Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Mitchell has been given the option of either paying his $$\$ 300$$ bill now or settling it for $$\$ 306$$ after 1 mo ( 30 days). If he chooses to pay after 1 mo, find the simple interest rate at which he would be charged.

Step-by-Step Solution

Answer

The simple interest rate at which Mitchell would be charged if he chose to pay after 1 month is 24%.

Step 1: Convert Time Period to Years

Since the time period given is 1 month (30 days), we'll convert it to years. There are 12 months in a year and 30 days in a month. Time (T) = 1 month * (1 year / 12 months) = \(\frac{1}{12}\) years

Step 2: Find the Simple Interest

The simple interest (I) can be found by subtracting the Principal amount (P) from the final amount (A): Simple Interest (I) = Final Amount (A) - Principal Amount (P) I = $$\$ 306 - \$ 300$$ I = $$\$ 6$$

Step 3: Solve for Rate of Interest (R)

Now, we have the simple interest, principal amount, and the time period. We can plug these values into the simple interest formula and solve for R: $$ 6 = (300 × R × \frac{1}{12}) / 100 $$ Next, we will multiply both sides by 12 to remove the fraction and then by 100 to isolate R: $$ R = (\frac{6 × 12 × 100}{300}) $$ Now, we can calculate the rate of interest: $$ R = 24 $$ So, the simple interest rate at which Mitchell would be charged if he chose to pay after 1 month is 24%.

Key Concepts

Simple Interest Calculation

Understanding how to calculate simple interest is crucial for managing personal finances and making informed decisions about loans and investments. In its most basic form, the simple interest formula is expressed as \( I = P \times R \times T \), where \( I \) is the interest earned or paid, \( P \) is the principal amount (the initial sum of money), \( R \) is the annual interest rate, and \( T \) is the time the money is invested or borrowed for, usually in years. In practice, the simple interest calculation allows you to predict how much extra money will be accrued on a loan or how much profit one can expect from an investment over a certain period.

For example, if Mitchell has a \(300 bill and decides to settle it later for \)306, the $6 difference represents the simple interest. To find the rate, we need to rearrange the formula to solve for \( R \), which gives us the annual interest rate that Mitchell's bill is effectively subject to. Always remember to ensure all units are consistent; in this case, the original amount (the principal), the interest charged, and the time period must align for the interest rate to be accurate. By understanding this process, students can apply the same principles to various financial situations involving simple interest.

Conversion of Time Periods in Interest Calculation

When calculating interest, it's essential to express the time period consistently with how the rate is quoted. For most interest calculations, the rate is an annual rate, meaning the time period \( T \) should be in years. This necessitates converting time periods such as months or days into a year fraction.

For instance, when dealing with a 1-month time frame as in Mitchell's case, we convert it to years since the interest rate is annual. There are 12 months in a year, so 1 month is equivalent to \( \frac{1}{12} \) or approximately 0.0833 years. A common mistake students make is not converting time periods, leading to incorrect interest amounts. It's also vital to realize that different months have different numbers of days, which can further complicate calculations for more precise interest assessments. Taking these conversion factors into account is fundamental in finance mathematics to ensure accurate calculations.

Finance Mathematics

Finance mathematics is an area of applied mathematics that focuses on financial markets, time-value of money, risk assessment, and investment strategies. It encompasses a range of topics from basic concepts like simple interest calculations to more complex topics such as compound interest, annuities, and amortization schedules.

For individuals like Mitchell, understanding finance mathematics can be the difference between making a financially sound decision and one that results in unnecessary expenses or lost opportunities. It's not just about calculating rates or converting periods; it's about using these numbers to guide real-world decisions and strategies.

In finance, mathematics is the language that allows us to quantify and compare the costs and benefits of different financial options. Whether it's deciding when to pay a bill, taking out a loan, or investing in the stock market, the principles of finance mathematics are the foundation of making educated and beneficial financial choices. This is why it is so important for students to grasp these concepts well, as they are applicable in everyday financial scenarios.