PreCalculus
Naturally Interesting
Name _________________________ #_____
Banks and investment companies offer a variety of accounts to help customers reach their financial
goals. These accounts may offer different rates of interest, based on the initial amount invested.
How much money would a person need to invest today in order to be a millionaire at age 65? How
long will it take an investment to double in value? These are questions financial advisers must be
able to answer for their clients.
In banking, principal refers to the amount of money invested or loaned. Interest is the amount
earned on invested money, or the fee charged for loaned money. The amount of interest received or
paid depends on three quantities: principal, interest rate, and time. Interest also varies according to
the method used to calculate it. In the following activities, you investigate how savings accounts earn
money.
________________________________________________________________________________
Mathematics Note
One method for determining the amount of interest earned or owed involves simple interest. In this
case, interest is paid or charged only on the original principal. The formula for calculating simple
interest, where I represents interest, P represents principal, r represents the interest rate time period,
and t represents the number of time periods, is I = Prt.
To use this formula, t must be expressed in the same units as time in the interest rate, r. For
example, if the interest rate is 5% per year, then t must be expressed in years. If $1000 is invested at
an annual interest rate of 5% for 3 years, the interest earned can be calculated as:
I = 1000(0.05)(3) = $150.
Most savings accounts pay interest not only on the original principal, but also on the interest earned
and deposited in any previous time periods. This is an example of compound interest. Each time
compound interest is calculated, the interest earned is added to the principal. This sum (the account
balance) becomes the new principal for the next interest calculation.
EXPLORATION 1
Suppose you deposit $500 in a savings account at 6% interest compounded annually. How much
money would you have in 5 years? Complete the table below.
Year
1
Account Balance
Develop an equation for calculating the total amount you
would have in your account at the end of n years.
Year 1:
2
3
Year 2:
4
5
Year 3:
Year n:
Year 20:
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Mathematics Note
When interest is compounded annually, the yearly account balances that result can be defined by the
following formula (assuming that no withdrawals are made and any interest earned is deposited in the
account):
Pt = P0(1 + r)t
Where Pt is the account balance after t years, P0 is the initial principal, r is the annual interest rate,
and t is the time in years.
For example, given an initial principal of $2000 and an annual interest rate of 4%, compounded
annually, the account balance after 12 years can be determined as follows:
P12 = 2000(1 + 0.04)12
≈ 3202.06
EXPLORATION 2
Now we will examine how the number of compoundings per year affects the amount of interest
earned in an account.
Complete the following table. Investigate how the number of compounding periods affects the
balance at the end of the year for $500 invested at 6% annually.
Compounding
Number per year
Account Balance
Annually
1
$530.00
Semiannually
Quarterly
Monthly
Daily
Hourly
By the minute
By the second
Predict the account balance after one year if interest is compounded continuously. _____________
Choose different values for the initial principal and the interest rate. Complete the chart below.
How is the balance affected
as the number of
compounding increase?
P0
r
Compoundings
1
2
4
12
365
8760
525600
31536000
Balance
Balance
When compounding interest c times per year for t years, the formula for the account balance after n
compounding periods is:
𝑟 𝑛
𝑟 𝑐𝑡
𝑃𝑛 = 𝑃0 (1 + ) = 𝑃0 (1 + )
𝑐
𝑐
where 𝑃𝑛 represents the principal after n compounding periods, 𝑃0 represents the initial principal, and
r is the annual interest rate. (Note: In this formula, n = ct).
For example, consider an initial investment of $1000 at an annual interest rate of 5%, compounded
quarterly. Assuming that no withdrawals are made and any interest earned is deposited in the
account, the principal after 3 years (or 4 x 3 = 12 compounding periods) can be calculated as follows:
𝑃12 = 1000 (1 +
0.05 12
) = $1160.75
4
EXPLORATION 3
With the development of calculators and computers, the determination of compound interest has
become quick and easy. This allows banks to compound interest on an account balance up to the
instant in which it is withdrawn. This method of calculating interest, known as compounding
continuously, mean that the number of compoundings per year approaches infinity.
As you have seen in previous explorations, the number of compoundings per year can affect the
balance of the savings account. What happens to this balance when the number of compoundings
increases without bound? In this exploration, you will investigate what happens as c, the number of
compoundings, changes for specific values of Po, r, and t.
Consider an investment of $1.00 at an annual interest rate of 100%, compounded continuously, for 1
year. Use the formula for account balance when interest in compounded c times a year to complete
the table for such an investment.
Number of Compoundings
per year (c)
1
Account Balance at the
end of the year ($)
As the number of compoundings per year
increases, what happens to the sequence
of account balances?
10
100
1000
10000
100000
1000000
10000000
100000000
Since P0 = 1, r = 1, and t = 1 in this situation, a formula for this data is:
In the context of this problem, c is a non-negative integer. However, this formula can be represented
more generally as the function:
1 𝑥
𝑓(𝑥) = (1 + )
𝑥
What are the domain and range of this function?
Graph the function below on your calculator. As x increases, what limiting value does the graph
appear to approach?
1 𝑥
The limiting value of 𝑓(𝑥) = (1 + 𝑥) is the irrational number e. The value of e can be represented
mathematically as:
Let’s explore how changing the interest rate affects the value of e.
n
1 𝑛
𝑃𝑛 = (1 + )
𝑛
2 𝑛
𝑃𝑛 = (1 + )
𝑛
3 𝑛
𝑃𝑛 = (1 + )
𝑛
Calculate e2 and e3.
1
10
100
1000
How do these
values compare to
the values in the
table?
10000
100000
1000000
10000000
100000000
𝑟 𝑐𝑡
At the beginning of this activity, we use the formula 𝑃𝑡 = 𝑃0 (1 + ) to calculate an account balance
𝑐
based on initial deposit, interest rate, number of compoundings, and time. If we know the account is
going to be compounded continuously, we can modify our account balance formula to:
Consider an initial investment of $500 at an annual interest rate of 6%, compounded continuously.
Assuming no withdrawals are made and any interest earned is deposited in the account, the account
balance after 5 years can be calculated as follows: P5 = 500e(0.06)(5) = $674.93