1. **Problem statement:** Ray took a 60-month loan of 12000 with monthly payments of 232 and no down payment. He wants to find the APR and the interest saved by paying off the loan early after 11 payments instead of 12. 2. **Formula for monthly payment of an installment loan:** $$P = \frac{r \times PV}{1 - (1+r)^{-n}}$$ where $P$ is the monthly payment, $r$ is the monthly interest rate (APR/12), $PV$ is the loan amount, and $n$ is the total number of payments. 3. **Given:** $P=232$, $PV=12000$, $n=60$. 4. **Find monthly interest rate $r$:** Rearranged formula: $$232 = \frac{r \times 12000}{1 - (1+r)^{-60}}$$ 5. **Solve for $r$ numerically (APR = 6% given):** Monthly rate $r = \frac{6}{100 \times 12} = 0.005$. 6. **Calculate total interest if paid fully over 60 months:** Total paid = $232 \times 60 = 13920$ Interest total = $13920 - 12000 = 1920$ 7. **Calculate remaining balance after 11 payments:** Remaining balance formula: $$B = PV \times (1+r)^n - P \times \frac{(1+r)^n - 1}{r}$$ where $n$ is number of payments made. Calculate balance after 11 payments: $$B = 12000 \times (1+0.005)^{60-11} - 232 \times \frac{(1+0.005)^{60-11} - 1}{0.005}$$ Calculate powers: $(1.005)^{49} \approx 1.28368$ Calculate numerator: $$232 \times \frac{1.28368 - 1}{0.005} = 232 \times \frac{0.28368}{0.005} = 232 \times 56.736 = 13162.75$$ Calculate balance: $$B = 12000 \times 1.28368 - 13162.75 = 15404.16 - 13162.75 = 2241.41$$ 8. **Interest saved by paying off early:** Interest if paid fully after 12 payments: Total paid after 12 payments = $232 \times 60 = 13920$ Interest saved = Total interest - interest paid so far - remaining balance Interest paid so far = $232 \times 11 = 2552$ Interest saved = $1920 - (2241.41 - (12000 - 2552))$ Calculate principal paid so far: $$12000 - 2241.41 = 9758.59$$ Interest paid so far: $$2552 - (12000 - 9758.59) = 2552 - 2241.41 = 310.59$$ Interest saved = $1920 - 310.59 = 1609.41$ **Final answers:** - APR = 6% - Interest saved by early payoff = 1609.41