Let's solve this problem step by step.

Step 1: Define Variables

Let the initial quantity of liquid $A$ in the can be $7x$ liters, and the initial quantity of liquid $B$ in the can be $5x$ liters. Thus, the total volume of the mixture is $7x + 5x = 12x$ liters.

Step 2: Mixture After Drawing Off 9 Liters

When 9 liters of the mixture is drawn off, the ratio of liquid $A$ and $B$ remains the same (7:5). So, the amount of liquid $A$ drawn off is: $\text{Amount of A drawn off} = \frac{7}{12} \times 9 = \frac{63}{12} = 5.25 \text{ liters}$ And the amount of liquid $B$ drawn off is: $\text{Amount of B drawn off} = \frac{5}{12} \times 9 = \frac{45}{12} = 3.75 \text{ liters}$

After removing 9 liters, the remaining amount of liquid $A$ is: $7x - 5.25 \text{ liters}$ And the remaining amount of liquid $B$ is: $5x - 3.75 \text{ liters}$

Step 3: Refilling with Liquid B

The can is refilled with 9 liters of liquid $B$, so the new quantity of liquid $B$ is: $(5x - 3.75) + 9 = 5x + 5.25 \text{ liters}$

Step 4: New Ratio After Refilling

The new ratio of liquid $A$ to liquid $B$ is given as $7:9$. Thus: $\frac{7x - 5.25}{5x + 5.25} = \frac{7}{9}$

Step 5: Solve the Equation

Now, cross-multiply to solve for $x$: $9(7x - 5.25) = 7(5x + 5.25)$ Expanding both sides: $63x - 47.25 = 35x + 36.75$ Simplifying: $63x - 35x = 36.75 + 47.25$ $28x = 84$ $x = 3$

Step 6: Initial Amount of Liquid A

The initial amount of liquid $A$ was $7x$ liters: $7x = 7 \times 3 = 21 \text{ liters}$

Final Answer

The initial amount of liquid $A$ in the can was 21 liters.

Would you like more details or have any questions?

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Related Questions:

1. How would the solution change if the ratio after refilling became 8:9 instead of 7:9?
2. What would happen if 12 liters were drawn off instead of 9 liters?
3. Can this problem be solved using a different method, such as algebraic substitution?
4. How would the initial quantities change if the ratio of A to B was initially 5:7 instead?
5. What is the effect of adding more than 9 liters of liquid B to the can after drawing off the mixture?
6. How could you verify the solution through a different logical approach?
7. What if the total volume of the can was limited, say to 15 liters? How would that affect the problem?
8. How would the problem change if liquid A evaporated at a constant rate during the process?

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Tip:

When solving ratio problems, carefully track the quantities before and after any changes to ensure accuracy in your final answer.