To determine how much flour Ayana should use for 1 cup of cocoa powder, we can set up a proportion based on the original recipe.
The original recipe calls for \( \frac{1}{2} \) cup of flour for \( \frac{1}{3} \) cup of cocoa powder. We can express this as a ratio:
\[ \text{Flour to Cocoa Ratio} = \frac{\frac{1}{2}}{\frac{1}{3}} = \frac{1/2}{1/3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} \]
This means for every \( \frac{1}{3} \) cup of cocoa, she uses \( \frac{3}{2} \) cups of flour.
Now we want to find out how much flour corresponds to 1 cup of cocoa powder. We know that:
1 cup of cocoa powder can be converted to a fraction that represents how many \( \frac{1}{3} \) cups of cocoa it contains:
\[ 1 \text{ cup} = 3 \times \left(\frac{1}{3} \text{ cup}\right) \]
Since 1 cup of cocoa is equal to 3 parts of \( \frac{1}{3} \) cup, we need to multiply the amount of flour for \( \frac{1}{3} \) cup of cocoa by 3:
\[ \text{Flour for 1 cup of cocoa} = 3 \times \left(\frac{1}{2} \text{ cup of flour}\right) = \frac{3}{2} \text{ cups of flour} \]
Therefore, Ayana should use \( \frac{3}{2} \) cups of flour, which is equal to 1.5 cups of flour, for 1 cup of cocoa powder.