Problem 97 - Elementary Algebra

Question

Problem: Find the simple interest earned on \(\$ 300\) invested for 5 years at an annual simple interest rate of \(4.5 \%\). Incorrect Answer: \(I=P R T\) $$ \begin{aligned} &I=(\$ 300)\left(\frac{4.5}{1 \text { year }}\right)(5 \text { years }) \\ &I=\$ 6750 \end{aligned} $$

Step-by-Step Solution

Answer

The simple interest earned is $67.50.

Step 1: Recall the simple interest formula

The formula for calculating simple interest is given by: \[ I = P \times R \times T \] where \( I \) is the interest, \( P \) is the principal amount, \( R \) is the annual interest rate (in decimal form), and \( T \) is the time in years.

Step 2: Convert the interest rate to a decimal

The given annual interest rate is 4.5%. Convert this percentage to a decimal by dividing by 100: \[ R = \frac{4.5}{100} = 0.045 \]

Step 3: Substitute the values into the formula

Substitute the given values into the simple interest formula: \[ P = 300, \ R = 0.045, \ T = 5 \] \[ I = 300 \times 0.045 \times 5 \]

Step 4: Calculate the interest

Perform the multiplication to find the interest: \[ I = 300 \times 0.045 \times 5 \] Calculate step-by-step: \[ 300 \times 0.045 = 13.5 \] \[ 13.5 \times 5 = 67.5 \] So, \[ I = 67.5 \]

Key Concepts

Simple Interest Formula

To find the simple interest on an investment, we use a specific formula. This formula helps us determine how much interest we will earn over a certain period. The simple interest formula is: \[ I = P \times R \times T \] Here, \[ I \] stands for interest, \[ P \] is the principal amount (the initial amount of money invested), \[ R \] is the annual interest rate expressed as a decimal, and \[ T \] is the time the money is invested for, in years. This formula is straightforward and only requires us to multiply three numbers together to get the interest earned.

Decimal Conversion

When working with percentages, you often need to convert them into decimal form to use in formulas. In our simple interest calculation, the interest rate is given as a percentage. The given interest rate is 4.5%. To convert this percentage into a decimal, you divide by 100. \[ R = \frac{4.5}{100} = 0.045 \] This step is crucial as using the percent form directly would lead to incorrect results. Remember, moving the decimal place two spots to the left is a quick way to convert percentages to decimals.

Interest Rate

The interest rate is an essential component of the simple interest formula. It represents the percentage of the principal that will be paid as interest each year. In our problem, the interest rate is 4.5% per year. After converting it to a decimal, it becomes 0.045. This decimal form makes it easier to multiply in the simple interest formula. It's important to ensure you have the correct decimal form to avoid errors in your calculations.

Multiplication Steps

After having all the values in the correct form, you can substitute them into the simple interest formula. Let's use our given problem values: principal \[ P = 300 \], rate \[ R = 0.045 \], and time \[ T = 5 \]. The formula we use is: \[ I = 300 \times 0.045 \times 5 \] First, multiply \[ 300 \times 0.045 \]. \[ 300 \times 0.045 = 13.5 \]. Next, multiply the result by 5. \[ 13.5 \times 5 = 67.5 \]. Therefore, the simple interest earned is \[ I = 67.5 \]. Breaking down the steps in the multiplication makes it easier to follow and helps ensure accuracy in your final answer.