Introductiong To The Concepts Of Organisational Behaviour Business Essay - Free Essay Example

Sample details Pages: 13 Words: 3971 Downloads: 8 Date added: 2017/06/26 Category Management Essay Type Narrative essay Did you like this example? As we are in the world of business organisation, there are many of circumstances that we are facing together may be it if human resource or environmental or to the growing competition in the business. Organisational Behaviours has critical vital role in the management.OB always deals with the application or knowledge of how individual or group of individual act in the management. Apart from this it also deals with human factors working in the management in the modern way of work to active the goal of the organisation.OB deals with the various factors of the management like methods of work, viewpoint and analysis to work done. Don’t waste time! Our writers will create an original "Introductiong To The Concepts Of Organisational Behaviour Business Essay" essay for you Create order Various authors have different views on Organisational Behaviour.Like, FRED LUTHANS defines OB as, The understanding, prediction and management of human behaviour in organisations .Organisation behaviour is the study of the structure, functioning and performance of organisation and the behaviour of group and individual within them, by DEREK PUGH. Porters perspectives on Behaviours in Organisations, understands one individuals behaviour is a challenging problem in and of itself . Organisation Behaviour is defined properly that it is the aspect that is prevalent to the organisations it is also a fact that all the organisation are taking effective methods for a smooth functioning in the organisation by taking various effective measures in terms of employees and trying to overcome their behaviour into an organisational behaviour which would beneficial to the organisation as well as the employees. Organisations are complex systems. In this century, progress depends heavily on the u nderstanding managing effective organisations.Nowdays, all organisation have realized that all the human behaviour in an organisation can be understood by studying applying the behavioural science frame work other discipline by exploring the various facts about human behaviour converting them to organisational behaviour in order to achieve to goal or aims of the organisation successfully. OB study involves with the different criteria, such as:- 1. Attitude 2. Interpersonal relationship 3. Performance. 4. Productivity. 5. Job satisfaction. 6. Commitment of employees. 7. Levels of organizational commitment and industrial relations. It also interprets with the organization in terms of people and group of people in the organization. The main purpose of OB is to achieve: A} Human objectives. B} Organizational objective. C} social objective. Organisational behaviour is so important for any organisation that gives opportunity to learn about human behaviour can be able to apply them in actual dealing with employees in an organisation its knowledge also helps to organize the employees nature behaviour. The following study refers the various aspects of Organisational Behaviour how to run a group or a team effectively by applying the theories of OB practically in various situation how to overcome the problems and solve them to earn the motive or result of the organisation. BRIEF INTRODUCTION TO THE CASE ON DALLAS BASED SOUTHWEST AIRLINES:- DALLAS SOUTHWEST AIRLINES are in filed of airlines from 35 years of profitability.It is also the largest airlines in United States ,by number of passengers carried.In last year 70 million passengers have flied through South West Airlines to near about 60 destinations around the United States.Passengers have found that SWA has very good range of fair for transportation,than the automobile transportation.For the sake of passengers SWA is willing to make in-flight meals,baggage transfers,and other frills for economically amenables wings. Due to its reinnovation in wings and other factors SWA remains a maverick in the industry.Southwest Air was founded in 1966 when a group of investors,including Rollin King,M.Lamar Muse,and Herbert D.Kelleher,pooled $560,000 to form the Air Southwest Company.It was incorparated in 1967,serving three cities within Texas:Dallas,Houston and San Antonio. By the end of 1971,Southwest owned four aircraft,which has offered hourly flights between Dallas and Houston,and some of to San Antonio and Houston.SWA has lots of controversy in it.In,1972 Dallas and Fort Worth and their Regional Airport Board filed suit to force the airlines to move from Love field to newly constructed Dallas-fort Worth regional airport,by which charging higher landing fees and rent there,which will help offset the cost of the expensive project.All the air lines have moved to the new airport.But South west air lines have refused to move there,due to not existence. In 1974, competitors of Southwest, began to move to Dallas-Fort Worth Airport, by giving cheaper, convenient airport. This was the new issue which SWA was facing. SWA has left its presence at Love field and its newly innovated facilities at Houstons Hobby Airport.This was due to strong commuter service and the border operations. South West Airlines is known for its innovation in operations and HR practices under the long standing CEO, Herbert D. Kelleher.He has brought many things to the management. Which has lead for the managements improvement.He has proved that management is the cornerstone of the organisation effectiveness,which integrates the activities that permeates every facet of the operations of the organistaion. An Overview Of The Case Study:- The SOUTH WEST AIRLINES is second leading position in airlines,which is famous for its Innovation in operations and its Human resources pratices.They have proved the statement in various sectors of management, and had integrated the this activities in the operations. Their philosophy followed by the management gave more importance to integrity and community as the backbone of theirs. They also have respected to their employees and there team. There business was not profit oriented but to satisfy their employess and customers.And they have proved it very clearly. The CEO, HERBERT D. KELLEHER has developed a new culture aimed at fun and employee satisfaction.He has also developed many oriented programs for employees like training and motivational programs.In work , environment was created very friendly and fun loving. Employees were treated as friends. By which their position was marked in the airline industry,due to there practices of HR and other. Since, going through many controve rsies they have overcome the problems and have solved them. The case study will show all the circumstances of the SWA and how they have over come them from various sectors of Management. The SWA Influence On Behaviour On:- INDIVIDUAL:- Organisation is madeof group of individuals. Wether he is acting as an isolated body or in group of people for the existence of the operations of the management. In SWA individual were in play full environment.They were not treated as employee. It was noticed that the needs of the individual and the demand of the organisation were in certain manner, due to the practices played by the management. It was the duty of the management to integrate individual and the organisation to provide a happy environment which permits satisfaction among the individuals. Which leds to the organisation goal very proudly. THE GROUP:- The group consist of various bodies in the organisation which works and performance together. It consist of group of one member or group of members.Group consist of two bodies Formal group and Informal group. Informal group sees the needs of the group or individual.In SWA group behaviours pressure can brought major influence on the organisation behaviour and performance. THE ORGANISTAION:- In SWA individual and groups use to together interact with both formal and informal.SWA has created a structure which establish relationship between this individual and group of individual.This structure has helped them to bring out their organisation goal. In this structure organisations processes are directed, planned and controlled. FACTORS THAT DETRMINE THE PERFORMANCE OF THE ORGANISATION 1) LEADERSHIP 2) MOTIVATION 3) CONTRIBUTION 4) ROLE OF INDIVIDUAL AND GROUPS LEADERSHIP Going through the various controversy and the aspects which brought the SWA to the decline process. They have outcome this problem by their various skills. THE CEO, of SOUTH WEST AIRLINES HERBERT D. KELLEHER has brought a new trend in the leadership style in those days where SWA was facing a lot of problem in the organisation. He has brought renovation in the business by way of thinking. HERBERT has brought innovation in the leadership style. He has traced the problems which were forcing SWA. By his leadership style he has found that management can solve these problems by their structure. He has built a structure in the organisation. Where in all processs use to Directed, Controlled and Planned. This structure consists of group of individuals and individual. The structure was in such a way that focuses the impact of organisation structure and design, and the pattern of management, on the behaviour of people within the organisation. South West Airlines has not the Human Resour ce Department; instead of they have People and Leadership Development Department. All others have tried to copy their style but they failed. The CEO of SWA said that they can copy us but cannot copy our people. The SWA was targeting HUMAN RESOURCES to overcome the problems. Employees were working in a flexible condition. By which they were not having in restriction and any stress. Employees were trained by various skills to improve their working ability. The CEO has arranged devised numerous employee oriented