Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Maya paid $$\$ 10,000$$ for a 7 -yr bond issued by a city. She received interest amounting to $$\$ 3500$$ over the life of the bonds. What rate of (simple) interest did the bond pay?

Step-by-Step Solution

Answer

The bond paid a simple interest rate of \(5\%\).

Step 1: Identify the given values

Principal amount, $$P = \$10,000$$. Time period, $$T = 7$$ years. Total Interest, $$I = \$3,500$$.

Step 2: Write the simple interest formula

The formula for calculating simple interest is: $$I = P * R * T$$ Where, $$I$$ is the total interest, $$P$$ is the principal amount, $$R$$ is the rate of interest, and $$T$$ is the time period.

Step 3: Rearrange the formula to find $$R$$

We will rearrange the formula to solve for $$R$$: $$R = \frac{I}{P * T}$$

Step 4: Substitute the values and calculate $$R$$

Now, let's substitute the given values and calculate $$R$$: $$R = \frac{\$3,500}{\$10,000 * 7}$$ $$R = \frac{3,500}{70,000}$$ $$R = 0.05$$ To convert the rate into a percentage, we multiply it by $$100$$. $$R = 0.05 * 100$$ $$R = 5\%$$ Hence, the bond paid a simple interest rate of $$5\%$$.

Key Concepts

Bond Calculation

Bonds are essentially a loan you give to entities such as governments or corporations, and in return, they pay you back with interest over a set period. The bond calculation involves understanding how much interest you are set to receive for lending your money over a specific term. In this case, Maya purchased a bond worth $10,000, and the bond was due over 7 years.

• The total interest she earned was $3,500 during this time.
• This interest payment is what makes bonds a potentially profitable investment choice.

Unlike some complex investments, bonds often use simple interest calculations, making it easier to predict your earnings.
You simply take the principal amount you invest, multiply it by the interest rate, multiply by the number of years, and you have your earnings. Understanding these principles makes evaluating the potential returns of a bond straightforward.

Interest Rate Formula

Calculating the interest rate for a bond involves using the simple interest formula, which provides a straightforward method to determine how much you earn from your investment. The formula is given as:\[ I = P \times R \times T \]Where:

• \( I \) is the total interest earned,
• \( P \) is the principal amount or initial investment,
• \( R \) is the rate of interest,
• \( T \) is the time period the money is invested for.

By rearranging this formula to solve for \( R \), we can determine the simple interest rate:\[ R = \frac{I}{P \times T} \]This formula is crucial in financial mathematics because it allows anyone to figure out what interest rate was applied to their investment, based on the amount they started with and how much they earned. With Maya's bond, we substitute the given values to find an interest rate of 5%.

Financial Mathematics

Financial mathematics is a core concept used in the world of finance to evaluate investments, calculate loans, and understand the value of money over time. Simple interest is one of the foundational concepts here.

• It is used when the interest earned over a set period is based only on the principal amount.
• Compared to compound interest, where interest is earned on previously accumulated interest, simple interest is easier to calculate and predict.

Understanding financial mathematics helps investors make informed decisions. It provides tools to determine how profitable an investment like a bond can be over time. It also enables investors to compare the potential returns of different financial products, ensuring they achieve their desired financial goals. In the case of simple interest, it's all about straightforward calculations that show how an investment is expected to perform.