To solve this problem, we need to analyze the water consumption before and after the arrival of the extra people and the leak. Here is the breakdown:

Step 1: Initial situation

The village has enough water to last 800 people for 60 days. Let the total amount of water in the village be $W$. The daily consumption for 800 people is:

$\text{Daily consumption for 800 people} = \frac{W}{60}$

Step 2: After 12 days

In the first 12 days, the village used some water. The total amount of water consumed in those 12 days is:

$\text{Water consumed in 12 days} = 12 \times \frac{W}{60} = \frac{W}{5}$

This means the remaining water after 12 days is:

$\text{Remaining water} = W - \frac{W}{5} = \frac{4W}{5}$

Step 3: After 200 more people arrive

Now there are $800 + 200 = 1000$ people in the village. The daily water consumption for 1000 people would initially be:

$\text{Daily consumption for 1000 people} = \frac{1000}{800} \times \frac{W}{60} = \frac{5W}{240}$

Step 4: Effect of the leak (20% increase in consumption)

Due to the leak, daily consumption increases by 20%, so the new daily consumption for 1000 people is:

$\text{New daily consumption} = 1.2 \times \frac{5W}{240} = \frac{6W}{240} = \frac{W}{40}$

Step 5: Time for remaining water to run out

The remaining water is $\frac{4W}{5}$, and the daily consumption is now $\frac{W}{40}$. The number of days the water will last is:

$\text{Number of days} = \frac{\text{Remaining water}}{\text{New daily consumption}} = \frac{\frac{4W}{5}}{\frac{W}{40}} = \frac{4W}{5} \times \frac{40}{W} = \frac{160}{5} = 32 \text{ days}$

Final Answer:

The water will last for 32 more days after the 12 days have passed.

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Would you like further clarification or details on any part of the solution?

Here are 5 related questions to think about:

1. What would happen if 100 more people arrived instead of 200?
2. How would the outcome change if the leak increased consumption by 10% instead of 20%?
3. How long would the water last if no additional people arrived but the leak still occurred?
4. What if the initial water supply was meant for 800 people for 90 days instead of 60?
5. How much water (in terms of $W$) is consumed in total after the additional people arrive?

Tip: When solving problems with increasing rates (like leaks or additional consumption), always calculate the updated rates carefully before proceeding to determine the remaining time or quantity.