Given that the amount of water is directly proportional to the amount of ammonia, we can express this relationship mathematically as:
\[ w = k \cdot a \]
where:
- \( w \) is the amount of water,
- \( a \) is the amount of ammonia,
- \( k \) is the constant of proportionality.
From the problem, we know:
- \( w = 5 \) gallons of water,
- \( a = 10 \) teaspoons of ammonia.
To find the constant of proportionality \( k \), we can rearrange the equation:
\[ k = \frac{w}{a} \]
Substituting the known values:
\[ k = \frac{5 \text{ gallons}}{10 \text{ teaspoons}} = \frac{5}{10} = 0.5 \]
This means the relationship between water and ammonia can be expressed as:
\[ w = 0.5 \cdot a \]
To express \( a \) in terms of \( w \), we can rearrange the equation:
\[ a = \frac{w}{0.5} = 2w \]
Now, if you have another amount of water, you can calculate the required amount of ammonia using the relationship \( a = 2w \).
If we need to find the answer in decimal form, the original ratio of water to ammonia is:
For 5 gallons of water, there are 10 teaspoons of ammonia, which can also be stated as:
\[ \frac{5 \text{ gallons of water}}{10 \text{ teaspoons of ammonia}} = 0.5 \text{ gallons per teaspoon} \]
In decimal form, the constant ratio or the relationship can simply be written as \( 0.5 \) gallons of water per teaspoon of ammonia.
Thus, the answer to the problem in decimal form showing the ratio of water to ammonia is:
0.5 (gallons of water per teaspoon of ammonia).