Alright, let's work through the problem step-by-step to find the proceeds of this non-interest-bearing promissory note.
### Step 1: Convert the Discount Period to Years
First, let's convert the discount period from months to years. Given that there are 12 months in a year:
[tex]\[ \text{Discount period in years} = \frac{55 \text{ months}}{12 \text{ months/year}} = 4.583333333333333 \text{ years} \][/tex]
### Step 2: Determine the Number of Compounding Periods
Given that interest is compounded semi-annually, we need to calculate the total number of compounding periods. There are 2 compounding periods per year (semi-annual):
[tex]\[ \text{Number of compounding periods} = 2 \text{ periods/year} \times 4.583333333333333 \text{ years} = 9.166666666666666 \][/tex]
### Step 3: Calculate the Discount Factor
Next, we need to calculate the discount factor using the formula for compound interest discounting:
[tex]\[ \text{Discount factor} = \left(1 + \frac{r}{m}\right)^{-n} \][/tex]
where:
- [tex]\( r \)[/tex] is the annual interest rate (0.099 or 9.9%),
- [tex]\( m \)[/tex] is the number of compounding periods per year (2),
- [tex]\( n \)[/tex] is the total number of compounding periods (9.166666666666666).
Substituting the given values:
[tex]\[ \text{Discount factor} = \left(1 + \frac{0.099}{2}\right)^{-9.166666666666666} = 0.6421861573826311 \][/tex]
### Step 4: Calculate the Proceeds
Finally, we calculate the proceeds by multiplying the face value of the promissory note by the discount factor:
[tex]\[ \text{Proceeds} = 7231.56 \times 0.6421861573826311 = 4644.00772828194 \][/tex]
### Summary
- Discount period in years: 4.583333333333333 years
- Number of compounding periods: 9.166666666666666
- Discount factor: 0.6421861573826311
- Proceeds: [tex]$4644.00772828194
Therefore, the proceeds of the non-interest-bearing promissory note are approximately $[/tex]4644.01.
