1. **Stating the problem:** Meg has a loan with an original amount and term. After 6 years (Jan 2016 to Jan 2022), the bank offers a reduced APR of 4.4% for the remaining 4 years (48 months). We need to find: (i) How much Meg still owes on Jan 1, 2022, by calculating the present value of the remaining repayments. (ii) The new monthly repayment amount over the remaining term at the reduced APR. 2. **Given data:** - Original loan amount: 464325.37 - Original term: 10 years (120 months) - Monthly repayment (previous): 3365.87 - Time passed: 6 years (72 months) - Remaining term: 4 years (48 months) - New APR: 4.4% annually 3. **Calculating the amount Meg still owes on Jan 1, 2022:** We use the present value of an annuity formula for the remaining repayments: $$ PV = R \times \frac{1 - (1 + i)^{-n}}{i} $$ where: - $R = 3365.87$ (monthly repayment) - $i = \frac{4.4\%}{12} = 0.0036667$ (monthly interest rate) - $n = 48$ (remaining months) 4. **Calculate the present value:** $$ PV = 3365.87 \times \frac{1 - (1 + 0.0036667)^{-48}}{0.0036667} $$ Calculate $(1 + 0.0036667)^{-48}$: $$ (1.0036667)^{-48} = \frac{1}{(1.0036667)^{48}} \approx \frac{1}{1.197} = 0.8353 $$ So, $$ PV = 3365.87 \times \frac{1 - 0.8353}{0.0036667} = 3365.87 \times \frac{0.1647}{0.0036667} = 3365.87 \times 44.91 = 151224.54 $$ 5. **Interpretation:** Meg still owes approximately 151224.54 on Jan 1, 2022. 6. **Calculating the new monthly repayment:** We use the amortization formula to find the new monthly repayment $R$: $$ R = PV \times \frac{i}{1 - (1 + i)^{-n}} $$ Substitute values: $$ R = 151224.54 \times \frac{0.0036667}{1 - (1.0036667)^{-48}} = 151224.54 \times \frac{0.0036667}{1 - 0.8353} = 151224.54 \times \frac{0.0036667}{0.1647} = 151224.54 \times 0.02226 = 3365.87 $$ 7. **Conclusion:** The new monthly repayment remains approximately 3365.87, consistent with the original monthly repayment. **Note:** The amount Meg has paid so far is the difference between the original loan amount and the amount she still owes. The present value calculation shows how much she still owes, not how much she has paid.