Category Archives: Maths homework

Mortgage

Question

You definitely reported on some of the considerations people make when picking mortgage. With the longer term, you may be able to afford a larger mortgage, which can be helpful in a state with high property costs like NJ.

For the example you shared, what was the initial cost of the home and down payment?  For each option, how much would you pay in total for the home, assuming you lived in it through the life of the loan?

Sample paper

Mortgage

The initial cost of the home is $300,000 and the down payment is $60,000. I opted to examine the 30-year fixed rate loan at 4.125% interest rate and the 15-year fixed rate loan at 3.5 % interest rate. The annual percentage rate (APR) gives the total cost of the mortgage. The total costs include the interest rate, broker fees, closing costs, and among other costs. The APR is thus a better figure in estimating the total costs of buying a home through a mortgage. For each option, I would pay different amounts depending on the APR rate. Under option for a 30-yeat fixed rate at an APR of 4.226%, I would pay $418,680 in total. For the 30-year fixed rate FHA at an APR of 4.778%, I would pay $477,000 in total. For the 15-year fixed rate, I would pay $308,880. Since I am looking for a 30-year mortgage option, the best option would be the 30-year fixed rate option at a rate of 4.226%. The total amount is $418,680.

Discussion Questions

  1. Define break-even point.
  2. Assume that the value of one discount point is $3,000. If by paying the $3,000 will lower the overall cost of the mortgage by $112.5, how long does it take to make up the cost of the point?
  • Would it be advisable to pay the extra $3,000 if you plan to stay in the house for less than 2 years? Why or why not?

 

The History of e IN Mathematics

The History of e IN Mathematics

Euler’s number, e, emerged during the renaissance period, which was characterized by emergence of new knowledge in Europe. During the 16th century, an important discovery was made; Michael Stifel, a German mathematician, expanded the knowledge on geometric progression and the respective exponents (Maor, 2009). Stifel, nonetheless, only examine integral values relating to exponents. In the 17th century, John Napier developed the idea of logarithms. His basic idea was that any positive number could be written in exponential form, and it would be possible to find the multiplication or division of the number by simply adding or subtracting the powers. In writing numbers to exponential form, a certain fixed number or the base is used. This was critical in making numerical computations easy. Napier’s logarithms decrease with each increase in numbers. This is due to the fact that in his system, Log 107= 0. In 1624 Henry Briggs, a professor of Geometry, published a revised system of logarithms (Maor, 2009). In this system, the logarithm of 1 was taken as 0, and the logarithm of 10 taken as powers of 10.

Napier seems to be the first person to have made note of the letter e in mathematics. In his work, Descriptio, he noted that the logarithm of 10 to the base e was 2.302585 (Maor, 2009). However, the critical question remains as to how the number e came into existence. The number e is derived from the mathematics of finance. In the calculation of compound interest, mathematicians were interested in knowing how long a sum of money would take to grow to a certain amount. In order to find the number e, the principal is compounded for 1 year, at the rate of 100%. As “n” (the number of times the principal is compounded) increases, the figure obtained increase but in less significant digits (Maor, 2009). In 1737, Euler proved that e is an irrational number. In his work Introductio, Euler emphasized on the role of the numbers e and ex in numerical analysis. In 1781, Euler estimated the number e to 16 places. Another important mathematician is Jacob Bernoulli, who attempted to establish the value of the limit of the equation n as n approaches infinity.

There are a number of interesting facts about e. The first interesting thing is that no matter how large the value of n (as it tends to infinity), the figure obtained will always be approximately 2.718281828. The number e is the limit of the equation n as n approaches infinity (Maor, 2009). Although it appears to have a repeating block of decimals, 1828, the number is irrational. It is non-repeating and non-terminating. The number e is not or cannot be a solution to any polynomial equation having integer coefficients.

The number e is applied in probability theory. Bernoulli successfully applied the number e in probability theory. This is known as the “misplaced envelope” problem. The number e is applied in measuring the decay rate of an exponential function (Pickover, 2009). This is critical in biological growth models that help in estimating the population of organisms that show a proportional rate of change to their number. The number e has numerous applications in calculus. For instance, it can help in finding the derivatives and in evaluation of limits. Its applications in differential equations are immense. Euler’s number is applicable to complex numbers (Pickover, 2009). The number helps in extending the use of ex even when the values of x fall under complex numbers. The number e is useful in giving different representations. For instance, it can be represented as an infinite product, a limit of a sequence, an infinite series, or as a stochastic representation.

Reference

Maor, E. (2009). E: The story of a number. Princeton, N.J: Princeton University Press.

Pickover, C. A. (2009). The math book: from Pythagoras to the 57th dimension, 250 milestones    in the history of mathematics. Sterling Publishing Company.

Auto Insurance Costs

Question

Read the article titled, “Auto Insurance Costs: Where Does Your State Rank?” located at http://www.cbsnews.com/8301-505145_162-40542496/auto-insurance-costs-where-does-your-state-rank/. Be prepared to discuss.

