To solve the problem, we start by establishing the relationship between water and ammonia in the cleaning mixture. We know that the amount of water (\(W\)) is directly proportional to the amount of ammonia (\(A\)). This can be expressed mathematically as:
\[ W = k \cdot A \]
where \(k\) is the constant of proportionality.
From the information provided, we know that when \(W = 5\) gallons, \(A = 10\) teaspoons. We can plug these values into the equation to find \(k\):
\[ 5 = k \cdot 10 \]
To find \(k\), we divide both sides by \(10\):
\[ k = \frac{5}{10} = 0.5 \]
Now that we have determined \(k\), the formula for \(W\) in terms of \(A\) becomes:
\[ W = 0.5 \cdot A \]
Next, we want to find out how many gallons of water are needed for \(25\) teaspoons of ammonia (\(A = 25\)). We can substitute \(A = 25\) into the equation:
\[ W = 0.5 \cdot 25 \]
Calculating \(W\):
\[ W = 12.5 \]
Thus, the amount of water needed for \(25\) teaspoons of ammonia is \(\boxed{12.5}\) gallons.