1. **Problem statement:** Yusniar borrows 10,000,000 with an annual nominal interest rate $j_{12} = 12\%$ and pays monthly installments of 250,000. We want to find: a. How many months are needed to fully repay the loan? b. The additional payment required in the last period using method 1 and method 2. 2. **Formula and rules:** The monthly interest rate $i$ is $\frac{j_{12}}{12} = \frac{12\%}{12} = 1\% = 0.01$. The loan amount $P = 10,000,000$. The monthly payment $R = 250,000$. The formula for the number of payments $n$ in an ordinary annuity is: $$R = P \times \frac{i(1+i)^n}{(1+i)^n - 1}$$ We solve for $n$: $$\frac{R}{P} = \frac{i(1+i)^n}{(1+i)^n - 1}$$ 3. **Calculate $n$:** Let $x = (1+i)^n$. Then: $$\frac{R}{P} = \frac{i x}{x - 1} \Rightarrow \frac{R}{P} (x - 1) = i x$$ $$\frac{R}{P} x - \frac{R}{P} = i x$$ $$\frac{R}{P} x - i x = \frac{R}{P}$$ $$x \left(\frac{R}{P} - i\right) = \frac{R}{P}$$ $$x = \frac{\frac{R}{P}}{\frac{R}{P} - i}$$ Substitute values: $$\frac{R}{P} = \frac{250,000}{10,000,000} = 0.025$$ $$i = 0.01$$ $$x = \frac{0.025}{0.025 - 0.01} = \frac{0.025}{0.015} = \frac{5}{3} \approx 1.6667$$ 4. **Solve for $n$:** $$x = (1+i)^n \Rightarrow n = \frac{\ln x}{\ln(1+i)} = \frac{\ln 1.6667}{\ln 1.01}$$ Calculate: $$\ln 1.6667 \approx 0.5108$$ $$\ln 1.01 \approx 0.00995$$ $$n = \frac{0.5108}{0.00995} \approx 51.34$$ So, Yusniar needs about 52 months (rounding up since partial month is not possible). 5. **Calculate the remaining balance after 51 months to find additional payment:** The balance after $k$ payments is: $$B_k = P(1+i)^k - R \frac{(1+i)^k - 1}{i}$$ For $k=51$: Calculate $(1+i)^{51} = 1.01^{51} \approx e^{51 \times 0.00995} = e^{0.507} \approx 1.660$. $$B_{51} = 10,000,000 \times 1.660 - 250,000 \times \frac{1.660 - 1}{0.01}$$ $$= 16,600,000 - 250,000 \times 66 = 16,600,000 - 16,500,000 = 100,000$$ 6. **Additional payment in last period:** a. **Method 1:** Pay the remaining balance $B_{51} = 100,000$ as additional payment. b. **Method 2:** Add the interest for one more month on the remaining balance: Interest for one month: $$I = B_{51} \times i = 100,000 \times 0.01 = 1,000$$ Total last payment: $$R + I = 250,000 + 1,000 = 251,000$$ **Final answers:** - a. Number of months needed: **52 months** - b. Additional payment in last period: - Method 1: **100,000** - Method 2: **251,000**