organisational practices, training and motivational programme. He has maintain all workers chain in such a way that a proper feedback will come from every part of the organisation. The CEO of SWA, has maintain a fun loving environment in the organisation. Due to his supreme leadership he has made faith in the workers, this was one of the most important factors of the organisation. But their were some obstacle that SWA was facing it. Workers were dependent thoroughly on the o rganisation. Since they were having all rights , due to flexible environment. In SWA if any chain broke down all work use to stop. This has brought critical debt in the organisation. But due to HERBERTS real innovation practices slowly this problem was solved,and once again he has proved that HUMAN RESOURCE factors can influence the organisations goal. All the air lines industries were afraid by the KELLEHERS innovative HR practises. Since leadership is inter related to motivation, interpersonal behaviour and the communication. A good leader knows all about the process and empowerment. Leadership is dynamic process. A leader has to be interpersonal not only with subordinate but also with the organisation. Good leadership helps to improve the team and teamwork. A leader is responsible for the organisation goal . And in South West Airlines leaders were knowing by this factor, which has helped them to achieve certain goals. IN SWA, leaders were allowed to take there personal decisio n at right time to get people to accept solutions. Like other organisation there was no cold war in the management and the leaders or workers. SWA has maintained coordination between workers and the management. This created a vital flow in the organisation. By there leadership style relation was maintain between people,managers have maintain with low level of involvement. SWA leaders has shown sympathy with people and given attention towards workers social need. This has helped for Team building. This case discusses how a leader can influence the HR practices of the whole organisation with way of his leadership style. KELLEHERS has also proved a proper decision at proper time can influence the Human resource. Due to his Ethics he has made a moral in leadership. He was likely know about the workers three basic social needs food, clothing, shelter. He behave not as a boss with them but as friend. Behaviour was the reason for workers understanding, management has brought a morali ty in the brains of the workers. The management has trust and value towards their employees. The management has created an ethical climate in the organisation. Proper direction was given. Employees were rewarded by their performance. A discussion among the employees(technical) and the managers and a feedback. Management was socially responsible for the welfare of the individuals. Spirituality awareness has action towards the organisation. Diversity was created amongst the employees by which new things can be created in the organisation. MOTIVATION:- According to Stephen P.Robbins, motivation is the willingness to exert high levels of effort towards organisation goals, conditioned by the efforts ability some individual need. In South West Air lines, when the firm was in going through various controversies workers have left their hopes. This created clashes between the organisation and the workers. Soon the organisation has brought a Motivation factor in the firm. Since workers have loosed their hopes, once again they were motivated to work in the safe chain. SWA has social contact at the work place with their workers. Workers were motivated by acknowledging their social needs and making them feel important among others. This what SWA has done to influence their employees. Workers were not having any uniform at workplace, they use dressed in shorts, polo shirts and sneakers. By which they were comfortable at work. They were encouraged to tell jokes and have fun loving job. By this reason SWA is known by fun-in-the-sky airline s. The CEO has always believed that the company is stronger if there is love in the atmosphere, rather than by the fear. This created a culture of responsibility, respect and accountability which comes from freedom. This made them most admired companies in the rest of the world. The only agenda they were following was employees are the first customers. SWA believed that if you treat your employees right they will treat the customers right. And if you treat your customers right then they will come again back and the shareholders are also happy. SWA is to send cards to all the employees working in the organisation on their birthday or anniversary of their employment. They use to have contests among the workers for fun loving. SWA has 7 elements in Motivation factor Strongest Set of Values. Values were given to the employees and their work. Values determine the behaviour of the employee at organisation. All workers behave according to these values. This created discipline and enforced across the company. IN SWA values are mandatory. Employees Come First SWA has provided a stable employment with equal opportunity. Employees are encouraged for improving the effectiveness. They have provided the same concern, respect, and caring attitudes within the organisation. They have given importance to every single job. All employees were given equal values and respect, by which created a strong feeling and mutual belief and trust. There is no doubt in the motivation of SWA. Rewards and Recognition Employees were known by the recognition and rewards that the organization provides them for their performance. They know what they will get instead of that for their exceptional work. SWA has given importance to all over the filed of organisation i.e. techniques, approach and device to recognize performance. SWA recognized all employees work directly to their personal accomplishments and rewarding and celebrating them in many different ways. This will create a continuous flow of chain in the organization consequently it will bring a happy and joyful environment. In SWA motivation is of secondary importance. Mission The mission of South West Airlines is dedication to the highest of Customer Service delivered with a sense of warmth, friendliness, individual pride, and Company spirit., HERBERT D. KELLEHER In SWA, employees motivation does not reside in its mission itself. Since SWA has framed their mission which helps employees to work sustainably. Employees know their mission very clearly and they know how to make possible. Hiring SWA is very careful about their selection of employees, there are concern about their employees work and their talent. SWA has a rigorous hiring procedure. South west Airlines has a significant decision about the which worker could be permanent, this procedure is very lengthy to get there right candidates. SWA has a very strong corporate culture with high performance in the oranisation. Distri buted Leadership In SWA, they have very strong leadership at the top and bottom level or throughout the organization. Leaders are responsible for the bottom level workers work concern. They are granted that everyone in the management is able to lead in significant ways. They are required to play leadership at different levels when needed. They are trained in such a way that there are skilled leads to do their duty. The management has appointed front line leaders for investigating the great deal. In SWA leaders are placed by their natural talent of motivating their subordinates. So SWA hire candidates who are not afraid at work or thrilled with the possibility of leading. Due to the changing environment in the business, SWA has adopted changes from time to time for the superior performance over the long run. Performance Management This is most important motivation element of SWA which conducts the performance of the management. SWA has three dimension of performance: E mployee well being Customer satisfaction Shareholder gain SWA has a clear vision on the needs of employees, they are the first customers of the organization. This has created hopes in minds of employees which created increase in performance. This in order has reached to a clear view of goal. The performance of worker is critical, so SWA has a element of tracking and rewarding of individual. Employees has clear vision, goals, roles and responsibilities of their work and management profit. 3) CONTRIBUTION: SWA has gone through various controversies they have fought with very courage and have made many ways. In these last consecutive years they have faced many controversies. May be of land or others government regulation, they have always managed it very properly. In this management has managed the workers by their skilled performance. And workers have also given their whole contribution to the organization. The CEO of South West Airlines has managed workers by their natural skills, right individual at right place. This helped the workers to work in fun loving environment. They dont have any regret towards the management. Leaders have encouraged them to do work with more performance and with skill. In decision making workers have contributed a lot, leader have motivated them in various field of their work. ROLE OF INDIVIDUAL AND GROUPS: In SWA, individual and groups have there vital role in the management. Due to the fun loving environment they are prompt at work, by the organisation goal. In SWA all jobs are allocated by the management, right person at right place. SWA has very high deal of importance on the hiring process to know people with their skills. They survey that how candidate can affect the overall operation. SWA has typical search for appropriate skills and experience. All decision is made by workers and management committees. Employees are allowed to make decision in the field. The management has made a good relation with the workers and leaders. Leaders have