Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the simple interest on a $$\$ 500$$ investment made for 2 yr at an interest rate of \(8 \% /\) year. What is the accumulated amount?

Step-by-Step Solution

Answer

The simple interest earned on the $$\$500$$ investment after 2 years at an 8% interest rate is $$\$80$$. Therefore, the accumulated amount is $$\$580$$.

Step 1: Identify the given values

In this problem, we are given: - Principal Amount (P) = $500 - Interest Rate (R) = 8% per year - Time (T) = 2 years

Step 2: Calculate the Simple Interest

To calculate the simple interest, we can use the formula: Simple Interest (SI) = \( \frac{Principal \times Interest Rate \times Time}{100} \) Substitute the given values into the formula: SI = \( \frac{500 \times 8 \times 2}{100} \)

Step 3: Solve for Simple Interest

Calculate the result of the equation: SI = \( \frac{8000}{100} \) SI = $80 So, the simple interest earned on the investment after 2 years is $80.

Step 4: Calculate the Accumulated Amount

To find the total accumulated amount, we can use the formula: Accumulated Amount = Principal Amount + Simple Interest Substitute the given values into the formula: Accumulated Amount = 500 + 80

Step 5: Solve for Accumulated Amount

Calculate the result of the equation: Accumulated Amount = $580 So, the accumulated amount after 2 years is $580.

Key Concepts

Interest Rate

The interest rate is a critical component when calculating simple interest. It refers to the percentage of the principal amount that is paid as interest over a specific period, usually one year. In simpler terms, if you borrow or invest money, the interest rate determines how much you will pay or earn as a percentage of the initial amount.

Understanding interest rates is essential for making informed financial decisions. Here are some key points about interest rates:

• It is expressed as a percentage and commonly represented per annum (yearly).
• A higher interest rate means a larger amount of interest earned or paid over time.
• In simple interest calculations, the rate remains fixed and does not compound.

In our example, the interest rate is 8% per year. This means for every $100 invested, you earn $8 per year.

Accumulated Amount

The accumulated amount is the total of the original principal amount plus the interest earned over a specific time period. It represents the end balance after interest has been applied to the initial investment or loan.

It's crucial to understand the accumulated amount because it shows the full value of your investment or loan payoff at the end of the interest period:

• The formula to find the accumulated amount in simple interest is: \( \text{Accumulated Amount} = \text{Principal} + \text{Simple Interest} \).
• This helps investors know exactly how much they will receive after the investment period.
• For borrowers, it helps understand the total amount to be repaid on a loan.

From our calculations, the accumulated amount after 2 years is \(580. This includes the original \)500 principal plus $80 earned as interest.

Mathematical Formula

Using a mathematical formula to calculate simple interest is essential to efficiently determine earnings or costs associated with borrowing or investing money. The formula for simple interest is straightforward and easy to use for quick calculations without complex financial tools.

Here are the basics of the simple interest formula:

• The formula is \( \text{SI} = \frac{P \times R \times T}{100} \), where \( P \) is the principal amount, \( R \) is the annual interest rate, and \( T \) is the time in years.
• This formula helps you calculate just the interest portion without using additional calculations.
• The simplicity of this formula makes it popular for short-term loans and clear investment projections.

Using the formula with the values given, we found the simple interest to be \(80 over 2 years for a \)500 investment at 8% interest. This formula is a tool to assess potential investments and understand growth over time.