Problem 69 - A Graphical Approach to College Algebra

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 \(3 \%\) annual interest for 4 years compounded (a) annually; (b) semiannually

Step-by-Step Solution

Answer

(a) Annual: $22,510.18; (b) Semiannual: $22,533.54.

Step 1: Identify the Formula

To find the future value of an investment using compound interest, we use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(A\) is the future value of the investment/loan, \(P\) is the principal investment amount (\$20,000 in this case), \(r\) is the annual interest rate (decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the number of years the money is invested for.

Step 2: Calculating for Annual Compounding

In this scenario, the interest is compounded annually, so \(n = 1\). We are given \(r = 0.03\), \(t = 4\), and \(P = 20000\). Substitute these values into the formula:\[ A = 20000 \left(1 + \frac{0.03}{1}\right)^{1 \times 4} \]\[ A = 20000 \times (1.03)^4 \]Calculate \((1.03)^4\) and multiply by \$20,000.

Step 3: Compute (1.03)^4

Calculate \((1.03)^4\):\[ 1.03^4 = 1.125509 \]Now multiply this by the principal amount to find \(A\):\[ A = 20000 \times 1.125509 = 22510.18 \]Thus, after 4 years with annual compounding, the account will have \$22510.18.

Step 4: Calculating for Semiannual Compounding

In this scenario, the interest is compounded semiannually, so \(n = 2\). Using the compound interest formula, substitute the given values:\[ A = 20000 \left(1 + \frac{0.03}{2}\right)^{2 \times 4} \]\[ A = 20000 \left(1 + 0.015\right)^8 \]\[ A = 20000 \times (1.015)^8 \]Now, calculate \((1.015)^8\) and multiply by \$20,000.

Step 5: Compute (1.015)^8

Calculate \((1.015)^8\):\[ 1.015^8 = 1.126677 \]Now multiply this by the principal amount to find \(A\):\[ A = 20000 \times 1.126677 = 22533.54 \]Thus, after 4 years with semiannual compounding, the account will have \$22533.54.

Key Concepts

compound interest formula

The compound interest formula is a key tool for understanding how your investments can grow over time. It allows us to calculate the future value of an investment based on regular compounding periods. The formula is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]\where:

• **\(A\)**: Future value of the investment.
• **\(P\)**: Principal amount or the initial investment.
• **\(r\)**: Annual interest rate expressed as a decimal.
• **\(n\)**: Number of times interest is compounded per year.
• **\(t\)**: Total number of years the money is invested.

The formula takes into account not just the initial principal and the interest rate, but also the frequency of compounding, which can significantly impact the total amount of return.

annual interest rate

Understanding the annual interest rate is crucial when dealing with compound interest calculations. The annual interest rate indicates how much money an investment will earn in one year before the effects of compounding are considered. **Conversion to Decimal** To use the interest rate in compound interest formulas, it's essential to convert the percentage into a decimal. For example, 3% becomes 0.03 in decimal form. This is done by dividing the percentage by 100. This conversion is important because the compound interest formula utilizes the rate as a decimal for accurate calculations. By applying the annual interest rate correctly, we can predict how quickly an investment will grow over time.

future value calculation

Calculating the future value of an investment using compound interest involves determining what the investment will be worth at a future date, given a specific interest rate and compounding frequency.**Step-by-Step Process**Let's break down the process using our example:

• Start with an initial investment \(P\) which is \$20,000 in our case.
• Determine the annual interest rate \(r\), converted to decimal, which is 0.03.
• Identify the number of compounding periods per year \(n\).
• Calculate for a set number of years \(t\), here, 4 years.
• Insert these values into the compound interest formula to find the future value.

By applying this step-by-step method, it becomes easier to determine the value an investment will reach, making future financial planning more straightforward.

compounding frequency

Compounding frequency refers to how often interest is applied to the principal balance of an investment within a year. The frequency is a significant factor because it can greatly affect the total amount of interest earned. **Types of Compounding Frequencies**

• **Annually**: Interest is compounded once per year.
• **Semiannually**: Interest is compounded twice per year.
• **Quarterly**: Interest is compounded four times per year.
• **Monthly**: Interest is compounded twelve times per year.

In examples:

• For annual compounding, the year-end value reflects the interest earned over a full year.
• With semiannual compounding, interest is calculated and added twice a year, thus increasing the effective annual return compared to simple annual compounding.

Choosing different compounding frequencies can lead to different future values, even with the same interest rate and time period, due to the nature of interest on interest.