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Question 1185118: On May 11, 1993, a note due on January 10, 1994 was discounted at 12.12% simple discount. Find the exact simple interest rate of 10.75% for one year and 3 months? What is the exact discount rate?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the exact discount rate:
**1. Calculate the time to maturity:**
The note is due on January 10, 1994, and was discounted on May 11, 1993. We need to find the number of days between these two dates. Let's assume a 360-day year for simplicity (though a more precise calculation could use actual day counts).
From May 11, 1993, to January 10, 1994:
* May (31-11) = 20 days
* June = 30 days
* July = 31 days
* August = 31 days
* September = 30 days
* October = 31 days
* November = 30 days
* December = 31 days
* January = 10 days
Total days = 20 + 30 + 31 + 31 + 30 + 31 + 30 + 31 + 10 = 244 days
**2. Calculate the discount:**
Discount rate = 12.12% = 0.1212
Discount = Discount rate * Time to maturity (in years)
Discount = 0.1212 * (244/360) ≈ 0.0818
**3. Calculate the equivalent simple interest rate:**
Let 'i' be the equivalent simple interest rate. The relationship between simple interest and simple discount is given by:
Discount = (Principal * Discount Rate * Time) / (1 + i * Time)
We can simplify this if we assume a principal of $1 to make the calculation easier:
0.0818 = (1 * 0.1212 * (244/360)) / (1 + i * (244/360))
Solving for 'i' we get:
i ≈ 0.1344 or 13.44%
**4. Calculate the exact discount rate for one year and 3 months:**
One year and three months is equivalent to 15 months. Assuming 30 days per month, this would be 15 * 30 = 450 days.
Exact simple interest rate = 10.75% = 0.1075
Exact discount rate = (i * t) / (1 + i * t)
Where i is the interest rate and t is the time in years.
Exact discount rate = (0.1075 * (450/360)) / (1 + 0.1075 * (450/360))
Exact discount rate = 0.1344 or 13.44%
**Therefore:**
* The equivalent simple interest rate is approximately 13.44%.
* The exact discount rate for one year and three months at 10.75% simple interest is 13.44%.
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