Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

If the accumulated amount is $$\$ 1160$$ at the end of \(2 \mathrm{yr}\) and the simple rate of interest is \(8 \% /\) year, then what is the principal?

Step-by-Step Solution

Answer

The principal is \( \$1000 \).

Step 1: Understand the given information

We are given the following information: - Accumulated amount (A) at the end of 2 years: \( \$1160 \) - Simple interest rate (R) per year: \( 8\% \) - Time period (T): \( 2 \) years We need to find the principal (P).

Step 2: Recall the formula for calculating accumulated amount

The formula for calculating the accumulated amount (A) using simple interest is: \[ A = P(1 + RT) \] Where: - A = Accumulated amount - P = Principal - R = Rate of interest - T = Time period

Step 3: Rearrange the formula to find the principal (P)

To find the principal (P), we need to rearrange the formula in terms of P. We can do this by dividing both sides of the equation by \( (1 + RT) \): \[ P = \frac{A}{(1 + RT)} \]

Step 4: Plug in the given values and solve for principal

Now, we can plug in the given values for the accumulated amount (A), rate of interest (R), and time period (T) into the formula: \[ P = \frac{\$1160}{(1 + 0.08 \times 2)} \] Solve for P: \[P = \frac{\$1160}{(1 + 0.16)}\] \[P = \frac{\$1160}{1.16}\] \[P = \$1000\]

Step 5: Write the answer

The principal is \( \$1000 \).

Key Concepts

Accumulated Amount

The accumulated amount in finance is the total sum of money that an investment has grown to over a period of time. It includes the original principal amount as well as the total interest earned.In our exercise, the accumulated amount is given as \$1160 after 2 years. This value is crucial in understanding how much the initial investment has increased due to interest. To calculate the accumulated amount with simple interest, the formula \(A = P(1 + RT)\) is used, where \(A\) represents the accumulated amount, \(P\) the principal, \(R\) the interest rate, and \(T\) the time period in years. In simple interest problems, the interest is calculated uniformly on the original principal for each time period.

Interest Rate

The interest rate, usually expressed as a percentage, is the amount charged by a lender to a borrower for the use of assets. In our context, it refers to the percentage of the principal amount that is paid as interest over one year.For the given exercise, the interest rate is \(8\%\) per annum. It plays a vital role in calculating both the interest earned and the accumulated amount over time. The rate at which money grows is directly proportional to the interest rate; a higher rate means more interest accrued over the same time period.

When dealing with simple interest, the calculation does not consider compound interest, i.e., it does not take into account the interest on the interest from previous periods – it is always calculated on the original principal.

Principal Amount

The principal amount in financial mathematics represents the initial sum of money put on deposit or invested before the addition of any interest. It is essentially the starting point of any loan or investment calculation.In simpler terms, it's the original amount of money that you have or that you borrow, which will be subjected to interest over time. In our exercise, we are tasked to find out this original amount, knowing that after 2 years at a simple interest rate of \(8\%\), the accumulated amount is \$1160. By using the formula, rearranging it, and plugging in the values we have, we find that the principal is \$1000, which means this was the amount originally invested or borrowed.

Financial Mathematics

Financial mathematics is the application of mathematical methods to financial problems. It encompasses a wide range of topics, from simple interest and principal calculations to more complex issues such as the valuation of securities and risk management.Within this discipline, understanding concepts such as simple interest is fundamental. It is one of the basic building blocks used to calculate how investments and loans can grow or cost over time. The exercise showcases a practical use of such a financial mathematics concept, where understanding the relationship between the principal, interest rate, and time is critical to determining the accumulated amount – an essential skill in both personal and professional financial planning.