Problem 53 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 500\) is invested at an interest rate of 3.75\(\%\) per year, compounded quarterly, find the value of the investment after the given number of years. $$ \begin{array}{llll}{\text { (a) } 1 \text { year }} & {\text { (b) } 2 \text { years }} & {\text { (c) } 10 \text { years }}\end{array} $$

Step-by-Step Solution

Answer

(a) \$518.95, (b) \$538.33, (c) \$717.99.

Step 1: Identify the Formula for Compound Interest

We use the compound interest formula to solve this problem. The formula is given by:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(A\) is the amount of money accumulated after \(n\) years, including interest. \(P\) is the principal amount (the initial amount), \(r\) is the annual interest rate (decimal), \(n\) is the number of times interest is compounded per year, and \(t\) is the time the money is invested for in years.

Step 2: Input Values for (a) 1 Year

Given that \(P = 500\), \(r = 0.0375\), \(n = 4\) (because interest is compounded quarterly), and \(t = 1\), we plug these values into the formula. \[ A = 500 \left(1 + \frac{0.0375}{4}\right)^{4 \times 1} \]

Step 3: Calculate (a) 1 Year Value

First, calculate \( \frac{0.0375}{4} = 0.009375 \) and then substitute it back into the formula:\[ A = 500 \left(1 + 0.009375\right)^4 \]Simplify and compute further:\[ A = 500 \times (1.009375)^4 \]Using a calculator, \( (1.009375)^4 \approx 1.037908 \), therefore:\[ A \approx 500 \times 1.037908 \approx 518.954 \]. So, the value after 1 year is approximately \$518.95.

Step 4: Input Values for (b) 2 Years

Using the same initial values of \(P = 500\), \(r = 0.0375\), and \(n = 4\), but changing \(t = 2\), we plug these new values into the formula:\[ A = 500 \left(1 + \frac{0.0375}{4}\right)^{4 \times 2} \]

Step 5: Calculate (b) 2 Years Value

Calculate \( A = 500 \left(1 + 0.009375\right)^8 \). Using a calculator, \( (1.009375)^8 \approx 1.076660 \), thus:\[ A \approx 500 \times 1.076660 \approx 538.33 \]. So, the value after 2 years is approximately \$538.33.

Step 6: Input Values for (c) 10 Years

For 10 years, the values are \(P = 500\), \(r = 0.0375\), \(n = 4\), and \(t = 10\). Substitute into the formula:\[ A = 500 \left(1 + \frac{0.0375}{4}\right)^{4 \times 10} \]

Step 7: Calculate (c) 10 Years Value

We calculate \( A = 500 \left(1 + 0.009375\right)^{40} \). Using a calculator, we find \( (1.009375)^{40} \approx 1.435974 \), thus:\[ A \approx 500 \times 1.435974 \approx 717.99 \].Therefore, the value after 10 years is approximately \$717.99.

Key Concepts

Investment Growth

Investment growth is the increase in value of an initial sum of money, known as the principal, over time. This growth occurs due to accumulated interest. Understanding how money grows is crucial in financial planning. It helps individuals plan their investments reliably to meet future financial goals.

• The initial amount, or principal, is the seed money that starts the investment journey.
• Growth happens because of two main components: the interest rate and the time invested.
• The longer the money stays invested, and the higher the interest rate, the more growth is likely to be seen.

Consider a scenario where you invest $500 at an interest rate of 3.75% per annum. As time progresses, your investment benefits from compound interest, increasing its value significantly from the initial amount. Whether you intend to leave your investment for 1 year or extend it to 10 years, your money's growth will reflect this duration.

Interest Rate Calculations

Interest is the fee paid for borrowing money or the income made from letting someone else use your money. Interest rate calculations determine how much interest your money will earn over a certain period.
The key formula for calculating compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• \( A \) is the final amount including the interest.
• \( P \) is the principal or starting amount.
• \( r \) is the annual interest rate expressed as a decimal.
• \( n \) is the number of compounding periods per year.
• \( t \) is the time in years.

To calculate the accumulated value after 1, 2, or 10 years, you adjust \( t \) for the respective years. The interest rate is divided by the number of compounding periods (in our case four for quarterly), which affects how quickly the interest grows. For the given exercise, substituting the known values simplifies the process, turning abstract numbers into tangible financial growth.

Quarterly Compounding

Quarterly compounding refers to the process of calculating interest four times a year, at each quarter. It accelerates investment growth compared to annually compounded interest, because it accounts for accumulated interest more frequently.
Understanding the benefits of quarterly compounding can significantly impact investment decisions:

• Investing with quarterly compounding means interest is calculated every three months.
• The interest earned each quarter begins to earn additional interest in subsequent periods.
• This results in higher returns over time compared to less frequent compounding intervals, like annually or semi-annually.

In our exercise, using quarterly compounding, interest is calculated every 3 months. This compounding frequency boosts the total value of the investment over the years. The formula adjusts for quarterly compounding by dividing the annual rate and multiplying the time by the number of periods, thus exploiting the full power of compounded growth. As seen from the calculations, each period incrementally increases the investment value faster than it would if compounded less frequently.