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The problem you've shared is an optimization problem that involves maximizing profit given certain constraints. Here's a step-by-step approach to solve it:

### Problem Restatement:
- **Budget Constraint:** The store owner can spend at most PHP 1,000,000 on PCs and laptops.
- **Costs:** 
  - Each PC costs PHP 15,000.
  - Each laptop costs PHP 20,000.
- **Profit:**
  - Each PC yields a profit of PHP 10,000.
  - Each laptop yields a profit of PHP 15,000.
- **Sales Constraints:**
  - At least 12 PCs must be sold, but no more than 30.
  - The number of laptops sold must be at most half the number of PCs sold.

### Variables:
Let \( x \) be the number of PCs sold and \( y \) be the number of laptops sold.

### Objective Function (Profit to Maximize):
\[ P(x, y) = 10,000x + 15,000y \]

### Constraints:
1. **Budget:** 
   \[ 15,000x + 20,000y \leq 1,000,000 \]
2. **PCs Sold:** 
   \[ 12 \leq x \leq 30 \]
3. **Laptops Sold:** 
   \[ y \leq \frac{x}{2} \]

### Solution Steps:

1. **Rewrite the Budget Constraint:**
   \[ 15x + 20y \leq 1000 \quad \text{(in thousands of PHP)} \]

2. **Express \( y \) from \( y \leq \frac{x}{2} \):**
   \[ y \leq \frac{x}{2} \]

3. **Substitute and Solve:**
   - Find combinations of \( x \) and \( y \) that satisfy all constraints and maximize the profit function \( P(x, y) = 10x + 15y \) (in thousands of PHP).

### Solving the Linear Programming Problem:

Let's manually check a few critical points:
1. \( x = 12 \):
   \[ y \leq \frac{12}{2} = 6 \]
   Budget constraint: 
   \[ 15(12) + 20(6) = 180 + 120 = 300 \leq 1000 \]
   Profit: 
   \[ P(12, 6) = 10(12) + 15(6) = 120 + 90 = 210 \text{ (in thousands of PHP)} \]

2. \( x = 30 \):
   \[ y \leq \frac{30}{2} = 15 \]
   Budget constraint: 
   \[ 15(30) + 20(15) = 450 + 300 = 750 \leq 1000 \]
   Profit: 
   \[ P(30, 15) = 10(30) + 15(15) = 300 + 225 = 525 \text{ (in thousands of PHP)} \]

Since the point \( (30, 15) \) satisfies all constraints and maximizes the profit, this is the optimal solution.

### Final Answer:
- The store owner should sell **30 PCs** and **15 laptops** to maximize profit.

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Would you like more details on any specific part of this solution or have any other questions?

### Related Questions:
1. How would the solution change if the cost of a laptop increases to PHP 25,000?
2. What if the store owner could spend PHP 1,200,000 instead of PHP 1,000,000?
3. How does the minimum requirement of selling 12 PCs affect the overall solution?
4. Can you generalize the solution for any given budget constraint?
5. What if the profit per laptop was lower than the profit per PC?

### Tip:
When dealing with linear programming problems, always check the constraints carefully, as they can significantly influence the optimal solution.

Each month a store owner Adler can spend at most PHP 1,000,000 on PCs and laptops. A PC costs the store owner PHP 15,000 and a laptop costs him PHP 20,000. Each PC is sold for a profit of PHP 10,000 while a laptop is sold for a profit of PHP 15,000. The store owner estimates that at least 12 PCs but no more than 30 are sold each month. He also estimates that the number of laptops sold is at most half the PCs. How many PCs and laptops should be sold to maximize the profit?

Solve an optimization problem involving maximizing profit for a store owner selling PCs and laptops under budget and sales constraints. Learn how to apply linear programming concepts and solve inequalities to find the optimal number of PCs and laptops to sell. Suitable for students in Grades 10-12.

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