Problem 51 - Algebra and Trigonometry

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. \(\begin{array}{llll}{\text { (a) } 5 \text { years }} & {\text { (b) } 10 \text { years }} & {\text { (c) } 15 \text { years }}\end{array}\)

Step-by-Step Solution

Answer

After 5 years: $11,604.41; after 10 years: $13,439.16; after 15 years: $15,561.12.

Step 1: Understand the Problem

We know the principal amount is $10,000, the interest rate is 3% per annum, compounded semiannually. We need to find the value of the investment for different years.

Step 2: Identify the Compound Interest Formula

The formula for compound interest is \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \( A \) is the amount, \( P \) is the principal amount, \( r \) is the annual interest rate (as a decimal), \( n \) is the number of times interest is compounded per year, and \( t \) is the number of years.

Step 3: Calculate for 5 Years

Using the formula, - \( P = 10000 \), \( r = \frac{3}{100} = 0.03 \), \( n = 2 \), \( t = 5 \).- Substitute these into the formula: \[A = 10000 \left(1 + \frac{0.03}{2}\right)^{2 \times 5}\]- Calculate:\[ A = 10000 \times \left(1 + 0.015\right)^{10} = 10000 \times 1.015^{10} \approx 11604.41 \]

Step 4: Calculate for 10 Years

Now for 10 years, - \( P = 10000 \), \( r = 0.03 \), \( n = 2 \), \( t = 10 \).- Use the formula: \[A = 10000 \left(1 + \frac{0.03}{2}\right)^{2 \times 10}\]- Calculate:\[ A = 10000 \times 1.015^{20} \approx 13439.16 \]

Step 5: Calculate for 15 Years

Finally for 15 years, - \( P = 10000 \), \( r = 0.03 \), \( n = 2 \), \( t = 15 \).- Use the formula: \[A = 10000 \left(1 + \frac{0.03}{2}\right)^{2 \times 15}\]- Calculate:\[ A = 10000 \times 1.015^{30} \approx 15561.12 \]

Key Concepts

Investment

When you decide to grow your money, the initial amount you set aside is your investment. In financial terms, this is often called the "principal." For example, if you start with $10,000, this amount is your starting point. Investments can grow over time with the help of interest.
This initial sum of money is crucial because it determines how much you might earn in the future. Choosing the right kind of investment, like saving accounts, stocks, or bonds, depends on your financial goals and risk tolerance.
Understanding the power of an investment is vital as it sets the foundation for future wealth growth.

Interest Rate

The interest rate is a percentage that shows how much you earn on your investment over a year. In our example, the interest rate is 3% annually. This rate determines how fast your money will grow.
Interest can be simple, calculated only on the principal, or compound, calculated on both the principal and previously earned interest. Small changes in the interest rate can significantly affect your investment's outcome.

• A higher interest rate leads to more earnings.
• A lower interest rate means slower growth potential.

As you can see, selecting the right interest rate is crucial when planning your finances.

Compounded Semiannually

Compounding refers to the process where your investment earns interest on both the initial amount and the accumulated interest over time. When an investment is compounded semiannually, it means the interest is calculated and added to the principal twice a year.
This process accelerates growth because as your investment grows, the interest per period increases due to the higher total value. Here's how it works:

• After six months, interest is added to the principal.
• After another six months, interest is calculated on the new total (principal + interest).

This compounding strategy can lead to higher returns compared to annual compounding.

Financial Mathematics

Financial mathematics is the study of applying mathematical methods to solve financial problems. It helps calculate interest, investments, loans, and other financial matters. In our context, we use the compound interest formula.
This formula, \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \], helps determine the future value of an investment given it is invested at a particular interest rate and compounding frequency.

• \( A \) is the amount after time \( t \).
• \( P \) is the principal investment.
• \( r \) is the annual interest rate.
• \( n \) is the number of times interest is compounded per year.
• \( t \) is the number of years the money is invested.

Mastering financial mathematics equips you to make informed investment decisions, thus optimizing your financial growth.