of the sweet almond nut itself comprises of the oil and therefore can be extracted in large amounts. In its pure form, it has little to no odor and has a faint, nutty scent. The oil has a long and extensive history, dating back to the Egyptians where it was used to strengthen hair and as a skin emollient. More than 50% of the oil comprises of monounsaturated fatty acids, making it suitable to be used for culinary purposes as it helps reduce blood cholesterol levels. The oil is also rich in minerals such as magnesium and the vitamins C and E. While the oil might be lightweight, it causes a brown stain to form when it comes into contact with clothes and bed sheets. Therefore, it should be strictly separated from such things. The oil has many established beneficial properties and is widely used in the cosmetic and food industry. It is also one of the most commonly used massage oils today. Strengthens the hair The almond oil provides essential minerals for normal and healthy hair growth. After continual application of the oil, the hair naturally grows thicker and stronger. It also promotes a lustrous, attractive shine on the hair when used in appropriate amounts. Coconut Oil Extraction: The extraction of oil from copra is one of the oldest seed crushing operations. In India and Sri Lanka copra is still crushed for oil extraction in the primitive chekkus as well as in rotary ghanis, expellers and hydraulic presses. The chekku is a fixed wooden or stone mortar inside which revolves on a hard wooden pestle. The pestle is attached to a long pole which is moved round via bullocks, donkey or by human labor. About 20 – 40 kg of copra can be handled by a chekku. Using coconut oil for hair maintenance may sound like an odd regimen, but it’s been proven to be effective. Coconut oil has been known to have a lot of benefits, both when consumed and applied. Specifically for hair and skin, it has been known to restore natural moisture resulting in shiny hair and a clear complexion. Studies have shown that this substance has the ability to penetrate the hair  shaft and to work its way through all layers of the hair strand. It helps reduce loss of protein, and aids in regaining your hair’s natural oil and moisture. More than keeping your hair healthy and shiny, coconut oil has a lot of other benefits as well. The use of coconut oil is not only natural and organic, but also cost-effective. 4. Culture We are committed to deliver 100% natural products, which are manufactured under stringent quality guidelines. Apart from commitment to quality and product authenticity, it is our compliance to timeliness, fair business practices and cost competency that has made us a preferred associate. 5. Goals: Become the market leader in Coconut and Almond based Hair Oil market in India till 2020. Collaborators: 1. Distributor: 2. Supplier: Local Supplier from Kerala, Tamil Nadu, Puducherry, Andhra Pradesh, Karnataka, Goa, Maharashtra, Odisha, West Bengal 3. Alliances: Alliances with retail out let, Online shopping Websites and smaller suppliers to create value for them and for the company. Climate or Context b.1. Political and regulatory environment that affect the market. b.2. Economic environment business cycles, inflation rate, interest rates and other issues of economic nature. b.3. Technological environment- new ways of satisfying needs, the impact of technology 4 P’s Of Marketing

Essay Sample on Mathematics The System of Linear Equations

Essay Sample on Mathematics The System of Linear Equations PATTERNS WITHIN SYSTEM OF LINEAR EQUATION A system of linear equation is basically dealt with in the algebra unit. It is a collection of the linear equations involving variables of the same set in the in the equations that are involved. For example a 2Ãâ€"2 system of linear equations includes: x + 2y=10 3x + 4y=15 Here in both the cases the equations only involve two variables that is x and y and no other variable is included. In the example of a 33 system of linear equations it mostly includes the variables x, y and z for example; 2x + y-z =11 x- 2y + 2z =-2 3x-y+2z =5 Where only the three variables are involved There are also various properties of the patterns of the linear systems. We will start with the consistency property. If the systems of the equations have common solutions, then they are said to be consistent. This therefore means that graphically the lines should be straight lines. The independence property is also termed as the linear independence. The systems of equations are usually independent since to start with, they are derived algebraically from others. For example the system 3x+4y =9 and 6x +8y =18. There are different ways of solving the systems of linear equations that includes; The elimination of variables The substitution of variables technique The row reduction method The crammers’ rule The matrix method In the mathematical field, the general linear equation in the x and y is Ax+By=C where both the A and B in the equation are not zeros. The y-intercept in the line is the y-coordinate of that point where graphically, the non-vertical line that is drawn