Sure, let's tackle the problem step by step.
1. Understanding the Principal Amount:
The total cost of materials and equipment is given as [tex]$6884.44.
2. Annual Interest Rate:
The interest rate provided is 8% per annum. We will convert this percentage into a decimal for calculations:
\( 8\% = \frac{8}{100} = 0.08 \).
3. Time Period:
The period for the loan is given as 17 months. To work with an annual interest rate, we'll convert the time period from months to years:
\[
\text{Time (in years)} = \frac{17 \text{ months}}{12 \text{ months/year}} = \frac{17}{12} \approx 1.4167 \text{ years}
\]
4. Formula for Compound Interest:
The compound interest \( CI \) can be calculated using the formula:
\[
CI = P \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)
\]
Where:
- \( P \) is the principal amount (\$[/tex]6884.44)
- [tex]\( r \)[/tex] is the annual interest rate (0.08)
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year (since it's compounded monthly, [tex]\( n = 12 \)[/tex])
- [tex]\( t \)[/tex] is the time in years ([tex]\(\frac{17}{12}\)[/tex])
5. Plugging in the Values:
Substituting the provided values into the formula:
[tex]\[
CI = 6884.44 \left( \left(1 + \frac{0.08}{12}\right)^{12 \cdot \left(\frac{17}{12}\right)} - 1 \right)
\][/tex]
6. Result:
After performing the calculations using the compound interest formula, the compound interest for the amount [tex]$6884.44 over 17 months at a rate of 8% per annum is approximately \( \$[/tex]823.27 \).
Thus, the compound interest on the amount over 17 months at an annual interest rate of 8% is approximately \$823.27.
