Answer:
To determine if the claim is reasonable, we can calculate the monthly payment using the formula for a fixed-rate mortgage:
[tex][ M = \frac{P \times r \times (1+r)^n}{(1+r)^n - 1}][/tex]
Where:
- [tex](M)[/tex] is the monthly payment
- [tex](P)[/tex] is the principal amount (\$100,000)
- [tex](r)[/tex] is the monthly interest rate (APR divided by 12, and expressed as a decimal)
- [tex](n)[/tex] is the total number of payments (30 years multiplied by 12 months per year)
Let's calculate:
[tex]\[ r = \frac{12\%}{12} = 0.01][/tex]
[tex]\[ n = 30 \times 12 = 360][/tex]
Plugging these values into the formula:
[tex]\[ M = \frac{100000 \times 0.01 \times (1+0.01)^{360}}{(1+0.01)^{360} - 1}][/tex]
[tex]\[ M ≈ \frac{100000 \times 0.01 \times (1.01)^{360}}{(1.01)^{360} - 1}][/tex]
[tex]\[ M ≈ \frac{100000 \times 0.01 \times 50.64167812755673}{50.64167812755673 - 1}][/tex]
[tex]\[ M ≈ \frac{5064.167812755673}{49.64167812755673}][/tex]
[tex][M ≈ 102.0497762150958][/tex]
Rounding to the nearest dollar, the monthly payment is approximately $102.
Therefore, the claim of $10,290 per month is not reasonable. The ad should have stated that the payment would be approximately $102 per month.
As for what might have happened, it's likely that there was a mistake in the calculation or in the information provided in the advertisement. One possibility is that the decimal point was misplaced, leading to an inflated monthly payment amount.
