We are given a problem involving a mixture of water and milk. Initially, the ratio of water to milk is $7:5$, and after some operations, the proportion of milk to water changes to $7:9$. Let's find out how much milk was in the container initially.

Step-by-Step Solution

1. Initial ratio and total amount: The initial ratio of water to milk is $7:5$. Let the total amount of liquid in the can be $x$ liters. Thus, the initial amount of milk and water can be expressed as: $\text{Amount of water} = \frac{7}{12}x \quad \text{(since water makes up 7 parts of the total 12 parts)}$ $\text{Amount of milk} = \frac{5}{12}x \quad \text{(since milk makes up 5 parts of the total 12 parts)}$

2. Removal of 9 liters of mixture: When 9 liters of the mixture are removed, it means that the same ratio $7:5$ is preserved in the removed mixture. Therefore, in the 9 liters removed: $\text{Amount of water removed} = \frac{7}{12} \times 9 = 5.25 \text{ liters}$ $\text{Amount of milk removed} = \frac{5}{12} \times 9 = 3.75 \text{ liters}$

3. Remaining amounts of water and milk: After removing the 9 liters, the remaining amounts of water and milk are: $\text{Remaining water} = \frac{7}{12}x - 5.25$ $\text{Remaining milk} = \frac{5}{12}x - 3.75$

4. Filling with water: The 9 liters removed are replaced entirely with water. So, the new amount of water becomes: $\text{New amount of water} = \frac{7}{12}x - 5.25 + 9 = \frac{7}{12}x + 3.75$

5. New ratio of milk to water: After this operation, the ratio of milk to water becomes $7:9$. Therefore, we can set up the following equation using the new amounts of milk and water: $\frac{\frac{5}{12}x - 3.75}{\frac{7}{12}x + 3.75} = \frac{7}{9}$

6. Solving the equation: Cross-multiply to get: $9\left(\frac{5}{12}x - 3.75\right) = 7\left(\frac{7}{12}x + 3.75\right)$ Expanding both sides: $9 \times \frac{5}{12}x - 9 \times 3.75 = 7 \times \frac{7}{12}x + 7 \times 3.75$ Simplify: $\frac{45}{12}x - 33.75 = \frac{49}{12}x + 26.25$ Multiply through by 12 to eliminate the fractions: $45x - 405 = 49x + 315$ Rearranging the terms: $45x - 49x = 315 + 405$ $-4x = 720$ $x = \frac{720}{-4} = -180$

7. Check for logical inconsistency: Upon reviewing, I made an algebraic mistake; however, correcting it