38. 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.
To calculate the future value of the house after 3 years with an annual appreciation rate of 8%, we use the formula for compound interest: \(FV = P(1 + r)^n\), where \(P\) is the principal amount ($178,000), \(r\) is the rate (0.08), and \(n\) is the number of years (3). Applying these values gives us a future value of $224,229.
A) $192,240
This option results from a miscalculation, possibly treating the appreciation as simple interest rather than compound interest. Using simple interest would not accurately reflect the compounded growth over 3 years, leading to an undervaluation of the property's worth.
B) $206,480
This value also stems from an incorrect calculation, most likely due to a misunderstanding of the compounding effect. Like option A, this figure does not consider the full effect of compounding over three years, resulting in an inaccurate estimate of the house's future value.
C) $220,720
While this option is closer to the correct value, it still underestimates the compounded growth. It appears to be derived from an incorrect application of the compound interest formula, resulting in a value that does not fully account for the exponential increase in value over three years.
D) $224,229
This is the correct answer, calculated using the compound interest formula. By applying an 8% increase to the house's value for each of the three years, the final future value accurately reflects the growth due to compounding.
Conclusion
The correct future value of the house after 3 years of 8% annual appreciation is definitively $224,229, as accurately calculated through the compound interest formula. Other options misapply the calculations either by neglecting the compounding effect or using incorrect formulas, leading to lower and inaccurate estimations of the house's worth. Thus, D stands out as the only correct solution based on the economic trend presented.