Let's break down the problem: James borrowed $7,000 today. - **First 1.5 years (18 months):** Interest rate = **3.2% compounded monthly** - **After 1.5 years:** Interest rate changes to **4.7% compounded semi-annually** - **Total period:** 8 years - **No payments are made during 8 years** We need to find: 1. **What amount is required to pay off the loan 8 years from today (no payments made till then)?** 2. **How much interest did James pay on the loan?** --- ### Step 1: Calculate the Amount Owed After 1.5 Years - Principal = $7,000 - Interest rate = 3.2% compounded monthly - Time = 1.5 years = 18 months The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( P = 7,000 \) - \( r = 0.032 \) - \( n = 12 \) - \( t = 1.5 \) \[ A_{1.5} = 7000 \left(1 + \frac{0.032}{12}\right)^{12 \times 1.5} \] \[ = 7000 \left(1 + 0.0026667\right)^{18} \] \[ = 7000 \times (1.0026667)^{18} \] \[ = 7000 \times 1.0488 \] \[ = \$7,341.60 \] --- ### Step 2: Calculate the Amount Owed After the Next 6.5 Years - New principal = \$7,341.60 - New interest rate = 4.7% compounded semi-annually - Time = 6.5 years \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( P = 7,341.60 \) - \( r = 0.047 \) - \( n = 2 \) - \( t = 6.5 \) \[ A_{8} = 7,341.60 \left(1 + \frac{0.047}{2}\right)^{2 \times 6.5} \] \[ = 7,341.60 \left(1 + 0.0235\right)^{13} \] \[ = 7,341.60 \times (1.0235)^{13} \] \[ = 7,341.60 \times 1.3356 \] \[ = \$9,803.30 \] --- ### **Answers** #### 1. Amount required to pay off the loan 8 years from today: \[ \boxed{\$9,803.30} \] #### 2. How much interest did James pay? \[ \text{Total interest} = \$9,803.30 - \$7,000 = \boxed{\$2,803.30} \] --- **Let me know if you need the step-by-step explanation or calculations!**