52. A house in a subdivision recently sold for $178,000. Sales records show that houses in this neighborhood have appreciated 8% per year for each of the past 3 years. If the economic trend continues at this rate, the house will be worth how much, on a compound basis, at the end of 3 years
Answer: D
The house will be worth $224,229 at the end of 3 years based on an 8% annual appreciation rate.
To calculate the future value of the house after 3 years with an annual appreciation of 8%, we apply the formula for compound interest: \(FV = P(1 + r)^n\). This results in \(FV = 178,000(1 + 0.08)^3\), which equals approximately $224,229.
A) 192,240
This option represents the future value of the house but is incorrect. The calculation for 3 years of appreciation at 8% does not yield this amount. Instead, it suggests a misunderstanding of the compound interest formula, as this figure appears to be a linear calculation rather than a compound one.
B) 206,480
This figure is also incorrect for the same reasons as option A. It does not take into account the compounding effect of the 8% annual appreciation over three years, which results in a much higher total value.
C) 220,720
Option C is close but still incorrect. This amount may arise from a miscalculation; it does not accurately reflect the compounded growth over three years at an 8% rate.
D) 224,229
This option is correct. The calculation confirms that after compounding the 8% annual growth over three years, the house's value will indeed reach $224,229, making this the accurate future value.
Conclusion
The correct answer is definitively option D, as it reflects the proper application of the compound interest formula for the specified appreciation rate over three years. The other options fail to account for the compounding effect, leading to lower and incorrect future value estimations.