1. **Problem Statement:** Evelyn and Marvin renegotiated their mortgage to a new term of five years with a fixed interest rate of 2.74% compounded semi-annually. They switched from monthly payments to bi-weekly payments, where the bi-weekly payment is half of their previous monthly payment. We need to find by how much the mortgage term was shortened due to this change. 2. **Given Data:** - Interest rate (annual nominal): $2.74\%$ compounded semi-annually (C/Y = 2) - Payments per year (P/Y): 26 (bi-weekly) - Present Value (PV): 382,272.92 - Monthly payment (PMT): 2,589.54 - Original term: 240 months (20 years) - New term found: 161 bi-weekly payments 3. **Formula and Concepts:** The mortgage term is shortened because making payments more frequently reduces the outstanding principal faster. The number of payments in the original term is $N_{original} = 240$ months. The new number of payments is $N_{new} = 161$ bi-weekly payments. Since bi-weekly payments are half the monthly payment, the total paid per year is slightly more frequent, accelerating principal repayment. 4. **Calculations:** - Convert original term to bi-weekly payments equivalent: $240$ months $= 20$ years $\times 12 = 240$ payments monthly. - Number of bi-weekly payments in 20 years: $20 \times 26 = 520$ bi-weekly payments. But the new term is only $161$ bi-weekly payments, so the mortgage is shortened by: $$\text{Payments shortened} = 520 - 161 = 359 \text{ bi-weekly payments}$$ Convert payments shortened to years and months: $$\text{Years shortened} = \frac{359}{26} = 13.81 \text{ years}$$ $$\text{Months shortened} = 0.81 \times 12 = 9.72 \approx 10 \text{ months}$$ So, the mortgage is shortened by approximately 13 years and 10 months. 5. **Summary:** By switching to bi-weekly payments at half the monthly payment amount, Evelyn and Marvin shortened their mortgage term by about 13 years and 10 months, significantly reducing the total interest paid and the loan duration. **Final answer:** The mortgage was shortened by approximately **13 years and 10 months**.