shared knowledge, goals, and mutual respect. Employees of SWA embrace their relation with one another, which allow them to work in coordination and effectively. Fun loving, creativity, individuality and empowerment is the back bone of the management. SWA has some high performance relationships practices:- Investing in frontline leadership Leading with credibility and caring Hiring and training for relational competence Using conflicts to build relationships Bridging the work/family divide Creating boundary spanners Keeping jobs flexible at the boundaries Measuring performance broadly Establishing partnership with unions Building relationship with suppliers All this practices made workers and management to work with courage and prominent to their goal and managements profit. OBSTACLES TO EFFECTIVE ORGANISATIONAL PERFORMANCE:- Communication Gap:- In south west airline, since employees were treated as the first customers then also there was a gap between management and the workers. Even though SWA has tried to overcome it but it was not totally solved. Workers have mutual understanding between their leaders and them, but not to the management. The CEO of SWA was hero between the employees then also some controversies were there. Cultural clash:- In SWA new employees were taking time to settle in the management. And older employees were not satisfied with the individual hired. Both were trained or educated in same environment, even though they their was clash between their culture. Newly acquisition has created this clashes. Soon SWA was trying to overcome. This can be overcome only when these workers are allowed to work in same routine. Or allowed to work in different shift. So this might over come the problem of cultural clash. Internal Threats:- SWA has faced many a internal threats. But the wast was when Kelleher loss his power and strategic direction, Due to the new entrants to the industry. In 2001, CEO has become chairman of the SWA, workers were not happy with that losing of their leader. And become the highly collaborative corporate culture. External threats:- When others copied the strategies of SWA the management was unhappy with this reason. Although SWA has a strong customer base and is diverse, then also were having challenges. Competition was large and competitors have reduced their fares. This affected SWA to make fares reduced. This can be solved by fare promotions in the market. Customers can made happy with new services in SWA. THE CHANGING NATURE OF MODERN WORK ORGANISATION:- The changing nature depends upon the people of the organisation. Environment causes mental and attitudes to the people in the organisation. Since the businesses are in global, the economy changes, increasing the business competitiveness in the market and more customers should be attract. There is no constant or flexibility in the market different changes have to made in the business. Newly technology and rapid spread of the market in the public and private sector and various impact of social and political have increased the attraction towards the business ethics. This has also increased the attention on the ethical behaviour of the decision taking of the leader and the staff. Due to which many of them have started to choose this ethics. The changing nature of the organisation at work placed pressure on the awareness and importance of the moderns ethics. Global competition has made employment guarantee demand management philosophy based on trust and teamwork. Workers are more res ponsible at work. Employees get a continuous learning and personal development. The technology has brought new innovation in the business, ways of work are structured again and again, and jobs are redesign has change the organisation. By this management has become diplomatic in decision taking. New ideas of motivation and development have engaged. Exchange of value has evolved. Workers are encouraged and permitted to achieve their goals as well as the managements goal. Since modern way of work has changed the nature of organisation, it has brought the emergence in the management. Many organisation have changed their organisation with the changing nature of organisation. Management have to take decision at the last, either the nature of organisation has changed the management is the cornerstone of organisational effectiveness. Conclusion:- South West Airlines has his own way of business in the market of airlines. Instead of many of controversies they have managed to work with more new decisions of there leader. The main thing in SWA is there leadership style. This has made them the rulers of airline industry in US. Management of SWA has made the statement true that management is the cornerstone of the effective organisation in various fields of the organisation whether it may be of leadership or motivating their workers or groups. Due to this they have managed to build loyalty in the customers mind. Since SWAs first customers are employees, employees love their job. They have a strategy to do always better, which has succeeded them in airline industry. Many of them have tried to copy them but have failed to do so. They can copy there management but cannot copy there employees. This made there workers unique in way of their efficiency of doing there work and prompt at work.

Electronic Medical Records Are Transforming The World Of...

In July of 2004, Tommy Thompson, U.S. Department of Health and Human Services Secretary, stated, [A]merica needs to move much faster to adopt information technology in our health care system...electronic health information will provide a quantum leap in patient power, doctor power, and effective health care. We can t wait any longer...† (hhs.gov). In the 12 years since Thompson’s statement, healthcare has been transformed by the beneficial adoption of electronic medical records (EMR) creating savings for healthcare organizations and reducing costs for practitioners and informaticists, as well as other professionals involved in the process. Electronic medical records have the potential to transform and develop healthcare in a multiplicity of ways over the coming years. According to Net Health, there are three different ways that specialized EMRs are transforming the world of healthcare today. The first way is the fact that the more accessible data exists in the healthcare industry, the easier it is to make a diagnosis (NetHealth). These diagnoses are not being used in just one setting, however, but in a conglomeration of medical care settings. These include clinics and treatments which are improving the quality of life for breast cancer patients, diabetes patients, chlamydia patients and even colorectal cancer patients (Kern). Electronic records are creating a huge and accessible database to reach information more quickly and more efficiently. As physicians and practices,Show MoreRelatedHealthcare Communication As A Result Of Mobile Health Technology921 Words   |  4 PagesHow healthcare communication as a result of M obile Health Technology Characterized as the â€Å"Digital Age,† modern technologies are flooding every aspect of our lives and completely transforming healthcare communication. 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Prices and Research

Questions: Visit the Australian Stock Exchange website, www.asx.com.au and from Prices and research drop-down menu, select Company information. Type in the ASX code CCL (Coca-Cola Amatil Limited), and find out details about the company. Your task will be to get the opening prices of a CCL share for every quarter from January 2001 to December 2015. If you are working with the monthly prices, read the values in the beginning of every Quarter (January, April, July, October) for every year from 2001 to 2015. It is part of the assignment task to test your ability to find the information from an appropriate website. If you are unable to do so, you may read the values from the chart provided below obtained from Etrade Australia. Obviously, reading from the chart will not be accurate and you may expect around 60 percent marks with such inaccuracy. After you have recorded the share prices, answer the following questions: a. List all the values in a table and then construct a stem-and-leaf display for the data. b. Construct a relative frequency histogram for these data with equal class widths, the first class being $4 to less than $6. c. Briefly describe what the histogram and the stem-and-leaf display tell you about the data. What effects would there be if the class width is doubled, which means the first class will be $4 to less than $8? d. What proportion of stock prices were above $10? Answers: a. The List of the share prices at the beginning of every quarter are given below: All units are in Dollars Year 1-Jan 1-Apr 1-Jul 1-Oct 2001 4.91 4.65 4.85 5.02 2002 6.37 6 6.3 5.28 2003 5.73 5.69 5.86 5.89 2004 6.35 6.89 7.02 7.36 2005 7.85 8.28 7.81 7.62 2006 7.68 7.28 6.85 6.96 2007 7.83 9.51 9.16 10.2 2008 9.36 8.41 7.72 8.05 2009 9.15 9.14 9.33 10.64 2010 10.93 11.22 11.47 11.14 2011 11.24 11.94 11.3 12.3 2012 11.55 12.45 13.93 13.45 2013 13.84 15.14 12.85 12.9 2014 11.69 9.25 9.31 9.15 2015 9.7 10.27 9.28 9.11 Next we display the data in stem and Leaf plot : 3 4 689 9 5 026788 16 6 0333889 25 7 023667888 28 8 024 (12) 9 111112233357 20 10 2269 16 11 12234569 8 12 3489 4 13 489 1 14 1 15 1 b. The histogram with the specified class intervals is given below: Class Interval Frequency 4-6 9 6-8 16 8-10 15 10-12 12 12-14 7 14-16 1 Total 60 The Histogram is shown below: c. After considering the stem and leaf diagram and the histogram plotted we can conclude that the distribution is nearly symmetric in nature. If we double class widths, then the diagram would not be good enough to judge the data and hence we could not give proper interpretations of the data. d. The proportion of stock Prices above 10 is 20/60=0.3333 (upto 4 decimal places).

