Are You Interested in Stretching Your Dollars?

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1 NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS NOVEMBER 00 Are You Interested in Stretching Your Dollars? Sour said, I really think that you can find a better way to become a millionaire. Why don t you put money into an investment that earns interest? For example, a savings account at my bank earns percent in interest each year. Since percent means out of one hundred, you can think of percent interest as earning pennies for every 00 pennies, or $0.0 on every dollar, that you save. It is like getting extra money for doing nothing. You will eventually have a million dollars without spending a million dollars. Besides, you have no guarantee that you will win the lottery. Pie responded, Yeah, but lots of people win. If I buy a lot of tickets, I ll have a better chance of winning. Plus I won t have to wait for the money to grow in a bank. After their discussion, Sour and his son sat down to determine the best method of achieving millionaire status. This activity is designed to help you discover what Sour and his son Pie found out.. How much interest would Pie earn if he put $0 in his dad s bank for a year?. Suppose that Pie started his account with one of the following amounts. Find the interest that each amount would earn at his dad s bank the first year. a) $0 c) $0 e) $00 b) $0 d) $00 f) $000. By doing some research, Pie and his dad found that different banks offered different interest rates for their savings accounts. Find the interest that $0 would earn the first year at each bank: a) First National % _ b) Yeast Banc 6% c) Corporate Trust 8.% Sour Dough was discussing future plans with his son Pie. Pie said, Dad, I want to be a millionaire, so I m going to buy lottery tickets until I win. Then I won t have to work. 0 0 The editors wish to thank Sandra M. Powers, College of Charleston, Charleston, SC 9, for the ideas used in this issue of Student Math Notes.

2 . What rule could you use to find the amount of interest earned in a year for any amount of money at any given rate?. Sour intended for Pie s money to grow in his savings account as he grew older. Complete the chart to see how much money Pie would have at the end of each year if he had invested $000. per Year (Interest Rounded to Nearest Cent) Interest Earned Ending $ $0.00 $00.00 $ $.60 $08.60 $ How much interest was earned during the third year? During the sixth year? During the tenth year? 7. Describe how the interest earned changes each year. Explain why the money grows the way that it does. NCTM Student Math Notes, November 00

3 The secret of the growth of a savings account is a process called compound interest, that is, calculating interest on interest that was previously earned. Another way to calculate compound interest is shown in the following chart. Year Rounded to Nearest Cent Ending $ $000.0 $00.00 $00.00 ($000.0).0, or $000.0 $08.60 $ a) Complete the preceding chart. b) Write a rule that enables you to determine the amount of money in the account after 8 years. After years. After 60 years. After n years. 9. a) What would $000 invested at 6 percent grow to after 8 years? b) After years? c) After 60 years? Compound interest is usually compounded more than once a year. Sometimes interest is compounded semiannually, quarterly, monthly, or even daily. NCTM Student Math Notes, November 00

4 How Compounded Annually Semiannually Quarterly Monthly Daily per Period 6 Number of Times Interest Is Added During a Year (every year) (every 6 months) (every months) (every month) 6 (every day) 0. Let s look at $000 at a percent interest rate, compounded semiannually. Complete the following chart. Number of Periods (Round to Nearest Cent) Ending at the End of Each Six-Month Period $ = % $000.0 $00.00 $00.00 = % ($000.0).0, or $000.0 $00.0 $00.0 = %. Write a formula for determining the ending amount if $000 is invested at percent per year compounded semiannually for 8 years. NCTM Student Math Notes, November 00

5 . Construct a chart for computing interest quarterly. Number of Periods (Round to Nearest Cent) Ending at the End of Each Three-Month Period $ = % $000.0 $00.00 $ Write a formula for determining the ending amount if $000 is invested at percent per year compounded quarterly for 8 years. The formula A = describes the value, A, of a savings account in which $000 is invested for years at 8 percent, with the interest compounded six times per year.. a) Write an expression for the value, A, of a savings account in which $000 is invested at 8 percent per year, compounded monthly for 0 years. b) What is the value of the account at the end of the 0-year period?. Pie Dough said to his dad Sour, All this interest stuff has been interesting, but I still think that I could do better in the lottery. Sour replied, Let s compare the two. Suppose that I gave you $000 to put in an investment that earns 0 percent interest per year and that you put $000 into the account each year until you had $,000,000 in the account. How long would you need to earn a million dollars? Including my $000, how much would you have to invest to earn it? a) Estimate the amount of time and money required to earn the $,000,000. NCTM Student Math Notes, November 00

