If 2 painters can paint 2 rooms in 2 hours how many painter is needed to paint 18 rooms in 6 hours?

Use algebraic expression 

Let's break down the problem using algebraic expressions.

### Step 1: Determine the rate of painting.
Given:
- 2 painters can paint 2 rooms in 2 hours.

First, calculate the rate at which the painters work together.

The rate of painting for the 2 painters is:
\[
\text{Rate of 2 painters} = \frac{2 \text{ rooms}}{2 \text{ hours}} = 1 \text{ room per hour}
\]

Since this rate is for 2 painters, the rate for 1 painter is:
\[
\text{Rate of 1 painter} = \frac{1 \text{ room per hour}}{2} = \frac{1}{2} \text{ room per hour}
\]

### Step 2: Set up the equation for the required number of painters.
Let \( x \) be the number of painters needed to paint 18 rooms in 6 hours.

The total work that needs to be done is 18 rooms, and they have 6 hours to do it. The rate at which these \( x \) painters need to work together is:
\[
\text{Rate of } x \text{ painters} = \frac{18 \text{ rooms}}{6 \text{ hours}} = 3 \text{ rooms per hour}
\]

### Step 3: Set up the equation.
We know that 1 painter can paint \(\frac{1}{2}\) room per hour, so \( x \) painters can paint:
\[
x \times \frac{1}{2} \text{ rooms per hour} = \frac{x}{2} \text{ rooms per hour}
\]

We want this to equal the required rate of 3 rooms per hour:
\[
\frac{x}{2} = 3
\]

### Step 4: Solve for \( x \).
Multiply both sides of the equation by 2 to solve for \( x \):
\[
x = 3 \times 2 = 6
\]

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

Would you like further details or have any questions? Here are 5 related questions you might find useful:

1. How would the number of painters change if you had more or fewer hours available?
2. What if the rate of painting for one painter was different—how would that affect the result?
3. Can you express the relationship between the number of rooms, painters, and time as a general formula?
4. What assumptions are we making in this problem?
5. How would you calculate the time needed to paint a different number of rooms with a fixed number of painters?

**Tip:** Always double-check the units you're working with when setting up and solving algebraic expressions, especially in word problems!

Learn how to calculate the number of painters needed to paint 18 rooms in 6 hours using algebraic expressions. Understand the concept of rate of work and solve the problem step-by-step. Suitable for students in Grades 6-8.

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