Student Explorations Do You Want to Be a Millionaire? in Mathematics Do You Want to Be a Millionaire? Sour Dough was discussing future plans with his son Pie, who said, “Dad, I want to be a millionaire, so I’m going to try to be a great singer or football player. If that doesn’t work out, I will keep buying lottery tickets until I win. Once I win, then I won’t have to work.” Sour answered, “Son, I think you can find a better way to become a millionaire. Why don’t you put money into an investment that earns interest? Did you know that the bank will pay—actually pay—you to hold your money? Here is how it works: be. You can eventually have a million dollars without ever having earned a million dollars. Think about it: An interestbearing savings account gives you a chance to become a millionaire. Besides, you have no guarantee that you will win the lottery.” “You deposit some money into the bank. The bank pays you interest—that is, money—on the amount of money that you deposit. So now you have the initial money you deposited plus the money from the interest you have earned. For example, a savings account at my bank earns 4.0 percent (4%) in interest each year. Because percent means out of one hundred, you can think of 4.0 percent interest as earning 4 pennies for every 100 pennies, or $0.04 on every dollar, that you deposit. It is like getting extra money for doing nothing. So, the more money you deposit in your account, the more interest you earn, and in turn, this will increase any savings you keep at the bank. The more you deposit, the greater the interest earned will 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.” Sour advised, “There may be many winners, but what you do not realize is that even more people are playing than winning. For each person who wins, millions—yes, millions—of people do not win. We only hear about those who win.” 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 Pie found out. Sandra M. Powers, College of Charleston, SC 29424 (original submission contributor) Copyright © 2012 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use only. 1 Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? 1. How much interest ( I ) would Pie earn for one year if he were to put $10.00 in the bank his dad uses? 5. Sour takes Pie to the bank, where they open a savings account and deposit $1,000.00. a. Complete table 1 to show how much money Pie would have at the end of each year if he deposits $1,000.00 and leaves it, plus the interest it earns, in the account for 10 years. 2. Suppose that Pie started his account with one of the following amounts. Find the interest that each amount would earn the first year at his dad’s bank. a. $20.00 b. Complete the graph below using the information from table 1. c. What is the difference in the amounts of interest earned between— • the first and second year? b. $42.50 • the second and third year? • the third and fourth year? c. $68.00 d. $300.50 3. By doing some research, Pie and his dad found that different banks offer different annual interest rates for their savings accounts. Find the interest that $40.00 would earn for the first year at each bank. d. Did you earn the same amount of interest each year? Why, or why not? Use table 1 or graph 1 to help justify your answer. e. Why do some people refer to this situation as watching your money grow? a. First National offers a rate of 2.0 percent. b. Ye Olde Banc offers a rate of 3.0 percent. c. Corporate Trust offers a rate of 2.5 percent. he secret of the growth in Pie’s savings account that T was shown is a process called compound interest; that is, calculating interest on interest that was previously earned. Table 2 shows another way to calculate compound interest. 6. a. Complete table 2. b. Using table 2, find a pattern that enables you to calculate the ending balance after 18 years. 4. Explain what you would do to find the amount of interest earned in a year for any amount of money at any given rate. • Calculate the ending balance after 25 years. 2 Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? Table 1. Investing $1,000.00 at 4.0 percent Beginning bank balance $1,000.00 $1,040.00 $1,081.60 Year 1 2 3 4 5 6 7 8 9 10 Interest rate per year 4.0% 4.0% Calculation (interest rounded to nearest cent) 0.04 3 1,000.00 0.04 3 1,040.00 Interest earned $40.00 $41.60 Ending balance $1,040.00 $1,081.60 Graph 1. Pie’s investment at 4.0 percent Ending Balance in $ 1,600 1,400 1,200 1,000 800 600 400 200 0 0 2 4 6 8 10 12 Year Table 2. Compound interest Year Beginning bank balance Interest rate per year 1 $1,000.00 4% 2 $1,040.00 4% Calculation (interest rounded to nearest cent) Interest earned Ending balance $1,000.00 3 1.041 $40.00 $1,040.00 ($1,000.00 3 1.04) 3 1.04, or $1,000.00 3 1.042 $41.60 $1,081.60 3 4 5 6 • Using your pattern, determine an equation that represents your ending balance after n years. 7. a. What would $1,000.00 invested at 6.0 percent grow to after 18 years? 3 b. What would $1,000.00 invested at 6.0 percent grow to after 25 years? c. What would $1,000.00 invested at 6.0 percent grow to after 60 years? Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? Compound interest can also be compounded more than once a year. Sometimes interest is compounded semi annually, quarterly, monthly, or even daily. These time frames are known as the period for the investment. Interestingly, banks use 360 for the number of days in a year instead of 365, in part because it made calculations easier in the era before the use of calculators and computers. 8. Complete table 3. b. What is the value of the account at the end of the 20-year period? 