Problem 14 - Prealgebra

Question

If you invest \(P\) dollars ( \(P\) is for principal) at simple interest rate \(R\) for \(T\) years, the amount of interest you will earn is given by the formula \(I=P \cdot R \cdot T .\) Find \(I\) $$P=\$ 2,000, R=\frac{6}{100}, \text { and } T=2 \frac{1}{2} \text { years }$$

Step-by-Step Solution

Answer

The interest earned is $300.

Step 1: Identify Given Values

First, we need to identify the values we have been given to solve the problem. We have:\[ P = 2000 \] the principal in dollars, \[ R = \frac{6}{100} = 0.06 \] the interest rate as a decimal, and \[ T = 2 \frac{1}{2} = 2.5 \] the time in years.

Step 2: Write the Simple Interest Formula

The formula for simple interest is given by: \[ I = P \cdot R \cdot T \]where \(I\) is the interest, \(P\) is the principal, \(R\) is the rate, and \(T\) is the time.

Step 3: Substitute Values into the Formula

Next, substitute the known values into the formula:\[ I = 2000 \times 0.06 \times 2.5 \]

Step 4: Perform the Multiplications

Calculate the interest by performing the multiplications step by step:First calculate \( 2000 \times 0.06 = 120 \).Then multiply the result by 2.5:\[ 120 \times 2.5 = 300 \]

Step 5: State the Final Answer

The amount of interest earned is \( I = 300 \) dollars.

Key Concepts

Principal in Simple Interest

The principal is the amount of money you initially invest or borrow before any interest is applied. Essentially, it is the starting point in any financial transaction involving interest. When calculating simple interest, the principal is denoted by the letter \(P\). In our example, we began with \(P = 2000\) dollars, meaning this is the original sum.Understanding the principal is crucial as it directly affects the total interest you earn or pay. Larger principal amounts usually lead to more interest accrued because you're earning interest on a larger initial sum. In simple interest calculations, your initial principal remains unchanged by interest—it doesn't grow; only your total earnings increase.

Understanding Interest Rate

The interest rate is a percentage that represents the cost of borrowing money or the return on investment. In simple interest calculations, it’s expressed as a decimal and is denoted by the symbol \(R\). For this exercise, the given rate was \(6\%\), which translates to \(0.06\) when used in calculations.Here’s why the rate matters:

• **Determines Earnings or Costs**: A higher rate means more interest earned or owed over time.
• **Applied Annually**: Simple interest assumes the rate is applied once per year, unlike compound interest, which may be applied more frequently.

Knowing how to convert a percentage into a decimal for use in formulas is essential for accurate calculations.

Time in Simple Interest Calculations

In simple interest calculations, time is measured in years and is denoted by \(T\). It is the duration for which the money is invested or borrowed. In the problem at hand, the time given was 2.5 years.Why does time matter?

• **Directly Affects Interest Earned/Owed**: The longer the time, the more interest accumulates. This is because interest is calculated using the formula \(I = P \cdot R \cdot T\), where time is a direct multiplier.
• **Easy Conversion**: Sometimes, you may need to convert months into years by dividing by 12 if the time isn't given in years initially. For instance, 30 months would be 2.5 years.

The simplicity of using years makes it straightforward to calculate the resulting interest when you know the time frames involved.