Let $x$ represent the number of ₱50 bills, and $y$ represent the number of ₱100 bills.

We are given two pieces of information:

1. The total value of the bills is ₱450. $50x + 100y = 450$

2. The number of ₱50 and ₱100 bills are whole numbers.

Now, let's solve for the values of $x$ and $y$:

Step 1: Simplify the equation

$50x + 100y = 450$ Divide the entire equation by 50: $x + 2y = 9$

Step 2: Solve for $x$ in terms of $y$

$x = 9 - 2y$

Step 3: Test possible values of $y$

Since both $x$ and $y$ must be non-negative integers, we will substitute different values of $y$ into the equation $x = 9 - 2y$.

• If $y = 0$, then $x = 9$.
• If $y = 1$, then $x = 7$.
• If $y = 2$, then $x = 5$.
• If $y = 3$, then $x = 3$.
• If $y = 4$, then $x = 1$.

Thus, the possible combinations of ₱50 and ₱100 bills that give a total of ₱450 are:

• 9 ₱50 bills and 0 ₱100 bills
• 7 ₱50 bills and 1 ₱100 bill
• 5 ₱50 bills and 2 ₱100 bills
• 3 ₱50 bills and 3 ₱100 bills
• 1 ₱50 bill and 4 ₱100 bills

Would you like more details or clarifications?

Here are five related questions to deepen your understanding:

1. What if the total value was ₱600 instead of ₱450?
2. How would the solution change if the wallet also contained ₱20 bills?
3. Can you find a way to represent the problem graphically?
4. How many combinations are possible if the number of bills is limited to 5?
5. Can this problem be solved using matrices?

Tip: Always simplify equations when possible to make the solving process faster and clearer.