Problem 75 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 10,000\) is invested at an interest rate of 10\(\%\) per year, compounded semiannully, 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: $16,288.95; after 10 years: $26,532.98; after 15 years: $43,219.42.

Step 1: Understanding the Problem

The problem involves calculating the future value of an investment using the compound interest formula. Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. Here, we will use the formula for compound interest, where the interest is compounded semiannually.

Step 2: Compound Interest Formula

The formula for compound interest is: \[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where:- \(A\) is the future value of the investment/loan, including interest.- \(P\) is the principal investment amount (the initial deposit or loan amount).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the number of years the money is invested or borrowed for.

Step 3: Applying to Part (a): 5 Years

For part (a), we have:- \(P = 10,000\)- \(r = 10\% = 0.10\)- \(n = 2\) (since interest is compounded semiannually)- \(t = 5\)Plug these values into the formula:\[A = 10,000 \left(1 + \frac{0.10}{2}\right)^{2 \times 5} = 10,000 \left(1 + 0.05\right)^{10} = 10,000 \times 1.62889463 \]After calculating, we find that \(A \approx 16,288.95\).

Step 4: Applying to Part (b): 10 Years

For part (b), the values are:- \(P = 10,000\)- \(r = 0.10\)- \(n = 2\)- \(t = 10\)Plug these values into the formula:\[A = 10,000 \left(1 + \frac{0.10}{2}\right)^{2 \times 10} = 10,000 \left(1 + 0.05\right)^{20} = 10,000 \times 2.65329771 \]After calculating, we find that \(A \approx 26,532.98\).

Step 5: Applying to Part (c): 15 Years

For part (c), the values are:- \(P = 10,000\)- \(r = 0.10\)- \(n = 2\)- \(t = 15\)Plug these values into the formula:\[A = 10,000 \left(1 + \frac{0.10}{2}\right)^{2 \times 15} = 10,000 \left(1 + 0.05\right)^{30} = 10,000 \times 4.32194238\]After calculating, we find that \(A \approx 43,219.42\).

Step 6: Conclusion

Based on our calculations, the values of the investment after different years, compounded semiannually, are as follows:- After 5 years: \(\(16,288.95\)- After 10 years: \(\)26,532.98\)- After 15 years: \($43,219.42\)

Key Concepts

Investment Growth

Investment growth through compound interest is like planting a tree and watching it grow taller every year. Unlike simple interest which only applies to the initial amount or principal, compound interest adds interest on interest, causing your investment to grow at an increasing rate over time. This is why investments can seem to "snowball" over the years, becoming much larger than they would with simple interest.

For example, if you invest $10,000 at an interest rate of 10% per annum compounded semiannually, your investment would grow more significantly than with simple interest. Each period, the interest is calculated on an ever-growing total: principal plus accrued interest. This process results in a better return on your investment, making compound interest a powerful tool for long-term wealth accumulation.

Interest Rate Calculation

Calculating the correct interest rate in a compound interest context is essential for determining investment growth. The annual interest rate needs to be divided by the number of compounding periods per year to find the rate per period. This is because compound interest assumes that interest is added multiple times per year.

Consider this example: With an annual interest rate of 10% and semiannual compounding, you would divide 0.10 by 2. This gives you an effective interest rate per period of 5% or 0.05. The number of periods is then multiplied by the number of years to find the total number of compounding periods. In essence, this calculation defines how frequently interest is added to the investment, impacting overall growth.

Future Value Formula

The future value formula helps to calculate how much an investment will be worth in the future, considering compound interest. The formula is given by: \[A = P \left(1 + \frac{r}{n}\right)^{nt}\]Where:

• \(A\) is the future value of the investment
• \(P\) is the principal or initial amount invested
• \(r\) is the annual interest rate (as a decimal)
• \(n\) is the number of times interest is compounded per year
• \(t\) is the time period in years

This formula accounts for both the principal and the interest that has accumulated over time. For instance, using this formula, a \(10,000 investment at a 10% annual rate, compounded semiannually over 5 years, gives a future value of about \)16,288.95. This reflects the power of compounding, as the interest impacts the total balance progressively.