Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 900\) at \(8 \%\) compounded quarterly for 3 years

Short answer

Answer: The final balance after 3 years is $1,141.42 and the total interest earned is $241.42.

Step-by-step solution

Understand the compound interest formula

The formula for compound interest is given by: \(A = P \cdot (1 + \frac{r}{n})^{nt}\) Where: \(A\) - The final balance after time \(t\). \(P\) - The initial principal amount. \(r\) - The annual interest rate (as a decimal). \(n\) - The number of times the interest is compounded per year. \(t\) - The number of years.

Convert the given data in the correct format

We have the following information: Principal amount, \(P = \$900\) Annual interest rate, \(r = 8\% = 0.08\) (converted to a decimal) Compounded quarterly, \(n = 4\) (since there are 4 quarters in a year) Time, \(t = 3\) years

Calculate the final balance

Using the compound interest formula, we can find the final balance after 3 years: \(A = 900 \cdot (1 + \frac{0.08}{4})^{(4 \cdot 3)}\) \(A = 900 \cdot (1+ 0.02)^{12}\) \(A = 900 \cdot (1.02)^{12}\) \(A = 900 \cdot 1.26824\) \(A = \$1,141.42\)

Calculate the total interest

To find the total interest earned, subtract the initial principal amount from the final balance: Interest = Final balance - Principal amount Interest = \(\$1,141.42 - \$900\) Interest = \(\$241.42\) The final balance after 3 years is \(\$1,141.42\) and the total interest earned is \(\$241.42.\)

Key concepts

Final Balance Calculation

The final balance calculation in the context of compound interest reflects the total amount of money accumulated after a certain period, accounting for the effects of compounding. The formula
\(A = P \cdot (1 + \frac{r}{n})^{nt}\)
is pivotal for calculating this balance. Here's how to break it down:

• Identify your initial investment or principal, \(P\), which is the starting amount before interest.
• Determine your annual interest rate, \(r\), and convert it to a decimal.
• Know the number of times the interest will compound yearly, \(n\).
• Decide the total number of years, \(t\), you'll be investing.

By plugging these values into the formula, you can calculate the future value of your investment.
In our exercise, the initial investment is \(900, with an 8% annual interest rate, compounded quarterly over 3 years. The calculated final balance, reflecting the compounded interest, amounts to \)1,141.42.
Analyzing this formula and the process can boost your understanding of how different factors like time and compounding frequency affect your final investment return.

Interest Rate Conversion

Interest rate conversion is an essential step in solving compound interest problems. The conversion entails transitioning the annual interest rate into a decimal format, turning a percentage into a number that can be used in mathematical equations.
Here are the steps:

• Start with your annual interest rate as a percentage.
• Divide the percentage by 100 to change it to a decimal. So, for an 8% interest rate, your calculation would be \(8 \/ 100 = 0.08\).

This conversion is crucial because the compound interest formula requires the interest rate to be in decimal form for accurate calculations. In the exercise example, we converted the 8% interest rate to 0.08 before using it in the formula. Proper conversion ensures precise computation of your investment's growth over time.

Compounding Frequency

Compounding frequency refers to how often interest is applied to the principal balance within a year. It's an integral part of the compound interest formula and has a significant impact on your final balance.

• The more frequently the interest is compounded, the greater the amount of interest accrued over time.
• Common compounding frequencies include yearly, semi-annually, quarterly, monthly, or even daily.
• The variable \(n\) in the compound interest formula represents this frequency.

To understand compounding frequency, let's look at our textbook example where the interest is compounded quarterly, which means \(n = 4\). The principal is compounded four times a year, which has been considered in the final balance calculation. This increased frequency can lead to a higher final balance compared to just compounding annually. It's essential to incorporate the correct compounding frequency to accurately evaluate an investment's growth potential.