1. **Problem statement:** Kendall borrows 100000 on January 1, 1993, to be repaid in 12 annual installments at an effective annual interest rate of 8%. The first annual payment is due January 1, 1994. Instead, Kendall opts to pay monthly installments equal to one-twelfth of the annual payment starting February 1, 1993. We need to find how many months it will take to pay off the loan. 2. **Step 1: Calculate the annual payment amount.** The loan is amortized over 12 years with annual payments at an effective annual interest rate of 8%. The formula for the annual payment $A$ on a loan $P$ with interest rate $i$ over $n$ years is: $$A = P \times \frac{i}{1 - (1+i)^{-n}}$$ where $P=100000$, $i=0.08$, $n=12$. 3. Calculate: $$A = 100000 \times \frac{0.08}{1 - (1.08)^{-12}}$$ First, compute $(1.08)^{-12} = \frac{1}{(1.08)^{12}} \approx \frac{1}{2.51817} = 0.3971$$ Then denominator: $$1 - 0.3971 = 0.6029$$ So: $$A = 100000 \times \frac{0.08}{0.6029} = 100000 \times 0.1327 = 13270$$ The annual payment is approximately 13270. 4. **Step 2: Calculate the monthly payment amount.** Monthly payment $m = \frac{A}{12} = \frac{13270}{12} \approx 1105.83$. 5. **Step 3: Determine the monthly interest rate equivalent to 8% effective annual rate.** The monthly interest rate $j$ satisfies: $$ (1 + j)^{12} = 1.08 $$ Taking 12th root: $$ 1 + j = (1.08)^{\frac{1}{12}} \approx 1.006434 $$ So: $$ j = 0.006434 $$ 6. **Step 4: Find the number of months $N$ to pay off the loan with monthly payments $m$ and monthly interest $j$.** The loan balance after $N$ months is zero, so: $$ 0 = P(1+j)^N - m \times \frac{(1+j)^N - 1}{j} $$ Rearranged: $$ P(1+j)^N = m \times \frac{(1+j)^N - 1}{j} $$ Multiply both sides by $j$: $$ P j (1+j)^N = m ((1+j)^N - 1) $$ Expand right side: $$ P j (1+j)^N = m (1+j)^N - m $$ Bring terms with $(1+j)^N$ to one side: $$ m (1+j)^N - P j (1+j)^N = m $$ Factor: $$ (1+j)^N (m - P j) = m $$ Divide both sides: $$ (1+j)^N = \frac{m}{m - P j} $$ Take natural logarithm: $$ N \ln(1+j) = \ln \left( \frac{m}{m - P j} \right) $$ So: $$ N = \frac{\ln \left( \frac{m}{m - P j} \right)}{\ln(1+j)} $$ 7. Substitute values: $$ P = 100000, m = 1105.83, j = 0.006434 $$ Calculate denominator: $$ m - P j = 1105.83 - 100000 \times 0.006434 = 1105.83 - 643.4 = 462.43 $$ Calculate fraction: $$ \frac{m}{m - P j} = \frac{1105.83}{462.43} \approx 2.39 $$ Calculate logarithms: $$ \ln(2.39) \approx 0.871, \quad \ln(1.006434) \approx 0.006414 $$ 8. Calculate $N$: $$ N = \frac{0.871}{0.006414} \approx 135.8 $$ 9. **Answer:** It will take approximately 136 months to pay off the loan with monthly payments equal to one-twelfth the annual payment. This is about 11 years and 4 months, which is less than the original 12 years because monthly payments start earlier and interest compounds monthly rather than annually.