Problem 19 - Algebra 1: Concepts and Skills

Question

You deposit \(\$1400\) in an account that pays 6% interest compounded yearly. Find the balance at the end of the given time period. $$20 years$$

Step-by-Step Solution

Answer

The balance at the end of 20 years will be approximately \$ 5028.82.

Step 1: Identify the Variables

We know that the principal amount \(P\) is \$1400, the annual interest rate \(r\) is 6% (expressed as 0.06 in decimal format), the number of times interest is compounded per year \(n\) is 1 (since it's compounded yearly), and the time period \(t\) is 20 years.

Step 2: Substitute the Variables into the Formula

Plugging these values into the formula, we get \(A = 1400(1 + 0.06/1)^(1*20)\).

Step 3: Solve for A

After simplifying the equation, it can be found that \(A = 1400 * (1.06)^(20)\). Solving this equation using a calculator or a computer, the future value of the investment is approximately \$ 5028.82.

Key Concepts

Principal Amount

The principal amount is the initial sum of money you deposit into an account before any interest starts to accumulate. It's the starting point from which you begin to grow your savings or investments. Understanding this is crucial because all interest calculations in the compound interest formula are based on the principal amount.

In the given exercise, the principal amount is noted as

• Amount: $1400

This is the deposit placed in an account with a specific interest rate, and it is the baseline figure upon which the interest calculations will be made over time. Having the correct principal amount is essential because even small changes in this figure can significantly affect the calculated future value.

Interest Rate

The interest rate is the percentage at which the principal amount grows per compounding period. In our context, it is compounded annually which means the interest adds to the principal once every year. An interest rate is usually expressed as a percentage, so it's crucial to convert it to decimal form when used in calculations.

For example, a 6% interest rate would be expressed as:

• Decimal Form: 0.06

This exercise uses a yearly compounded interest rate of 6%. Thus, every year, the principal amount increases by 6%, building upon itself and leading to compound growth. The higher the interest rate, the faster your principal will grow over time, making it an influential component of future value.

Future Value

Future value is the amount of money an initial investment will grow into after interest has been applied over a certain period. It includes the principal amount plus any compounded interest accrued. Calculating future value is a way to estimate what your current investments will be worth in the future.

In our example, the formula for future value is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:

• A: Future Value
• P: Principal Amount (\(1400 in our case)
• r: Interest Rate (0.06 as a decimal)
• n: Number of times interest is compounded per year (1 for yearly)
• t: Time period in years (20 years for the exercise)

By plugging the known values into the formula, we get:\[ A = 1400 \times (1.06)^{20} \]Solving this results in the future value of approximately \)5028.82. This is the amount the original principal would grow into over 20 years at a 6% yearly compounded interest rate.