Nothing Goes to Waste free essay sample

All my life, I identified myself as a soccer player. I grew up in a town that prided itself on an outstanding soccer program. Everyone around me had the same interest in it: my brother, sisters, parents, and friends. My life revolved around this sport, and I was successful at it. I played on competitive teams, but I also participated in sports like basketball, gymnastics, skiing, and swimming. As I got older, I focused more exclusively on soccer. Soccer was my all-year sport, that is, until I got cut. Going into my sophomore year, being cut from the team was a huge disappointment, especially since I was now separated from my teammates and friends. I had two choices: (1) to feel sorry for myself, or (2) to make something positive out of a negative experience. I convinced myself to find another sport in which to channel my energy. As a freshman, I was on the varsity swim team, but never took it as seriously as soccer. We will write a custom essay sample on Nothing Goes to Waste or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page So, during soccer season I could always be found in the pool. I participated in a club swim team with the thought that practice would help make me a better swimmer. When high school swim season came around, I made the team and tremendously improved my skills. I advanced two levels and pushed myself at practices and meets more than I ever had in the past. As the season continued, I qualified for Sectionals which meant that I was competing at an even higher level. A few days prior to Sectionals, I became very ill and had to forfeit my spot to another team member who was healthy and could compete. It was tough to relinquish my spot, especially because I worked so hard for this, but I knew that the following year I would re-engage my effort to make my goal of Sectionals again. By the end of my sophomore swim season, my coach gave me the award for â€Å"Rising Star.† One person each year receives this award for their resilience, attitude, sportsmanship, and being a positive asset to the team. Though I was not able compete at Sectionals because of my sudden illness, this award showed that my hard work and diligence did not go to waste. When junior year arrived, I was training with my club swim team, still preparing for winter swim team. Given the setback I endured the previous year, I was determined to make sure that I could not only make Sectionals but qualify for States. My sights were set very high. I practiced and prepared myself mentally to swim my best, and I did, qualifying for Sectionals and States. Over time, I realized that my failure to make the soccer team turned into successful opportunities in swimming. I gained another close knit community, broadened my interests, met new friends, and improved my abilities. By reaching out of my comfort zone, I was able to realize my potential in swimming, but more particularly, as a person. Swimming in Sectional and States meets was a symbol of my perseverance over the last two years. No longer did I define myself as a soccer player, but as a swimmer. I could play soccer here and there, but my idea of success had changed. Success is not making a certain team or playing a certain sport; it is setting a personal goal and striving to reach it. Success doesn’t mean that I must focus all of my energy in one area. I must have diversification in my life, because being almost exclusive in one area limits my potential and opportunities, which will again cause me to fail.

Music Literature I Baroque Flashcard

Music Literature I Baroque

Complete Guide to Integers on ACT Math (Advanced)

Complete Guide to Integers on ACT Math (Advanced) SAT / ACT Prep Online Guides and Tips Integers, integers, integers (oh, my)! You've already read up on your basic ACT integers and now you're hankering to tackle the heavy hitters of the integer world. Want to know how to (quickly) find a list of prime numbers? Want to know how to manipulate and solve exponent problems? Root problems? Well look no further! This will be your complete guide to advanced ACT integers, including prime numbers, exponents, absolute values, consecutive numbers, and roots- what they mean, as well as how to solve the more difficult integer questions that may show up on the ACT. Typical Integer Questions on the ACT First thing's first- there is, unfortunately, no â€Å"typical† integer question on the ACT. Integers cover such a wide variety of topics that the questions will be numerous and varied. And as such, there can be no clear template for a standard integer question. However, this guide will walk you through several real ACT math examples on each integer topic in order to show you some of the many different kinds of integer questions the ACT may throw at you. As a rule of thumb, you can tell when an ACT question requires you to use your integer techniques and skills when: #1: The question specifically mentions integers (or consecutive integers) It could be a word problem or even a geometry problem, but you will know that your answer must be in whole numbers (integers) when the question asks for one or more integers. (We will go through the process of solving this question later in the guide) #2: The question involves prime numbers A prime number is a specific kind of integer, which we will discuss later in the guide. For now, know that any mention of prime numbers means it is an integer question. A prime number a is squared and then added to a different prime number, b. Which of the following could be the final result? An even number An odd number A positive number I only II only III only I and III only I, II, and III (We'll go through the process of solving this question later in the guide) #3: The question involves multiplying or dividing bases and exponents Exponents will always be a number that is positioned higher than the main (base) number: $4^3$, $(y^5)^2$ You may be asked to find the values of exponents or find the new expression once you have multiplied or divided terms with exponents. (We will go through the process of solving this question later in the guide) #4: The question uses perfect squares or asks you to reduce a root value A root question will always involve the root sign: √ $√36$, $^3√8$ The ACT may ask you to reduce a root, or to find the square root of a perfect square (a number that is equal to an integer squared). You may also need to multiply two or more roots together. We will go through these definitions as well as how all of these processes are done in the section on roots. (We will go through the process of solving this question later in the guide) (Note: A root question with perfect squares may involve fractions. For more information on this concept, look to our guide on fractions and ratios.) #5: The question involves an absolute value equation (with integers) Anything that is an absolute value will be bracketed with absolute value signs which look like this: | | For example: $|-43|$ or $|z + 4|$ (We will go through how to solve this problem later in the guide) Note: there are generally two different kinds of absolute value problems on the ACT- equations and inequalities. About a quarter of the absolute value questions you come across will involve the use of inequalities (represented by or ). If you are unfamiliar with inequalities, check out our guide to ACT inequalities (coming soon!). The majority of absolute value questions on the ACT will involve a written equation, either using integers or variables. These should be fairly straightforward to solve once you learn the ins and outs of absolute values (and keep track of your negative signs!), all of which we will cover below. We will, however, only be covering written absolute value equations in this guide. Absolute value questions with inequalities are covered