Problem 39 - Elementary Algebra

Question

Use the formula \(i=P n\) to reach a solution. (Objective A) Find the annual interest rate if \(\$ 560\) in interest is earned when \(\$ 3500\) was invested for 2 years.

Step-by-Step Solution

Answer

The annual interest rate is 8%.

Step 1: Identify Given Values

We are given that the principal, \( P = 3500 \), the interest \( i = 560 \), and the time \( n = 2 \) years. We need to find the interest rate, \( r \), which is not directly in the formula, but can be found through the relationship of interest, principal, and time.

Step 2: Rearrange the Formula

The formula for simple interest is \( i = P n r \). To find \( r \), we rearrange the formula to solve for \( r \), yielding \( r = \frac{i}{P n} \).

Step 3: Substitute the Given Values

Substitute the known values into the formula: \( r = \frac{560}{3500 \times 2} \).

Step 4: Perform the Calculation

Calculate the interest rate: \( r = \frac{560}{7000} = 0.08 \).

Step 5: Convert to Percentage

Convert the decimal to a percentage by multiplying by 100: \( r = 0.08 \times 100 = 8\% \).

Key Concepts

Interest Rate Calculation

Calculating the interest rate is often a simple process when you follow the straightforward steps connected to simple interest. The goal is to determine the rate at which your investment grows over a certain period. In the context of simple interest, the formula we typically use is given by:

\( i = P n r \)

where:

• \(i\) is the interest earned or paid.
• \(P\) is the principal, or the original amount of money invested or loaned.
• \(n\) is the time period for which the money is invested or borrowed, often measured in years.
• \(r\) is the interest rate expressed as a decimal.

To find the interest rate, you need to rearrange the formula to solve for \(r\), given the principal \(P\), interest \(i\), and time \(n\):

\( r = \frac{i}{P n} \)

By following this rearrangement and substituting in the known values, you can easily calculate the interest rate as a decimal. To convert it to a percentage, a simple multiplication by 100 is needed.

Principal and Interest

Understanding the concepts of principal and interest is fundamental to comprehending how loans and investments work.

The principal refers to the original sum of money that is either invested in a savings account or borrowed from a lender. This is the baseline amount before any interest is applied.

Consider an example where you invest \(\\(3500\). This amount is your principal. Now, the interest is the extra money earned (or paid, in the case of a loan) over time, as a percentage of the principal. For example, if after two years you earn \(\\)560\) in interest, this is the additional money your investment has generated.

Why are these concepts critical? Because understanding them helps you grasp how different interest rates and time frames can affect the final amount of your investment or debt. It's always important to remember that the larger the principal, or the longer the investment period, the more interest you can potentially accumulate.

Mathematical Formulas

Mathematical formulas play a crucial role in calculating financial concepts like simple interest. They provide a structured way to approach problems. Below, we focus on the formula for simple interest:

\( i = P n r \)

To effectively use this formula, you'll need to comprehend what each variable represents:

• \(i\), the interest: This is what you earn or owe, based on the principal and the interest rate over time.
• \(P\), the principal: The base amount placed in or taken out of savings or a loan.
• \(n\), the time: Typically represented in years, this is how long the principal is invested or borrowed.
• \(r\), the rate: Expressed as a decimal, it indicates the percentage of interest applied annually.

To find any unknown in this context, like the interest rate, you may need to rearrange the formula. This mathematical manipulation provides flexibility, allowing you to solve for the variable that you are interested in finding. Learning to adjust formulas based on the known values is a fundamental skill in mathematics, ensuring you're prepared to tackle a range of problems using logical steps.