Let's break down the problem step-by-step.

1. Determine total days and cost per bottle:

   • Bill drinks one bottle of milk per day for a year, so he drinks 365 bottles.
   • Each bottle costs $1, so without considering any discounts, he would pay $365.

2. Calculate the free bottles:

   • For every 10 lids collected, Bill gets one free bottle.
   • Since he drinks 365 bottles, he initially buys 365 bottles and therefore collects 365 lids.
   • With 365 lids, the number of free bottles he can initially get is: $\frac{365}{10} = 36 \text{ free bottles, with 5 lids remaining}$
   • After getting 36 free bottles, he has 5 leftover lids.

3. Account for additional free bottles from subsequent lids:

   • Bill will drink these 36 free bottles, earning an additional 36 lids.
   • Adding these 36 lids to his leftover 5 gives him 41 lids.
   • From these 41 lids, he can get an additional: $\frac{41}{10} = 4 \text{ free bottles, with 1 lid remaining}$

4. Final round of free bottles:

   • Drinking the 4 additional free bottles will yield 4 more lids.
   • Adding these 4 to his 1 remaining lid gives him 5 lids, which is insufficient for another free bottle.

5. Calculate total bottles Bill paid for:

   • Bill drank 365 bottles in total.
   • He received 36 + 4 = 40 bottles for free.
   • Therefore, he paid for: $365 - 40 = 325 \text{ bottles}$

6. Calculate total cost:

   • Since each bottle costs $1, the total amount Bill pays is: $325 \times 1 = 325 \text{ dollars}$

Final Answer:

Bill will pay $325 over the course of the year.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What if each bottle cost $1.50? How would the solution change?
2. If he could get a free bottle after collecting 8 lids, what would be the new cost?
3. How would the solution change if the year had 366 days?
4. What if Bill decides to drink 2 bottles of milk per day?
5. How would the answer change if he could get a free bottle every 12 lids instead of 10?

Tip: When solving such problems, break down the total into separate steps (paid and free items) to keep the calculations manageable and accurate.