Problem 69 - 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. \(\$ 20,000\) invested at \(1 \%\) annual interest for 4 years compounded (a) annually; (b) semiannually

Step-by-Step Solution

Answer

(a) $20812.08; (b) $20814.84

Step 1: Understand the formula for compound interest

The future value of an investment compounded periodically is given by the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after \( n \) years, including interest, \( P \) is the principal amount (the initial amount), \( r \) is the annual interest rate (as a decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time the money is invested for in years.

Step 2: Calculate for Annual Compounding

Here, the conditions are as follows: \( P = 20000 \), \( r = 0.01 \), \( n = 1 \), \( t = 4 \). Substitute these values into the formula.\[ A = 20000 \left(1 + \frac{0.01}{1}\right)^{1 \times 4} = 20000 \times (1.01)^4 \]Now calculate \((1.01)^4\), which equals approximately \(1.04060401\).Therefore, \( A = 20000 \times 1.04060401 \approx 20812.08 \).

Step 3: Calculate for Semiannual Compounding

Now, consider the conditions for semiannual compounding: \( P = 20000 \), \( r = 0.01 \), \( n = 2 \), \( t = 4 \). Plug these into the formula.\[ A = 20000 \left(1 + \frac{0.01}{2}\right)^{2 \times 4} = 20000 \times (1.005)^8 \]Calculate \((1.005)^8\), which is approximately \(1.04074203\).Thus, \( A = 20000 \times 1.04074203 \approx 20814.84 \).

Key Concepts

Annual Compounding

When discussing compound interest, a common type is annual compounding. This means that the interest is calculated and added to the principal amount once a year. Let’s break it down with an example.
Suppose you invest \(20,000 at an interest rate of 1% per year for four years. With annual compounding, the compound interest formula simplifies since the interest is applied once per year. In this case, the number of compounding periods, denoted by \( n \), is 1.

• The principal \( P \) is \)20,000.
• The annual interest rate \( r \) is 0.01 (or 1%).
• The time \( t \) is 4 years.

Substituting these values into the compound interest formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), we have: \[ A = 20000 \left(1 + \frac{0.01}{1}\right)^{1 \times 4} \] By calculating \((1.01)^4\), we find it equals approximately 1.04060401, leading to a final amount \( A \approx 20812.08 \). This showcases that annually compounding interest accumulates in a consistent, straightforward manner each year, increasing your investment.

Semiannual Compounding

Semiannual compounding differs from annual compounding in that interest is compounded twice a year. This means the interest rate is split into two equal parts and applied every six months. Let’s see how this works with our example.
Imagine the same initial investment of \(20,000 at an annual interest rate of 1%, but this time compounded semiannually.

• The principal \( P \) is \)20,000.
• The annual interest rate \( r \) is 0.01 (or 1%).
• The time \( t \) is 4 years.
• Since we are compounding semiannually, \( n \) becomes 2.

Inserting these values into the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), it becomes: \[ A = 20000 \left(1 + \frac{0.01}{2}\right)^{2 \times 4} \]By evaluating \((1.005)^8\), we discover it equals approximately 1.04074203, resulting in a final amount \( A \approx 20814.84 \). Due to the more frequent application of interest, semiannual compounding slightly improves the growth of your investment compared to annual compounding.

Interest Rate Calculations

Interest rate calculations are fundamental when dealing with compound interest. Understanding how they influence the future value of investments is critical. The key here is to properly apply the interest rate when calculating compounding.When using the compound interest formula, you have to adjust the annual interest rate \( r \) according to the compounding frequency \( n \). This adjustment is done by dividing the annual interest rate by the number of compounding periods per year.

• For **annual compounding**, since the interest is compounded once a year, the adjustment is simple: you just apply the annual rate as it is.
• For **semiannual compounding**, you must divide the annual rate by 2, as the interest is compounded twice a year. Hence, each compounding period, the interest rate applied is half of the annual rate.

The formula used for calculating compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where each component interacts to define the growth of the investment.
This crucial relationship between the interest rate, compounding frequency, and growth highlights the powerful effect even a minor change in these inputs can have on financial outcomes.