in our guide to ACT inequalities. We will go through all of these questions and topics throughout this guide in the order of greatest prevalence on the ACT. We promise that your path to advanced integers will not take you a decade or more to get through (looking at you, Odysseus). Exponents Exponent questions will appear on every single ACT, and you'll likely see an exponent question at least twice per test. Whether you're being asked to multiply exponents, divide them, or take one exponent to another, you'll need to know your exponent rules and definitions. An exponent indicates how many times a number (called a â€Å"base†) must be multiplied by itself. So $3^2$ is the same thing as saying 3*3. And $3^4$ is the same thing as saying 3*3*3*3. Here, 3 is the base and 2 and 4 are the exponents. You may also have a base to a negative exponent. This is the same thing as saying: 1 divided by the base to the positive exponent. For example, 4-3 becomes $1/{4^3}$ = $1/64$ But how do you multiply or divide bases and exponents? Never fear! Below are the main exponent rules that will be helpful for you to know for the ACT. Exponent Formulas: Multiplying Numbers with Exponents: $x^a * x^b = x^[a + b]$ (Note: the bases must be the same for this rule to apply) Why is this true? Think about it using real numbers. If you have $3^2 * 3^4$, you have: (3*3)*(3*3*3*3) If you count them, this give you 3 multiplied by itself 6 times, or $3^6$. So $3^2 * 3^4$ = $3^[2 + 4]$ = $3^6$. $x^a*y^a=(xy)^a$ (Note: the exponents must be the same for this rule to apply) Why is this true? Think about it using real numbers. If you have $3^5*2^5$, you have: (3*3*3*3*3)*(2*2*2*2*2) = (3*2)*(3*2)*(3*2)*(3*2)*(3*2) So you have $(3*2)^5$, or $6^5$ If $3^x*4^y=12^x$, what is y in terms of x? ${1/2}x$ x 2x x+2 4x We can see here that the base of the final answer is 12 and $3 *4= 12$. We can also see that the final result, $12^x$, is taken to one of the original exponent values in the equation (x). This means that the exponents must be equal, as only then can you multiply the bases and keep the exponent intact. So our final answer is B, $y = x$ If you were uncertain about your answer, then plug in your own numbers for the variables. Let's say that $x = 2$ $32 * 4y = 122$ $9 * 4y = 144$ $4y = 16$ $y = 2$ Since we said that $x = 2$ and we discovered that $y = 2$, then $x = y$. So again, our answer is B, y = x Dividing Exponents: ${x^a}/{x^b} = x^[a - b]$ (Note: the bases must be the same for this rule to apply) Why is this true? Think about it using real numbers. ${3^6}/{3^4}$ can also be written as: ${(3 * 3 * 3 * 3 * 3 * 3)}/{(3 * 3 * 3 * 3)}$ If you cancel out your bottom 3s, you’re left with (3 * 3), or $3^2$ So ${3^6}/{3^4}$ = $3^[6 - 4]$ = $3^2$ The above $(x * 10^y)$ is called "scientific notation" and is a method of writing either very large numbers or very small ones. You don't need to understand how it works in order to solve this problem, however. Just think of these as any other bases with exponents. We have a certain number of hydrogen molecules and the dimensions of a box. We are looking for the number of molecules per one cubic centimeter, which means we must divide our hydrogen molecules by our volume. So: $${8*10^12}/{4*10^4}$$ Take each component separately. $8/4=2$, so we know our answer is either G or H. Now to complete it, we would say: $10^12/10^4=10^[12−4]=10^8$ Now put the pieces together: $2x10^8$ So our full and final answer is H, there are $2x10^8$ hydrogen molecules per cubic centimeter in the box. Taking Exponents to Exponents: $(x^a)^b=x^[a*b]$ Why is this true? Think about it using real numbers. $(3^2)^4$ can also be written as: (3*3)*(3*3)*(3*3)*(3*3) If you count them, 3 is being multiplied by itself 8 times. So $(3^2)^4$=$3^[2*4]$=$3^8$ $(x^y)3=x^9$, what is the value of y? 2 3 6 10 12 Because exponents taken to exponents are multiplied together, our problem would look like: $y*3=9$ $y=3$ So our final answer is B, 3. Distributing Exponents: $(x/y)^a = x^a/y^a$ Why is this true? Think about it using real numbers. $(3/4)^3$ can be written as $(3/4)(3/4)(3/4)=9/64$ You could also say $3^3/4^3= 9/64$ $(xy)^z=x^z*y^z$ If you are taking a modified base to the power of an exponent, you must distribute that exponent across both the modifier and the base. $(2x)^3$=$2^3*x^3$ In this case, we are distributing our outer exponent across both pieces of the inner term. So: $3^3=27$ And we can see that this is an exponent taken to an exponent problem, so we must multiply our exponents together. $x^[3*3]=x^9$ This means our final answer is E, $27x^9$ And if you're uncertain whether you have found the right answer, you can always test it out using real numbers. Instead of using a variable, x, let us replace it with 2. $(3x^3)^3$ $(3*2^3)^3$ $(3*8)^3$ $24^3$ 13,824 Now test which answer matches 13,824. We'll save ourselves some time by testing E first. $27x^9$ $27*2^9$ $27*512$ 13,824 We have found the same answer, so we know for certain that E must be correct. (Note: when distributing exponents, you may do so with multiplication or division- exponents do not distribute over addition or subtraction. $(x+y)^a$ is not $x^a+y^a$, for example) Special Exponents: It is common for the ACT to ask you what happens when you have an exponent of 0: $x^0=1$ where x is any number except 0 (Why any number but 0? Well 0 to any power other than 0 equals 0, because $0^x=0$. And any other number to the power of 0 = 1. This makes $0^0$ undefined, as it could be both 0 and 1 according to these guidelines.) Solving an Exponent Question: Always remember that you can test out exponent rules with real numbers in the same way that we did in our examples above. If you are presented with $(x^3)^2$ and don’t know whether you are supposed to add or multiply your exponents, replace your x with a real number! $(2^3)^2=(8)^2=64$ Now check if you are supposed to add or multiply your exponents. $2^[2+3]=2^5=32$ $2^[3*2]=2^6=64$ So you know you’re supposed to multiply when exponents are taken to another exponent. This also works if you are given something enormous, like $(x^19)^3$. You don’t have to test it out with $2^19$! Just use smaller numbers like we did above to figure out the rules of exponents. Then, apply your newfound knowledge to the larger problem. And exponents are down for the count. Instant KO! Roots Root questions are fairly common on the ACT, and they go hand-in-hand with exponents. Why are roots related to exponents? Well, technically, roots are fractional exponents. You are likely most familiar with square roots, however, so you may have never heard a root expressed in terms of exponents before. A square root asks the question: "What number needs to be multiplied by itself one time in order to equal the number under the root sign?" So $√81=9$ because 9 must be multiplied by itself one time to equal 81. In other words, $9^2=81$ Another way to write $√{81}$ is to say $^2√{81}$. The 2 at the top of the root sign indicates how many numbers (two numbers, both the same) are being multiplied together to become 81. (Special note: you do not need the 2 on the root sign to indicate that the root is a square root. But you DO need the indicator for anything that is NOT a square root, like cube roots, etc.) This means that $^3√27=3$ because three numbers, all of which are the same (3*3*3), are multiplied together to equal 27. Or $3^3=27$. Fractional Exponents If you have a number to a fractional exponent, it is just another way of asking you for a root. So $4^{1/2}= √4$ To turn a fractional exponent into a root, the denominator becomes the value to which you take the root. But what if you have a number other than 1 in the numerator? $4^{2/3}$=$^3√{4^2}$ The denominator becomes the value to which you take the root, and the numerator becomes the exponent to which you