question 7

what if you were to instead by a cheaper house, say for $300,000 (instead of $400,000), the difference in payment would be about $700 per month.

assume a loan of $300,000 at 7% for 30 years with an extra $700 per month, how much interest will you save?

Question

question 7

what if you were to instead by a cheaper house, say for $300,000 (instead of $400,000), the difference in payment would be about $700 per month.

assume a loan of $300,000 at 7% for 30 years with an extra $700 per month, how much interest will you save?

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Amortization Formula", "Mortgage Interest Calculation", "Loan Interest Savings" ], "new_concepts": [], "current_concepts": [ "Amortization Formula", "Mortgage Interest Calculation", "Loan Interest Savings", "Extra Principal Payments" ] } ## Calculate standard monthly payment Using the *Amortization Formula* knowledge point $$ \begin{aligned} P &= 300000\ r &= \frac{0.07}{12}\ n &= 360\ M &= P \frac{r(1+r)^n}{(1+r)^n - 1} \approx 1995.91 \end{aligned} $$ ## Calculate total standard interest Using the *Mortgage Interest Calculation* knowledge point $$ \begin{aligned} T_{\text{std}} &= 1995.91 \times 360 = 718527.60\ I_{\text{std}} &= 718527.60 - 300000 = 418527.60 \end{aligned} $$ ## Determine accelerated payoff schedule Using the *Amortization Formula* knowledge point $$ \begin{aligned} M_{\text{new}} &= 1995.91 + 700 = 2695.91\ 300000 &= 2695.91 \frac{1 - (1+r)^{-N}}{r}\ (1+r)^{-N} &\approx 0.35089\ -N \ln(1+r) &\approx \ln(0.35089)\ N &\approx 180.14 \text{ months} \end{aligned} $$ ## Calculate total accelerated interest Using the *Mortgage Interest Calculation* knowledge point $$ \begin{aligned} T_{\text{acc}} &= 180 \times 2695.91 + 371.74 = 485635.54\ I_{\text{acc}} &= 485635.54 - 300000 = 185635.54 \end{aligned} $$ ## Calculate total interest saved Using the *Loan Interest Savings* knowledge point $$ \begin{aligned} \text{Savings} &= I_{\text{std}} - I_{\text{acc}}\ \text{Savings} &= 418527.60 - 185635.54 = 232892.06 \end{aligned} $$ Assume a loan of $300,000 at 7% for 30 years with an extra $700 per month, how much interest will you save? \$232,892.06 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Amortization Schedule" ] }

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Question

Explanation:

Calculate standard monthly payment

Calculate total standard interest

Determine accelerated payoff schedule

Calculate total accelerated interest

Calculate total interest saved

Answer: