 # Do You Want to Be a Millionaire? Student Explorations in Mathematics

```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.
and Pie found out.
Sandra M. Powers, College of Charleston, SC 29424 (original submission contributor)
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
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
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
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
= 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?
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.
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
\$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
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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
(\$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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