take the number under the root sign. Distributing Roots $√xy=√x*√y$ Just like with exponents, roots can be separated out. So $√30$ = $√2*√15$, $√3*√10$, or $√5*√6$ $√x*2√13=2√39$. What is the value of x? 1 3 9 13 26 We know that we must multiply the numbers under the root signs when root expressions are multiplied together. So: $x*13=39$ $x=3$ This means that our final answer is B, $x=3$ to get our final expression $2√39$ $√x*√y=√xy$ Because they can be separated, roots can also come together. So $√5*√6$ = $√30$ Reducing Roots It is common to encounter a problem with a mixed root, where you have an integer multiplied by a root (for example, $4√3$). Here, $4√3$ is reduced to its simplest form because the number under the root sign, 3, is prime (and therefore has no perfect squares). But let's say you had something like $3√18$ instead. Now, $3√18$ is NOT as reduced as it can be. In order to reduce it, we must find out if there are any perfect squares that factor into 18. If there are, then we can take them out from under the root sign. (Note: if there is more than one perfect square that can factor into your number under the root sign, use the largest one.) 18 has several factor pairs. These are: $1*18$ $2*9$ $3*6$ Well, 9 is a perfect square because $3*3=9$. That means that $√9=3$. This means that we can take 9 out from under the root sign. Why? Because we know that $√{xy}=√x*√y$. So $√{18}=√2*√9$. And $√9=3$. So 9 can come out from under the root sign and be replaced by 3 instead. $√2$ is as reduced as we can make it, since it is a prime number. We are left with $3√2$ as the most reduced form of $√18$ (Note: you can test to see if this is true on most calculators. $√18=4.2426$ and $3*√2=3*1.4142=4.2426$. The two expressions are identical.) We are still not done, however. We wanted to originally change $3√18$ to its most reduced form. So far we have found the most reduced expression of $√18$, so now we must multiply them together. $3√18=3*3√2$ $9√2$ So our final answer is $9√2$, this is the most reduced form of $3√{18}$. You've rooted out your answers, you've gotten to the root of the problem, you've touched up those roots.... Absolute Values Absolute values are quite common on the ACT. You should expect to see at least one question on absolute values per test. An absolute value is a representation of distance along a number line, forward or backwards. This means that an absolute value equation will always have two solutions. It also means that whatever is in the absolute value sign will be positive, as it represents distance along a number line and there is no such thing as a negative distance. An equation $|x+4|=12$, has two solutions: $x=8$ $x=−16$ Why -16? Well $−16+4=−12$ and, because it is an absolute value (and therefore a distance), the final answer becomes positive. So $|−12|=12$ When you are presented with an absolute value, instead of doing the math in your head to find the negative and positive solution, you can instead rewrite the equation into two different equations. When presented with the above equation $|x+4|=12$, take away the absolute value sign and transform it into two equations- one with a positive solution and one with a negative solution. So $|x+4|=12$ becomes: $x+4=12$ AND $x+4=−12$ Solve for x $x=8$ and $x=−16$ Now let's look at our absolute value problem from earlier: As you can see, this absolute value problem is fairly straightforward. Its only potential pitfalls are its parentheses and negatives, so we need to be sure to be careful with them. Solve the problem inside the absolute value sign first and then use the absolute value signs to make our final answer positive. (By process of elimination, we can already get rid of answer choices A and B, as we know that an absolute value cannot be negative.) $|7(−3)+2(4)|$ $|−21+8|$ $|−13|$ We have solved our problem. But we know that −13 is inside an absolute value sign, which means it must be positive. So our final answer is C, 13. Absolutely fabulous absolute values are absolutely solvable. I promise this absolutely. Consecutive Numbers Questions about consecutive numbers may or may not show up on your ACT. If they appear, it will be for a maximum of one question. Regardless, they are still an important concept for you to understand. Consecutive numbers are numbers that go continuously along the number line with a set distance between each number. So an example of positive, consecutive numbers would be: 5, 6, 7, 8, 9 An example of negative, consecutive numbers would be: -9, -8, -7, -6, -5 (Notice how the negative integers go from greatest to least- if you remember the basic guide to ACT integers, this is because of how they lie on the number line in relation to 0) You can write unknown consecutive numbers out algebraically by assigning the first in the series a variable, x, and then continuing the sequence of adding 1 to each additional number. The sum of five positive, consecutive integers is 5. What is the first of these integers? 21 22 23 24 25 If x is our first, unknown, integer in the sequence, so you can write all four numbers as: $x+(x+1)+(x+2)+(x+3)+(x+4)=5$ $5x+10=5$ $5x=105$ $x=21$ So x is our first number in the sequence and $x=21$: This means our final answer is A, the first number in our sequence is 21. (Note: always pay attention to what number they want you to find! If they had asked for the median number in the sequence, you would have had to continue the problem with $x=21$, $x+2=$median, $23=$median.) You may also be asked to find consecutive even or consecutive odd integers. This is the same as consecutive integers, only they are going up every other number instead of every number. This means there is a difference of two units between each number in the sequence instead of 1. An example of positive, consecutive even integers: 10, 12, 14, 16, 18 An example of positive, consecutive odd integers: 17, 19, 21, 23, 25 Both consecutive even or consecutive odd integers can be written out in sequence as: $x,x+2,x+4,x+6$, etc. No matter if the beginning number is even or odd, the numbers in the sequence will always be two units apart. What is the largest number in the sequence of four positive, consecutive odd integers whose sum is 160? 37 39 41 43 45 $x+(x+2)+(x+4)+(x+6)=160$ $4x+12=160$ $4x=148$ $x=37$ So the first number in the sequence is 37. This means the full sequence is: 37, 39, 41, 43 Our final answer is D, the largest number in the sequence is 43 (x+6). When consecutive numbers make all the difference. Remainders Questions involving remainders are rare on the ACT, but they still show up often enough that you should be aware of them. A remainder is the amount left over when two numbers do not divide evenly. If you divide 18 by 6, you will not have any remainder (your remainder will be zero). But if you divide 19 by 6, you will have a remainder of 1, because there is 1 left over. You can think of the division as $19/6 = 3{1/6}$. That extra 1 is left over. Most of you probably haven’t worked with integer remainders since elementary school, as most higher level math classes and questions use decimals to express the remaining amount after a division (for the above example, $19/6 = 3$ remainder 1 or 3.167). But you may still come across the occasional remainder question on the ACT. How many integers between 10 and 40, inclusive, can be divided by 3 with a remainder of zero? 9 10 12 15 18 Now, we know that when a division problem results in a remainder of zero, that means the numbers divide evenly. $9/3 =3$ remainder 0, for example. So we are looking for all the numbers between 10 and 40 that are evenly divisible by 3. There are two ways we can do this- by listing the numbers out by hand or by taking the difference of 40 and 10 and dividing that difference by 3. That quotient (answer to a division problem) rounded to the nearest integer will be the number of integers divisible by 3. Let's try the first technique first and list out all the numbers divisible by 3 between 10 and 40, inclusive. The first integer after 10 to be evenly divisible by 3 is 12. After that, we can just add 3 to every number until we either hit 40 or go beyond 40. 