Let's break down the problem step by step: --- ### **Problem Recap** You have a loan of \$2,500 at 6% annual interest compounded monthly. You will pay off the loan by making monthly payments for 10 years. #### **Questions:** a) How much is the payment amount for 10 years? b) How much total interest will you pay? --- ## **Step 1: Identify the Formula** For an ordinary annuity (typical loan payments): \[ PMT = \frac{PV \cdot r}{1 - (1 + r)^{-n}} \] Where: - \( PMT \) = monthly payment - \( PV \) = present value (loan amount) = \$2,500 - \( r \) = monthly interest rate = annual rate / 12 = .06/12 - \( n \) = total number of payments = years × 12 = 10 × 12 = 120 --- ## **Step 2: Plug in the Values** ### **a) Find the monthly payment** \[ r = \frac{.06}{12} = .005 \] \[ n = 10 \times 12 = 120 \] \[ PV = \$2,500 \] \[ PMT = \frac{250 \cdot .005}{1 - (1 + .005)^{-120}} \] First, calculate the denominator: \[ 1 + .005 = 1.005 \] \[ (1.005)^{-120} = \frac{1}{(1.005)^{120}} \] Calculate \( (1.005)^{120} \): \[ (1.005)^{120} \approx e^{120 \cdot \ln(1.005)} \approx e^{120 \cdot .0049875} \approx e^{.5985} \approx 1.8194 \] \[ (1.005)^{-120} \approx \frac{1}{1.8194} \approx .5494 \] Now the denominator: \[ 1 - .5494 = .4506 \] Now plug everything in: \[ PMT = \frac{250 \cdot .005}{.4506} = \frac{12.5}{.4506} \approx 27.74 \] **Final answer for (a):** \[ \boxed{\$27.74} \] --- ### **b) Find the total interest paid** First, total payments over 10 years: \[ \text{Total Paid} = PMT \times n = 27.74 \times 120 = \$3,328.80 \] Total interest: \[ \text{Total Interest} = \text{Total Paid} - \text{Principal} \] \[ \text{Total Interest} = 3,328.80 - 2,500 = \$828.80 \] **Final answer for (b):** \[ \boxed{\$828.80} \] --- ## **Summary Table** | Part | Answer | |------|-------------| | a | \$27.74 | | b | \$828.80 | --- ### **Explanation** - You used the ordinary annuity formula to find the monthly payment. - You found the total paid and subtracted the original loan to get total interest. Let me know if you need a breakdown or want to see the calculations differently!