A wallet contains ₱50 and ₱100 bills amounting ₱450. How many of each kind will produce the given amount? 

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.

A wallet contains ₱50 and ₱100 bills amounting to ₱450. How many of each kind will produce the given amount?

Learn how to solve the problem of determining the number of ₱50 and ₱100 bills in a wallet that totals ₱450. This solution covers linear equations, simplification, and finding non-negative integer solutions. Suitable for students in Grades 6-8.

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