Problem 18 - Prealgebra

Question

The problems that follow involve compound interest. Compound Interest Suppose \(\$ 1,200\) is invested in a savings account that pays \(6 \%\) compounded semiannually. How much is in the account at the end of \(1 \frac{1}{2}\) years?

Step-by-Step Solution

Answer

Approximately $1311.27 is in the account after 1.5 years.

Step 1: Understand the Compound Interest Formula

The compound interest formula is given by \( A = P(1 + \frac{r}{n})^{nt} \), where \( A \) is the amount of money accumulated after \( n \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (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: Define the Variables

Given in the problem: \( P = 1200 \), \( r = 6\% = 0.06 \), \( n = 2 \) (since it's compounded semi-annually), and \( t = 1.5 \) years. We need to find \( A \), the amount in the account at the end of \( 1.5 \) years.

Step 3: Substitute the Values into the Formula

Substitute the known values into the compound interest formula: \[ A = 1200 \left( 1 + \frac{0.06}{2} \right)^{2 \times 1.5} \] which simplifies to \[ A = 1200 \left( 1 + 0.03 \right)^3 \].

Step 4: Calculate the Expression

Simplify and calculate the expression: \( 1 + 0.03 = 1.03 \). Raise this to the power of 3: \( 1.03^3 \approx 1.092727 \). Multiply this result by the initial principal: \( A = 1200 \times 1.092727 \approx 1311.27 \).

Step 5: Interpret the Result

The accumulated amount in the savings account at the end of \(1.5\) years is approximately \( \$1311.27 \).

Key Concepts

Understanding the Compound Interest Formula

To solve problems involving compound interest, you first need to understand the compound interest formula. This formula allows you to calculate the future value of an investment based on an initial principal amount, interest rate, and compounding frequency.
The mathematical expression is given by:

• \( A = P \left( 1 + \frac{r}{n} \right)^{nt} \)

Here:

• \( A \) represents the total accumulated amount, including both the principal and the interest earned.
• \( P \) is the principal amount or initial sum of money invested.
• \( r \) denotes the annual interest rate in decimal form.
• \( n \) is the number of times interest is compounded per year.
• \( t \) stands for the time in years that the money is invested or borrowed.

This formula is fundamental to exploring how different factors, like interest rates and compounding frequency, can influence the future value of an investment.
Understanding these components is crucial for performing accurate calculations and making informed financial decisions.

Semiannual Compounding Explained

In the context of compound interest, compounding frequency plays a significant role in determining how much interest will accrue over time.
Semiannual compounding means that interest is added to the principal twice a year. Therefore, every six months, the amount of money you have in the account increases at the specified interest rate divided by two.
For our scenario:

• The annual interest rate \( r \) is 6%, which equates to 0.06 when converted to decimal form.
• Since compounding occurs semiannually, we set \( n = 2 \).

Thus, each compounding period applies an interest rate of \( \frac{0.06}{2} = 0.03 \).
This split results in the interest being calculated more frequently, helping the investment grow slightly faster than if it were compounded annually, harnessing the power of compounding for additional returns.

The Process of Investment Growth

Investment growth occurs when the interest earned on the principal is reinvested, leading the investment to grow exponentially over time instead of in a linear fashion.
The beauty of compound interest lies in this reinvestment:

• After the first compounding period, the earned interest becomes part of the new principal.
• In subsequent periods, interest is earned on both the original principal and the previously accumulated interest.

In the given example, after each six-month period, the interest accrued is added to the initial investment of \( \$1200 \), allowing the entire amount to earn interest in the next period.
This concept underscores why compound interest can be a powerful driver of wealth growth over time, as it leverages the snowball effect of reinvesting earnings back into the principal again and again.

Calculation of Accumulated Amount

Calculating the accumulated amount involves substituting known values into the compound interest formula and performing the arithmetic operations:
Given the problem's values:

• \( P = \\(1200 \)
• Annual interest rate \( r = 0.06 \)
• Compounding semiannually \( n = 2 \)
• Time \( t = 1.5 \) years

Substitute these into the formula:

• \( A = 1200 \left( 1 + \frac{0.06}{2} \right)^{2 \times 1.5} \)
• Further simplify: \( A = 1200 \left( 1 + 0.03 \right)^3 \)
• Calculate: \( A = 1200 \times 1.092727 \approx 1311.27 \)

The resulting \(\\)1311.27\) represents how much the investment is worth after accounting for the effects of compound interest over \(1.5\) years with semiannual compounding.
This calculation illustrates how to translate theoretical concepts into practical financial insights by showing how an initial investment can grow due to interest accrual.