14. Compare the values of a savings account in which $4,000.00 is invested at 9.0 percent per year compounded for 20 years: a. annually 9. Let’s look at what happens to $1,000.00 that is invested at an annual interest rate of 4.0 percent compounded semiannually (that is, twice a year). Complete table 4. b. semiannually (twice per year) 10. Complete table 5 for computing interest quarterly. Begin with the current year for period 1. c. quarterly 11. W rite an equation for determining the ending amount if $1,000.00 is invested at 4.0 percent per year compounded quarterly for 18 years. d. monthly e. daily The formula to calculate compound interest A = P (1+ nr )nt describes the value, A, of a savings account in which P, the principal, is invested for t years at r percent (the annual rate using decimal notation) with the interest compounded n times per year. r 12. Using the formula A = P (1+ n )nt, explain how the equation you created in question 11 connects to the formula to calculate compound interest. 13. a. Write an expression for value A of a savings account in which $4,000.00 is invested at 9.0 percent per year compounded monthly for 20 years. f. After 5 years, in which of the following circum stances would you have more money? • investing $600.00 at 3.0 percent interest compounded annually • starting with $0 and adding $10.00 a month that earns 3.0 percent interest compounded monthly for 5 years g. Deposit $440.00 at 3.0 percent for 2 years compounded annually, and then deposit $700.00 at the same rate compounded annually, for an additional 3 years. What is the amount of interest you earn? One evening Pie came home excited about a credit card offer. Because he and his dad had already been talking about credit, interest rates, and the like, Pie had some questions to ask about the offer. 4 Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? Table 3. Compounding periods How often interest is compounded Interest rate per period Annually Semiannually 2 Number of times interest is added during a year = 0.04 = 4.0% 1 (every year) = 0.02 = 2.0% 2 (every 6 months) Quarterly 4 (every 3 months) Monthly 12 (every month) Daily 360 (every day) Table 4. Interest compounded semiannually 1 Beginning bank balance $1,000.00 Interest rate 2.0% 2 $1,020.00 2.0% 3 $1,040.40 2.0% Period Calculation (interest rounded to nearest cent) $1,000.00 3 1.021 ($1,000.00 3 1.02) 3 1.02, or $1,000.00 3 1.022 Ending balance $1,020.00 $1,040.40 4 5 6 7 8 Table 5. Interest compounded quarterly 1 Beginning bank balance $1,000.00 2 $1,010.00 Period Interest rate 1.0% Year Calculation (interest rounded to nearest cent) $1,000.00 3 1.011 Ending balance $1,010.00 3 4 5 6 7 8 The credit card offer contained the following terms: Credit limit $2,000.00, with an introductory rate of 15.0 percent APR (Annual Percentage Rate, interest that is charged for “borrowing” money, or using your credit card) for the first 6 months as a promotion. After the first 6 months—when the promotional period ends—the APR becomes 24.0 percent. 5 Pie asked his dad if APR was the same as the interest from the simple interest and compounded interest that they had discussed earlier. So that his son would better understand the potential impact of APRs, Sour asked Pie to calculate the cost of using this credit card. Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? 15. a. Imagine that Pie used the new credit card to go shopping and charge the following: $75.00 to Sneaker Locker, $120.00 to Jersey Outlet, $52.00 for dinner at the Math Garden Bistro, and $210.00 to the Polynomial Coat Factory. Determine the total amount that his credit card would be charged. b. Suppose Pie paid a minimum fee of $25.00 a month at 15.0 percent APR. • Use the formula for compounding interest (monthly) to determine what his new balance would be (including the 15.0 percent interest) at the end of the first month. • What would be the total interest paid over the time period? d. But remember, if Pie does not make his total payment on time in the seventh month, he will be given a higher APR. Use the simulator and change the time span to 6 months to determine how much Pie would owe at this point. • What would be the total remaining after 6 months? • What would the remaining balance be after he pays $25.00? • How much would the use of the credit card have cost Pie up to this point (the current cost minus the amount charged)? • How much of the amount of money he spent using his credit card would Pie have actually paid? • What would be the cost of using his credit card for the first month? e. Now take the remaining total that Pie would owe after 6 months and change the APR to 24.0 percent. Assume that he continues to pay the minimum of $25.00. • Use the simulator (you may need to round your total) to determine how long it would take Pie to pay off the new total. c. Using the compound interest simulator at http:// illuminations.nctm.org/ActivityDetail.aspx?ID=172 (be sure to select the Credit Card tab), determine the following: •P aying $25.00 per month, how long would it take Pie to pay off $457.00 at 15.0 percent APR? 6 • What would be his total interest paid for this amount? Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? f. What would be Pie’s total cost for using the credit card? • How long would it take to pay off the credit card? more and more like A = P(e)rt where A is the amount, P is the principal, e is equivalent to approximately 2.71828 (e commemorates Leonard Euler, who discovered this important constant), r is the interest rate, and t is the time in years. This process, in which we look at what happens to a function as a value gets closer and closer either to a specific value or to infinity, is known as taking the limit of the function. The function gets closer and closer to our new formula as the value of n gets larger and larger. This formula is used to calculate interest that is compounded continuously. 