To determine the proceeds of the note in both scenarios, we will use the relevant formulas for interest calculations.
a. Note with Interest Rate of 5%
The interest on the note can be calculated using the formula:
\[ \text{Interest} = \text{Face Amount} \times \text{Rate} \times \text{Time} \]
In this context, the face amount is $81,600, the interest rate is 5% (0.05), and the time is 90 days. Since we're using a 360-day year, time in years will be:
\[ \text{Time in years} = \frac{90}{360} = 0.25 \]
Now, calculate the interest:
\[ \text{Interest} = 81,600 \times 0.05 \times 0.25 = 81,600 \times 0.0125 = 1,020 \]
Now, to find the proceeds of the note (the amount of cash received), subtract the interest from the face amount:
\[ \text{Proceeds} = \text{Face Amount} - \text{Interest} = 81,600 - 1,020 = 80,580 \]
Proceeds of the note (with interest): $80,580
b. Note Discounted at 5%
For a discounted note, the formula for the proceeds is:
\[ \text{Proceeds} = \text{Face Amount} - \text{Discount} \]
Where the discount is calculated as follows:
\[ \text{Discount} = \text{Face Amount} \times \text{Rate} \times \text{Time} \]
Using the same face amount, rate, and time as before:
\[ \text{Discount} = 81,600 \times 0.05 \times 0.25 = 1,020 \]
Now calculate the proceeds:
\[ \text{Proceeds} = 81,600 - 1,020 = 80,580 \]
However, since the proceeds are calculated before the duration starts, to get the actual cash received, we calculate on the face amount directly:
The formula for the proceeds when discounted is:
\[ \text{Proceeds} = \text{Face Amount} \times (1 - \text{Rate} \times \text{Time}) \]
Thus:
\[ \text{Proceeds} = 81,600 \times (1 - 0.05 \times 0.25) = 81,600 \times (1 - 0.0125) = 81,600 \times 0.9875 \]
Calculating the above gives:
\[ \text{Proceeds} = 81,600 \times 0.9875 = 80,580 \]
Proceeds of the note (discounted): $80,580
Summary of Results:
a. Proceeds with interest: $80,580
b. Proceeds when discounted: $80,580
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