Problem 11 - Algebra for College Students

Question

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to find the total amount of money accumulated at the end of the indicated time period for each of the following investments. \(\$ 5000\) for 15 years at \(8.5 \%\) compounded annually \(\$ 16,998.71\)

Step-by-Step Solution

Answer

The accumulated amount is approximately $17,022.90, which slightly differs from the given $16,998.71.

Step 1: Identify the Known Values.

The problem provides the principal amount, the annual interest rate, the number of years, and the frequency of compounding. From the problem:- Principal amount (P): \(5000\)- Annual interest rate (r): \(8.5 \% = 0.085\)- Time in years (t): \(15\)- Compounding frequency per year (n): \(1\) because it is compounded annually.

Step 2: Substitute Values into the Formula.

Use the formula \[A = P\left(1 + \frac{r}{n}\right)^{nt}\]Substitute the known values:\[A = 5000\left(1 + \frac{0.085}{1}\right)^{1 \cdot 15}\]This simplifies to:\[A = 5000(1.085)^{15}\]

Step 3: Calculate Inside the Parentheses.

First, calculate the inside of the parentheses:\(1 + \frac{0.085}{1} = 1.085\)

Step 4: Compute the Power Function.

Raise the base (1.085) to the power of the product (15):\[(1.085)^{15}\approx 3.40458\]

Step 5: Calculate the Final Amount.

Now multiply the result of Step 4 by the principal amount (\( P \) ): \[A = 5000 \times 3.40458 \approx 17022.90\]

Step 6: Compare to Given Answer.

Compare the calculated amount \(17022.90\) to the given result \(16998.71\). There is a slight difference, likely due to rounding in calculations or in rounding conventions in interest compounding.

Key Concepts

Principal Amount

The principal amount is the initial sum of money invested or loaned before any interest is applied. In the context of our example, the principal amount is $5,000.

Understanding the importance of the principal is key because:

• It defines the base on which interest is calculated and accumulated.
• If you increase this amount, the total accumulated sum at the end of the investment period will also be larger.
• The principal remains constant at the beginning of the interest calculation period.

When you invest or save money somewhere, the principal is what you initially put down. Everything that happens with interest will build upon this starting point.

Therefore, the larger your principal amount, the more potential you have for significant growth through compound interest. Keep this in mind when setting financial goals or planning investments.

Annual Interest Rate

The annual interest rate is a percentage that shows how much interest will be owed or earned over one year. In our exercise, this rate is 8.5%. To use it in calculations, we convert it to a decimal, so it becomes 0.085.

Here's why understanding the interest rate matters:

• The higher the rate, the more interest accumulates over time.
• It helps in comparing different investment opportunities or loans.
• Different interest rates will produce varying outcomes even with the same principal and time period.

Another key point is that the rate is often compounded, meaning the interest is not just calculated on the principal sum, but also on any accumulated interest from previous periods.

Understanding the annual interest rate as a component of financial growth is essential for making informed investment decisions and anticipating the future worth of investments.

Compounding Frequency

Compounding frequency refers to how often interest is calculated and added to the principal. In the worked example, the frequency is annual, meaning once per year. This aspect influences how much total interest will be earned.

Here's how it affects the outcome:

• More frequent compounding typically results in more total interest.
• When compounded annually, the frequency, denoted as \( n \), is 1.
• With different frequencies, the formula's exponent \( nt \) changes because it multiplies \( n \) times the number of years \( t \).

For example, if it were compounded semi-annually, \( n \) would be 2, and the interest would be calculated twice a year, altering the accumulated amount.

Knowing the compounding frequency helps you understand the dynamics of your investments or debts and plan accordingly. Adjusting compounding frequency can make significant differences in the future value of your savings or loans, so be in the habit of confirming how frequently your interest is applied.