Let's solve each part of the problem step-by-step:
### Part (a): Find the Proceeds
1. Identify the given values:
- Note value: \[tex]$7,600
- Discount rate: 6.6% (or 0.066 in decimal)
- Loan duration: 260 days
- Days in a year: 360
2. Calculate the total discount:
The discount amount is calculated using the formula:
\[
\text{Discount} = \text{Note Value} \times \text{Discount Rate} \times \left(\frac{\text{Loan Days}}{\text{Days in Year}}\right)
\]
Substituting in the given values:
\[
\text{Discount} = 7600 \times 0.066 \times \left(\frac{260}{360}\right)
\]
3. Simplify the fraction within the formula:
\[
\frac{260}{360} = 0.7222\overline{2}
\]
4. Calculate the discount:
\[
\text{Discount} = 7600 \times 0.066 \times 0.7222\overline{2} \approx 362.27
\]
5. Calculate the proceeds:
The proceeds are the note value minus the discount:
\[
\text{Proceeds} = \text{Note Value} - \text{Discount}
\]
Substituting the values:
\[
\text{Proceeds} = 7600 - 362.27 = 7237.73
\]
Thus, the proceeds are \$[/tex]7,237.73.
### Part (b): Find the Effective Rate Charged by the Bank
1. Identify the values needed:
- Discount: \[tex]$362.27
- Proceeds: \$[/tex]7,237.73
- Loan duration: 260 days
2. Calculate the effective annual interest rate:
The effective rate can be calculated using the formula:
[tex]\[
\text{Effective Rate} = \left( \frac{\text{Discount}}{\text{Proceeds}} \right) \times \left( \frac{\text{Days in Year}}{\text{Loan Days}} \right) \times 100\%
\][/tex]
Substitute the known values into the formula:
[tex]\[
\text{Effective Rate} = \left( \frac{362.27}{7237.73} \right) \times \left( \frac{360}{260} \right) \times 100\%
\][/tex]
3. Calculate the fraction part inside the formula:
[tex]\[
\frac{362.27}{7237.73} \approx 0.05007
\][/tex]
4. Simplify the second fraction:
[tex]\[
\frac{360}{260} \approx 1.3846
\][/tex]
5. Combine these fractions to find the effective rate:
[tex]\[
\text{Effective Rate} = 0.05007 \times 1.3846 \times 100 \approx 6.93
\][/tex]
6. Round the effective rate to the nearest tenth percent:
Thus, the effective rate is 6.9%.
Therefore:
- The proceeds are \$7,237.73.
- The effective rate charged by the bank is 6.9%.
