LSP 120 Final Review

Percentages and Rates

1.  The following table list the 2009 average circulation of ten magazines with the largest circulation in the US.

a. How many times larger is the circulation of AARP The Magazine greater than the circulation of National Geographic?

Answer. The circulation of AARP The Magazine is 5.22 times larger than the circulation of National Geographic. The calculation is 24,554,819/4,708,307 » 5.22.

b. By how many percent is the circulation of Reader's Digest larger than the circulation of Family Circle?

Answer. The circulation of Reader's Digest is 107% larger than the circulation of Family Circle. The calculation is (8,158,652 - 3,932,510)/3,932,510  » 1.07 = 107%.

c. The magazine that had the largest circulation in the United Kingdom was the entertainment magazine Skymag.  Its 2009 circulation was 7,545,510.  By how many percent is its circulation larger than the circulation of the National Geographic?

Answer. Skymag has a circulation 60% larger than National Geographic.  The calculation is (7,545,510 - 4,708,307)/4,708,307  » .60 = 60%.

d. In 2005 the magazine TV Guide was ranked 4th in the nation with a circulation of 8,211,581. Just four years later it had fallen to 22nd place with a circulation of 2,934,969. By how many percent did the circulation drop from 2005 to 2009?

Answer. The circulation of TV Guide dropped 64%.  The calculation is (2,934,969 -8,211,581)/8,211,581  » .64 = 64%.

2.  In the period January 2000 to March 2000, stock for Sapient Corporation, a consulting company that helps businesses put their operations on the internet, dropped 51% and then rose by 49%.  By how many percent did price change over the whole period?

Answer. Starting at 100%, the stock dropped to 49% of its original value.  It then increase by 49% of the 49%, i.e., 0.49*0.49 = 0.2401 or 24.01% of the original price. Thus in the end it was 49% + 24.01% or 73.01% of its original value.   It was down 26.99%.

3. Cigarette use in the US has decreased over the last 20 years, even though the rate is still relatively high.  In 1998, 277 of every 1000 individuals in the US over the age of 12 reported that they were smoking, a 28% decrease since 1985.  What was the rate in 1985?

Answer.

x - (0.28*x) = 277

x*(1 - 0.28) = 277

x(0.72) = 277

x = 277/0.72  » 385

The answer is approximately 385 of every 1000 individuals over the age of 12.

4.  a.  At DePaul University, minority students represented 31% of the enrollment in 2001. If there were 21,363 students enrolled in 2001, about how many of these students are minority students?

6622.53 or about 6622 students

b.  Chicago’s population grew from 2.78 million in 1990 to 2.90 million in 2000.  By how many percent did it grow?

 4.3% increase

c.  The January 14, 2002 issue of TIME magazine reported that Wal-Mart’s 3^rd quarter revenues for 2001 were 600% more than its No. 2 competitor, K-Mart.  What were K-Mart’s revenues during this period if Wal-Mart’s revenues were $48 billion? 

 $6.86 billion

d.  In 2006, 261.9 million people in the United States were born in the United States, and the rest, 37.5 million were foreign born.  What percentage of the population of the US was foreign born?

 12.5% foreign born

e.  Over the last four years, DePaul’s freshman class has grown by 4.8%.  What is the Fall 2005 freshman enrollment if the 2009 enrollment was 3,041 students? 

2902 students

5.  In 2001, 6,113 applicants were accepted to DePaul. This represents about a 72% acceptance rate. Of the students who were accepted, 2,050 students enrolled at the university.

1. How many students in 2001 applied for admission at DePaul?
2. What percent of the students who were accepted to DePaul actually enrolled at the University?

a.  8490 students applied
b.  33.54% of students who were accepted enrolled

6.  The November 16, 2009 issue of the New York Times  reported that there were 671,616 foreign students enrolled at US colleges and universities in 2008-09 — representing an 8% increase from the previous year. It was the largest percentage increase in more than 25 years. 

a.       How many foreign students were studying in US colleges and universities during 2007-2008?  

b.      The greatest number (103,260) of foreign students came from India.  What percent of the foreign students studying in the US were from India?  

c.       China currently has the second largest number of foreign students studying in the US, however they are rapidly catching up to India. The number of foreign students from China in 2007-2008 was 81,413. That number grew by 21% over the next year. How many foreign students were from China during 2007-2008?

