To break $100 using only 50 bills, you need to ensure the denominations of the bills add up to exactly $100, and there are exactly 50 bills total. This can be achieved by using a combination of $1 and $2 bills. Here's how:

Solution:

Let:

• $x$ = the number of $1 bills
• $y$ = the number of $2 bills

Equations:

1. $x + y = 50$ (You have 50 bills total)
2. $1x + 2y = 100$ (The total value is $100)

Solving the system of equations:

From the first equation: $x = 50 - y$

Substitute $x = 50 - y$ into the second equation: $1(50 - y) + 2y = 100$ $50 - y + 2y = 100$ $50 + y = 100$ $y = 50$

Now substitute $y = 50$ into $x + y = 50$: $x + 50 = 50$ $x = 0$

This solution indicates there are 50 $2 bills and no $1 bills.

Verification:

• Total bills: $50 + 0 = 50$ ✅
• Total value: $50 \times 2 + 0 \times 1 = 100$ ✅

Final Answer:

Use 50 $2 bills to break $100 into exactly 50 bills.

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Would you like more details or have any questions? Here are 5 related questions to expand understanding:

1. How would the solution change if $5 bills were allowed?
2. Can you break $100 using 50 bills if coins are included?
3. What if the total number of bills was 60 instead of 50?
4. How would the solution differ for other totals, like $200 using 50 bills?
5. Can you create an equation-solving strategy for more general cases like this?

Tip: Always verify the solution by checking both the total number of items and the total value.