Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the simple interest on a $$\$ 1000$$ investment made for 3 yr at an interest rate of \(5 \% /\) year. What is the accumulated amount?

Step-by-Step Solution

Answer

The simple interest earned on the $1000 investment over 3 years at a 5% interest rate is $150. The accumulated amount after 3 years is $1150.

Step 1: Understand the simple interest formula

The formula to calculate simple interest is as follows: I = P × R × T where I = simple interest P = principal amount (initial investment) R = interest rate per period (per year in this case) T = number of periods (number of years in this case)

Step 2: Apply the formula to the given exercise

In this exercise, we have: P = 1000 (initial investment) R = 5% (interest rate per year) T = 3 years Let's plug these values into the formula: I = 1000 × 0.05 × 3

Step 3: Calculate the simple interest

Now we will calculate the interest earned on this investment by solving the equation: I = 1000 × 0.05 × 3 I = 1000 × 0.15 I = 150 So the simple interest earned on this investment over 3 years is $150.

Step 4: Calculate the accumulated amount

To find the accumulated amount, we will add the simple interest earned to the initial investment: accumulated amount = principal amount + simple interest accumulated amount = 1000 + 150

Step 5: Find the final accumulated amount

Now we just need to add the numbers: accumulated amount = 1000 + 150 accumulated amount = 1150 Thus, the accumulated amount after 3 years is $1150.

Key Concepts

Principal Amount

The principal amount is simply the initial sum of money that you invest or deposit, before any interest is applied. It's the foundation of any investment or loan agreement. In the context of simple interest, the principal amount is crucial because it directly influences the total interest earned or owed over time.

For our example, the principal amount is \( \$1000 \), which is the initial investment sum. It remains constant since simple interest does not compound or add onto itself during the investment period. When calculating simple interest, the principal amount is multiplied by the interest rate and the time period to determine total interest.

Interest Rate

The interest rate is a percentage that represents how much extra money is earned on an investment or paid on a loan per period. It is expressed as a percentage of the principal amount and reflects the cost of borrowing or the reward for investing money.

In simple interest calculations, the interest rate is important because it directly determines the earnings or costs associated with the principal amount. In our example, the interest rate is \( 5\% \) per year. This means that each year, \( 5\% \) of the principal amount (\$1000) is added as interest. For calculations, convert the percentage to a decimal, so \( 5\% \) becomes \( 0.05 \). This conversion helps in applying the rate in mathematical formulas more easily.

Accumulated Amount

The accumulated amount represents the total value of an investment at the end of the interest period. This includes both the original principal amount and any interest earned. In simple terms, it's the sum you get after combining the principal amount and the interest.

For the example provided, the accumulated amount is calculated by adding the principal amount (\\(1000) to the simple interest earned (\\)150). The result is \\(1150. The accumulated amount provides a clear picture of the outcome of the investment or loan over the specified time period.

• Principal Amount: \\)1000
• Simple Interest: \\(150
• Accumulated Amount: \\)1150

Thus, understanding the accumulated amount helps investors or borrowers to comprehend the overall returns or costs after a given period.