Problem 63 - 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. 6% compounded monthly

Step-by-Step Solution

Answer

The total amount in the bank account after 5 years is approximately $2007.98.

Step 1: Identify Given Values

Obtain the values from the problem needed for the formula. This includes: initial deposit (P) $1500, interest rate (r) 6%, time in years (t) 5 years and number of times compounded per year (n) 12 times.

Step 2: Convert Rate to Decimal

Convert the interest rate from a percentage into a decimal by dividing by 100. So, \(r=6/100=0.06\).

Step 3: Substituting values in the formula

Substitute the values into the compound interest formula \(A = P(1 + r/n)^{nt}\), which becomes \(A = 1500(1 + 0.06/12)^{12*5}\).

Step 4: Calculate the Value

Calculate the total amount in the bank account after 5 years which is approximately $2007.98.

Key Concepts

Interest Rate

Interest rate is a fundamental concept in finance, reflecting the cost of borrowing money or the reward for saving. In the context of our exercise, the interest rate is 6%.

**Understanding Interest Rates** means knowing how they affect the growth of your money. When you deposit money in a bank account, the interest rate determines how much extra money you'll earn over time. For instance:

• A higher interest rate will speed up the growth of your savings.
• A lower interest rate will slow down this growth.

For our problem, the interest rate is applied to an initial deposit, helping the money to grow over a specified period.

To use it in calculations, the annual interest rate must be converted to a decimal. In our solution, 6% becomes 0.06 when divided by 100. This decimal form is key to applying the compound interest formula.

Compounding Frequency

Compounding frequency is the number of times interest is calculated and added to the initial deposit within a year. This is a crucial factor in determining how quickly your savings grow. In typical scenarios, interest can be compounded annually, semi-annually, quarterly, monthly, daily, or even continuously.

Our exercise uses a **monthly compounding frequency**. This means that the interest is calculated and added to the balance every month. This impacts the total growth because:

• More frequent compounding results in more frequent interest calculations.
• With each calculation, interest is earned on the previously added interest, accelerating growth.

The number of compounding periods per year (n) is 12 for monthly compounding. This frequent addition of interest is beneficial because it contributes to a faster accumulation of wealth compared to less frequent compounding, such as annually.

Initial Deposit

The initial deposit, also known as the principal (P), is the starting amount of money placed into a savings or investment account. In our given problem, the initial deposit is $1500.

**The Role of Initial Deposit** in compound interest calculations is pivotal because:

• It is the base amount on which interest is calculated.
• The larger the initial deposit, the more interest generated over time.

When you initially deposit money, it sets the foundation for future growth. Over time, the interest earned on the initial deposit can significantly increase the total amount in the account, especially with frequent compounding and significant interest rates.

Understanding how the initial deposit interacts with the interest rate and compounding frequency helps you to systematically predict how your savings will grow in the future.