Problem 71 - Intermediate Algebra

Question

The equation \(I(r)=500 r\) expresses the amount of simple interest earned by an investment of \(\$ 500\) for 1 year as a function of the rate of interest \((r)\). Compute \(I(0.04), I(0.06), I(0.075)\), and \(I(0.09)\).

Step-by-Step Solution

Answer

The amounts are $20, $30, $37.5, and $45 respectively.

Step 1: Understanding the Formula

The formula given is \( I(r) = 500r \), which calculates the simple interest for one year on an investment of $500, where \( r \) is the interest rate expressed as a decimal.

Step 2: Calculate I(0.04)

Substitute \( r = 0.04 \) into the formula: \( I(0.04) = 500 \times 0.04 = 20 \). So, the simple interest for \( r = 0.04 \) is $20.

Step 3: Calculate I(0.06)

Substitute \( r = 0.06 \) into the formula: \( I(0.06) = 500 \times 0.06 = 30 \). So, the simple interest for \( r = 0.06 \) is $30.

Step 4: Calculate I(0.075)

Substitute \( r = 0.075 \) into the formula: \( I(0.075) = 500 \times 0.075 = 37.5 \). So, the simple interest for \( r = 0.075 \) is $37.5.

Step 5: Calculate I(0.09)

Substitute \( r = 0.09 \) into the formula: \( I(0.09) = 500 \times 0.09 = 45 \). So, the simple interest for \( r = 0.09 \) is $45.

Key Concepts

Understanding Interest Rate

Interest rate is a fundamental concept in finance. It's expressed as a percentage and determines how much interest your investment will earn over a period of time. In the formula given, the interest rate is represented by the variable \( r \) which is used as a decimal.

• For instance, an interest rate of 4% is expressed as 0.04 in this context.
• Converting percentages to decimals is done by dividing the percentage by 100.

Understanding and comparing different interest rates is crucial because a higher interest rate means more earnings on an investment. For students learning about this concept, it’s important to recognize that the interest rate directly influences the simple interest calculation.

Basics of Investment Math

Investment math helps you calculate how much money you'll make when you invest your principal amount at a certain interest rate. In our problem, the principal is \$500, and we are calculating how much interest this will generate over one year for different interest rates.

• Simple interest can be calculated using the formula: \( I = P \times r \times t \), where \( P \) is the principal, \( r \) is the rate, and \( t \) is the time in years.
• In this context \( t = 1 \), simplifying our formula to \( I = P \times r \).

The beauty of investment math is that it allows investors to understand growth over time and helps in making informed financial decisions. By comparing results from various interest rates, you can optimize and choose the best investment options.

Working with Algebraic Functions

Algebraic functions like \( I(r) = 500r \) are crafted to solve real-world problems by expressing relationships between different quantities. Here, the relationship is between the interest earned and the rate of interest for a fixed principal over a given time.

• Functions help us replace manual calculations with a systematic formula, making it easier to compute results for multiple scenarios.
• By plugging different values of \( r \) into the function, you can quickly find interest for different rates.

In this particular exercise, we substitute values into the function to calculate the simple interest efficiently. Understanding algebraic functions enables students to handle complex financial calculations with higher accuracy and efficiency.