17. a. Use the formula above to calculate the total amount of investing a principal of $4,000.00 at 9.0 percent per year compounded for 20 years. • What percentage of the original amount would Pie end up paying in interest? b. How does your answer above compare to the answers in question 14? g. P erhaps now you can see why Sour was concerned about using credit cards. It took Pie almost 2 years to pay off his charges, and that was only if he did not use the credit card again during the same 2-year period. Use the simulator to determine how much Pie would have to pay each month to pay the entire total in 6 months or less. 18. a. Using Sour’s first suggestion to invest, estimate the amount of time and money required to earn the $1,000,000.00. h. In your opinion, what are the advantages and dis advantages of making purchases with credit cards? b. Using a spreadsheet to complete and extend table 6, determine the actual time and money required. Note that you cannot use the compound-interest formula, because you are adding money to the account. 16. In question 14, we used the compound interest formula to find the money saved that was compounded for several periods such as annually, semiannually, quarterly, monthly, and daily. Describe any patterns you see. Some banks compound interest every hour, every minute, every second, every microsecond. That is, they compound continuously. If we look at what happens in the formula for compound interest as the number of time periods becomes larger and larger (because we are decreasing the length of time more and more), we can see that the formula looks 7 c. Pie’s dad said, “Suppose that instead of investing $1,000.00 a year, I gave you that amount to buy different $1.00 lottery tickets. You continued to spend $1,000.00 a year for the same number of years that my investment would take to earn $1 million.” How much would Pie spend on $1.00 lottery tickets? Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? d. “Did you know that 13,983,816 different lottery tickets are possible?” Pie’s father asked. The probability of winning the lottery is the number of tickets purchased divided by the number of tickets possible. What are Pie’s chances of winning the lottery? Can you … • use the “rule of 72” to estimate the amount of time necessary for an amount of money to double if the annual interest rate is known? • explain how compound interest is similar to exponential growth? • compare the properties of linear and exponential functions? Did you know that … e. Using mathematics to support your argument, write a paragraph advising Pie whether he should invest his money or play the lottery. • the populations of countries can be projected by using concepts similar to compounding interest? • health officials can make projections of bacterial counts using concepts similar to compounding interest? • compound interest can be calculated using recursion? • the value of the irrational number e can be determined by the formula used to calculate investing $1.00 at 100.0 percent interest compounded continuously? Mathematical content Simple interest; Compound interest; Exponents; Exponential growth; Functions Student Explorations in Mathematics is published electronically by the National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502. The five issues per year appear in September, November, January, March, and May. Pages may be reproduced for classroom use without permission. Editorial Panel Chair: Co-Editor: Editorial Panel: Field Editor: Board Liaison: Editorial Manager: Production Editor: Production Specialist: 8 Mary Lou Metz, Indiana University of Pennsylvania, [email protected] Cheryl Adeyemi, Virginia State University, [email protected] Darshan Jain, Adlai E. Stevenson High School, Lincolnshire, IL, [email protected] Larry Linnen, University of Colorado–Denver, [email protected] Sharon McCready, Department of Education, Nova Scotia, Canada, [email protected] Anthony Stinson, Clayton State University, Morrow, GA, [email protected] Ed Nolan, Montgomery County Public Schools, Rockville, MD, [email protected] Latrenda Knighten, Melrose Elementary, Baton Rouge, LA, [email protected] Beth Skipper, NCTM, [email protected] Luanne Flom, NCTM Rebecca Totten, NCTM Student Explorations in Mathematics, March 2012 Do You Want to Be a Millionaire? Table 6. Compounding interest and adding more money Year 9 Beginning balance Interest rate Interest earned Additional deposits ($1,000.00 per year) Ending balance 1 $ 5,000.00 10.0% $ 500.00 $1,000.00 $ 6,500.00 2 $ 6,500.00 10.0% $ 650.00 $1,000.00 $ 8,150.00 3 10.0% $1,000.00 4 10.0% $1,000.00 5 10.0% $1,000.00 6 10.0% $1,000.00 7 10.0% $1,000.00 8 10.0% $1,000.00 9 10.0% $1,000.00 10 10.0% $1,000.00 11 10.0% $1,000.00 12 10.0% $1,000.00 13 10.0% $1,000.00 14 10.0% $1,000.00 15 10.0% $1,000.00 16 10.0% $1,000.00 17 10.0% $1,000.00 18 10.0% $1,000.00 19 10.0% $1,000.00 20 10.0% $1,000.00 21 10.0% $1,000.00 22 10.0% $1,000.00 23 10.0% $1,000.00 24 10.0% $1,000.00 25 10.0% $1,000.00 26 10.0% $1,000.00 27 10.0% $1,000.00 28 10.0% $1,000.00 29 10.0% $1,000.00 30 10.0% $1,000.00 31 10.0% $1,000.00 32 10.0% $1,000.00 33 10.0% $1,000.00 34 10.0% $1,000.00 35 10.0% $1,000.00 36 10.0% $1,000.00 37 10.0% $1,000.00 38 10.0% $1,000.00 39 10.0% $1,000.00 40 10.0% $1,000.00 41 10.0% $1,000.00 42 10.0% $1,000.00 43 10.0% $1,000.00 44 10.0% $1,000.00 45 10.0% $1,000.00 Student Explorations in Mathematics, March 2012

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