Let the surface area of the smaller dog house be \( S \) and the surface area of the larger dog house be \( L \).
According to the problem, the larger house is 12 times the size of the smaller house: \[ L = 12S \]
To find out how much smaller the surface area of the small house is compared to that of the big house, we can calculate the difference in their surface areas: \[ L - S = 12S - S = 11S \]
Now, we want to express how much smaller the surface area of the small house is as a fraction of the larger house's surface area: \[ \text{Fraction} = \frac{L - S}{L} = \frac{11S}{12S} \]
This simplifies to: \[ \text{Fraction} = \frac{11}{12} \]
Thus, the surface area of the small house will be \(\frac{11}{12}\) the size of the big house. Please note that this means the small house is smaller by the fraction \(\frac{11}{12}\) compared to the total surface area of the large house.