Problem 92 - College Algebra and Calculus: An Applied Approach

Question

Solve for the indicated variable. Investment at Compound Interest Solve for \(P\) in \(A=P\left(1+\frac{r}{n}\right)^{n t}\).

Step-by-Step Solution

Answer

The solution for \(P\) is \(P=\frac{A}{\left(1+\frac{r}{n}\right)^{n t}}\).

Step 1: Identify the existance of single variable

Firstly, locate \(P\) within the given formula which is \(A=P\left(1+\frac{r}{n}\right)^{n t}\).

Step 2: Isolate the P variable

The target is to get \(P\) alone on one side of the equation. To achieve this, both sides of the equation should be divided by the exponential term \( \left(1+\frac{r}{n}\right)^{n t} \) to isolate \(P\). This yields the equation \(P=\frac{A}{\left(1+\frac{r}{n}\right)^{n t}}\).

Step 3: Final Answer

Now, the variable \(P\) has been written in terms of the other variables, which is the desired outcome. Thus, the final answer is \(P=\frac{A}{\left(1+\frac{r}{n}\right)^{n t}}\).

Key Concepts

Algebraic Manipulation

In order to isolate a particular variable in an equation, such as isolating \(P\) in the compound interest formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\), algebraic manipulation is crucial. This involves rearranging the equation to express one variable in terms of others. This step can seem complex, but breaking it down into smaller parts makes it simpler.

To isolate \(P\), we need to ensure \(P\) is alone on one side of the equation. We achieve this by dividing both sides of the equation by the term \( \left(1+\frac{r}{n}\right)^{n t} \).

By performing the division, we derive the formula:

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

This indicates that \(P\) can be calculated if you know \(A\), \(r\), \(n\), and \(t\).

Why is this important? Algebraic manipulation allows us to make predictions, solve for unknowns, and understand relationships between different components of equations. It is a foundational tool in algebra and a necessary skill for tackling a variety of mathematical problems.

Exponential Functions

An exponential function is a mathematical function in the form \(f(x) = a \cdot b^x\), where \(b\) is a constant and \(x\) is an exponent. In the context of compound interest, the exponential growth is characterized by the equation \(\left(1+\frac{r}{n}\right)^{n t}\).

This particular function is an exponentiation which depicts how the money grows over time based on certain interest rates.
Understanding exponential behavior is essential, because in compound interest, the value of the investment increases by a fixed percentage over consistent intervals.

The exponent \(n \cdot t\) also plays a crucial role:

• \(n\) - the number of times compounding happens in a year
• \(t\) - the number of years the money is invested or borrowed for

Thus, \(nt\) represents the total compounding periods, magnifying the effects of accumulative growth.

By understanding exponential functions, we gain insights into how quickly investments can grow or decay over time, which is foundational in both mathematics and finance.

Investment Calculations

Investment calculations involve finding out present and future values of investments to make informed financial decisions. For example, solving \(A=P\left(1+\frac{r}{n}\right)^{n t}\) helps determine the initial investment needed (\(P\)) given a specific future goal (\(A\)). Understanding these calculations can empower investors to set realistic financial goals and plan effectively.

Key terms include:

• \(A\) - Future value of the investment
• \(P\) - Principal, or the initial amount invested
• \(r\) - Annual interest rate (in decimal)
• \(n\) - Number of compounding periods per year
• \(t\) - Number of years

To compute these correctly, investors need to substitute known values into the equation and solve for the desired variable. Proper calculation helps in assessing several financial scenarios, comparing investment pathways, and understanding the time value of money.

Through mastery of investment calculations, individuals can better manage their finances and ensure they meet their personal and financial goals.