Sarah wants to borrow a loan at \( 9.6 \% \), compounded quarterly. She wants to pay off this loan by making quarterly payments of \( \$ 480 \). a) If she wants to pay off the loan in 4 years, how much can Sarah borrow? Show your work. b) How much interest will Sarah pay? Show your work.

Sure, let's solve the problem step by step. ### **Given:** - **Annual Interest Rate (APR):** 9.6% - **Compounding Frequency:** Quarterly - **Quarterly Payment:** \$480 - **Loan Term:** 4 years --- ### **Part a) Determining How Much Sarah Can Borrow** **Objective:** Calculate the present value (initial loan amount) that Sarah can borrow given her payment plan. **Step 1: Determine the Quarterly Interest Rate and Total Number of Payments** 1. **Quarterly Interest Rate (r):** \[ r = \frac{\text{Annual Interest Rate}}{\text{Number of Quarters per Year}} = \frac{9.6\%}{4} = 2.4\% \text{ per quarter} \] \[ r = 0.024 \text{ (in decimal)} \] 2. **Total Number of Quarterly Payments (n):** \[ n = \text{Number of Years} \times \text{Number of Quarters per Year} = 4 \times 4 = 16 \text{ quarters} \] **Step 2: Use the Present Value of an Annuity Formula** The present value \( PV \) of an annuity (series of equal payments) is calculated as: \[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] where: - \( P \) = Quarterly payment (\$480) - \( r \) = Quarterly interest rate (0.024) - \( n \) = Total number of payments (16) **Step 3: Plug in the Values** \[ PV = 480 \times \left( \frac{1 - (1 + 0.024)^{-16}}{0.024} \right) \] **Step 4: Calculate the Values Inside the Parentheses** 1. Calculate \( (1 + r)^{-n} \): \[ (1 + 0.024)^{-16} = (1.024)^{-16} \approx 0.677 \] 2. Subtract this from 1: \[ 1 - 0.677 = 0.323 \] 3. Divide by \( r \): \[ \frac{0.323}{0.024} \approx 13.458 \] **Step 5: Multiply by the Quarterly Payment** \[ PV = 480 \times 13.458 \approx 6,469.44 \] **Answer:** Sarah can borrow approximately **\$6,469.44**. --- ### **Part b) Calculating the Total Interest Paid** **Objective:** Determine the total interest Sarah will pay over the life of the loan. **Step 1: Calculate the Total Amount Paid** \[ \text{Total Payments} = \text{Quarterly Payment} \times \text{Total Number of Payments} = 480 \times 16 = \$7,680 \] **Step 2: Subtract the Initial Loan Amount to Find Interest** \[ \text{Total Interest} = \text{Total Payments} - \text{Initial Loan Amount} = 7,680 - 6,469.44 = \$1,210.56 \] **Answer:** Sarah will pay a total of approximately **\$1,210.56** in interest. --- ### **Summary:** - **a)** Sarah can borrow **\$6,469.44**. - **b)** She will pay **\$1,210.56** in interest over 4 years.