Problem 52 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 2500\) is invested at an interest rate of 2.5\(\%\) per year, compounded daily, find the value of the investment after the given number of years. $$ \begin{array}{llll}{\text { (a) } 2 \text { years }} & {\text { (b) } 3 \text { years }} & {\text { (c) } 6 \text { years }}\end{array} $$

Step-by-Step Solution

Answer

2 years: $2628.09; 3 years: $2694.43; 6 years: $2896.25.

Step 1: Understand the Compound Interest Formula

The formula for compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after \( n \) years, including interest. \( P \) is the principal amount (\$2500), \( r \) is the annual interest rate (written as a decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time in years.

Step 2: Identify Each Parameter for Calculations

For our problem: \( P = 2500 \), \( r = 0.025 \) (since 2.5\% equals 0.025 as a decimal), \( n = 365 \) because the interest is compounded daily, and we'll be calculating values for \( t = 2 \), \( t = 3 \), and \( t = 6 \) years.

Step 3: Calculate for 2 Years

Plug the values for \( t = 2 \) into the formula: \[ A = 2500 \left(1 + \frac{0.025}{365}\right)^{365 \times 2} \]. Calculating inside the parentheses first gives us a value of approximately 1.0000685. Raising this to the power of 730 (\( 365 \times 2 \)) results in approximately 1.0512, and multiplying by 2500 gives \( A \approx 2628.09 \).

Step 4: Calculate for 3 Years

Use the formula with \( t = 3 \): \[ A = 2500 \left(1 + \frac{0.025}{365}\right)^{365 \times 3} \]. The value inside the parentheses remains the same (approximately 1.0000685). Raised to the power of 1095 (\( 365 \times 3 \)), it becomes approximately 1.0778. Multiplying by 2500 results in \( A \approx 2694.43 \).

Step 5: Calculate for 6 Years

Substitute \( t = 6 \) into the formula: \[ A = 2500 \left(1 + \frac{0.025}{365}\right)^{365 \times 6} \]. With the same value in the parentheses, raised now to the power of 2190 (\( 365 \times 6 \)), you get approximately 1.1585. Multiplying by 2500 gives \( A \approx 2896.25 \).

Step 6: Compile the Results

After calculating each scenario, the values are: (a) After 2 years, approximately \\(2628.09; (b) After 3 years, approximately \\)2694.43; (c) After 6 years, approximately \$2896.25.

Key Concepts

Investment Calculation

Investment calculation is a fundamental concept in finance that involves determining the future value of an initial investment after a certain period. This requires understanding key variables like the principal amount, interest rates, and the duration of investment.

• When calculating investments, a critical factor is the type of interest applied, such as compound or simple interest.
• Compound interest, which is more common, will typically yield higher returns as it accumulates on both the initial principal and prior interest.
• Understanding the frequency of compounding, whether daily, monthly, or annually, is crucial as it significantly impacts the investment's growth.

Whether saving for retirement, purchasing an asset, or securing future funds, mastering investment calculation is essential for evaluating financial goals.

Interest Rate

The interest rate is the percentage at which your investment grows annually. It represents the cost of borrowing money or the reward for saving.

• In our example, the interest rate is 2.5% per annum, reflecting a low-risk, conservative investment option.
• An easy way to convert a percentage to a decimal is by dividing the interest rate by 100, so 2.5% becomes 0.025.
• It is important to consider how often the interest is applied, or compounded, as this affects how quickly your investment grows.

Interest rates vary across different financial products, and understanding them helps in making informed investment choices.

Compound Interest Formula

The compound interest formula is a mathematical equation used to calculate the amount of interest earned on an initial investment over time. It is expressed as:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]

• \(A\) is the final amount, including the principal and the interest.
• \(P\) stands for the principal amount, or the initial sum of money invested.
• \(r\) represents the annual interest rate (as a decimal), and \(n\) is the number of times interest is compounded per year.
• \(t\) is the investment period expressed in years.

By substituting the given parameters into the formula, you can compute the future value of the investment, providing insight into its potential growth.

Principal Amount

The principal amount is the initial sum of money you start with in an investment. In this example, the principal amount is $2500, serving as the base for calculating growth via compound interest.

• The larger the principal, the more interest you can potentially earn, making it an essential component in investment decisions.
• The principal remains fixed unless additional contributions are made, affecting the total return.
• Tracking changes to the principal through withdrawals or additional deposits can alter investment outcomes.

Understanding the role of the principal amount helps in making strategic financial decisions and optimizing investment results.