Problem 96 - Intermediate Algebra

Question

Investments. An amount \(P\), invested at a simple interest rate \(r\) will grow to an amount \(A\) in \(t\) years according to the formula \(A=P(1+r t) .\) Solve for \(P .\) Suppose a man invested some money at \(5.5 \%\). If after 5 years, he had \(\$ 6,693.75\) on deposit, what amount did he originally invest?

Step-by-Step Solution

Answer

The man initially invested \$5,250.

Step 1: Identify Known Values

The problem provides us with the final amount \(A\), the interest rate \(r\), and the time \(t\). From the problem, we know: \(A = \$6,693.75\), \(r = 5.5\% = 0.055\), and \(t = 5\) years.

Step 2: Write Down the Formula for Simple Interest

We use the formula for final amount in simple interest: \(A = P(1 + rt)\), where \(P\) is the principal amount we want to find.

Step 3: Rearrange the Formula to Solve for \(P\)

To isolate \(P\), we rearrange the formula: \(P = \frac{A}{1 + rt}\). This allows us to solve for \(P\) using the known values of \(A\), \(r\), and \(t\).

Step 4: Plug in the Known Values

Substitute the known values into the equation: \(P = \frac{6,693.75}{1 + (0.055 \times 5)}\).

Step 5: Calculate the Denominator

Compute the denominator: \(1 + (0.055 \times 5) = 1 + 0.275 = 1.275\).

Step 6: Compute the Principal Amount \(P\)

Divide the final amount by the computed denominator to find \(P\): \(P = \frac{6,693.75}{1.275} = 5,250\).

Step 7: Interpret the Result

The initial investment \(P\) that amounts to \\(6,693.75 after 5 years at a 5.5% interest rate is \\)5,250.

Key Concepts

Principal Amount

The principal amount is the initial sum of money that you invest or borrow, before any interest is added. Understanding this concept is crucial in financial mathematics as it forms the baseline for calculating the future value of investments. Consider the principal amount as the foundation of your financial calculations. It's like the seed money you start with in an investment. Knowing the principal helps in planning and comparing different financial strategies.
A clear understanding of the principal is essential when you're determining how much money you'll need to gather or save for a future purchase or investment. The principal is crucial because any interest calculated will be based on this original amount. This means if your principal is larger, the absolute amount of interest you gain or owe will be larger as well.

Interest Rate

The interest rate is a percentage that represents the cost of borrowing money or the reward of investing money over a specific period. It's often expressed as an annual percentage, but it can also apply to different periods such as monthly or quarterly. When we talk about interest rates in simple interest calculations, we're using a straightforward concept where the interest is calculated only on the initial principal amount. For example, with a 5.5% rate, you multiply the principal by 0.055 to find the annual interest earned or owed. Understanding the interest rate is important because it directly affects how quickly an investment grows or a debt increases. A higher interest rate can significantly boost your returns on investments, while the same higher rate can increase costs when you're borrowing.

Investment Formula

In financial mathematics, the investment formula for simple interest is a tool used to calculate the amount of money an investment will grow to over time. The formula is expressed as: \[ A = P(1 + rt) \] where \(A\) is the amount on deposit after time \(t\), \(P\) is the principal amount, \(r\) is the interest rate as a decimal, and \(t\) is the time period in years.This formula is helpful because it allows you to find out how much your investment will be worth after a certain period. It can be rearranged to solve for any component, such as the principal, interest rate, or time, as long as the other values are known. The straightforward nature of the formula makes it easy to use and understand, which is why it's a staple in introductory financial mathematics.

Financial Mathematics

Financial mathematics is a field of applied mathematics that analyzes and solves financial problems using mathematical theories and calculations. It combines concepts from algebra, probability, and calculus to manage portfolios, determine pricing models, and assess risks and gains. One essential aspect of financial mathematics is understanding the growth of money over time, whether it's through lending, investing, or saving. In our example, simple interest represents a basic form of financial mathematics. It helps you calculate how money compounds over time using established formulas like the one in this exercise. Studying financial mathematics gives you the tools to effectively manage personal and corporate finances. It enhances your ability to make informed financial decisions and plan for future investments. By mastering these concepts, you can optimize your investment strategies and better understand the complexities of the financial markets.