Answer: $27,000
Explanation
Divide the contractor's old price over the price per square foot to get
12000/500 = 24
The old area being remodeled was 24 square feet. So it could be a 8 by 3 section, or perhaps a 6 by 4 section. It turns out it doesn't matter what the dimensions are since they all lead to the same final answer.
If the linear scale factor is 1.5, then the area scale factor is (1.5)^2 = 2.25
The old 24 square foot area is now 24*2.25 = 54 square feet.
If you're not sure about this then consider a 6 by 4 rectangle. Multiply each value by 1.5 to get 1.5*6 = 9 and 1.5*4 = 6, so the new area is 9*6 = 54; notice how 54/24 = 2.25 to show the area scale factor.
Anyways the new area is 54 square feet. Multiply this with the price per square foot to get: 54*500 = $27,000
Note: 27000/12000 = 2.25 which isn't a coincidence.
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In short, square 1.5 to get (1.5)^2 = 2.25 then 2.25*12000 = $27,000 which is the final answer. Keep in mind that this is an estimate.
