Let's analyze the given data step-by-step to determine at which interest rate the bank makes the most profit.
1. First, we have the initial loan amount, which is [tex]\( \$10,000 \)[/tex], and this amount remains constant regardless of the interest rate.
2. We are given total payments for each interest rate over a five-year period:
- For an interest rate of [tex]\(5\%\)[/tex]:
[tex]\[
\text{Total paid} = \$11,322.00
\][/tex]
- For an interest rate of [tex]\(10\%\)[/tex]:
[tex]\[
\text{Total paid} = \$12,750.00
\][/tex]
- For an interest rate of [tex]\(15\%\)[/tex]:
[tex]\[
\text{Total paid} = \$14,274.00
\][/tex]
- For an interest rate of [tex]\(18\%\)[/tex]:
[tex]\[
\text{Total paid} = \$15,234.00
\][/tex]
3. The profit made by the bank is the total amount paid by the borrower minus the initial loan amount. Let's calculate the profit for each interest rate:
- For [tex]\(5\%\)[/tex]:
[tex]\[
\text{Profit} = \$11,322.00 - \$10,000.00 = \$1,322.00
\][/tex]
- For [tex]\(10\%\)[/tex]:
[tex]\[
\text{Profit} = \$12,750.00 - \$10,000.00 = \$2,750.00
\][/tex]
- For [tex]\(15\%\)[/tex]:
[tex]\[
\text{Profit} = \$14,274.00 - \$10,000.00 = \$4,274.00
\][/tex]
- For [tex]\(18\%\)[/tex]:
[tex]\[
\text{Profit} = \$15,234.00 - \$10,000.00 = \$5,234.00
\][/tex]
4. Now, we compare these profits to determine at which interest rate the bank makes the most money:
- Profit at [tex]\(5\%\)[/tex]: \[tex]$1,322.00
- Profit at \(10\%\): \$[/tex]2,750.00
- Profit at [tex]\(15\%\)[/tex]: \[tex]$4,274.00
- Profit at \(18\%\): \$[/tex]5,234.00
5. Clearly, the profit is highest when the interest rate is [tex]\(18\%\)[/tex].
Therefore, banks make the most money and take the most risk with an interest rate of [tex]\( 18\% \)[/tex].
