Problem 64 - Precalculus : Building Concepts and Connections

Question

For an initial deposit of \(\$ 1500\) , find the total amount in a bank account after 5 years for the interest rates and compounding frequencies given. 3% compounded quarterly

Step-by-Step Solution

Answer

The total amount in the bank account after 5 years is approximately $1,733.01

Step 1: Define Given Variables

The problem gives several variables. The principal amount \( P \) is $1500, the annual interest rate \( r \) is 3% or 0.03 in decimal, the time \( t \) is 5 years, and the compounding frequency \( n \) is quarterly or 4 times per year.

Step 2: Substitute Variables into Compound Interest Formula

Next, substitute the defined values into the compound interest formula \( A = P (1 + r/n )^{nt} \).This becomes \( A = 1500 (1 + 0.03/4 )^{4 * 5} \).

Step 3: Calculate the Total Amount after 5 Years

Now, calculate the amount using the given values. After solving, the total amount in the bank account after 5 years will be found.

Key Concepts

Principal Amount

The principal amount is the initial sum of money deposited or invested in a financial account. It is the starting point for calculating interest, whether it is simple or compound. In this exercise, the principal amount \( P \) is \\(1500\\).
For any investment or savings plan, understanding the principal amount is vital because it determines the base that grows with interest accumulation.

• When compounded, the principal amount can significantly increase over time.
• It's essential to verify this amount to ensure calculations are correct from the beginning.

In financial contexts, always ensure you know your starting investment, as it impacts the outcome of your savings strategy.

Interest Rate

The interest rate is a percentage that defines how much interest will be added to the principal amount over a specified period. In this case, it is 3% per year, expressed as a decimal \( r = 0.03 \).
This rate determines the growth rate of your investment or savings over time. A higher interest rate generally means more earned interest.

• Interest rates can be annual, semi-annual, or for shorter periods.
• Expressing the rate as a decimal is essential for calculations in formulas.

In practices like loans or savings accounts, always be mindful of the interest rate as it directly affects the growth or cost of the financial product.

Compounding Frequency

Compounding frequency refers to how often the accrued interest is added to the principal amount. In this example, interest is compounded quarterly, meaning that it is calculated and added four times a year \( n = 4 \).
The compounding frequency can significantly affect the total interest earned or paid.

• Interest can be compounded yearly, semi-annually, quarterly, or even daily.
• More frequent compounding results in more effective interest accumulation.

Hence, understanding the compounding frequency helps in maximizing or minimizing the future value of the investment or loan.

Time Period

The time period is the length of time the money is invested or borrowed. In our exercise, the time \( t \) is 5 years.
It plays a critical role in calculating compound interest, as it is a component of the exponential factor in the formula.

• The longer the time period, the more interest will accumulate.
• Ensure you always match the time period with the interest rate period in calculations.

Knowing the timeframe of your investment helps in planning and estimating how your savings will grow or what a loan will cost over time.