Problem 76 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 4000\) is borrowed at a rate of 16\(\%\) interest per year, compounded quarterly, find the amount due at the end of the given number of years. (a) 4 years (b) 6 years (c) 8 years

Step-by-Step Solution

Answer

Amount due for: (a) 4 years = $7424.46; (b) 6 years = $10432.20; (c) 8 years = $14656.30.

Step 1: Understanding the Formula

The compound interest formula to calculate the amount due 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, \( n \) is the number of times interest is compounded per year, and \( t \) is the number of years.

Step 2: Identifying Values for the Formula

For all parts of this problem, \( P = 4000 \), \( r = 0.16 \), and \( n = 4 \) since interest is compounded quarterly.

Step 3: Calculating for 4 Years

Substitute the values for 4 years into the formula: \( t = 4 \). Calculate \( A = 4000 \left( 1 + \frac{0.16}{4} \right)^{4 \times 4} \). This simplifies to \( A = 4000 (1.04)^{16} \). Compute the power and multiply by 4000.

Step 4: Calculating for 6 Years

Substitute the values for 6 years into the formula: \( t = 6 \). Calculate \( A = 4000 \left( 1 + \frac{0.16}{4} \right)^{4 \times 6} \). This simplifies to \( A = 4000 (1.04)^{24} \). Compute the power and multiply by 4000.

Step 5: Calculating for 8 Years

Substitute the values for 8 years into the formula: \( t = 8 \). Calculate \( A = 4000 \left( 1 + \frac{0.16}{4} \right)^{4 \times 8} \). This simplifies to \( A = 4000 (1.04)^{32} \). Compute the power and multiply by 4000.

Step 6: Final Amounts

From the calculations: For 4 years, \( A \approx 7424.46 \); for 6 years, \( A \approx 10432.20 \); for 8 years, \( A \approx 14656.30 \).

Key Concepts

Compound Interest Formula

The compound interest formula is a mathematical tool used to calculate the total amount of money accumulated after a specified period of time, based on the principal and the interest rate. It is represented as: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]In this formula:

• A stands for the total amount after interest has been applied.
• P is the principal amount or the original sum of money invested or borrowed.
• r represents the annual interest rate (expressed as a decimal).
• n is the number of times the interest is compounded per year.
• t indicates the number of years the money is invested or borrowed.

The formula compounds the interest periodically rather than as a simple rate applied annually. This means that interest earns more interest, making it a powerful tool for growing money over time.

Compounded Quarterly

Compounding interest quarterly means calculating the interest four times a year. Each quarter, a portion of the yearly interest rate is applied, leading to a situation where the interest earned itself earns interest in subsequent periods. This is an important feature of compound interest because:

• Frequency: Since the interest is compounded four times a year, the yearly rate is divided by 4.
• Growth: Earnings are reinvested faster compared to compounding annually, which can significantly increase the amount over time, especially with higher rates.

In our exercise, interest calculated quarterly means using \( n = 4 \) in the formula. This increases the compounding effects, suggesting why the final amounts are noticeably larger than the initial principal.

Annual Interest Rate

The annual interest rate is a percentage that indicates how much interest you will earn or pay over one year. It's crucial in the compound interest formula because it determines how quickly your investment grows or debt increases. To incorporate it into calculations:

• The rate must be converted into a decimal, so a 16% annual interest rate becomes \( r = 0.16 \).
• During the quarterly compounding, this rate is divided by 4 to reflect 4 periods a year, each applying \( \frac{0.16}{4} \) or 0.04 interest.

Understanding and applying the annual interest rate correctly is critical, as small incremental increases lead to much larger sums over time due to the power of compounding.

Principal Amount

The principal amount is the original sum of money that is invested or borrowed before any interest is applied. It is denoted by \( P \) in the compound interest formula. Understanding this concept involves:

• Baseline: Everything builds from this initial amount, making it crucial in calculating the compound interest.
• Multiplicative factor: The compounded interest works on multiplying this sum, affecting the total significantly over longer periods.

In our problem, the principal amount is \( \$4000 \), a starting point for all the calculations. This initial value, when subjected to compounding mechanisms, emphasizes the power and potential of compound interest to transform small investments into significant financial gains over time.