Problem 78 - Algebra and Trigonometry

Question

Nancy wants to invest \(\$ 4000\) in saving certificates that bear an interest rate of 9.75\(\%\) per year, compounded semiannully. How long a time period should she choose to save an amount of \(\$ 5000 ?\)

Step-by-Step Solution

Answer

Nancy should choose a period of about 2.36 years to save $5000.

Step 1: Identify Key Values

In this exercise, identify the initial investment (principal) as \( P = 4000 \), the future value \( A = 5000 \), the interest rate as \( r = 0.0975 \), and the compounding frequency as semiannually, so \( n = 2 \). We want to solve for time \( t \).

Step 2: Use the Compound Interest Formula

The formula for compound interest is given by \( A = P \left(1 + \frac{r}{n}\right)^{nt} \). Substitute the known values into the equation: \( 5000 = 4000 \left(1 + \frac{0.0975}{2}\right)^{2t} \).

Step 3: Simplify and Solve for \( t \)

Begin by dividing both sides by 4000 to isolate the term involving \( t \): \( 1.25 = \left(1 + \frac{0.0975}{2}\right)^{2t} \). Calculate \( 1 + \frac{0.0975}{2} = 1.04875 \). Then, the equation becomes \( 1.25 = 1.04875^{2t} \).

Step 4: Take the Natural Logarithm of Both Sides

Apply the logarithm to both sides to solve for \( t \): \( \ln(1.25) = \ln(1.04875^{2t}) \). Simplifying, you have \( \ln(1.25) = 2t \cdot \ln(1.04875) \).

Step 5: Solve for \( t \)

Rearrange the equation to solve for \( t \): \( t = \frac{\ln(1.25)}{2 \cdot \ln(1.04875)} \). Calculate \( t \) using a calculator to find \( t \approx 2.356 \).

Step 6: Interpret the Result

Because the compounding is semiannual, the time period corresponds to approximately 2.356 years. This means Nancy should choose a period a little over 2 years to achieve her financial goal.

Key Concepts

Investment

Investment is the act of using money to purchase assets or assets expected to generate returns. This is a critical financial tool to grow wealth over time. When you decide to invest, you start with a principal amount, like the $4000 in Nancy's case.

Investments can take many forms, including:

• Stocks
• Bonds
• Real Estate
• Savings Accounts
• Savings Certificates (like in Nancy's situation)

Each type of investment has its own risk level and potential returns. The key to a good investment is often a balance between risk and return that meets an individual’s financial goals.

Before investing, one should consider the following factors:

• Time horizon: How long are you planning to keep your money invested?
• Risk tolerance: How much risk can you afford to take?
• Financial goals: What do you want to achieve with your investment?

In the exercise, Nancy’s goal is to grow her principal of $4000 into $5000, making her financial future more secure.

Interest Rate

Interest rate is the percentage at which your investment grows over a specified period. In the scenario provided, Nancy's investment in savings certificates has an interest rate of 9.75% annually.

Interest rates can be of different types:

• Fixed: The rate remains constant throughout the investment term.
• Variable: The rate can change based on economic conditions.

Understanding interest rates is essential because they directly impact how fast your investment grows. A higher interest rate typically means a faster growth rate for your investment.

Interest can be compounded in different ways:

• Annually
• Semiannually
• Quarterly
• Monthly
• Daily

This exercise involves semiannual compounding. This means the interest is calculated and added to the principal twice a year. Compounding frequency affects the total amount of interest earned, as compounding more frequently results in more interest accumulation.

Logarithm

A logarithm is a mathematical tool used to solve equations involving exponentials. It helps you isolate the exponent in an equation, making it ideal for solving compound interest problems like Nancy's.

In simpler terms, a logarithm tells you what power you need to raise a base number to obtain a certain value. For instance, if you know the result of an exponentiation and the base, the logarithm gives you the exponent.

For Nancy's situation, we use the natural logarithm (denoted as \( \ln \)) to find the time \( t \). This involves taking the log of both sides of the equation:

• \( \ln(1.25) = \ln(1.04875^{2t}) \)

This simplifies to:

• \( \ln(1.25) = 2t \cdot \ln(1.04875) \)

The calculation helps determine how many years Nancy needs to invest to reach her goal. Using a calculator, she finds \( t \approx 2.356 \), indicating she needs a little over two years to achieve her desired future value. Understanding logarithms can vastly simplify solving for unknowns in equations involving growth or decay.