A property has a total value of $750,000, including a site value of $200,000. The improvements have an effective age of 10 years and a total life of 50 years, with straight-line depreciation and no other loss. What is the cost new of the improvements?
Correct Answer
B) $687,500
(750000−200000)÷(1−10÷50) = 687500. Common errors: not removing the site value before dividing gives $937,500; multiplying by the percent good instead of dividing by it gives $440,000; dividing by the depreciation fraction instead of the percent good gives $2,750,000.
Why This Is the Correct Answer
Option B is correct because the improvement contribution is $750,000 − $200,000 = $550,000. Since contribution equals cost new × (1 − 10/50) = cost new × 0.80, cost new is $550,000 ÷ 0.80 = $687,500. Checking: $687,500 × 0.80 = $550,000, plus $200,000 site equals $750,000.
Why the Other Options Are Wrong
Option A: $440,000
Option A, $440,000, multiplies by the percent good instead of dividing: $550,000 × 0.80 = $440,000. Solving the model for cost new requires $550,000 ÷ 0.80 = $687,500.
Option C: $937,500
Option C, $937,500, does not remove the site value before dividing: $750,000 ÷ 0.80 = $937,500. The site is not depreciated in this model, so subtract it first: ($750,000 − $200,000) ÷ 0.80 = $687,500.
Option D: $2,750,000
Option D, $2,750,000, divides by the depreciation fraction instead of the percent good: $550,000 ÷ (10/50) = $550,000 ÷ 0.20 = $2,750,000. The contribution is cost new times 0.80, so divide by 0.80 to get $687,500.
Strip site, then divide
To work backward, strip off the site value, then divide the remainder by the percent good. Multiply back to check.
How to use: Use this when a question gives total value and asks for an unknown cost new.
Exam Tip
After solving, multiply your cost new by the percent good and add the site to confirm it returns the stated total.
Common Mistakes to Avoid
- -Leaving site value in before dividing
- -Multiplying by percent good when solving for cost new
- -Dividing by the depreciation fraction instead of percent good
Concept Deep Dive
Analysis
This question tests working the age-life model in reverse. First remove site value from total value to isolate the depreciated improvement contribution, then divide by the percent good, which is 1 − 10/50 = 0.80. The wrong choices come from skipping the site step or using the wrong operation or fraction.
Background Knowledge
In the age-life method, depreciation is effective age divided by total economic life, and the remaining fraction is often called percent good. Depreciated improvement contribution equals cost new times percent good. Rearranging this relationship lets an appraiser solve for cost new when the contribution is known.
Real-World Application
An appraiser testing a market extraction model knows a sale's total price and its land value. Working the age-life relationship backward shows what cost new the sale implies, which can be compared with cost service estimates.
More Cost Approach Questions
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