what is the rate when the maturity value is 8801, the interest is $81.20 and the time is 77/360 days
The correct answer and explanation is:
To find the rate (annual interest rate), we can use the simple interest formula: Interest=Principal×Rate×Time\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}
However, we are given:
- Maturity Value (MV) = $8,801
- Interest (I) = $81.20
- Time (T) = 77/360 (in years)
We are solving for the Rate (R).
Step 1: Find the Principal
Since: Maturity Value=Principal+Interest\text{Maturity Value} = \text{Principal} + \text{Interest}
Then: Principal=8,801−81.20=8,719.80\text{Principal} = 8,801 – 81.20 = 8,719.80
Step 2: Use the Simple Interest Formula
Interest=Principal×Rate×Time\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} 81.20=8,719.80×R×7736081.20 = 8,719.80 \times R \times \frac{77}{360}
Step 3: Solve for R
R=81.20×3608,719.80×77R = \frac{81.20 \times 360}{8,719.80 \times 77} R=29,232671,425.46≈0.0435R = \frac{29,232}{671,425.46} \approx 0.0435 Rate=0.0435 or 4.35%\text{Rate} = 0.0435 \text{ or } \boxed{4.35\%}
Explanation (300 Words):
In financial mathematics, the interest rate reflects the percentage of the principal that is paid as interest over a specific time period, typically one year. When given a maturity value, interest earned, and time, we can reverse-engineer the interest formula to find the rate.
The maturity value includes both the original principal and the interest accrued. So, we first subtract the interest from the maturity value to determine the actual principal amount that was invested or loaned. In this case, the maturity value is $8,801 and the interest earned is $81.20, leaving a principal of $8,719.80.
Next, we apply the simple interest formula: I=P×R×TI = P \times R \times T
This is rearranged to solve for the rate: R=IP×TR = \frac{I}{P \times T}
Because the time is provided as a fraction of a year (77 days out of 360, using the banker’s rule), we multiply accordingly. Substituting the values into the formula gives us: R=81.20×3608,719.80×77≈0.0435R = \frac{81.20 \times 360}{8,719.80 \times 77} \approx 0.0435
Thus, the interest rate is approximately 4.35% annually.
This method is commonly used in financial calculations involving treasury bills, short-term loans, and bank notes, especially when using 360-day financial years.