Problem 61 - 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 annually

Step-by-Step Solution

Answer

The total amount in the bank account after 5 years will be \$2002.54.

Step 1: Identify the Known Values

The known values are: the initial principal deposit \( P = $1500 \), the annual interest rate \( r = 6\% = 0.06 \) (after converting the percentage into a decimal), the number of times the interest is compounded in a year \( n = 1 \) (since it is compounded annually), and the duration of time in which the money is left in the account \( t = 5 \) years.

Step 2: Use the Formula to Compute the Total Amount

The formula for compound interest is \( A = P(1 + r/n)^{nt} \). Substituting all the known values: \( A = $1500 * (1 + 0.06/1)^{(1*5)} \).

Step 3: Compute the Total Amount

Perform the calculation: \( A = $1500 * (1 + 0.06)^5 = $1500 * 1.06^5 \) . Calculation yields \( A = $2002.54 \). This is the total amount in the account after 5 years.

Key Concepts

Annual Interest Rate

When you think about growing your savings or investments, the annual interest rate plays a huge role. This rate tells you how much your initial amount, or principal, will grow each year. For example, in the exercise, an annual interest rate of 6% means that the money is expected to increase by 6% over the whole year. But what does 6% really mean in calculations?

First, you need to convert the percentage to a decimal for use in formulas. For a 6% interest rate, you divide by 100 to get 0.06. This decimal is what you use in interest formulas, such as the compound interest formula. Having a higher annual interest rate usually means your money grows faster, but it also depends on how often the interest is applied. Make sure to consider this rate carefully when planning your investments.

Principal Deposit

The principal deposit is the starting point of any investment or savings account. It's the initial amount you put into the account, hoping it will grow over time. In our example exercise, the principal deposit is $1500. Think of the principal as the seed you plant.

• The larger the principal, the more potential you have to earn from interest.
• This is because interest is typically calculated as a percentage of the principal.

To get the most from your principal, consider factors like the interest rate and how often interest is compounded. Remember, your principal is the foundation of your future earnings.

Compounding Frequency

Compounding frequency indicates how often the interest is calculated and added to the account. This concept is crucial in determining how much your money will grow over time. In our exercise, the compounding frequency is annual, meaning the interest is calculated once a year.

• Annual compounding means the interest is calculated once per year.
• More frequent compounding (e.g., quarterly or monthly) can lead to higher returns.

The compound interest formula, \( A = P(1 + r/n)^{nt} \), displays the importance of compounding frequency. Here, \( n \), the frequency, affects how often interest gets added to the principal.

Understanding this factor helps in realizing how your money will grow. For long-term savings, frequent compounding can make a noticeable difference!