 Your professor loves sandwiches. Every day, he goes to his favorite deli and orders a sandwich for $6.40. His favorite deli only takes cash however. On his first trip he breaks a $20 and keeps the $13.60 change to use next time. When he runs out of money, he will break another $20.  Solve the following. A) How many orders before the 3rd bill. B) How many orders before he uses all the change exactly. C) How Much did this cost him total

Let's break this down step by step:

### A) **How many orders before the 3rd bill?**

1. Each sandwich costs **$6.40**.
2. When he breaks a $20 bill, he gets **$13.60** in change (since $20 - 6.40 = 13.60$) to use for the next orders.

He uses this change for subsequent orders until he runs out of money. Let's calculate how many sandwiches he can buy before needing to break a new bill.

- After the first purchase, he has **$13.60** remaining.
- He uses this change to buy sandwiches until the change is less than $6.40 (the cost of one sandwich).

\[
\text{Remaining Change after each order:}
\]
1. First sandwich: $ 13.60 - 6.40 = 7.20 $ (after first use of change)
2. Second sandwich: $ 7.20 - 6.40 = 0.80 $ (after second use of change)

At this point, he doesn't have enough money to buy another sandwich. So, he needs to break another $20 to buy the next sandwich.

### Sandwich orders before breaking the first bill:

- He breaks the first $20 before the **3rd sandwich**.

- Repeat the same process for the next $20 bills. Each time he breaks a $20 bill, he buys **2 sandwiches with the remaining change**.

Thus, the pattern is that every time he breaks a bill, he can buy **2 sandwiches using the change**, and he needs to break a new bill for the next sandwich after that.

**How many orders before the 3rd bill?**

1. For the first $20: he buys **3 sandwiches** (1 after breaking the bill, 2 with the remaining change).
2. For the second $20: he buys **3 sandwiches**.
   
Therefore, after **6 sandwiches**, he needs to break the third bill. So, **6 orders** before breaking the third bill.

---

### B) **How many orders before he uses all the change exactly?**

For this, we consider how the professor will eventually use the exact amount of change without any leftover.

- Each time he breaks a $20 bill, he can use it for **3 sandwiches** (the third one requiring the next bill).
- So, after 2 sandwiches using the change, he has **$0.80 left**.
- After the third sandwich, he breaks another bill.

The key point here is that since $0.80 is not divisible by $6.40, he will never use the change exactly with the pattern given. He'll always have some leftover amount when he breaks a bill.

**Answer: There is no point where he uses all the change exactly.**

---

### C) **How much did this cost him in total?**

Let's assume he continues this process indefinitely, buying sandwiches and breaking $20 bills.

1. For every **3 sandwiches**, he spends exactly **$20**.

Thus, the cost for every 3 sandwiches is $20.

---

Let me know if you'd like more details, and here are some related questions to consider:

1. How many total sandwiches can he buy with $60?
2. If each sandwich costs $7 instead, how would the calculations change?
3. What if the deli offers a loyalty program, where the 10th sandwich is free?
4. How many sandwiches can he buy if he uses $100 in total?
5. What if he tipped the deli $0.50 for each sandwich he bought?

**Tip**: In problems involving cycles or repetitive actions, identifying patterns early can simplify complex calculations.

Your professor loves sandwiches. Every day, he goes to his favorite deli and orders a sandwich for $6.40. His favorite deli only takes cash however. On his first trip he breaks a $20 and keeps the $13.60 change to use next time. When he runs out of money, he will break another $20. Solve the following. A) How many orders before the 3rd bill. B) How many orders before he uses all the change exactly. C) How Much did this cost him total.

Learn how to calculate the number of sandwich orders before breaking a $20 bill and using all the change. This problem involves basic arithmetic operations and pattern recognition. Suitable for students in Grades 5-7.

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