Problem 7 - Algebra 2

Question

Use the following information. Simple interest is calculated using the formula \(I=p r t,\) where \(p\) represents the principal in dollars, \(r\) represents the annual interest rate, and \(t\) represents the time in years. Find the simple interest I given each set of values. \(p=\$ 1800, r=6 \%, t=4\) years

Step-by-Step Solution

Answer

The simple interest is $432.

Step 1: Convert Percentage to Decimal

The interest rate given is 6%. To use it in the formula, convert this percentage into a decimal by dividing by 100. Thus, \( r = \frac{6}{100} = 0.06 \).

Step 2: Use the Simple Interest Formula

Substitute the given values into the simple interest formula \( I = prt \). Here, \( p = 1800 \), \( r = 0.06 \), and \( t = 4 \).

Step 3: Calculate the Interest

Now calculate the simple interest using the formula: \[ I = 1800 \times 0.06 \times 4 \].

Step 4: Complete the Calculation

Multiply the values: first multiply 1800 and 0.06 to get 108. Then multiply 108 by 4 to get the final interest amount: \( I = 432 \).

Key Concepts

Interest Rate Conversion

To accurately calculate simple interest, the interest rate must be in a decimal format. This conversion is essential because percentage values cannot be directly used in mathematical equations. Conversion involves dividing the percentage by 100.
For example, an interest rate of 6% would be converted to a decimal as follows:

• Take the percentage value: 6%
• Divide by 100: 6 ÷ 100 = 0.06

By converting to a decimal, calculations become straightforward and adaptable for insertion into formulas. This step ensures precision, particularly important in financial calculations where exactness is crucial.

Principal Amount

The principal amount refers to the initial sum of money that is loaned or invested, excluding any interest or additional funds. Understanding the principal is important because it serves as the basis for calculating interest.
Let's consider an example with a principal of $1800:

• This is the starting point, unaffected by any earnings or fees.
• Interest is earned or paid on the principal over time.

Grasping the concept of the principal helps in understanding how investments grow or loans become larger with the accrual of interest.

Interest Calculation

Interest calculation using the simple interest formula involves straightforward math but requires precision: The formula for simple interest is: \[ I = p \, r \, t \]where:

• \( I \) is the simple interest.
• \( p \) is the principal amount (e.g., $1800).
• \( r \) is the annual interest rate in decimal form (e.g., 0.06).
• \( t \) is the time in years (e.g., 4 years).

In our example, substitute the given values into the formula: \[ I = 1800 \times 0.06 \times 4 \]
Following the calculation steps reveals the interest earned: \( I = 432 \). This result signifies the total earnings from the investment based on the given rate and time period.

Algebraic Formulas

Algebraic formulas make financial calculations efficient and effective by providing a structured approach. In the context of simple interest, the formula \( I = prt \) serves as an algebraic model.
This model allows:

• Flexible substitution of variables depending on available data
• Adjustment for various principal amounts, rates, and times

For instance, algebra can solve for any missing variable if the others are known, making it a powerful tool for financial planning and comparisons.
Understanding these formulas is crucial for analyzing financial scenarios, predicting future outcomes, and making informed decisions in both personal and professional financial contexts.