To find the unit rate of soda to pineapple juice in the punch recipe, we need to express the ratio in terms of a single cup of pineapple juice.
Let's denote:
- \( x \) as the amount of pineapple juice in cups,
- \( y \) as the amount of soda in cups.
From the recipe, we know that for every \( x \) cups of pineapple juice, there are \( y \) cups of soda.
If the recipe specifies \( \frac{1}{4} \) cup of pineapple juice for every \( N \) cups of soda, we can express the ratio as follows:
\[ \text{Ratio} = \frac{y}{x} = \frac{N}{\frac{1}{4}} \]
To find the unit rate of soda to pineapple juice, we would express it as:
\[ \text{Unit Rate (soda to pineapple juice)} = \frac{y}{x} \]
Assuming you clarified that for \( N \) cups of soda, there is \( \frac{1}{4} \) cup of pineapple juice; for each cup of pineapple juice, there are \( 4N \) cups of soda.
So the unit rate of soda to pineapple juice is:
\[ \text{Unit Rate} = \frac{4N}{1} = 4N \text{ cups of soda for 1 cup of pineapple juice} \]
In conclusion, the unit rate of soda to pineapple juice in the punch is \( 4N \), where \( N \) is the number of cups of soda used corresponding to \( \frac{1}{4} \) cup of pineapple juice.