Compound Interest of Exponential Growth of the Population | MATH 113, Exams of Mathematics

Download Compound Interest of Exponential Growth of the Population | MATH 113 and more Exams Mathematics in PDF only on Docsity! 1 Announcement Math 113 section 21-26 Instructor: M.Xiao •The Fourth Hour Exam: Dec. 4, Friday in class. Nov. 16, 2009 Help sessions Tuesday and Thursday 6:30-7:45p.m. Location: agriculture building (AG) 152. Course web page: http://kalman.math.siu.edu/~mxiao/math113.html Compound Interest (Exponential Growth) If an original population (P) grows exponentially with effective Rate r, at the end of n compounding periods the population will be nrPA )1( += Aè accumulated population (or principal) Pè beginning population (or principal, the initial balance) rè rate of interest (written as a decimal) per compounding period nè number of compounding periods of investment Compounded quarterly: Interest is determined 4 times a year and the principle is updated accordingly at that time. Compounded monthly: Interest is determined 12 times a year and the principle is updated accordingly at that time. Example: Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year. Determine the amount in the account after 10 years with the compounding period: a) Annual compounding. b) Quarterly compounding. c) Monthly compounding. Solution: nrPA )1( += a) The annual rate of 10% gives r=0.1, after 10 years the account has 74.2593$)10.1(1000$)10.01(1000$ 1010 ==+ b) r=010/4=0.025, and after 10 years (40 quarters) the account contains 06.2685$)025.01(1000$ 40 =+ c) r=0.10/12=0.008333. The amount in the account after 10 years (120 months) is 04.2707$)008333.01(1000$ 120 =+ Effective Rate Per Period P PrP n −+ )1( Effective rate per n period = = 1)1( −+ nr Aè accumulated population (or principal) Pè beginning population (or principal, the initial balance) rè rate of interest (written as a decimal) per compounding period nè number of compounding periods of investment APR and APY The nominal rate for a year is called the annual (percentage) rate (APR). The effective rate is called the annual (equivalent) yield or annual percentage yield (APY). Old Exam Question What is the APY for 5.3% compounded quarterly? A) 5.3% B) 5.4% C) 5.5% D) 5.6% 1)1( −+= nrAPY %41.50541.01)4/053.01( 4 ==−+=APY 2       −+ = r r dA n 1)1( Savings Formula: Example: An individual saves $100 per month, deposited directly into her credit union account on payday, the last day of the month. The account earns 6% per year, compounded monthly. How much will she have at the end of 5 years, assuming that the credit union continuous to pay the same interest rate?       −+ = r r dA n 1)1( where d is a uniform deposit per period (deposited at the end of the period). r is the interest rate per period. A is the accumulated amount. Solution: In this case d=$100, the monthly interest rate is 0.06/12. 00.6977$ 12 06.0 1) 12 06.01( 100$ 60 =           −+ =A Answer: Old Exam Question The Chavez family has decided to save up for a new spa. They want To save $10,000 in five years. They find a savings account for w hich Interest was compounded monthly at 8.2%. How much will they have To deposit each month to meet this goal? A) $54 B) $97 C) $135 D) $256       −+ = r r dA n 1)1( The monthly rate is 8.2%/12=0.0068333333 )86117227.73( 006833333.0 1)006833333.1(000,100$ 512 dd =      −= × 38.135$ 86.73 100000$ ==d The Consumer Price Index (CPI) The official measure of inflation is the consumer price index (CPI), prepared by the Bureau of Labor Statistics. Here we describe and use the CPI-U, the index for all urban consumers, which covers about 80% of the U.S. population and is usually referred to in newspaper and magazine articles. Each month, the Bureau of Labor Statistics determines the average cost of a “market basket” of goods, including food, housing, transportation, clothing, and other items. It compares this cost to the cost of the same goods in a base period. The base period used to construct the CPI-U for other years is calculated by using the proportion CPI for other year cost of market basket in other year = 100 Cost of market basket in base year Comparing Costs Cost in year A CPI for year A Cost in year B CPI for year B = Example: The average cost of a Madison house in 1976 is $38, 323. What is the average cost of a home in Madison in 2003? Cost in 2003 CPI for 2003 Cost in 1976 CPI for 1976 = Cost in 2003 188.5 $38, 323 56.9 = Thus Cost in 2003 = 958,126$313.3323,38$ 9.56 5.188 323,38$ =×=× Old Exam Question Betty bought a house in 1987 for $99,000 and sold it in 2001. If the 1987 CPI is 113.6 and the 2001 CPI is 177.7, how much would the house be worth in 2001 dollars? A) $134,921 B) $165,157 C) $102,356 D)$154,862 Cost in 2001 CPI for 2001 Cost in 1987 CPI for 1987 = Cost in 2001 177.7 99000 113.6 = Cost in 2001 = 79.154861$6.113 7.177 99000$ =×