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# Barbara &amp; Allen's Compound Interest

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Introduction

Angela C. Rosario Mr. Thomas IB Mathematics SL I (3) 24 March 2009 Practice IB Internal Assessment Introduction: The purpose of interest is for a bank to pay an individual for the use of their money. Interest therefore represents one's return on the investment. To calculate n interest compoundings per year, one must utilize the formula: 1. Alan invests \$1000 at an interest rate of 12% per year. Copy and complete Table 1 which shows A, the value of the investment in dollars after t years, assuming that the interest is compounded yearly. One must utilize the formula for Compound Interest in order to determine the answer. In the case of Alan, his principle (P) amount of money is \$1000, and he collects interest at a rate (r) of 12% per year. To determine the value of the investment in dollars after t years in order to satisfy Table 1, the formula necessary is: A= 1000 ( 1+ .12/1)(1)(t) where P= \$1000, r= 0.12, and n=1 because interest is being compounded annually or once a year, and t= the number of years the money is present in the account. t=0, A= 1000( 1+ 0.12/1)(1)(0) = 1000(1+ 0.12) 0= 1000(1.12) 0= 1000(1)= 1000 t=1, A= 1000( 1+ 0.12/1)(1)(1) = 1000(1+ 0.12) 1= 1000(1.12) 1= 1000(1.12)= 1120 t=2, A= 1000( 1+ 0.12/1)(1)(2) = 1000(1+ 0.12) 2= 1000(1.12) 2= 1000(1.2544)= 1254.40 t=5, A= 1000( 1+ 0.12/1)(1)(5) = 1000(1+ 0.12) 5= 1000(1.12) 5� 1000(1.76234)� 1762.34 t=10, A= 1000( 1+ 0.12/1)(1)(10) = 1000(1+ 0.12) 10= 1000(1.12) 10� 1000(3.10585)� 3105.85 t=20, A= 1000( 1+ 0.12/1)(1)(20) = 1000(1+ 0.12) 20= 1000(1.12) ...read more.

Middle

Thereafter, she stops adding \$200 at the start of each year but leaves her investment in the bank to earn interest at 12% per year. Table 3 shows C, the value of the investment in dollars after t years, assuming that the interest is compounded yearly. Table 3 Barbara's Investment of \$200 with an Additional \$200 Added to the Sum the First Five Years at an Interest Rate of 12% Compounded Yearly t 0 1 2 5 10 20 30 C 200.00 424.00 674.88 1423.04 2507.88 7789.09 24191.72 Draw a new table showing the value D, Barbara's investment in dollars after t years, assuming that the interest in compounded monthly. Label it Table 4. To calculate Barbara's investment in dollars with a monthly compounding, one must substitute 12 for the value of n because there are 12 months in a year. In addition to changing the value of n, one must account for the \$200 added to the sum for the first five years, causing the formula for the first five years of her investment appear as follows: D= [P(1+0.12/12)(12)(t)]+200 where the additional \$200 is added to the final sum of the end of year interest collected. Thus, the formula for Barbara's investment is: D= [200(1+0.12/12)(12)(t)] + 200= [200(1.01)12t] + 200 however, for the first five years, one must substitute 1 for t for the first five years because of the additional \$200 Barbara adds to her investment. Also, one should not start adding \$200 until after t= 0, because it only consists of the primary \$200 invested. When: t= 0, D= 200(1.01)(12)(0) = 200(1.01)0= 200(1) ...read more.

Conclusion

� t= 14, A=1000(1.12)14 � 1000 (4.887112285)� 4887.11 2009 � t= 15, A=1000(1.12)15� 1000 (5.473565759)� 5473.57 2010 � t= 16, A=1000(1.12)16 � 1000 (6.13039365)� 6130.39 Table 6 Alan's Investment of \$1000 at an Interest Rate of 12% Compounded Yearly in the Interval of Years 2000 to 2010* Year Amount 2000 1973.82 2001 2210.68 2002 2475.96 2003 2773.08 2004 3105.85 2005 3478.55 2006 3895.98 2007 4363.49 2008 4887.11 2009 5473.57 2010 6130.39 *Investment started in 1994 *Final answers rounded to the nearest hundredth in order to comply with the general money standard of cents. *Final answers rounded to the nearest hundredth in order to comply with the general money standard of cents. The reason why Barbara's investment is worth more than Alan's, even though she began with only \$200 in her account while Alan started with \$1000, is because Barbara collected interest for four years longer than Alan, and furthermore added money to her investment for the first five years. Due to Barbara's addition of \$200 to her final investment at the end of each year for the first five years, she collected a new principal. At the end of the first five years of her investment, instead of having \$200 as her principal, which would have been the case had she done a single investment, Barbara thus had the larger principal of \$1423.04. With this as the new principal for Barbara during her investment for the following years, Barbara was able to collect more interest than Alan, making her investment worth more than Alan's by \$1658.71, or approximately 21.3%. As a result, the larger one's principal is, the more interest will be collected, creating a larger final investment. ?? ?? ?? ?? Rosario 1 ...read more.

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