Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 35,000\) invested at \(1.2 \%\) annual interest for 3 years compounded (a) annually; (b) quarterly

Step-by-Step Solution

Answer

For annual compounding, the amount is $36,267; for quarterly compounding, it's $36,284.

Step 1: Identify the compound interest formula

The general formula for compound interest is given by \( A = P \left( 1 + \frac{r}{n} \right)^{nt} \), where \( A \) is the amount after time \( t \), \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years.

Step 2: Calculate for annual compounding (a)

For annual compounding, \( n = 1 \). Substitute \( P = 35,000 \), \( r = 0.012 \) (since 1.2% = 0.012 as a decimal), \( n = 1 \), and \( t = 3 \) into the formula:\[A = 35000 \left( 1 + \frac{0.012}{1} \right)^{1 \times 3} = 35000 \left( 1 + 0.012 \right)^3 = 35000 \times 1.012^3\]

Step 3: Simplify for annual compounding

Calculate \( 1.012^3 \):\[1.012^3 = 1.0362\]So, \( A = 35000 \times 1.0362 = 36,267 \).

Step 4: Calculate for quarterly compounding (b)

For quarterly compounding, \( n = 4 \). Substitute \( P = 35,000 \), \( r = 0.012 \), \( n = 4 \), and \( t = 3 \) into the formula:\[A = 35000 \left( 1 + \frac{0.012}{4} \right)^{4 \times 3} = 35000 \left( 1 + 0.003 \right)^{12} = 35000 \times 1.003^{12}\]

Step 5: Simplify for quarterly compounding

Calculate \( 1.003^{12} \):\[1.003^{12} = 1.0367\]So, \( A = 35000 \times 1.0367 = 36,284 \).

Key Concepts

Annual Compounding

When we talk about annual compounding, what we mean is that interest is calculated once per year. This is a straightforward method where the interest earned each year adds to the initial principal amount. The formula used is part of the standard compound interest formula:

• The principal ( \( P \) ) is the initial amount of money invested or borrowed.
• The rate ( \( r \) ) is the annual interest rate expressed as a decimal.
• The number of times interest is compounded per year ( \( n \) ) equals 1 for annual compounding.
• The time ( \( t \) ) is the duration the money is invested or borrowed, in years.

In the given problem, we have \( P = 35,000 \)\\( r = 0.012 \) and\( t = 3 \).Putting these values into the formula gives:\[ A = 35000 \left( 1 + \frac{0.012}{1} \right)^{1 \times 3} = 35000 \times 1.012^3 \]After calculating \( 1.012^3 \), we get \( 1.0362 \) and multiplying it by \( 35,000 \), we find the total amount to be \( 36,267 \). This result shows how interest accumulates over time with annual compounding.

Quarterly Compounding

Quarterly compounding means that the interest is calculated four times a year. This method can grow your investment faster than annual compounding because interest is added more frequently. For quarterly compounding, the variables in the compound interest formula change slightly:

• The principal ( \( P \) ) remains the initial amount, unchanged.
• The rate ( \( r \) ) is still the annual interest rate in decimal form.
• Here, the number of times interest is compounded per year ( \( n \) ) is \( 4 \).

Using the same problem data, we substitute into the formula:\[ A = 35000 \left( 1 + \frac{0.012}{4} \right)^{4 \times 3} = 35000 \times 1.003^{12} \]Upon calculating \( 1.003^{12} \), we get \( 1.0367 \). Then multiplying by \( 35,000 \), the total amount becomes \( 36,284 \).Quarterly compounding provides a bit more cumulative interest over three years compared to annual compounding due to the more frequent addition of interest.

Compound Interest Formula

The compound interest formula is a fundamental concept in finance and helps determine how investments grow over time. It is expressed as:\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]This formula allows us to calculate the future value (\( A \)) of an investment:

• The principal ( \( P \) ) is the starting amount.
• The rate ( \( r \) ) is the annual interest rate converted to a decimal.
• The compounding frequency ( \( n \) ) is how often interest is calculated and added each year.
• The time period ( \( t \) ) is the length of time the money is invested.

What makes compound interest so powerful is that each time interest is added, it also earns interest afterward. This is why more frequent compounding typically results in more interest earned. Understanding this formula supports making better financial decisions for investments and savings.