1. **State the problem:** We need to calculate the total interest paid on a mortgage loan of amount $271,626$ with annual interest rate $7\%$ over $30$ years, with monthly payments. 2. **Identify parameters:** - Principal $P = 271626$ - Annual interest rate $r = 7\% = 0.07$ - Loan term $t = 30$ years - Number of payments per year $n = 12$ 3. **Calculate monthly interest rate:** $$ i = \frac{r}{n} = \frac{0.07}{12} \approx 0.0058333 $$ 4. **Calculate total number of payments:** $$ N = n \times t = 12 \times 30 = 360 $$ 5. **Calculate monthly mortgage payment using formula:** $$ M = P \times \frac{i(1 + i)^N}{(1 + i)^N - 1} $$ Calculate $(1 + i)^N$: $$ (1.0058333)^{360} \approx 10.677 $$ Calculate numerator: $$ 0.0058333 \times 10.677 \approx 0.0623 $$ Calculate denominator: $$ 10.677 - 1 = 9.677 $$ Calculate monthly payment $M$: $$ M = 271626 \times \frac{0.0623}{9.677} \approx 271626 \times 0.00644 = 1748.06 $$ 6. **Calculate total amount paid over 30 years:** $$ \text{Total payment} = M \times N = 1748.06 \times 360 = 629301.60 $$ 7. **Calculate total interest paid:** $$ \text{Interest} = \text{Total payment} - P = 629301.60 - 271626 = 357675.60 $$ **Final answer:** The total interest paid on the mortgage over 30 years is $357675.60$ rounded to the nearest hundredth.