a.  621,867 foreign students 

b.  15.37% from India

c.   98,510 from China in 2007-2008

7.  The December 11, 2002 issue of Sports Illustrated reported that the federal spending for Olympic Games held in the United States was $610 million for the 1996 Atlanta Games and $1.5 billion for the 2002 Salt Lake City Games. (The figures have been adjusted for inflation.)

a.       How many times more federal money was spent on the Salt Lake City Games than the Atlanta Games?

b.       By what percent  is the federal spending for the Salt Lake City Games more than the Atlanta Games?

c.       By what percent is the federal spending for the Atlanta Games less than the Salt Lake City Games?

 a.  2.46 times more

 b.  146% 

 c. -59.33%

Consumer Price Index

8. Carefully describe a) what the CPI is, b) how it is constructed, and c) what the meaning of actual index numbers is (e.g., the CPI in 1999 was 166.6; what does 166.6 mean?)

The CPI is number used to measure the changing value of money over time. It is published by the US Bureau of Labor Statistics.  Economists select an imaginary "market basket" of goods and services that represents the buying patterns of most people.  Every month they collect price data on the items in the basket and essentially compute the value of the entire basket. The CPI is literally the ratio of the price of the basket in a given year to the price of the basket in a chosen base period multiplied by 100.  For example, the 166.6 CPI in 1999 means that the price of the standard market basket is 1.666 times as much as it was in the base period (1982-84).  It means that in 1999 the same goods and services cost about 66.6% more than they did in 1982-84.   Another interpretation of the index number 166.6 is the following: a person would have to pay $166.60 to buy the same goods and services that a person could buy for $100.00 in the base period on average. For more information, read the CPI Tutorial. 

1. Using the CPI, add a column to the table in which you calculate defense spending for these years in constant 2002 dollars, and cut and paste the table into your word document. 

   You need to paste in the CPI values in column C. The needed Excel formula in column D is  =B5*179.9/C5 or =B5*$C$47/C5. The top of the table is:

   Year	Spending	CPI	Spending in 2002$
   1960	53.5	29.6	325.2
   1961	55.3	29.9	332.7
   1962	57.9	30.2	344.9
   1963	58.9	30.6	346.3
   1964	60.5	31.0	351.1
   1965	56.3	31.5	321.5
   1966	64.1	32.4	355.9
   1967	78.1	33.4	420.7
   1968	88.9	34.8	459.6
   1969	90.2	36.7	442.2
   1970	90.4	38.8	419.1

2. Create an XY graph showing spending in constant 1999 dollars for the period 1960-2002.

3. In a well written paragraph describe the graph you made in c.

Defense spending was about 300 million dollars in 1960. It rose slightly in the early Sixties before dropping to it absolute minimum during the period 297 million in 1965.  It then rose dramatically, achieving a local maximum of 426 million in 1968, the height of the Vietnam War.  Defense dropped fairly rapidly from 1968 until 1974.  It then remained fairly constant (at about the 320 million level) until 1981. During the Reagan administration in the early Eighties, defense spending shot up, and defense spending was at it highest point in the entire period in 1986 (456 million). Spending leveled off a bit in the late Eighties but then plunged in the Nineties. In the late 1990s, it remained relatively constant.  After 9/11/2001, defense spending increased dramatically.  The overall picture shows two massive arms buildups, one corresponding to the Vietnam War and the other to the Reagan Administration's Strategic Defense Initiative and other Cold War efforts.  We are perhaps at the start of another arms buildup.

Since the value of the dollar decreases over time due to inflation, it is hard to compare spending from 1985 to 2002.  You cannot conclude that the actual dollars spent increased that the value of the spending increased.  The value of the dollar is not the same in both years.

new CPI/old CPI * old actual $ = 179.9/107.6*279 = 466.5.  1985 spending in 2002 constant dollars is $466.5 billion.  This is much higher than the actual spending in 2002.  This means defense spending grew less than inflation over the 17 years.  Defense spending was very high in the mid-1980's.

(466.5-399.5)/399.5 = 16.7%

Trendlines and Graphs

11.  Below is a trendline graph of the number cellular telecommunications subscribers from 1994 - 2003.

a.  Based on what you learned about creating effective graphs, list two concerns about the above graph.

• the source is not listed

• the title does not address the 4-Ws

b.  Using your equation, predict when their were no cell phone subscribers.  How much faith do you have in your prediction?  Based on the data from 1994 through 2003, there were no cell phones in 1993.  Using your knowledge about cell phones, the equation is not a good fit for the data (when predicting backwards) because there were cell phones in 1993.

c. Using your equation, predict the number of subscribers in 2030.  How much faith do you have in your prediction? Based on date from 1994 through 2003, there will be approximately 573,933,000 in 2030.  Once again, practical knowledge dictates whether or not we have faith in the prediction.  Since technology changes so quickly, 2030 is too far out for a prediction.  Also, 573,933,000 far exceeds the current U.S. population.