either manually or graphically intersects the y-axis. Also, the x-intercept is the point where the non horizontal line crosses the x-axis. Therefore the most general equation for a line with slope m and the y-intercept passing through b as the y intercept is written as y= mx + b. Therefore, one can easily find the slope and at the same time the y-intercept of any line. For example finding the slope and the y-intercept for 4x+5y=40 Solution: first and foremost, solve the equation for y to put it in the slope intercept format 4x+5y=20 5y=20-4x y=4-4/5x y=-4/5x+4 therefore the slope m=-4/5 and the y intercept is b=4 Consider this 2Ãâ€"2 system of linear equations 4x+3y=7 3x-2y=9 When we examine our first equation 4x+3y=7, there is a pattern in the constants of the equations used. Here 4 is the constant associated with the variable x and it therefore precedes the variable x. Also 3 is a constant that is preceding the variable y and the equation results to 7. In the second equation, 3x-2y=9, the constant 3 precedes our variable x and the constant -2 precedes the variable y making the equation to result to 9. It is also clear that in the two equations, the constants both have a difference of one. Solving the equations simultaneously, we first multiply the first equation by 3 and then multiply the second equation by 4 in order to eliminate the variable x and solve for the variable y. The equation then becomes; 12x+9y=21 12x-8y=36 17y=-15 Therefore solving the equation yields y=-15/17. Putting the value of x in any of the solution to obtain the value of x; 4(x) +3(-15/17) = 21 X=41/17 Graphically the system of equation is solved as This is first done by putting the two equations in the form of y=mx+b. The solution of the equations is by observing the point of intersection of the two lines that are plotted graphically. In this system of equation the solution therefore is (41/17, -15/17) Consider this 2Ãâ€"2 system of linear equations x+2y=3 and 2x-y=-4 The two equations are linear because the unknowns only appear to the first power, no unknown in the denominator of a fraction is in the equations and there are no products of unknowns. Therefore, the most general linear equation is a11x1 + a12x2 ++ a1nxn=b1 a21x1 +a22x2++a2nxn=b2 am1x1+ am2x1 +..+amnxn=bm m With unknowns x1, x2..xn and coefficients a1, a2an . In x+2y=3, the constant is 3 and the unknowns are x and y whereby x= 3-2y and y= (3-x) à ·2. The gradient of the linear equation is -1/2 and the y- intercept is 3/2. The gradient is negative therefore it is negatively sloped. In 2x-y =-4, the constant is -4 and the unknowns are x and y where x= (y-4) à ·2 and y=2x+4. The gradient of the line is 2 and the y- intercept is 4. The gradient is positive and therefore positively sloped. Each of the unknown variables can be solved using the graphical calculator in the matrix calculation area [1 2: 3] and [2 -1: -4] The inverse is The solution therefore is; 1 2: x =3 2 -1: y =-4 1/5 2/5 3 =x 2/5 -1/5 -4 =y The graphic calculator here was used during this step to solve the matrix equation, normally if the equation is Ax= B then the solution is x= A-1B X=-1 y=2 The first function from the graph is sloped from left to right that is it is negatively sloped since the gradient is negative and the second equation is sloped from right to left since the gradient is positive. From the graph the solution of the equations is x and y2. This is read directly from the graph where the two lines intersect. In short, the solution to the system of equation is unique in that there is only one solution set to the system of equations and the solution satisfies the individual equations in the system of equations. Therefore, when x=-1 and y=2. Then -1+ (2Ãâ€"2) =3 and (2Ãâ€"-1) 2 = -4 which satisfy the equations that are given. Another example of linear equations is x+2y =3 3x-5y=9 This is a system of equations since it contains more than one equation. The solution set to the system of linear equation is the set of numbers n and m such that if we let x=n and y=m then we will obtain the result of the right hand side of the equation. For instance ax+by =c, if x=n and y=m then we obtain the result c given a and b are known constants. Each of the unknown variables can be solved using the graphical calculator in the matrix calculation area 1 2: x =3 3 -5: y =9 The inverse of the equation is -5/11 -2/11 -3/11 1/11 The solution therefore is, -5/11 -2/11 3 =x -3/11 1/11 9 =y Therefore xand y The graph of the two equations is as shown below This is so since (3 1) + (2 0) =3 and (33)-(50) = 9 as proven from the equation. Now consider the 2 2 system below x2y=4 5x-y=1/5 In the first equation, x2y=4 f(x) = x/8. Therefore (0, 0) In the second equation, 5x-y=1/5 then f(x) = 5x-1/5. The gradient is 5 and the y intercept is -1/5 The matrix of the equation therefore is [1/8 -1: 0] [5 -1: 1/5] Each of the unknown variables can be solved using the graphical calculator in the matrix calculation area -39/8 39/8: 0 = x -195/8 39/64: 1/5 = y Consider the following two by two system of equation y + 2x=7 y + x/2=3 In the first equation, the line is negatively