Problem 8 - 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. \(p=\$ 31,000, r=2 \frac{1}{2} \%, t=18\) months

Step-by-Step Solution

Answer

The simple interest is $1162.50.

Step 1: Convert the Interest Rate

The interest rate is given as 2 1/2%, which can be converted to a decimal by dividing by 100. So, \( r = 2.5\% \) becomes \( r = \frac{2.5}{100} = 0.025 \).

Step 2: Convert Time to Years

The time is given as 18 months. Since there are 12 months in a year, \( t = \frac{18}{12} = 1.5 \) years.

Step 3: Apply the Simple Interest Formula

Use the formula \( I = p \times r \times t \) with the given values: \( p = 31000 \), \( r = 0.025 \), and \( t = 1.5 \). Substitute these into the formula: \( I = 31000 \times 0.025 \times 1.5 \).

Step 4: Calculate the Simple Interest

Perform the multiplication: \( 31000 \times 0.025 = 775 \). Then multiply by 1.5: \( 775 \times 1.5 = 1162.5 \). Thus, the simple interest \( I \) is $1162.50.

Key Concepts

Interest Rate

Understanding interest rate is crucial for calculating simple interest. The interest rate represents the percentage of the principal charged as interest for a particular period. In this context, it's annual, which means it corresponds to a one-year period.

• When the interest rate is given in percentages, you need to convert it into a decimal before inserting it into the simple interest formula.
• To convert a percentage to a decimal, divide the percentage value by 100. For instance, converting 2.5% to decimal results in 0.025.

Performing this conversion correctly ensures that your calculations for interest are precise.

Time Conversion

Time conversion plays a vital role in calculating simple interest, especially when time is not directly provided in years. Interest calculations require time to be in years.

• If time is specified in months, you need to convert this to years.
• There are 12 months in a year, so simply divide the number of months by 12 to get the time in years. For example, 18 months equals 1.5 years.

Accurate time conversion is essential for ensuring the simple interest calculation is correctly scaled to the intended period.

Interest Formula

The simple interest formula is the heart of calculations involving simple interest. It is expressed as: \( I = p \times r \times t \). Here's a simple breakdown of the formula:

• \(I\) represents the interest earned or paid.
• \(p\) is the principal, which is the initial amount of money loaned or invested.
• \(r\) stands for the interest rate, which is converted into decimal form as shown earlier.
• \(t\) represents time in years.

To compute the interest, multiply the values of the principal, rate, and time together. This approach provides a straightforward way to understand financial growth or cost over a defined period. Applying this formula helps across various financial scenarios, from loans to investments.

Financial Calculations

Financial calculations are critical for making informed financial decisions. Simple interest calculations are among the fundamental financial computations used in everyday scenarios. They allow stakeholders to see how much interest will accrue on a principal over time.

• The computation involves basic algebraic operations with emphasis on the accurate conversion of interest rates and time.
• Understanding financial calculations can help in evaluating costs of loans or understanding gains from investments.

Mastering the art of financial calculations not only aids in performing interest computations but also builds a foundation for understanding more complex financial products.