“Data Description” (Note: Please respond to one [1] of the following two [2] bulleted items)
From the e-Activity, the table shows Average Insurance Costs by State. Select two (2) states that are of interest to you. Next, speculate on three (3) possible reasons why the states you have chosen would have a difference in average insurance costs.
From the e-Activity, select ten (10) states and calculate the mean and standard deviation for average insurance costs.  Next, calculate the mean and standard deviation for average insurance costs for all 51 states. Compare and contrast the means and standard deviations for the ten (10) states you selected and all fifty one (51) states.

Auto Insurance Costs

The state and the federal law state that all car owners should insure their vehicle accident related risks.  Insurance companies or government agencies largely focus on making an arrangement with the car owners to provide a guarantee of compensation for particular losses as defined in the customer policy bought by a client.  However, the client has an obligation of paying monthly premiums as calculated by the auto insurance companies (Kofman and Nini, 2012). However, traffic tickets, accidents, and even credit score highly determine a clients auto insurance rate. This study will attempt to identify the reasons for different average insurance costs in different states.

According to an article on Auto Insurance Costs: Where Does Your State Rank published back in 2015, different states have different average insurance costs owing to factors such as the state laws and the judicial system in a state.  According to this article, Michigan had the highest average insurance costs with an average rate of $2,541 followed by Louisiana with an average of $2,453 and Vermont had the least average cost of $995 (Cbsnews.com, 2017).  One of the major causes of this variation is the presence of uninsured drivers in the states which are a violation of the state law.   According to the article, there is a direct correlation between the number of uninsured drivers and average costs.  For example, Michigan and Louisiana have 17 % and 12 % uninsured drivers respectively.  On the same note, the level of competition among auto insurance is another major factor that contributes to this variation.  The article goes on to state that Vermont has a high competition among auto insurance companies than both Michigan and Louisiana reducing this average cost (Cbsnews.com, 2017). Finally, the level and degree of protection provided by the state are a major contributor to this cost.  For example, Michigan guarantees unlimited personal injury protection, unlike other states

References

Cbsnews.com. (2017). Auto Insurance Costs: Where Does Your State Rank?. [online] Available             at:http://www.cbsnews.com/news/auto-insurance-costs-where-does-your-state-rank/         [Accessed 24 Jul. 2017].

Kofman, P. and Nini, G. (2012). Do Insurance Companies Possess an Informational Monopoly?               Empirical Evidence From Auto Insurance. Journal of Risk and Insurance, 80(4),   pp.1001-1026.

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Data Representation

Data Representation

Question

Visit one (1) of the following newspaper’s Websites: Wall Street Journal, Washington Post, USA Today, or New York Times. Select a line graph, pie chart, or bar chart on the Website. Be prepared to discuss.Graphical Representations of Data” Please respond to the following:
From the e-Activity, discuss the data that the graph is used to visually represent. Speculate on why the editor chose the graph, as opposed to merely reporting the data.

Sample paper

Data Representation

The wall street journal has been one of the outstanding America business focused journal for decades. The journal is often published six days a week by Dow Jones & Company, where it focuses on business and economic news as well as luxury and lifestyle magazines.  Research shows that it is the largest newspaper in the US by circulation.  Notably, the newspaper is popular for its use of graphs to represent financial and economic data of the country and other countries in other regions.  For example,  the journal uses the Dow Jones Industrial Average to show the stock market ratio and some of the indices in the market (Bace, 2016). This study will attempt to identify and analyze one of the graphs used in the journal.

The graph above represents the daily market index of the Dow Jones Industrial Average which closes at 1650. A stock market index shows the measurement of a part of the stock market.  The index is often calculated from the values of the chosen stocks, particularly the weighted average of the stocks.  The blue line in the graph represents the market index value while the red line shows the simple moving average often regarded as the change in the market index.

Different analysts and authors use graphs for differevnt reasons to drive their message across to their readers and audiences.  However, the primary reason that drove the editor to use this graph is to simplify it to make it easy for the readers and audiences to process and understand. It is always difficult for people to understand and manage large amounts of data and figures, but a graph makes it easy to simply and analyze these large amounts of data at a glance (Powell & Hopkins, 2015). Therefore, the editor wanted to make it easy for the readers to understand the market performance of the stocks.

References

Bace, E. (2016). STOCK RETURNS AND LEVERAGE: ANALYSIS OF THE DOW JONES INDUSTRIAL AVERAGE, 2000-2015. Proceedings of the 23rd International Academic Conference, Venice. doi:10.20472/iac.2016.023.017

Powell, J., & Hopkins, M. (2015). Graph databases and how to use them. A Librarian’s Guide to Graphs, Data and the Semantic Web, 197-207. doi:10.1016/b978-1-84334-753-8.00022-1