sloped since y= f(x) = -2x+7. The gradient is -2 which means that the change in x compared to the change in y is -2. The y intercept is 7 and therefore when x=0, y=7 In the second equation, the equation line is also negatively sloped. The equation y= f(x) =-x/2 +3. The gradient is -1/2 and the y- intercept is 3 meaning when x=0, y=3 Each of the unknown variables can be solved using the graphical calculator in the matrix calculation area Therefore the solution to the equation is x=8/3 and y=5/3 The graph of f(x) =-2x+7 and f(x) =-x/2+3 are as follows: From the graph both the linear equations are negatively sloping but that of f(x)= -2x+1 is steeper than that of –x/2 + 3. The equations above are linear equations that results to linear curves and therefore two equations are enough to solve the equations. This two equations result into a square matrix. In the multiplication of matrices for instances, if A is an np matrix and B is a pm, then AB is the product of A and B denoted by AB and AB will be an nm matrix. That is AB exists if and only if number of columns of A is equal to the number of rows of B and that it should be noted that ABBA therefore the matrices do not commute. Therefore in solving the equation for example Ax=B, to find the values of x, the inverse of A is first found then multiplied by B. That is x= A-1B If A is a square matrix, we can find another matrix B known as the inverse of the matrix such that AB=BA=I. The inverse however can be a right inverse or a left inverse most commonly denoted as A-1. If AB=BA=I then A-1= B. Therefore if B exists then A is said to be invertible and non singular matrix. If B does not exist then A is said to be singular matrix. For the solution to be found in the equations, the matrix involved should be non singular. Theorem If A is a non singular matrix then A-1 is unique Proof Let A-1 =B then AB=BA=I Suppose B is not unique, then there exist C such that CB Then CA=AC=I But B=IB therefore (AC) B= (CA) B and thus C (AB) =CI =C Therefore B=C, a contradiction that B is unique An example of invertible matrix can be solved by looking at the following question. Solve the equations -4x-2y=8 and 6x+3y=12 The matrix of the equation is a 22 matrix and therefore the equation can written in the augmented form as shown Any matrix is said to be in reduced row echelon form if it satisfies the following conditions: Any row of all zeros appear at the bottom If a row does not consist of all zeros then its first non zero entity is called a leading 1 and it is one If any two successive rows the leading 1 of the lower is further to the right of the leading 1 of the highest row If a column contains a leading 1 then all the other entries are zero However, each of the unknown variables can be solved using the graphical calculator in the matrix calculation area But the inverse of the matrix does not exist since in fact the determinant is zero (-43) (6-2) =0. Therefore there is no solution to the equation above. Graphically, the lines to the equations are parallel and never intersect therefore there is no solution to the problems From the graph, it is clear that the two lines are parallel and are never to intersect and therefore this means that the equations do not have a solution. In some cases, the system will have many solutions in the algebraic sense, however geometrically, the lines will collide and look like there is only one line and therefore all the points along that line are indeed solutions to the equation. For example the equation -2x+y=8 and -4x+2y= 16 matrix to the equation is One of the solutions is x=0 and y=-4 and many other solutions. The matrix does not have an inverse as shown by the graphical calculator in the matrix calculation area since the determinant of the matrix is zero. This means that the solution of the equation is not one hence the equation has many solutions. In general, given any systems of linear equation with two unknown solution, the two lines will graphically intersect at one point. The point of intersection is the solution to the systems of linear equations. Also, the lines can be parallel to each other meaning that the system does not have any solution and finally the lines can collide and the solution to the system is not unique that is there are many solutions to the system of linear equations. The system with many solutions can be presented in the graph below using the equation given above. Note that: -4x+2y=16 2 {(-2x+y) =8} and therefore one equation is a multiple of the other which basically means that the equation is more or less the same. Remember that a system such as 2x-3y=7 and x+7y=11 can be written in the form = It is normally represented as Am=b where A= , b= and m= If b=0, then the system Am= b has m=0 as a trivial solution but if A-1 exists then m=0 is the only solution to the system. There is also the possibility of graphing those equations with piecewise defined functions. For instance there are functions such as |x| = In our example, we will graph one of the most commonly used piecewise defined functions. f (x) = In this case, the entire function is considered