Question

Two bank accounts are opened at the same time. The first has a principal of $$ 1000$ in an account earning 5 % compounded monthly. The second has a principal of 2000 in an account earning 4 % interest compounded annually. Determine the number of years, to the nearest tenth, at which the account balances will be equal.

Short answer

Approximately 35.0 years

Step-by-step solution

Define the compound interest formulas

The compound interest formula is given by where: - \(A\) is the amount of money accumulated after n years, including interest. - \(P\) is the principal amount. - \(r\) is the annual interest rate (decimal). - \(n\) is the number of times that interest is compounded per year. - \(t\) is the number of years.

Set up equations for each account

For the first account: - Initial principal \(P_1 = 1000\) - Annual interest rate \(r_1 = 0.05\) - Compounded monthly \(n_1 = 12\) - We use the formula: to get For the second account: - Initial principal \(P_2 = 2000\) - Annual interest rate \(r_2 = 0.04\) - Compounded annually \(n_2 = 1\) - We use the formula: to get

Set the equations equal to each other

To find when balances are equal we set equal to and solve for \(t\):

Solve the equation

To find \(t\), isolate \(t\) in the equality then: Using logarithms: Equate the exponentiation, further solve for \(t\):

Calculate the value of t

Using a calculator to approximate: Therefore, \(t\approx 35.0\) years.

Key concepts

Annual Interest Rate

The annual interest rate is the percentage increase on the principal amount in one year. It's expressed as a decimal for calculations. For example, a 5% annual interest rate is represented as 0.05. The annual interest rate influences how much money will grow over time.
It's crucial to understand that the interest rate can vary depending on how frequently the interest is compounded. Compounding means calculating interest on both the initial principal and the accumulated interest from previous periods.

Compound Annually

When interest is compounded annually, it means the interest is added to the principal once per year. The formula used in this case is:
\[A = P(1 + r/n)^(nt)\]
However, when compounded annually, \(n=1\) simplifying the formula to:
\[A = P(1 + r)^t\]
In the provided exercise, the second bank account earns a 4% annual interest compounded yearly:
\(P_2 = 2000\), \(r_2 = 0.04\), \(n_2 = 1\)
So the formula becomes:
\[A = 2000(1 + 0.04)^t\]

Monthly Compounding

Monthly compounding means the interest is added twelve times a year, once at the end of every month. This method increases how fast the amount grows compared to annual compounding. The formula in such cases is:
\[A = P \bigg(1 + \frac{r}{n} \bigg)^{nt}\]
In the exercise, the first account uses monthly compounding with:\(P_1 = 1000\), \(r_1 = 0.05\), \(n_1 = 12\).
So, the formula we use becomes:
\[A = 1000 \bigg(1 + \frac{0.05}{12}\bigg)^{12t}\]
Thanks to more frequent compounding, the interest grows quicker even if the annual rate is the same.

Principal Amount

The principal amount is the initial sum of money put into an account or investment. Think of it as the starting point. In our exercise, the principal amounts are \(1000 and \)2000 for the two respective accounts. These principals will grow based on their respective interest rates and compounding frequencies.
When calculating compound interest, we factor in the principal amount to see how it will grow over time. Adjusting the principal impacts the final amount significantly. Always remember:

• A higher principal means more money to earn interest on.
• The principal remains the base for all interest calculations.