Problem 28 - College Algebra

Question

\(A=P(1+r t)\) is used to fi \(\mathrm{d}\) the principal amount \(P\) deposited, earning \(r \%\) interest, for \(t\) years. Use this to fi \(\mathrm{d}\) what principal amount \(P\) David invested at a \(3 \%\) rate for 20 yr if \(A=\$ 8,000\).

Step-by-Step Solution

Answer

David invested \$5000.

Step 1: Write Down the Given Information

We are provided with the formula \(A = P(1 + rt)\) to find the principal amount \(P\). We know the following values: \(A = 8000\), \(r = 3\% = 0.03\), and \(t = 20\).

Step 2: Substitute the Values into the Formula

Substitute \(A = 8000\), \(r = 0.03\), and \(t = 20\) into the formula \(A = P(1 + rt)\). This gives us the equation:\[ 8000 = P(1 + 0.03 imes 20) \]

Step 3: Simplify the Equation

Calculate the value inside the parentheses:\(1 + 0.03 \times 20 = 1 + 0.6 = 1.6 \)So, the equation becomes:\[ 8000 = P \times 1.6 \]

Step 4: Solve for \(P\)

To solve for \(P\), divide both sides of the equation by 1.6:\[ P = \frac{8000}{1.6} \]

Step 5: Calculate the Value of \(P\)

Perform the division to find \(P\):\[ P = 5000 \]David invested \$5000.

Key Concepts

Principal Amount

Understanding the principal amount is essential when dealing with simple interest calculations. The principal amount is the initial sum of money that you invest or borrow. In the context of investments, it represents the starting money that begins to grow as interest accumulates over time. For someone like David, who aims to grow his savings, identifying the correct principal amount determines the potential size of his future wealth.

Calculating the principal involves knowing the final amount (after interest), the interest rate, and the duration of the investment. In David's case, the final amount he achieved after his investment period was $8000. Using the formula provided, you can isolate the principal by rearranging and solving the equation for the variable \(P\). This way, you can ascertain how much David originally put aside.

Interest Rate

The interest rate is a pivotal piece of the picture in understanding simple interest. It dictates how quickly or slowly your investment grows over time. In simple interest calculations, the interest rate is expressed as a percentage and signifies how much interest is earned per year per unit of the principal amount.

For David, the interest rate was 3%, which translates to earning 0.03 parts of his principal as interest each year. To use the interest rate effectively in calculations, it's essential to convert it into decimal form, as seen in the formula application. This conversion is crucial as it aligns the values appropriately for mathematical operations.

• Conversions: Divide the percentage by 100 -> 3% becomes 0.03.
• Usage: Applies directly in the formula \(A = P(1 + rt)\).

A solid grasp on manipulating interest rates can significantly affect the outcome of financial planning and decision-making.

Investment Duration

Investment duration is another fundamental aspect when using the simple interest formula. It defines the period over which the money remains invested or borrowed. The longer the investment duration, the more interest accrues, meaning you'll witness a more substantial growth in your initial investment.

In David's situation, the duration was 20 years. This lengthy timeframe allowed the interest to multiply over each year, contributing to the final increased amount of $8000.

• Time Factor: Represented by \( t \) in the formula.
• Calculation: The interest multiplies yearly over the specified duration.

Analyzing investment duration helps gauge the effectiveness of the investment strategy and aligns with the individual's financial goals, needing careful consideration when planning future investments.