Problem 31 - College Algebra

Question

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). An account is opened with an initial deposit of \(\$ 6,500\) and earns \(3.6 \%\) interest compounded semi-annually. What will the account be worth in 20 years?

Step-by-Step Solution

Answer

The account will be worth approximately $13,347.75 in 20 years.

Step 1: Understand the Compound Interest Formula

The compound interest formula is given as \(A(t) = P\left(1 + \frac{r}{n}\right)^{nt}\), where \(A(t)\) 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: Identify the Known Values

We know that the principal amount \(P = 6500\), the annual interest rate \(r = 0.036\), the interest is compounded semi-annually so \(n = 2\), and time \(t = 20\) years.

Step 3: Substitute the Values into the Formula

Plug the known values into the formula: \[A(20) = 6500\left(1 + \frac{0.036}{2}\right)^{2 \times 20}\].

Step 4: Calculate the Compound Interest

First, calculate \(\frac{0.036}{2} = 0.018\). Next, calculate \(1 + 0.018 = 1.018\). Then, find the exponent: \(2 \times 20 = 40\). Now compute \(1.018^{40}\). Finally, multiply by 6500 to find the final amount.

Step 5: Compute the Final Amount

Calculating further, \(1.018^{40} \approx 2.0535\). Multiply this result by 6500 to get the final account value. \[A(20) = 6500 \times 2.0535 \approx 13,347.75\].

Key Concepts

Principal Amount

The principal amount is the foundation of any investment or savings strategy. It refers to the initial deposit or investment made, before any interest accrues. In this exercise, the principal amount is $6,500. It's the starting point in the compound interest formula:

• Initial capital that grows over time with interest.
• Acts as a baseline to calculate future earnings.

Understanding the principal is crucial as it determines the initial value on which interest calculations are based. The higher the principal amount, the more potential there is for earnings through interest compounding.

Annual Interest Rate

The annual interest rate is an essential factor in determining how much extra money your investment will earn each year. It's usually expressed as a percentage and in this scenario, it's 3.6%. This rate tells you how much interest you'll earn on your principal amount across a year.

• Expressed as a decimal in calculations, e.g., 3.6% becomes 0.036.
• Determines the growth speed of your investment.

A higher annual interest rate means more earnings, making it a critical part of any savings or investment plan. Understanding this rate helps you predict how your investments will grow year over year.

Compounded Semi-Annually

Compounding is a powerful process that can significantly boost the value of an investment over time. When compounded semi-annually, interest is added to the principal twice a year, which is a common strategy to accelerate growth. Here's how it works:

• Interest is calculated and added to the principal twice each year.
• This frequency leads to more interest-on-interest compared to yearly compounding.

Semi-annual compounding means the rate is divided by 2, and the number of compounding periods is doubled, leading to faster accumulation of wealth compared to annual intervals.

Exponent Calculation

The exponent in the compound interest formula plays a critical role in calculating future value. It determines how many times the interest is applied to the principal across the total investment period. Here's a breakdown:

• Exponent calculated as the product of the number of compounding periods per year and the number of years.
• Represents the number of times interest adds to itself, increasing the total value.

In this exercise, the exponent is calculated as 40 (2 periods per year times 20 years), and is an essential step for determining how much the investment grows over time. Understanding this can help predict how quickly the original investment will expand.