Exponential Modeling and Solving Exponential Equations

12. Carbon dioxide emissions from the burning of fossil fuels is almost surely a cause of global warming.  One of the difficulties in negotiating a treaty for the reduction of carbon dioxide emissions is the discrepancy in emission levels between developing and developed nations.  For example, in 1999, Brazil emitted 89 million metric tons of carbon into the atmosphere, while Canada, a much less populous country, emitted 151 million metric tons.  Brazil’s emissions, however, are growing at 4.1% per year, while Canada’s are growing at 0.8%.

1. In Excel, make an exponential model for Brazil’s and Canada’s carbon emissions.

 Answer:

2. Using your model, predict when Brazil’s emissions will exceed Canada’s.

 Answer. Extending the model, in a., Brazil’s emissions will exceed Canada’s in 2016.

3. How much confidence you have in your prediction in b?

Answer. I have only moderate confidence.  17 years is quite far from the existing data.  On the other hand, I have complete confidence that Brazil’s emissions will exceed Canada’s in the not so distant future.  I am not sure that Brazil will be able to keep up a 4.1% annual rate of percentage increase for so long.

4. Find formula for the exponential function which models Brazil’s emissions (using 1999 as year 0).

 Answer.  f(x) = 89 * (1.041) ^ x     or written more traditionally f(x)  = 89(1.041)^x.

5. Using the formula for Brazil’s model, predict Brazil’s emissions in 2010.

 Answer. 2010 is 11 years after 1999.  f(11) = 89 * (1.041) ^ 11 »  138.5 metric tons.

13. In 1994, 24 samples from the archeological site at Oslonki, Poland were dated using carbon 14.  The archeological site is significant for the large quantities of copper implements and jewelry found there.  One of the samples had 47.8% of the carbon 14 that would normally be present in a living organism.  Approximately, how old is the sample?  (Carbon 14 decays approximately 1.202% every 100 years.) Please include the Excel table you used to calculate your answer; if you didn't use an Excel table to calculate your answer, briefly describe your calculation.

The Excel table is long, so I'll only paste in the beginning.   The sample is about 6100 years old.

14.  In 1990, the population of the town Erehwon was 5000 and has been increasing by 2.9% every year.  The population of the town Exalpon, Erehwon’s nearest neighbor, was 4500 and increasing by 3.2% every year.  When, will the population of Exalpon exceed Erehwon?

In 37 years.

Loans

19.  You have decided to buy a 2003 Chevy Cavalier. The total cost with tax, title, license and optional equipment is $20,500. You will make a down payment equal to 10% of the total cost of the car. The remaining amount will be financed.

You have two financing options.

Option 1 - Get $2,500 cash back. This amount will reduce the amount you need to finance. You will be able to get a 5 year car loan on the remaining amount with a 6.0% interest rate from your bank.

Option 2 - Get no cash back and finance the entire amount. The loan will be for 3 years with a 0.0% interest rate from GMAC financing.

a. What is the amount of money that you will need to borrow under Option 1? ($15,950)
b. Make an amortization table for Option 1 and verify that your ending balance after 60 months is zero. Paste the first five lines (months 0 - 4) in your Word document. What is your monthly payment? What is the total amount paid over the term on the loan?

 ($308.76, $18,501.49)
c. How much will you need to borrow if you choose option 2? What is your monthly payment?($18,450, $512.50)
d. Which financing option should you choose? Explain.  You can decide which is more important, the payment amount or the total amount you pay over the duration of the loan.

Credit Cards

20.  You only use your credit card in the case of an emergency. You charge $4,000 in auto repairs to credit card which has an annual interest rate of 11.99%. What should your monthly payment be if you would like to pay off your balance in full after two years?   $188.28

21.  You make a purchase for $3,501 on your credit card which has an annual interest rate of 17.99%. If you pay the minimum payment of 2% per month (not less than $25), what is your balance after 4.5 years, assuming you make no other purchases with your credit card? $2,669.57

22.  You make a purchase for $2,723 on your credit card which has an annual interest rate of 15.99%. The credit card company offers a minimum payment of 3% per month (not less than $25). How many years will it take your balance to equal $10.00 assuming your make no other purchases with your credit card?

between and 114 and 115 months