Bill drinks one bottle of milk every day. The milk costs 1 dollar per bottle. If he collects 10
lids from the milk bottles, he can get one free bottle of milk. How much will Bill pay for the
milk until exactly one year after he starts drinking? Assume: He always drinks the free milk
on the day after he collects 10 caps, and one year is 365 days.

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.

Bill drinks one bottle of milk every day. The milk costs 1 dollar per bottle. If he collects 10 lids from the milk bottles, he can get one free bottle of milk. How much will Bill pay for the milk until exactly one year after he starts drinking? Assume: He always drinks the free milk on the day after he collects 10 caps, and one year is 365 days.

Explore how to calculate the total cost for Bill's milk consumption over a year, with free bottles earned for every 10 bottle lids collected. This problem covers arithmetic concepts suitable for Grades 5-7, including multiplication, division, and problem-solving with constraints.

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