as one function in who’s the domain is the real numbers. APPLICATION OF LINEAR FUNCTIONS There are times when the solutions for the complicated functions cannot easily be obtained. This lead to the use of the linear equations that is the equations that are to only one degree to be used in the approximation of the complicated functions since they gives some little bit of accuracy and the linear functions are easy to work with. This is basically known as linearization. This is mostly used in conjunction with the differential functions. Here if the function, normally denoted as f is differentiable at x=a, then in this case the approximation function denoted as L(x) = f (a) +f’ (a) (x-a) is now what is known as the linearization of the function f at a. For example we will try to find the linearization of the function f(x) = at x=0 The above graph is now the linearization of the function at x=o and x=3. We now know that f’(x) =1/2(1+x)-1/2. We will therefore see that f (0) =1 and also that f’ (0) =1/2. Therefore this concludes that the linearization will therefore be: L(x) = 1+1/2(x-0) = 1+x/2 These are some of the general applications of the linear equations/functions and many other that are dealt with at the higher level of the course work Theorem in the solution of system of equation If A is invertible, then there is only one solution to Am=b which is the unique solution Proof Let w be any solution such that wA-1b That is Aw=b but since A is invertible A-1 exists that is Aw=b Therefore multiplying both sides to the left with A-1 we have A-1Aw = A-1b I w = A-1b w = A-1b which is a contradiction and therefore A-1b is the only solution to the system Next there are equations that are to the second degree and the linear equations are used to find the gradients at particular points through the use of the tangent line and the normal lines to the equations that are being considered in this case. These equations mostly include the parabolas and other quadratic equations among others. Though our main interest is not the parabolas and such equations, the linear equations are particularly used here to serve various mathematical purposes. The parabola for instance is a set that usually consist of all points in a plane that is equal in distance sense from a point that is given and also a given line. Mostly the parabolas will have a graph of equations of the form y= ax2+bx+c. We will for instance plot a graph of y=x2. In this case the graph is a simple graph that is curved in u shape. But mathematically, we may want to find the gradient of the graph at particular points. We will therefore use the current technology for graph plotting to plot both the graphs as shown; The tangent line is used to find the gradient of the curve at that particular point. The graph shown is a curve with the equation y=x2. The axis of the parabola is the y-axis that is it is the axis of symmetry.lso, the vertex of the parabola as seen from the graph is at the origin. The parabola is seen to open upwards when the values of the constant are positive and increasing and open up downwards if the values of the constant are negatively increasing. Now if we consider the 33 matrix system, there are 3 variables that are involved, we will concentrate on the variables x, y and z. For instance, let as consider the matrix below 2x+y-z=11: Here the constants are 2 that precede the variable x, 1 that precedes the variable y and -1 that precedes the variable z. The system can be solved using the usual matrix method, the elimination method or the use of a three dimension matrices. When we deal with the matrix method the graphing calculator here is used to find the inverse of the matrix. The system of equations can be basically being written as; M X=A Using the graphing calculator to find the inverse of the matrix will yield X= M-1A X= And therefore the solution to the equation becomes X = In this type of system, there are also the possibility of obtaining a unique solution, the; possibility of many solution and the option of no solution. The possibility of many solutions or no solution is as a result of having a singular matrix that is a matrix with a zero determinant. For example looking at the following system of solution x+2y+3z=4 4x+6y+8z=10 2x+y=-1 The determinant of the matrix is zero and therefore there can be the case where there are many solutions and graphically in a three dimension graph, the lines are common or the case where there are is no solution and the lines are parallel to each other. We can also use our technology to create a family of linear equations that are usually similar in characteristics. On the same set of the axis, we usually display the equations and evaluate them mathematically. The family of curves will include several lines which usually have a wide range of equations. This can be represented as; The family of linear equations above all have different gradients fro negative to zero to positive. In a 33 matrix, the solution can also be obtained geometrically and algebraically. This is so because the graph of the equations can be plotted in the graph especially with