1. **Stating the problem:** You want to buy a $218,000 home, pay 10% down, and take out a 30-year loan for the rest. We are asked to find: a) The loan amount. b) Monthly payments at 6% interest. c) Monthly payments at 7% interest. 2. **Calculate the loan amount:** Down payment = 10% of $218,000 = $$218000 \times 0.10 = 21800$$ Loan amount = Purchase price - Down payment $$218000 - 21800 = 196200$$ 3. **Monthly payment formula:** The monthly payment for a loan is calculated by: $$M = P \frac{r(1+r)^n}{(1+r)^n - 1}$$ where $P$ = loan amount, $r$ = monthly interest rate = annual rate / 12, $n$ = total number of payments = years \times 12. 4. **Calculate monthly payment at 6% interest:** Annual interest rate = 6% = 0.06 Monthly interest rate: $$r = \frac{0.06}{12} = 0.005$$ Number of payments: $$n = 30 \times 12 = 360$$ Plug values into formula: $$M = 196200 \times \frac{0.005(1+0.005)^{360}}{(1+0.005)^{360} - 1}$$ Calculate $(1+0.005)^{360}$: $$ (1.005)^{360} \approx 6.022575$$ Now, $$M = 196200 \times \frac{0.005 \times 6.022575}{6.022575 - 1} = 196200 \times \frac{0.0301129}{5.022575} = 196200 \times 0.0059961 \approx 1175.25$$ 5. **Calculate monthly payment at 7% interest:** Annual interest rate = 7% = 0.07 Monthly interest rate: $$r = \frac{0.07}{12} \approx 0.0058333$$ Number of payments: $$n = 360$$ Calculate $(1+0.0058333)^{360}$: $$ (1.0058333)^{360} \approx 10.6771$$ Now, $$M = 196200 \times \frac{0.0058333 \times 10.6771}{10.6771 - 1} = 196200 \times \frac{0.062284}{9.6771} = 196200 \times 0.006436 \approx 1262.83$$ **Final answers:** a) Loan amount = $196,200 b) Monthly payment at 6% = $1175.25 c) Monthly payment at 7% = $1262.83