12, 15, 18, 21, 24, 27, 30, 33, 36, 39 If we count all the numbers more than 10 and less than 40 in our list, we wind up with 10 integers that can be divided by 3 with a remainder of zero. This means our final answer is B, 10. Alternatively, we could use our second technique. $40−10=30$ $30/3$ $=10$ Again, our answer is B, 10. (Note: if the difference of the two numbers had NOT be divisible by 3, we would have taken the nearest rounded integer. For example, if we had been asked to find all the numbers between 10 and 50 that were evenly divisible by 3, we would have said: $50−10=40$ $40/3$ =13.333 $13.333$, rounded = 13 So our final answer would have been 13. And you can always test this by hand if you do not feel confident with your answer.) Prime Numbers Prime numbers are relatively rare on the ACT, but that is not to say that they never show up at all. So be sure to understand what they are and how to find them. A prime number is a number that is only divisible by two numbers- itself and 1. For example, 13 is a prime number because $1*13$ is its only factor. (13 is not evenly divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, , or 12). 12 is NOT a prime number, because its factors are 1, 2, 3, 4, 6, and 12. It has more factors than just itself and 1. 1 is NOT a prime number, because its only factor is 1. The only even prime number is 2. Standardized tests love to include the fact that 2 is a prime number as a way to subtly trick students who go too quickly through the test. If you assume that all prime numbers must be odd, then you may get a question on primes wrong. A prime number x is squared and then added to a different prime number, y. Which of the following could be the final result? An even number An odd number A positive number I only II only III only I and III only I, II, and III Now, this question relies on your knowledge of both number relationships and primes. You know that any number squared (the number times itself) will be an even number if the original number was even, and an odd number if the original number was odd. Why? Because an even * an even = an even, and an odd * an odd = an odd ($2*2=4$ $3*3=9$). Next, we are adding that square to another prime number. You’ll also remember that an even number + an odd number is odd, an odd number + an odd number is even, and an even number + an even number is even. Knowing that 2 is a prime number, let’s replace x with 2. $2^2=4$. Now if y is a different prime number (as stipulated in the question), it must be odd, because the only even prime number is 2. So let’s say $y=5$. $4+5=$. So the end result is odd. This means II is correct. But what if both x and y were odd prime numbers? So let’s say that $x=3$ and $y=5$. So $3^2=9$ and 9+5=14$. So the end result is even. This means I is correct. Now, for option number III, our results show that it is possible to get a positive number result, since both our results were positive. This means the final answer is E, I, II, and III If you forgot that 2 was a prime number, you would have picked D, I and III only, because there would have been no possible way to get an odd number. Remembering that 2 is a prime number is the key to solving this question. Another prime number question you may see on the ACT will ask you to identify how many prime numbers fall in a certain range of numbers. How many prime numbers are between 20 and 40, inclusive? Three Four Five Six Seven This might seem intimidating or time-consuming, but I promise you do NOT need to memorize a list of prime numbers. First, eliminate all even numbers from the list, as you know the only even prime number is 2. Next, eliminate all numbers that end in 5. Any number that ends is 5 or 0 is divisible by 5. Now your list looks like this: 21, 23, 27, 29, 31, 33, 37, 39 This is much easier to work with, but we need to narrow it down further. (You could start using your calculator here, or you can do this by hand.) A way to see if a number is divisible by 3 is to add the digits together. If that number is 3 or divisible by 3, then the final result is divisible by 3. For example, the number 23 is NOT divisible by 3 because $2+3=5$, which is not divisible by 3. However 21 is divisible by 3 because $2+1=3$, which is divisible by 3. So we can now eliminate 21 $(2+1=3)$, 27 $(2+7=9)$, 33 $(3+3=6)$, and 39 $(3+9=12)$ from the list. We are left with 23, 29, 31, 37. Now, to make sure you try every necessary potential factor, take the square root of the number you are trying to determine is prime. Any integer equal to or less than a number's square root could be a potential factor, but you do not have to try any numbers higher. Why? Well let’s take 36 as an example. Its factors are: 1, 2, 3, 4, 6, 9, 12, 18, and 36. But now look at the factor pairings. 1 36 2 18 3 12 4 9 6 6 (9 4) (12 3) (18 2) (36 1) After you get past 6, the numbers repeat. If you test out 4, you will know that 9 goes evenly into your larger number- no need to actually test 9 just to get 4 again! So all numbers less than or equal to a potential prime’s square root are the only potential factors you need to test. And, since we are dealing with potential primes, we only need to test odd integers equal to or less than the square root. Why? Because all multiples of even numbers will be even, and 2 is the only even prime number. Going back to our list, we have 23, 29, 31, 37. Well the closest square root to 23 and 29 is 5. We already know that neither 2 nor 3 nor 5 factor evenly into 23 or 29. You’re done. Both 23 and 29 must be prime. (Why didn't we test 4? Because all multiples of 4 are even, as an even * an even = an even.) As for 31 and 37, the closest square root of these is 6. But because 6 is even, we don't need to test it. So we need only to test odd numbers less than six. And we already know that neither 2 nor 3 nor 5 factor evenly into 31 or 37. So we are done. We have found all of our prime numbers. So your final answer is B, there are four prime numbers (23, 29, 31, 37) between 20 and 40. A different kind of Prime. Steps to Solving an ACT Integer Question Because ACT integer questions are so numerous and varied, there is no set way to approach them that is entirely separate from approaching other kinds of ACT math questions. But there are a few techniques that will help you approach your ACT integer questions (and by extension, most questions on ACT math). #1. Make sure the question requires an integer. If the question does NOT specify that you are looking for an integer, then any number- including decimals and fractions- are fair game. Always read the question carefully to make sure you are on the right track. #2. Use real numbers if you forget your integer rules. Forget whether positive, even consecutive integers should be written as x+(x+1) or x+(x+2)? Test it out with real numbers! 6, 8, 10 are consecutive even integers. If x=6, 8=x+2, and 10=x+4. This works for most all of your integer rules. Forget your exponent rules? Plug in real numbers! Forget whether an even * an even makes an even or an odd? Plug in real numbers! #3. Keep your work organized. Like with most ACT math questions, integer questions can seem more complex than they are, or will be presented to you in strange ways. Keep your work well organized and keep track of your values to make sure your answer is exactly what the question is asking for. Got your list in order? Than let's get cracking! Test Your Knowledge 1. 2. 3. 4. 