To determine how much the couple can spend on the home at different interest rates, we use the formula for the present value of an annuity to calculate the loan amount they can afford. The total price of the home will be the loan amount plus the down payment.

The present value formula for a loan is:

$P = \frac{M \times (1 - (1 + r)^{-n})}{r}$

Where:

• $P$ = Present value (loan amount)
• $M$ = Monthly payment
• $r$ = Monthly interest rate (annual rate divided by 12)
• $n$ = Total number of payments (years $\times$ 12)

Finally, the total home price is the loan amount plus the down payment.

(a) $6.6\%$ Compounded Monthly

1. $r = \frac{6.6\%}{12} = 0.0055$
2. $n = 25 \times 12 = 300$
3. $M = 1800$

Substitute into the formula:

$P = \frac{1800 \times (1 - (1 + 0.0055)^{-300})}{0.0055}$

Calculating step-by-step:

$(1 + 0.0055)^{-300} = (1.0055)^{-300} \approx 0.16586$ $1 - 0.16586 = 0.83414$ $P = \frac{1800 \times 0.83414}{0.0055} \approx 272,392.36$

Add the down payment to determine the home price:

$\text{Home Price} = 272,392.36 + 19,000 = 291,392.36$

(b) $7.2\%$ Compounded Monthly

1. $r = \frac{7.2\%}{12} = 0.006$
2. $n = 25 \times 12 = 300$
3. $M = 1800$

Substitute into the formula:

$P = \frac{1800 \times (1 - (1 + 0.006)^{-300})}{0.006}$

Calculating step-by-step:

$(1 + 0.006)^{-300} = (1.006)^{-300} \approx 0.16528$ $1 - 0.16528 = 0.83472$ $P = \frac{1800 \times 0.83472}{0.006} \approx 250,416.00$

Add the down payment to determine the home price:

$\text{Home Price} = 250,416.00 + 19,000 = 269,416.00$

Final Answers:

(a) $6.6\%$: $\mathbf{291,392.36}$
(b) $7.2\%$: $\mathbf{269,416.00}$

Let me know if you want additional details or further explanations!

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5 Related Questions:

1. How does increasing the down payment impact the maximum home price?
2. What would happen to the loan amount if the loan term were reduced to 20 years?
3. How would biweekly payments affect the affordability calculations?
4. What if the interest rates decreased by 1%? How much more could the couple afford?
5. How does compounding frequency (monthly vs. annually) influence affordability?

Tip:

Always account for other potential costs like property taxes and insurance when budgeting for a home loan!