If 2 painters can paint 2 rooms in 2 hours, how many painters do you need to paint 18 rooms in 6 hours?

To solve this problem, let's break it down step by step.

### Step 1: Understand the rate of painting.
- **2 painters** can paint **2 rooms** in **2 hours**. 
- This means the rate of painting per painter is \( \frac{2 \text{ rooms}}{2 \text{ painters} \times 2 \text{ hours}} = \frac{1 \text{ room}}{2 \text{ hours}} \) per painter.

### Step 2: Calculate the total work needed.
- You need to paint **18 rooms** in **6 hours**.

### Step 3: Determine the number of painters required.
- Let's denote the number of painters required as \( n \).
- The rate of work required is \( \frac{18 \text{ rooms}}{6 \text{ hours}} = 3 \text{ rooms per hour} \).

Each painter can paint at a rate of \( \frac{1 \text{ room}}{2 \text{ hours}} = 0.5 \text{ rooms per hour} \).

- To paint 3 rooms per hour, the number of painters \( n \) needed is:
  \[
  n \times 0.5 \text{ rooms per hour} = 3 \text{ rooms per hour}
  \]
  Solving for \( n \):
  \[
  n = \frac{3 \text{ rooms per hour}}{0.5 \text{ rooms per hour}} = 6 \text{ painters}
  \]

### Conclusion:
You need **6 painters** to paint 18 rooms in 6 hours.

Would you like more details on any part of this calculation? Here are some related questions:

1. How do you determine the rate of work for a group of workers?
2. What is the general formula for solving work-rate problems?
3. How would the solution change if the painters had different efficiencies?
4. Can you explain the concept of work done in such problems?
5. How do you solve similar problems involving machines instead of people?

**Tip:** In work-rate problems, the key is to find the rate at which work is completed and use it to calculate the required resources (like workers or time).

Discover how to solve a work rate problem involving 2 painters painting 18 rooms in 6 hours. This problem uses the concept of rate of work, proportional relationships, and equations to determine the number of painters needed. Suitable for high school students and those preparing for advanced mathematical problem-solving.

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