Problem 12 - 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. \(\$ 7500\) for 20 years at \(9.5 \%\) compounded semiannually \(\$ 47,997.93\)

Step-by-Step Solution

Answer

The accumulated amount is \($47,997.93\).

Step 1: Identify the given values

First, we need to identify the values given in the problem. We have:- Principal amount \(P = \$7500\)- Annual interest rate \(r = 9.5\% = 0.095\)- Number of times interest is compounded per year \(n = 2\) (since it's semiannually)- Time in years \(t = 20\)

Step 2: Substitute values into the formula

Next, substitute the identified values into the compound interest formula:\[A = P\left(1+\frac{r}{n}\right)^{nt}\]\[A = 7500\left(1+\frac{0.095}{2}\right)^{2 \times 20}\]

Step 3: Calculate the compound interest factor

Calculate \(\frac{r}{n}\):\(\frac{0.095}{2} = 0.0475\)Now compute \(1 + \frac{r}{n}\):\(1 + 0.0475 = 1.0475\)

Step 4: Compute the exponent

Calculate the exponent term \(nt\):\(n \times t = 2 \times 20 = 40\)

Step 5: Raise the base to the calculated exponent

Raise the result from Step 3 to the power calculated in Step 4:\(1.0475^{40}\).

Step 6: Multiply by the principal amount

After computing \(1.0475^{40}\), multiply this value by the principal \(P\):\(A = 7500 \times 1.0475^{40}\).This calculation results in approximately \($47,997.93\), matching the provided final amount.

Key Concepts

Understanding Principal Amount

The principal amount is the starting point in any investment or savings scenario. It represents the initial sum of money that you invest or save. In the context of calculating compound interest, the principal is the base upon which interest is computed.
In the exercise above, the principal amount, denoted as \( P \), is \$7,500. This is the money initially invested or deposited, and it is crucial because it determines the magnitude of the amount of interest that will be added over time. The larger the principal, the higher the total interest accrued will be.
Always remember:

• Principal Amount \( P \) is initial investment or savings.
• Directly affects the total interest and final amount.
• Key component of the compound interest formula \( A = P\left(1+\frac{r}{n}\right)^{nt} \).

Grasping the Interest Rate

Interest rate is a percentage that determines how much interest is added to your principal amount over time. It influences how quickly your savings or investment grows.
The interest rate, \( r \), in our example is 9.5\%, which must be converted into a decimal form for calculations (0.095). This rate is crucial as it dictates the growth rate of your investment.
Key points about interest rate:

• Expressed as a percentage, but converted to a decimal for computing.
• Higher rates lead to faster growth in your investment.
• Affects the compound interest via the formula \( A = P\left(1+\frac{r}{n}\right)^{nt} \).

The Role of Exponentiation in Compound Interest

Exponentiation is a mathematical operation where a number is multiplied by itself a certain number of times, defined by an exponent. In the compound interest formula, exponentiation determines how the interest gets compounded over multiple periods.
In this exercise, the exponent \( nt \) equates to \( 40 \), formed from \( n \times t \), where \( n \) is the number of compounding periods per year and \( t \) is the total number of years. This step shows the power of compound interest, where interest not only applies to the principal but also on the accumulated interest from previous periods.
Important aspects of exponentiation:

• Raised to \( nt \) to reflect overall compounding over time.
• Illustrates the 'compounding of the compounding', showing an exponential growth in interest.
• Critical in calculating future values using the compound interest formula.