1. **Problem statement:** You want to buy a 218000 home, pay 10% down, and take a 30-year loan for the rest. 2. **Calculate loan amount:** Down payment is 10% of 218000: $$ 0.10 \times 218000 = 21800 $$ Loan amount = home price - down payment: $$ 218000 - 21800 = 196200 $$ 3. **Monthly payment formula:** For a loan amount $P$, monthly interest rate $r$, and number of payments $n$: $$ M = P \frac{r (1+r)^n}{(1+r)^n - 1} $$ Where: - $P$ is the loan principal - $r$ is monthly interest rate (annual rate divided by 12) - $n$ is total number of payments (years times 12) 4. **Calculate monthly payment for 6% interest:** - Annual rate = 6%, so monthly rate: $$ r = \frac{0.06}{12} = 0.005 $$ - Number of payments: $$ n = 30 \times 12 = 360 $$ - Compute payment: $$ M = 196200 \times \frac{0.005 (1+0.005)^{360}}{(1+0.005)^{360} - 1} $$ Calculate numerator and denominator: - $$ (1+0.005)^{360} = (1.005)^{360} \approx 6.022575 $$ - Numerator: $$ 0.005 \times 6.022575 = 0.0301129 $$ - Denominator: $$ 6.022575 - 1 = 5.022575 $$ - Fraction: $$ \frac{0.0301129}{5.022575} \approx 0.005996 $$ - Monthly payment: $$ 196200 \times 0.005996 \approx 1176.15 $$ 5. **Monthly payment for 7% interest:** - Annual rate = 7%, so monthly rate: $$ r = \frac{0.07}{12} = 0.0058333 $$ - Payments: $$ n = 360 $$ - Compute payment: $$ M = 196200 \times \frac{0.0058333 (1+0.0058333)^{360}}{(1+0.0058333)^{360} - 1} $$ Calculate powers: - $$ (1+0.0058333)^{360} = (1.0058333)^{360} \approx 8.1364 $$ - Numerator: $$ 0.0058333 \times 8.1364 = 0.047477 $$ - Denominator: $$ 8.1364 - 1 = 7.1364 $$ - Fraction: $$ \frac{0.047477}{7.1364} \approx 0.006654 $$ - Monthly payment: $$ 196200 \times 0.006654 \approx 1305.48 $$ **Final answers:** - a) Loan amount = 196200 - b) Monthly payment at 6% = 1176.15 - c) Monthly payment at 7% = 1305.48