To determine how much smaller the surface area of the small dog house is compared to the big dog house, we first need to denote some variables and establish the relationship between the sizes of the dog houses.
Let's call the surface area of the larger dog house \( A \). According to the problem, the smaller dog house is \( \frac{1}{2} \) the size of the larger dog house. This means that if we denote the surface area of the smaller dog house as \( A_{small} \), we have:
\[ A_{small} = \frac{1}{2} A \]
Next, we want to find how much smaller the surface area of the small house is compared to the big house. This is given by the difference in their surface areas:
\[ \text{Difference} = A - A_{small} \]
Substituting the expression for \( A_{small} \):
\[ \text{Difference} = A - \frac{1}{2} A \]
Now we can simplify the expression:
\[ \text{Difference} = A - 0.5A = \frac{1}{2} A \]
Thus, the surface area of the small house is \( \frac{1}{2} A \) smaller than that of the big house.
In conclusion, the small house will be half the surface area of the big house, or in other words, the surface area difference is:
\[ \frac{1}{2} \times A \]
This means the small house's surface area is \( A_{small} = \frac{1}{2} A \), and the difference in surface area is also \( \frac{1}{2} A \), confirming our calculations.