5. Answers: C, D, B, F, H Answer Explanations: 1. We are tasked here with finding the smallest integer greater than $√58$. There are two ways to approach this- using a calculator or using our knowledge of perfect squares. Each will take about the same amount of time, so it's a matter of preference (and calculator ability). If you plug $√58$ into your calculator, you'll wind up with 7.615. This means that 8 is the smallest integer greater than this (because 7.616 is not an integer). Thus your final answer is C, 8. Alternatively, you could use your knowledge of perfect squares. $7^2=49$ and $8^2=64$ $√58$ is between these and larger than $√49$, so your closest integer larger than $√58$ would be 8. Again, our answer is C, 8. 2. Here, we must find possible values for a and b such that $|a+b|=|a−b|$. It'll be fastest for us to look to the answers in order to test which ones are true. (For more information on how to plug in answers, check out our article on plugging in answers) Answer choice A says this equation is "always" true, but we can see this is incorrect by plugging in some values for a and b. If $a=2$ and $b=4$, then $|a+b|=6$ and $|a−b|=|−2|=2$ 6≠ 2, so answer choice A is wrong. We can also see that answer choice B is wrong. Why? Because when a and b are equal, $|a−b|$ will equal 0, but $|a+b|$ will not. If $a=2$ and $b=2$ then $|a+b|=4$ and $|a−b|=0$ $4≠ 0$ Now let's look at answer choice C. It's true that when $a=0$ and $b=0$ that $|a+b|=|a−b|$ because $0=0$. But is this the only time that the equation works? We're not sure yet, so let's not eliminate this answer for now. So now let's try D. If $a=0$, but b=any other integer, does the equation work? Let's say that $b=2$, so $|a+b|=|0+2|=2$ and $|a−b|=|0−2|=|−2|=2$ $2=2$ We can also see that the same would work when $b=0$ $a=2$ and $b=0$, so $|a+b|=|2+0|=2$ and $|a−b|=|2−0|=2$ $2=2$ So our final answer is D, the equation is true when either $a=0$, $b=0$, or both a and b equal 0. 3. We are told that we have two, unknown, consecutive integers. And the smaller integer plus triple the larger integer equals 79. So let's find our two integers by writing the proper equation. If we call our smaller integer x, then our larger integer will be $x+1$. So: $x+3(x+1)=79$ $x+3x+3=79$ $4x=76$ $x=19$ Because we isolated the x, and the x stood in place of our smaller integer, this means our smaller integer is 19. Our larger integer must therefore be 20. (We can even test this by plugging these answers back into the original problem: $19+3(20)=19+60=79$) This means our final answer is B, 19 and 20. 4. We are being asked to find the smallest value of a number from several options. All of these options rely on our knowledge of roots, so let's examine them. Option F is $√x$. This will be the square root of x (in other words, a number*itself=x.) Option G says $√2x$. Well this will always be more than $√x$. Why? Because, the greater the number under the root sign, the greater the square root. Think of it in terms of real numbers. $√9=3$ and $√16=4$. The larger the number under the root sign, the larger the square root. This means that G will be larger than F, so we can cross G off the list. Similarly, we can cross off H. Why? Because $√x*x$ will be even bigger than $2x$ and will thus have a larger number under the root sign and a larger square root than $√x$. Option J will also be larger than option F because $√x$ will always be less than $√x$*another number larger than 1 (and the question specifically said that x1.) Remember it using real numbers. $√16$ (answer=4) will be less than $16√16$ (answer=64). And finally, K will be more than $√x$ as well. Why? Because K is the square of x (in other words, $x*x=x^2$) and the square of a number will always be larger than that number's square root. This means that our final answer is F, $√x$ is the least of all these terms. 5. Here, we are multiplying bases and exponents. We have ($2x^4y$) and we want to multiply it by ($3x^5y^8$). So let's multiply them piece by piece. First, multiply your integers. $2*3=6$ Next, multiply your x bases and their exponents. We know that we must add the exponents when multiplying two of the same base together. $x^4*x^5=x^[4+5]=x^9$ Next, multiply your y bases and their exponents. $y*y^8=y^[1+8]=y^9$ (Why is this $y^9$? Because y without an exponent is the same thing as saying $y^1$, so we needed to add that single exponent to the 8 from $y^8$.) Put the pieces together and you have: $6x^9y^9$ So our final answer is H, 6x9y9 Now celebrate because you rocked those integers! The Take-Aways Integers and integer questions can be tricky for some students, as they often involve concepts not tested in high school level math classes (have you had reason to use remainders much outside of elementary school?). But most integer questions are much simpler than they appear. If you know your way around exponents and you remember your definitions- integers, consecutive integers, absolute values, etc.- you’ll be able to solve most any ACT integer question that comes your way. What’s Next? You've taken on integers, both basic and advanced, and emerged victorious. Now that you’ve mastered these foundational topics of the ACT math, make sure you’ve got a solid grasp of all the math topics covered by the ACT math section, so that you can take on the ACT with confidence. Find yourself running out of time on ACT math? Check out our article on how to keep from running out of time on the ACT math section before it's pencil's down. Feeling overwhelmed? Start by figuring out your ideal score and work to improve little by little from there. Already have pretty good scores and looking to get a perfect 36? Check out our article on how to get a perfect ACT math score written by a 36 ACT-scorer. Want to improve your ACT score by 4 points? 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Book Review on the Gospel according to Mattew Essay

Book Review on the Gospel according to Mattew - Essay Example Just like Paul who had a Hebrew name Shaoul, Matthew had Levi as his original name. Matthew, son of Alpheus (Mark 2:14) came from Galilee. However, he is thought to have lived in Antioch, Syria. He collected taxes for Herod Antipas, a Jewish leader, ruler of Galilee and Peraea between 4 BCE and 39 CE.1 Just after following Jesus, Matthew held a feast in his house where tax collectors and sinners recline at the table with the Christ and his disciples. This is where Jesus drew protest from the Pharisees. The author, however, had a great influence on the development of Christianity. It based most of his writing on Mark’s gospel.2 The Gospel of Matthew is the first book in the 27 books of the New Testament. In the New Testament, the following books are Matthew, Mark, Luke, John, The Acts, Romans, 1 Corinthians, 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 Thessalonians, 2 Thessalonians, 1 Timothy, 2 Timothy, Titus, Philemon, Hebrews, James, 1 Peter, 2 Peter, 1 John, 2 John , 3 John, Jude, Revelation. Aside from one of the first twelve disciples of Jesus (Matthew 9:1; 10:1-4) and an eye-witness, Matthew records more of Jesus teaching concerning Gods heavenly kingdom than the other writers, for example the entire Sermon on the Mount. Matthew, compared to the first few men (Peter, Andrew, James, and John) whom Jesus chose (who were fishermen), was skilled that he handled accounts of figures being a tax collector and obviously equipped with the pen. More so, he chose to leave his ludicrous post to be with Jesus and become an evangelist. It is to be recalled when a certain man holding many riches wants to be one of Christ’s disciples. Jesus told this man â€Å"If you want to be perfect, go sell your belongings and give to the poor and you will have treasure in heaven, and come be my follower.† However, the man refused to let go of his belongings and went away grieved. (Matthew: 19:21-22). Quoted on Papias, one