Question

True or False? If true, prove it. If false. find the true answer. If you invest \(\$ 500\), an annual rate of interest of \(3 \%\) yields more money in the first year than a \(2.5 \%\) continuous rate of interest.

Short answer

True; 3% annual yields more than 2.5% continuous interest.

Step-by-step solution

Calculate simple interest for 3% annual rate

To find the amount of money yielded by a 3% annual rate of interest, use the formula for simple interest: \[ A = P(1 + rt) \]where \(P\) is the principal amount (\\(500), \(r\) is the annual interest rate (0.03 for 3%), and \(t\) is the time in years (1 year). Plugging in the values:\[ A = 500(1 + 0.03 \times 1) = 500(1.03) = 515 \]So, the amount after one year with a 3% annual rate of interest is \\)515.

Calculate continuous interest for 2.5% rate

To find the amount with a 2.5% continuous interest rate, use the formula for continuous compounding: \[ A = Pe^{rt} \]where \(P\) is the principal amount (\\(500), \(r\) is the continuous interest rate (0.025 for 2.5%), and \(t\) is the time in years (1 year). Plugging in the values:\[ A = 500e^{0.025 \times 1} = 500e^{0.025} \]Using the approximate value of \(e^{0.025} \approx 1.02532\), we find:\[ A = 500 \times 1.02532 \approx 512.66 \]So, the amount after one year with a 2.5% continuous rate of interest is approximately \\)512.66.

Compare the results

Compare the amount yielded by the two different interest rates: - 3% simple annual interest yields \\(515.- 2.5% continuous interest yields approximately \\)512.66.Since \\(515 is greater than \\)512.66, the statement is true.

Key concepts

Understanding Simple Interest

Simple interest is a method to calculate the interest you earn on an investment at a fixed percentage over time. It's straightforward and relies on a simple formula:\[ A = P(1 + rt) \]Where:

• \( A \) is the total amount after interest.
• \( P \) is the principal or initial amount (\\(500 in our case).
• \( r \) is the annual interest rate expressed as a decimal (0.03 for 3%).
• \( t \) is the time period in years (1 year here).

The beauty of simple interest is its predictability. You calculate the interest based only on the original amount, so it doesn't change year-to-year.
In our example, after one year, the investment grows to \\)515. This happens because you add a fixed \$15, calculated by multiplying the principal \( 500 \) by the rate 0.03.

The Basics of Continuous Compounding

Continuous compounding is a bit more complex compared to simple interest. Instead of being calculated at fixed intervals, the interest is compounded continuously. This sounds challenging, but the formula helps clarify:\[ A = Pe^{rt} \]Where:

• \( A \) is the amount after interest.
• \( P \) is the principal amount (\\(500 in the exercise).
• \( r \) is the continuous interest rate as a decimal (0.025 for 2.5%).
• \( t \) is the time in years (1 year).

One special element is the mathematical constant \( e \), approximately 2.71828, which represents continuous growth. Applying it to our exercise, the result is approximately \\)512.66 after one year. This calculation shows continuous growth is effective over longer periods but may yield slightly less over short terms compared to simple interest.

Interest Rate Comparison

Comparing different interest rates requires a clear understanding of how each grows your investment over a set period.

• Simple interest, with a 3% annual rate, resulted in \\(515 after one year.
• Continuous compounding, at a rate of 2.5%, yielded approximately \\)512.66.

In this situation, the 3% simple annual interest rate yields more than the 2.5% continuous interest for a one-year term. This happens because, over short periods, a higher simple rate can outpace a slightly lower continuous rate. For those looking to maximize short-term gains, a higher simple interest rate may be more beneficial. However, continuous compounding can become advantageous over longer times, potentially surpassing simple interest due to its exponential nature.