If two painters working together can paint two rooms in two hours, how many painters working at the same constant rate are required to paint 18 rooms in 6 hours?

Introduction / Context:
This problem is a classic work and time question. It involves understanding how work rate scales with the number of workers and the time available. By determining the rate at which painters complete rooms, we can then scale up to find how many painters are required to complete a larger amount of work within a given time. These questions help in understanding proportional reasoning and basic work rate concepts.

Given Data / Assumptions:

• Two painters can paint two rooms in two hours.
• All painters work at the same constant rate.
• We need to paint 18 rooms in 6 hours.
• We must find the number of painters required.

Concept / Approach:
The key idea is to compute the work rate per painter per hour. First, calculate the total painter-hours used in the initial situation and divide by the number of rooms completed to get rooms per painter-hour. Then, determine the total number of painter-hours needed to paint 18 rooms and divide this by the time allowed to find how many painters are required. This approach treats work as proportional to the product of number of workers and time.

Step-by-Step Solution:
Step 1: Two painters paint two rooms in two hours. Step 2: Total painter-hours used = number of painters * time = 2 * 2 = 4 painter-hours. Step 3: In these 4 painter-hours, 2 rooms are completed. Step 4: Therefore, the rate is 2 rooms / 4 painter-hours = 0.5 rooms per painter-hour. Step 5: Now we need to paint 18 rooms. Step 6: Total painter-hours required for 18 rooms = 18 / 0.5 = 36 painter-hours. Step 7: The time available is 6 hours. Step 8: Number of painters needed = total painter-hours / time = 36 / 6 = 6 painters. Step 9: Hence, 6 painters can complete 18 rooms in 6 hours at the same rate.

Verification / Alternative check:
With 6 painters working for 6 hours, total painter-hours are 6 * 6 = 36. At a rate of 0.5 rooms per painter-hour, the number of rooms completed is 36 * 0.5 = 18 rooms. This exactly matches the target number of rooms, confirming that 6 painters are sufficient and correctly calculated.

Why Other Options Are Wrong:
Option (a) 2 painters: They would provide only 2 * 6 = 12 painter-hours, resulting in 12 * 0.5 = 6 rooms, which is too few.
Option (c) 4 painters: They provide 4 * 6 = 24 painter-hours, giving 24 * 0.5 = 12 rooms, still less than 18.
Option (d) None: This suggests no option is correct, but we have shown that 6 painters works exactly, so this is incorrect.

Common Pitfalls:
A common mistake is to assume a direct multiple without computing the rate or to misinterpret how to scale workers and time. Some learners may incorrectly divide rooms by painters or time without relating everything through a consistent rate. Clearly computing painter-hours and using them as a unit of work helps avoid such confusion.

Final Answer:
The required number of painters is 6 painters.