Rosie is building two dog houses, one for each of her dogs. One dog is bigger than the other, so she decides to make one house 1/2 the size of the other. How much smaller will the surface area of the small house be compared to that of the big house? Write your response as a fraction.

Answer 1

Let the surface area of the larger dog house be denoted as \( S \). According to the problem, the surface area of the smaller dog house is \( \frac{1}{2} S \).

To find the difference in surface area between the large and small houses, we calculate:

\[ \text{Difference in surface area} = S - \frac{1}{2} S \]

This simplifies to:

\[ = S - \frac{1}{2} S = \frac{2}{2} S - \frac{1}{2} S = \frac{1}{2} S \]

Now, to express how much smaller the surface area of the small house is compared to that of the big house as a fraction of the larger house's surface area, we set up the ratio:

\[ \text{Fraction smaller} = \frac{\text{Difference}}{S} = \frac{\frac{1}{2} S}{S} \]

This simplifies to:

\[ = \frac{1}{2} \]

Thus, the surface area of the small house is \(\frac{1}{2}\) the surface area of the large house. Therefore, the surface area of the small house is \(\frac{1}{2}\) smaller compared to the surface area of the large house.

In conclusion, the answer is:

\[ \frac{1}{2} \]

Answer 2

wrong

Answer 3

Let’s clarify the problem and recalculate the difference in surface area.

Let the surface area of the larger dog house be \( S \). According to the problem, the surface area of the smaller dog house is half of that, which is:

\[ \text{Surface area of small house} = \frac{1}{2} S \]

To find how much smaller the surface area of the small house is compared to the large house, we can calculate the difference in their surface areas:

\[ \text{Difference} = S - \frac{1}{2} S = \frac{1}{2} S \]

Now, we want to express this difference as a fraction of the surface area of the large house \( S \):

The fraction by which the small house's surface area is smaller than that of the large house is:

\[ \text{Fraction smaller} = \frac{\text{Difference in area}}{\text{Surface area of large house}} = \frac{\frac{1}{2} S}{S} = \frac{1}{2} \]

So, my original response was actually correct. The surface area of the small house is \(\frac{1}{2}\) smaller than that of the large house when expressed as a fraction of the larger house's surface area.

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