Problem 28 - Intermediate Algebra

Question

Two years ago Daniel invested some money at \(8 \%\) interest compounded annually. Today it is worth $$\$758.16$$. How much did he invest two years ago?

Step-by-Step Solution

Answer

Daniel initially invested approximately $650.00.

Step 1: Understand the Formula

The future value of an investment compounded annually is calculated using the formula: \( A = P(1 + r)^n \), where \( A \) is the future value, \( P \) is the principal amount invested, \( r \) is the annual interest rate, and \( n \) is the number of years. Here, \( A = 758.16 \), \( r = 0.08 \), and \( n = 2 \).

Step 2: Set Up the Equation

Substitute the given values into the compound interest formula: \( 758.16 = P(1 + 0.08)^2 \).

Step 3: Simplify Inside the Formula

Calculate \( (1 + 0.08)^2 \):\[ (1 + 0.08)^2 = 1.08^2 = 1.1664. \] So the equation becomes: \( 758.16 = P imes 1.1664 \).

Step 4: Solve for the Principal Amount

Rearrange the equation to solve for \( P \):\[ P = \frac{758.16}{1.1664}. \]

Step 5: Calculate the Principal Amount

Perform the division to find \( P \):\[ P = \frac{758.16}{1.1664} \approx 650.00. \]Thus, Daniel initially invested approximately \$650.00.

Key Concepts

Investment Calculation

Investment calculation is a valuable tool for predicting how much your initial investment will grow over time. It's essential for anyone planning their finances or aiming to maximize their wealth. The process involves understanding the effects of compound interest, which can significantly increase the amount of money in your account over a period.

Compound interest means that interest is calculated on the initial principal and also on accumulated interest from previous periods. This type of interest accelerates wealth build-up as the investment duration increases.

• The longer you leave your money invested, the more it grows due to compounding.
• Knowing how to calculate future values helps you make informed financial decisions.

Understanding these calculations helps in setting realistic financial goals and choosing the right investment strategies to achieve them.

Principal Amount

The principal amount is the initial sum of money that Daniel originally invested. It represents the base figure on which the interest is calculated. In the context of the exercise, the goal is to determine how much Daniel invested two years ago, given the future value and the interest rate.

Calculating the principal allows investors to compare different investment options by their initial requirements and future potential. To find the initial principal using the compound interest formula, rearrange the formula to solve for "P." You already have the future value, interest rate, and time period; plug these into the formula to isolate and calculate the principal. This shows you how much your original investment was worth before any interest accrued.

Annual Interest Rate

The annual interest rate is the percentage increase on the money invested per year, a critical component in determining the future value of investments. It shows how much the investment grows each year and is usually compounded annually.

In Daniel’s scenario, the interest rate is 8%, meaning every year, the investment increases by 8% of its value at the beginning of that year.

• The rate impacts how quickly your investment grows.
• A higher annual rate leads to a larger future value, assuming all other factors remain constant.

Understanding the annual interest rate helps investors assess which investment provides the best return over the book, considering the risk associated with different rates.

Future Value Formula

The future value formula is a mathematical expression that helps determine how much a current investment will be worth in the future, given a certain rate of interest over a specified period. It’s expressed as: \[ A = P(1 + r)^n \]

Where:

• \( A \) is the future value of the investment.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (expressed as a decimal).
• \( n \) is the number of years the money is invested for.

This formula allows you to calculate how your investment will grow over time, providing a future value of \$758.16 in Daniel's example. By substituting known values into the equation and solving for the principal, you can deduce the initial investment amount.

The future value formula is vital for financial planning, as it helps individuals project their savings growth and plan accordingly for large expenses or retirement. Understanding how to use this formula empowers you to make strategic investment choices.