1. **State the problem:** Dazzline took a loan of 56000 at an annual interest rate of 3.47% compounded monthly and makes monthly payments of 1868. We need to find how many payments she must make to fully pay off the loan. 2. **Identify variables:** - Principal $P = 56000$ - Annual interest rate $r = 0.0347$ - Monthly interest rate $i = \frac{r}{12} = \frac{0.0347}{12} = 0.0028917$ - Monthly payment $M = 1868$ 3. **Formula for number of payments $n$ to amortize a loan:** $$ n = \frac{\log\left(\frac{M}{M - P i}\right)}{\log(1 + i)} $$ 4. **Calculate $n$:** $$ n = \frac{\log\left(\frac{1868}{1868 - 56000 \times 0.0028917}\right)}{\log(1 + 0.0028917)} = \frac{\log\left(\frac{1868}{1868 - 161.8352}\right)}{\log(1.0028917)} = \frac{\log\left(\frac{1868}{1706.1648}\right)}{\log(1.0028917)} $$ Calculate numerator: $$ \log\left(1.0953\right) = 0.0410 $$ Calculate denominator: $$ \log(1.0028917) = 0.001252 $$ So, $$ n = \frac{0.0410}{0.001252} = 32.75 $$ 5. **Interpretation:** Since payments must be whole, round up to the next whole number: $$ n = 33\text{ payments} $$ 6. **Calculate the size of the final payment:** The final payment will be less than the regular payment because the loan will be almost paid off after 32 payments. Calculate the balance after 32 payments using the formula for remaining balance $B_n$: $$ B_n = P(1+i)^n - M \frac{(1+i)^n - 1}{i} $$ Calculate $B_{32}$: $$ B_{32} = 56000(1.0028917)^{32} - 1868 \times \frac{(1.0028917)^{32} - 1}{0.0028917} $$ Calculate $(1.0028917)^{32}$: $$ (1.0028917)^{32} = 1.0963 $$ Calculate numerator of fraction: $$ 1.0963 - 1 = 0.0963 $$ Calculate fraction: $$ \frac{0.0963}{0.0028917} = 33.31 $$ Calculate balance: $$ B_{32} = 56000 \times 1.0963 - 1868 \times 33.31 = 61352.8 - 62230.3 = -877.5 $$ Since balance is negative, it means the loan is fully paid before the 33rd payment. Calculate balance after 31 payments: $$ (1.0028917)^{31} = 1.0934 $$ $$ B_{31} = 56000 \times 1.0934 - 1868 \times \frac{1.0934 - 1}{0.0028917} = 61150.4 - 1868 \times 32.3 = 61150.4 - 60336.4 = 814 $$ So after 31 payments, balance is 814. The 32nd payment will be the final payment and will be: $$ \text{Final payment} = 814 \times (1 + i) = 814 \times 1.0028917 = 816.4 $$ **Final answers:** - Number of payments: 33 - Final payment size: 816.4