Problem 51 - Precalculus Mathematics for Calculus

Question

Compound Interest If \(\$ 10,000\) is invested at an interest rate of \(3 \%\) per year, compounded semiannually, find the value of the investment after the given number of years. (a) 5 years (b) 10 years (c) 15 years

Step-by-Step Solution

Answer

After 5 years: $11,609. After 10 years: $13,449. After 15 years: $15,569.

Step 1: Identify the Compound Interest Formula

We will use the compound interest formula, which 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 (initial investment), \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time in years.

Step 2: Determine Values for Variables

Given:- Principal (\( P \)) = 10,000 - Annual interest rate (\( r \)) = 3% or 0.03 as a decimal - Compounded semiannually (\( n \)) = 2 times a year

Step 3: Calculate for 5 years

Using the formula \( A = 10000 \left(1 + \frac{0.03}{2}\right)^{2 \times 5} \), calculate:\( A = 10000 (1.015)^{10} = 10000 \times 1.1609 = 11609 \). The value after 5 years is approximately \( \$ 11609 \).

Step 4: Calculate for 10 years

Apply the formula for 10 years:\( A = 10000 \left(1 + \frac{0.03}{2}\right)^{2 \times 10} \), compute:\( A = 10000 (1.015)^{20} = 10000 \times 1.3449 = 13449 \). The value after 10 years is approximately \( \$ 13449 \).

Step 5: Calculate for 15 years

Use the formula for 15 years:\( A = 10000 \left(1 + \frac{0.03}{2}\right)^{2 \times 15} \), then:\( A = 10000 (1.015)^{30} = 10000 \times 1.5569 = 15569 \). The value after 15 years is approximately \( \$ 15569 \).

Key Concepts

Compound Interest Formula

The compound interest formula is a powerful tool that helps calculate the growth of an investment over time when interest is added to the principal balance at regular intervals. This formula is useful because it provides a clear picture of how much an investment will be worth in the future. The formula is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• A is the future value of the investment, including interest.
• P is the principal investment amount (the initial deposit).
• r is the annual interest rate (in decimal form, so 3% becomes 0.03).
• n is the number of times interest is compounded per year.
• t is the number of years the money is invested.

By inputting the values for the principal, rate, and time, this formula lets you estimate how much your initial investment will grow with compound interest.

Semiannual Compounding

Semiannual compounding refers to the process of calculating interest twice a year. It's one of several compounding frequencies that could include daily, monthly, quarterly, or annually. When interest is compounded semiannually:

• The interest is added to the principal two times a year.
• Each compounding period adds interest according to the formula provided, which gives a compounding frequency \( n \) of 2.

For example, with an annual interest rate of 3%, the effective rate per period would be 1.5% because 3% ÷ 2 = 1.5%. In the formula, this is reflected as:\[ A = P \left(1 + \frac{0.03}{2}\right)^{2t} \]Over 5, 10, or 15 years, semiannual compounding significantly boosts the total interest earned compared to simple annual compounding.

Investment Growth

Investment growth under compound interest happens as interest is earned on both the initial principal and the accumulated interest from previous periods. This is what makes compound interest so beneficial to long-term investments. With time:

• The amount of interest earned each period increases because the base amount — the principal plus accumulated interest — grows.
• The effects of compounding are more pronounced over longer periods, meaning investments grow at a faster rate.

For instance, investing $10,000 at 3% interest compounded semiannually results in the following amounts:

• After 5 years, approximately $11,609.
• After 10 years, approximately $13,449.
• After 15 years, approximately $15,569.

Thus, the longer the investment period, the greater the total returns, illustrating the substantial impact of compound interest on investment growth.