Jose took out a loan for $\$ 6800$ that charges an annual interest rate of $8.7 \%$, compounded monthly. Ans list of financial formulas. (a) Find the amount owed after one year, assuming no payments are made. Do not round any intermediate computations, and round your answer to the nearest cent. $\$ \square$ (b) Find the effective annual interest rate, expressed as a percentage. Do not round any intermediate computations, and round your answer to the nearest hundredth of a percent. $\square \%$
Solution
Step 1 ：Given that Jose took out a loan for $6800 that charges an annual interest rate of 8.7%, compounded monthly. We are asked to find the amount owed after one year, assuming no payments are made, and the effective annual interest rate.
Step 2 ：We can use the formula for compound interest, which is \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.
Step 3 ：Substituting the given values into the formula, we get \(A = 6800(1 + \frac{0.087}{12})^{12*1}\).
Step 4 ：Calculating the above expression, we find that the amount owed after one year, assuming no payments are made, is $7415.77.
Step 5 ：To find the effective annual interest rate, we can use the formula \(EAR = (1 + \frac{i}{n})^{nt} - 1\), where \(i\) is the nominal interest rate (in decimal), \(n\) is the number of compounding periods per year, and \(t\) is the number of years.
Step 6 ：Substituting the given values into the formula, we get \(EAR = (1 + \frac{0.087}{12})^{12*1} - 1\).
Step 7 ：Calculating the above expression, we find that the effective annual interest rate is 9.06%.
Step 8 ：So, the final answers are: \(\boxed{\$7415.77}\) is the amount owed after one year, assuming no payments are made, and \(\boxed{9.06\%}\) is the effective annual interest rate.
