Let's denote the surface area of the larger dog house as \( S \). According to the problem, the smaller dog house is \( \frac{1}{2} \) the size of the larger one.
Thus, the surface area of the smaller dog house will be:
\[ S_{\text{small}} = \frac{1}{2} S \]
To find how much smaller the surface area of the small house is compared to the big house, we can compare the surface area of the small house to that of the big house:
\[ \text{Decrease in surface area} = S - S_{\text{small}} = S - \frac{1}{2} S = \frac{1}{2} S \]
Now, we can express the surface area of the small house as a fraction of the surface area of the big house. Specifically, the surface area of the small house compared to the big house is:
\[ \frac{S_{\text{small}}}{S} = \frac{\frac{1}{2} S}{S} = \frac{1}{2} \]
Therefore, the surface area of the small house will be \(\frac{1}{2}\) the size of the big house. Thus, the surface area of the small house will be:
\[ \frac{1}{2} \text{ the size of the big house.} \]
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