the current technologies and calculators and it can be done algebraically through various methods which include the elimination methods and the current modern methods. Therefore the 33 matrix can be dealt with in the same manner as the 22 matrix. There are many ways of proving mathematical theorems and terms such as the contradiction method, proving by induction and many others in the above matrix we have used the contradiction method. In the 33 matrix, we are going to basically see how to prove by induction the conjectures that are involved. Conjectures in mathematics are some of the propositions and they are easily not disapproved since they are believed to be true For instance the sequence an= n (n-1) is the sequence such that a1= 1*0 a2= 2*1 : : an= n*(n-1) When we sum up the sequence of the first n numbers we obtain a series and therefore sn= a1+a2++an. Therefore; S1=0 S2=2 Sn= Sn-1+an Next, the difference between successive sums is made until the constant term in the series is obtained so long as the nth term n0, This will result to a polynomial of the third degree in order for the constant terms to be obtained in that the equation for the series will therefore be Sn = Ax3+Bx2+Cx+D where A,B,C and D are constants that are and xâ ± ¤ Now replacing x in the equation with the natural numbers 1,2,3,4,5.. we get the A+B+C+D=0 8A+4B+2C+D=2 27A+9B+3C+D=8 64A+16B+4C+D=20 This is a four equation system since there are four different unknown variables. Therefore we will use the graphing calculator to find the solutions to the unknown variables 1 1 1 1: 0 8 4 2 1: 2 27 9 3 1: 8 64 16 4 1: 20 We will find that A=1/3, B=0, C=-1/3 and D=0 Sn=1/3 x3-1/3 x The graph for the equation is therefore as follows Proving the equation by the induction method therefore will be (for n0); For n=1: =1/3*13 1/3* 1 =0 For n=3: =1/3*33 -1/3*3 =8 For n=5: =1/3*53 1/3*5 =40 Therefore we can assume that the equation is true for all values of natural number that is n0, We therefore assume that the equation is true for n=k Therefore for n=n+1, Sk+1= Sk+ (k+1)*k =1/3k3 -1/3k +k2 +k =1/3(k+1)3 -1/3(k+1) Since the expression is true for n=k+1 is true, the equation is true by induction. In the mathematical sense, a function of a polynomial p is normally written as p(x) =anxn+an-1+.+a1 x+ a0. In this case the n are non negative integers and the a’s are the coefficients of the polynomial itself. Usually all the polynomials have the domain of (-, ). In this case we can say that the linear functions themselves are polynomials of degree one while the quadratic functions are polynomials of the second degree and so on. As with our 33 matrix, the polynomial involved was a cubic function of the third degree. For instance the polynomial y=84-143-92+11x-1 Linear Algebraic Equations A teacher is looking for the best option in purchasing school supplies for a classroom. Company A is offering a discount for every dollar amount spent; Company B is offering a higher discount for every dollar spent above $20. Determine which company will offer a better price based upon the dollar amount the teacher spends on the school. In this scenario, it mostly involves the computation of the purchase of the school inventories at a cheaper price. Inventory generally is the stock of raw materials, work in progress units, finished goods, consumables and spare parts being held in store at a given time period. There are different kinds and groups of inventories that includes; movement inventories which are inventories on transit from one point to another, safety stock or the buffer stock which are the inventories that must always be maintained in the store so as to meet the unexpected demand, cyclical inventory, anticipatory inventory and the decoupling inventory. However, in this scenario we will focus on how to purchase inventory while at the same time using the mathematical knowledge to reduce inventory related costs. Here, the teacher is looking for the best option in purchasing school supplies for a classroom and therefore the best option is the option with reduced costs. The customer also has to ensure that tho ugh the goods are purchased at a cheaper price, they are of the best and desirable quality. The supplier of the goods should also be in a position to supply goods to the customer when they are needed both in the short term period and in the long term period and in time as to the date of the specifications. Therefore the customer has to look deep into these needs before making the decision on where to make their orders. The hypothetical customer, the teacher in this case has to make an informed decision based on questions such as how many units to order at that time, how often should the school supplies be made, how many orders are to be placed in that particular year and this is mainly done to reduce cost. In this scenario therefore we will focus mainly on two cost options that are for Company A which is offering a discount for every dollar spent. This is where a constant rate of discount for every dollar spent. This is where a constant rate of discount is offered irrespective of the number of