Problem 81 - College Algebra and Calculus: An Applied Approach

Question

For \(\$ 1500\) to grow to more than \(\$ 1890\) in 3 years, what must the simple interest rate be?

Step-by-Step Solution

Answer

The simple interest rate required for $1500 to grow to more than $1890 in 3 years is approximately 8.67%.

Step 1: Identify Given Values

First, identify the given values from the exercise. The initial principal \(P\) is $1500, the final total amount is $1890, and the time \(T\) in years is 3.

Step 2: Calculate Interest

Next, calculate the total interest obtained. This is found by subtracting the initial principal from the final amount: \(I = $1890 - $1500 = $390\).

Step 3: Rearrange Simple Interest Formula

In order to find the rate, the formula for simple interest needs to be rearranged to solve for rate \(R\), this can be written as: \(R = I / PT\).

Step 4: Solve for Rate

Now, substitute the given values into the rearranged formula: \(R = $390 / ($1500*3) = 0.0867 \). This is the decimal form of the rate.

Step 5: Convert to Percentage

Finally, to express the rate as a percentage, which is the usual way interest rates are represented, multiply the decimal by 100: \(0.0867 * 100 = 8.67\%\). This is the simple interest rate required.

Key Concepts

Interest Rate Calculation

Interest rate calculation is a fundamental concept in finance, especially important when dealing with simple interest. Simple interest refers to the additional amount earned or paid on a principal over a period of time. Calculating the interest rate is an essential skill, as it helps determine the profitability or cost of an investment or loan.

To find the interest rate using simple interest, you can use the formula:
\[ R = \frac{I}{P \times T} \]
Where:

• \( R \) is the interest rate.
• \( I \) is the interest earned or paid.
• \( P \) is the principal amount.
• \( T \) is the time period in years.

Once you have the rate in decimal form, it is often converted into a percentage by multiplying by 100. This step makes the interest rate easier to interpret and apply in practical scenarios, as percentages are more intuitive when discussing rates of change or growth. Understanding how to calculate and interpret the interest rate is crucial for making informed financial decisions.

Principal Amount

The principal amount is the initial money invested or borrowed before any interest is applied. It's a foundational element in finance, serving as the base upon which interest is calculated. Knowing the principal amount is vital for determining how much interest will accrue over time.

Let's break it down further:

• The principal is denoted as \( P \) in financial formulas.
• In our original exercise, the principal amount was \$1500.
• It is the value from which any earned interest will be calculated.

Understanding the principal is not only important for interest calculations but also for assessing your total financial commitment when entering into loans or investments. The principal tells us the magnitude of the initial financial transaction, setting a baseline for how much growth (through investments) or cost (through loans) to expect.

Time in Years

Time in years plays a significant role when calculating simple interest, as it directly impacts how much total interest is accumulated. The concept of time helps determine the duration over which the principal is invested or borrowed, crucial for understanding the full effect of interest.

Here are some key points to remember:

• Time, denoted as \( T \), is measured in years in the context of simple interest calculations.
• In the provided exercise, the time period was 3 years.
• The longer the time, the more interest will be accrued on the principal.

Using time appropriately is essential for accurate calculations and projections in financial planning. It allows you to forecast how much interest will accumulate and helps in making strategic decisions about investments and loans. Recognizing the role of time can make a significant difference in financial outcomes.