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A 10l tank and a pall contained a total of 8346 ml of water. After 314 ml water was poured from the pail into the tank the tank had 12 times as much water as the pail . How much water was there in the tank at first?

To solve this problem, we can use algebra. Let's break it down step by step.

### Problem Recap:
- A 10L tank and a pail together contain a total of 8346 ml of water.
- After 314 ml is poured from the pail into the tank, the tank has 12 times as much water as the pail.
- We need to find out how much water was in the tank at first.

### Step-by-Step Solution:

1. **Define Variables:**
   - Let \( x \) be the initial amount of water in the tank (in ml).
   - Let \( y \) be the initial amount of water in the pail (in ml).

2. **Total Water Equation:**
   \[
   x + y = 8346 \text{ ml}
   \]

3. **After Pouring Water:**
   - The pail loses 314 ml, so the water in the pail is now \( y - 314 \) ml.
   - The tank gains 314 ml, so the water in the tank is now \( x + 314 \) ml.
   - According to the problem, the tank now has 12 times as much water as the pail:
   \[
   x + 314 = 12(y - 314)
   \]

4. **Solving the System of Equations:**

   Start with the equation:
   \[
   x + 314 = 12(y - 314)
   \]
   Expand and simplify:
   \[
   x + 314 = 12y - 3768
   \]
   Rearrange to isolate \( x \):
   \[
   x = 12y - 4082
   \]

   Now, substitute \( x = 12y - 4082 \) into the total water equation \( x + y = 8346 \):
   \[
   (12y - 4082) + y = 8346
   \]
   Simplify and solve for \( y \):
   \[
   13y - 4082 = 8346
   \]
   \[
   13y = 12428
   \]
   \[
   y = \frac{12428}{13} = 956 \text{ ml}
   \]

   Now that we know \( y = 956 \) ml, substitute it back to find \( x \):
   \[
   x = 12(956) - 4082 = 11472 - 4082 = 7390 \text{ ml}
   \]

### Final Answer:
- The initial amount of water in the tank was **7390 ml**.

Would you like further details or have any questions? Here are some related questions for further practice:

1. What would the scenario look like if 500 ml had been transferred instead of 314 ml?
2. How much water would be left in the pail if another 200 ml were poured into the tank?
3. What if the tank initially contained 8000 ml? How would the problem change?
4. Can you express the relationship between the water in the tank and the pail as a single equation after the pouring?
5. What would be the situation if the total amount of water was different, say 9000 ml?

**Tip:** When solving problems involving total quantities and relative changes, always set up clear equations that represent the relationships given in the problem. This helps in isolating the variables effectively.

Learn how to solve a problem involving the distribution of water between a tank and a pail. Understand the algebraic approach to finding initial water quantities and relationships after pouring water. Suitable for Grades 7-9.

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