units purchased and it is commonly known as a single discount. We will assume that the unit price of each product that is to be purchased is $5 and that the discount for every dollar spent is 5 percent (5%). In this scenario, the discount offered is for every dollar that will be spent and no conditions as to the amount and the limit of expenditure. The second option is for Company B which is offering a higher discount for every dollar spent above $20. This therefore guarantees the teacher discount after spending $20 in the purchase of school supplies which will be a much higher discount than that of the purchase of the goods worth $20. In this case the teacher will get a discount similar to that of the single discount up to the expenditure of $20 and later the discount is increased accordingly. We will therefore assume that the unit purchase price is $5 and that every dollar spent to the expenditure of $20 is 5%, with more dollars spent, the discount increases to 7.5%. In our scenario, the demand should be known in advance with certainty and will remain constant within the relevant range. The algebraic equations to represent the cost of each option are: Company A: Offers a discount for every dollar spent Here there are many cost related to the purchase of the school supplies which includes the purchase cost, the ordering cost, the holding cost and in some cases the shortage cost. However we will only focus on the purchase cost and ignore all the other related cost inorder to come up with the required linear equations. Let’s assume further that the teacher purchases x units of the school supplies Unit price = $5 Discount =5% Let the total cost=y Total purchase cost =$5 * (100% -5%) *x Total purchase cost = $5 *0.95 *x = 4.75x Company B: Offers a higher discount of 7.55 for every dollar spent above $20. The teacher here should know that for the first $20 spent, the discount is 5% and above the expenditure of $20, the discount increases to 7.5%. This will probably lure customers desire to purchase more but we will try evaluating the two equations. The equation for company B is therefore as follows Let’s assume that the teacher purchases x units of the school supplies The discount for the first 4 purchases of the school supplies =5% i.e ($20/5) Unit price =$5 Let the total cost=y The discount for the purchase of more than 4 =7.5% Total purchase cost = ($5*4*0.95) + ($5*0.925)(x-4) Total purchase cost = $19+4.625x –$18.5 =$0.5+4.625x The equation of company A is used since the company only offers a single discount for all the purchases that are made by the customer. Therefore the discount will be distributed equally. For Company B, there is a constant in the algebraic equation since in the purchase of the first $20 items, the discount is 5% that is it is constant and since the customer has to purchase more than this to gain the discount of 7.5% then that part of the equation will vary with the extra units purchased. The solution to the equation can be done through several ways such as elimination method, substitution method or the graphical method; the equations are normally written as y= 4.75x y= 0.5+4.625x We are going to solve the equation using the substitution method. Since in the first equation y=4.75x, we will substitute this to the second equation.4.75x= 0.5+4.625x and we therefore solve the equation mathematically. In this equation, the solution is x=4. Where the total costs will be the same. But with the increase in the purchase of the school supplies, the total cost will be higher for the purchase related to Company A than that of the purchase from Company B. Also, with the decrease in the purchase of the school supplies, the cost purchases from Company A are less than that of Company B. The graph of the two scenarios can be represented as follows. Though the cost associated in the two scenarios are close, there is a negligible difference as a result of the discounts. It is therefore correct to conclude that if the teacher is in need of less than four units of purchase, it is advisable to purchase the school supplies from Company A, if the teacher wants to purchase 4 units of item, this can be done from any company and if it is more than 4 company, it is cost effective to purchase from Company B. At the lower levels of purchase, presence of discount appears attractive for the company with a single discount for any unit of purchase made. That is, there is no constant related to the purchase of goods in the algebraic equation.. However beyond a certain level of purchase, taking up a discount results into a net increase in total cost in the company using a single discount method. In the other case for company with an increased discount after purchase of some discount, the purchase of many items become cost effective in this company. Therefore the teacher should ensure that he or she takes up the least quantity required to qualify for the highest discount in order for the total cost to be less than that of company A. Since the principle of discount states that only the least quantity required to qualify for the discount should be purchased.