Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

To help finance the purchase of a new house, the Abdullahs have decided to apply for a shortterm loan (a bridge loan) in the amount of $$\$ 120,000$$ for a term of 3 mo. If the bank charges simple interest at the rate of \(12 \% /\) year, how much will the Abdullahs owe the bank at the end of the term?

Step-by-Step Solution

Answer

The Abdullahs will owe the bank $123,600 at the end of the 3-month term after taking a bridge loan of $120,000 at a simple interest rate of 12% per year.

Step 1: Write down the given values

We are given: - Principal (P) = $120,000 - Interest rate (R) = 12% per year - Time (T) = 3 months (1/4 of a year)

Step 2: Convert the time to a fraction of a year

Since the loan is for 3 months, we need to convert it into a fraction of a year. To do this, divide the number of months by 12 (the number of months in a year). T = 3/12 = 1/4

Step 3: Calculate the simple interest

To calculate the simple interest (I), we can use the formula: I = P * R * T where P = Principal amount R = Interest rate (in decimal form) T = Time (in years) First, we will convert the interest rate (R) from a percentage to a decimal by dividing by 100. So, 12% as a decimal is 0.12. Now, substitute the values of P, R, and T in the formula: I = $120,000 * 0.12 * (1/4)

Step 4: Perform the calculations

Now, we need to calculate the product of the numbers: I = \(120,000 * 0.12 * (1/4) = \)120,000 * 0.12 * 0.25 = $3,600

Step 5: Calculate the total amount owed

The total amount owed by the Abdullahs at the end of the term is the sum of the principal amount and the interest. Total amount owed = Principal amount + Interest amount Total amount owed = \(120,000 + \)3,600 = $123,600 So, the Abdullahs will owe the bank $123,600 at the end of the 3-month term.

Key Concepts

Applied Mathematics

Applied mathematics plays a crucial role in solving real-world problems, such as calculating the costs associated with loans and investments. When the Abdullahs decide to take out a bridge loan to finance their new house, they are encountering a practical application of mathematical principles.

Converting time periods and rates, and then using these conversions in formulas to compute the total amount owed, showcases how mathematics can be applied to financial situations. It’s not just about understanding the formula; it’s about knowing how to manipulate it to fit different scenarios—like changing a 3-month period into a fraction of a year for the interest rate calculation.

Financial Mathematics

Financial mathematics is the backbone of many transactions and decisions in the economic world. The simple interest calculation is a fundamental concept in this field and recognizes the cost of borrowing money over time.

In the case of the Abdullahs' loan, financial mathematics allows us to compute the interest that will accumulate over a short term, based on a given principal amount, an interest rate, and a time period. This area of study enables individuals and businesses to plan their finances, budget for the future, and understand the implications of the financial choices they make.

Interest Rate Calculation

Interest rate calculation is a vital process in understanding loans and investments. Simple interest is calculated by multiplying the principal amount, the interest rate, and the time period of the loan or investment. What’s important is recognizing that the interest rate must be expressed as a decimal and the time as a yearly fraction for the formula to work.

The Abdullahs' example elaborates on how simple interest can be calculated when borrowing money. The simplicity of the formula makes it easy to apply, which is especially useful for short-term loans or when the interest does not compound. By mastering this computation, students can make informed financial decisions and better comprehend how interest affects the total amount they will owe or earn.