Two painters can complete two rooms in two hours. Assuming a constant work rate and equal efficiency, how many painters are needed to paint 18 rooms in 6 hours?

Introduction / Context:
This question is a typical work and time problem. It requires you to understand how work rate scales with the number of workers and the amount of time available. The painters are assumed to work at a constant rate, and you must find how many painters are required to complete a larger job in a given time based on their initial performance on a smaller job.

Given Data / Assumptions:

• Two painters can complete two rooms in two hours.
• We assume all painters work at the same rate and share the total work equally.
• The goal is to paint 18 rooms in 6 hours.
• We must find the number of painters required to achieve this target.

Concept / Approach:
The central concept is that work done equals rate multiplied by time, and total rate is proportional to the number of painters. From the initial information, we can compute how many painter hours are required per room. Then, for the larger task of 18 rooms, we determine the total painter hours needed. Finally, we divide by the available time to find how many painters are required. This is a scaling problem where both time and crew size can vary.

Step-by-Step Solution:
Step 1: From the statement, 2 painters complete 2 rooms in 2 hours. Step 2: Compute the total painter hours for this job: 2 painters × 2 hours = 4 painter hours. Step 3: These 4 painter hours are used to paint 2 rooms, so each room requires 4 / 2 = 2 painter hours. Step 4: Now consider the new task: 18 rooms must be painted. Step 5: Total painter hours needed for 18 rooms is 18 × 2 = 36 painter hours. Step 6: The total time available is 6 hours. Step 7: Let P be the number of painters required. Then P × 6 hours must equal 36 painter hours. Step 8: Solve the equation P × 6 = 36, which gives P = 36 / 6 = 6. Step 9: Therefore, 6 painters are needed to complete 18 rooms in 6 hours at the same work rate.

Verification / Alternative check:
Verify the answer by checking the rate of painting per painter. From the original case, two painters do two rooms in two hours, so combined they paint one room per hour. Hence, each painter paints half a room per hour. With six painters, the rate becomes 6 × 0.5 = 3 rooms per hour. Over 6 hours, they complete 3 × 6 = 18 rooms, which matches the requirement. This confirms that six painters suffice.

Why Other Options Are Wrong:
If only 2 painters worked for 6 hours, they would contribute 12 painter hours, enough for only 6 rooms. Four painters in 6 hours would give 24 painter hours, enough for 12 rooms. Eight painters in 6 hours would give 48 painter hours, which is more than needed for 18 rooms, so that crew size is unnecessarily large. Ten painters would also overshoot the requirement. Only 6 painters provide exactly 36 painter hours and thus match the needed work precisely.

Common Pitfalls:
Students often misread the initial relation and assume that two painters can complete one room in two hours, which leads to incorrect painter hour calculations. Some also confuse total hours with painter hours and do not multiply by the number of workers correctly. Always compute the total painter hours for a job, then scale linearly for different numbers of rooms and workers. This systematic method helps avoid confusion and ensures correct answers.

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