Problem 56 - College Algebra Essentials

Question

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{n}\) to solve \(.\) Round answers to the nearest cent. Suppose that you have \(\$ 6000\) to invest. Which investment yields the greater return over 4 years: \(8.25 \%\) compounded quarterly or \(8.3 \%\) compounded semiannually?

Step-by-Step Solution

Answer

To determine the better investment option, compute the total amount of each investment using the compound interest formula. Compare the two results and the investment that yields a higher amount should be chosen.

Step 1: Identify the Parameters for Each Investment

Firstly, identify the components needed in our formula for each investment based on their conditions: Principal Amount (P), Rate of Interest (r), Number of Times Interest Applied per Time Period (n), and Number of Time Periods (t). \n\nFor the first investment of 8.25% compounded quarterly:\n \(P_1 = \$6000\), \n \(r_1 = 8.25%\), convert this percentage to a decimal by dividing by 100, so \(r_1 = 0.0825\), \n \(n_1 = 4\), as it is compounded quarterly meaning 4 times a year, and \n \(t_1 = 4\), the total time is 4 years. \n\nFor the second investment of 8.3% compounded semiannually: \n \(P_2 = \$6000\), \n \(r_2 = 8.3%\), again convert this percentage to a decimal by dividing by 100, so \(r_2 = 0.083\), \n \(n_2 = 2\), as it is compounded semiannually or 2 times a year, and \n \(t_2 = 4\), the total time is 4 years.

Step 2: Substitute the Parameters into the Compound Interest Formula

Now, plug the values identified in step 1 into the compound interest formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) for each investment. \n\nFor the first investment:\n \(A_1 = P_1\left(1+\frac{r_1}{n_1}\right)^{n_1 t_1}\), \nwhich simplifies to, \n \(A_1 = 6000 \left(1+\frac{0.0825}{4}\right)^{4 \times 4}\). \n\nFor the second investment:\n \(A_2 = P_2\left(1+\frac{r_2}{n_2}\right)^{n_2 t_2}\), \nwhich simplifies to, \n \(A_2 = 6000 \left(1+\frac{0.083}{2}\right)^{2 \times 4}\).

Step 3: Calculate the Total Amount for Each Investment

Calculate the total amount \(A_1\) and \(A_2\) for each investment by evaluating the expressions from step 2. This will give us the total amount of money yielded by each investment after 4 years. \n\nSolving for \(A_1\) and \(A_2\) will give us the total amount for each investment.

Step 4: Compare the Total Amounts

Once you have the total amounts, compare them. The investment that yields the higher total amount is the better investment over 4 years.

Key Concepts

Investment Comparison

When comparing investments, it is essential to examine which option yields a higher return over a set period. Here's how you can effectively compare two different investment scenarios:

• Identify the principal amount: This is the initial amount of money you are investing, in this case, $6000 for both options.
• Determine the interest rate: Compare the interest rates provided. An interest rate is a percentage charged on the total invested principal over a specified period.
• Consider compounding frequency: Investigate how frequently the interest is compounded, such as quarterly or semiannually.

By evaluating these factors, you can decipher which investment will be more profitable in the long term. Make sure you account for the compounding effect, as it plays a crucial role in accruing interest.

Interest Rate

The interest rate is a critical component when considering investments. It dictates how much interest your principal will earn over time. In general, a higher interest rate can lead to greater profits, as long as it is compounded effectively.

• The rate is usually expressed as a percentage, like 8.25% or 8.3% in the example exercise.
• To work with the interest rate mathematically, convert it into a decimal form by dividing by 100. For example, 8.25% becomes 0.0825.

Comprehending how the interest rate works is essential, as it directly influences the amount of money you earn from your investment.

Time Periods in Investments

Time plays a significant role in how investments grow, particularly when discussing compounding interest. In the exercise, the time period is set at 4 years for both cases.

• Time period refers to the duration for which the investment is made, often affecting the total accumulated interest.
• A longer time frame generally leads to higher returns due to the accumulating effect of compound interest.

Always consider how long you are willing to invest your money, as the duration impacts the risk involved and the return you can expect.

Quarterly Compounding

Quarterly compounding refers to the process of calculating and adding interest four times a year. This method can impact the total amount more significantly than annual compounding due to the frequency at which interest is compounded.

• The compounding period is every 3 months, translating to an annual frequency of 4 times.
  
• For the quarterly compounding formula, the interest rate is divided by 4, and the time is multiplied by 4 within the compound interest formula: \[ A = P\left(1+\frac{r}{n}\right)^{nt}\]

This method allows interest to be calculated more frequently, increasing the amount of times your investment earns interest throughout the year.

Semiannual Compounding

Semiannual compounding occurs twice a year, every 6 months, and is another option for accumulating interest. It is essential to understand how this method affects investment growth compared to other compounding frequencies.

• The interest rate is divided by 2 while the time period is also adjusted by multiplying by 2 in the formula: \[ A = P\left(1+\frac{r}{n}\right)^{nt} \]
• This frequency still offers more opportunities for the interest to add up than annual compounding, but less than quarterly.
  

Being aware of these differences ensures you can make a more informed decision about which investment is likely to yield better returns.