6 b) Determine the actual time and money required by using a spreadsheet to complete and extend the following table. Note that you cannot use the compound-interest formula, because you are adding money to the account. Interest Rate Interest Earned Additional Deposits ($000 per Year) Ending $000 0% $00 $000 $600 $600 0% $60 $000 $80 $80 0% $000 0% $000 c) Pie s dad said, Suppose that I gave you $000 to buy $ lottery tickets. Then instead of investing $000 a year, you spent $000 a year to buy lottery tickets. You continue to spend $000 a year for the same number of years that my investment would take to earn $ million. How much would Pie spend on $ lottery tickets? d) Did you know that,979, different lottery tickets are possible? Pie s father asked. The probability of winning the lottery is the number of tickets bought divided by the number of tickets possible. What are Pie s chances of winning the lottery? e) Write a paragraph advising Pie whether he should invest his money or play the lottery. Use mathematics to support your argument. Can you use the rule of 7 to estimate the amount of time necessary for an amount of money to double if the annual interest rate is known? explain how interest compounding is related to exponential growth? compare the properties of linear and exponential functions? Did you know that the populations of countries can be projected by using concepts similar to the compounding of interest? that health officials can make projections of bacterial counts using concepts similar to the compounding of interest? that interest compounding can be calculated using recursion? that the value of the irrational number e can be determined by the formula used to calculate investing $ at 00 percent interest compounded continuously? Mathematical content simple interest, compound interest, exponents, exponential growth, functions 6 NCTM Student Math Notes, November 00

7 Bibliography Media Clips. Mathematics Teacher 9 (December 00): 70. Metz, James. Activities: Seeing How Money Grows. Mathematics Teacher 9 (April 00): Answers Are You Interested in Stretching Your Dollars? Continued. $0.0. a) $0.80 b) $.0 c) $.60 d) $.00 e) $8.00 f) $0.00. First National $0.80 Yeast Banc $.0 Corporate Trust $.0 P.. amount of interest per year = amount of money annual interest rate, or I = r. per Year (Interest Rounded to Nearest Cent) Interest Earned Ending $ $00.00 $08.60 $.86 $69.8 $6.6 $6. $.9 $68.6 $ $0.00 $.60 $.6 $.99 $6.79 $8.67 $0.6 $.6 $.7 $6.9 $00.00 $08.60 $.86 $69.8 $6.6 $6. $.9 $68.6 $.0 $ $.6, $8.67, $ The amount of interest increases because the interest is calculated on previous interest earned, as well as on the beginning balance. 8. a) Year Rounded to Nearest Cent Ending $ $000.0 $00.00 $00.00 $08.60 ($000.0).0, or $000.0 $000.0 $08.60 $.86 Slight differences in ending balances in answers and 8 are caused by round-off error. $.86 $000.0 $69.86 $69.86 $000.0 $6.6 b) $ , $000.0, $ , $000.0 n NCTM Student Math Notes, November 00 7

8 9. a) $ = $8. 0 b) $ = $9.87 c) $ = $, $ $00.00 $00.0 $06. $08. = % = % = % = % = % Number of Periods (Round to Nearest Cent) $000.0 ($000.0).0, or $000.0 $000.0 $000.0 $000.0 Ending at the End of Each Six-Month Period $00.00 $00.0 $06. $08. $0.08. $ A sample chart follows: Number of Periods (Round to Nearest Cent) Ending at the End of Each Three-Month Period $ = % $000.0 $00.00 $00.00 = % $000.0 $00.0 $00.0 = % $000.0 $00.0 $00.0 = % $000.0 $00.60 $00.60 = % $000.0 $0.0.. a) 7 $000.0 $07.0 b) $9,707.. a) Student estimations will vary. b) To make at least $,000,000, you would need to invest $,000. Forty-nine years would be needed. c) He will spend $0,000 on lottery tickets. d) $ , = 0.%.,979, e) Sample answer: Pie should invest the money. He will have $,000,000 after 9 years if the interest rate stays at 0 percent. However, he has less than 0. percent chance of winning the lottery after 9 years and after spending $0,000 on lottery tickets. NCTM STUDENT MATH NOTES is published exclusively on the NCTM Web site, by the National Council of Teachers of Mathematics, 906 Association Drive, Reston, VA 09. The five issues a year appear in September, November, January, March, and May. Pages may be reproduced for classroom use without permission. Editor: Terry Souhrada, University of Montana, Missoula, MT Editorial Panel: Hope Florence, College of Charleston, Charleston, SC 9 Micah Fogel, Illinois Mathematics and Science Academy, Aurora, IL Cathryn Hund, St. Thomas Aquinas High School, Overland Park, KS 66 Frederick Creed, Parkview High School, Lilburn, GA 007 Board Liaison: Susan K. Eddins, Illinois Mathematics and Science Academy, Aurora, IL 6006 Editorial Coordinator: Joan Armistead, jarmistead@nctm.org Production Editor: Nancy Green Production Specialist: Rebecca Totten Keyboarding Specialist: Jody s 8 NCTM Student Math Notes, November 00