A room has length 25 m, breadth 12 m, and height 10 m. The four walls and the ceiling (but not the floor) are to be painted. Painter A can paint 200 m² in 5 days, while Painter B can paint 250 m² in 2 days. If A and B work together at their respective constant rates, in how many days will they complete painting the four walls and the ceiling?

Introduction:
This problem combines geometry (surface area of a cuboid) with work and time concepts. First we compute the total area that must be painted and then work out how long two painters, working together at different rates, will take to finish the job.

Given Data / Assumptions:

• Room dimensions: length l = 25 m, breadth b = 12 m, height h = 10 m.
• Surfaces to be painted: four walls and ceiling (no floor).
• Painter A paints 200 m² in 5 days.
• Painter B paints 250 m² in 2 days.
• Both painters work at constant rates and can work together.

Concept / Approach:
Step one is to compute the total area to be painted. The four walls of a cuboid have area 2h(l + b). The ceiling has area l * b. Next, we find each painter's daily painting rate and add them to get a combined rate. Time required equals total area divided by total rate.

Step-by-Step Solution:
Area of four walls = 2h(l + b) = 2 * 10 * (25 + 12) = 20 * 37 = 740 m². Area of ceiling = l * b = 25 * 12 = 300 m². Total area to paint = 740 + 300 = 1040 m². Painter A's rate = 200 m² / 5 days = 40 m² per day. Painter B's rate = 250 m² / 2 days = 125 m² per day. Combined rate = 40 + 125 = 165 m² per day. Time required = total area / combined rate = 1040 / 165 days. Compute 1040 / 165: 165 * 6 = 990, remainder = 50. So time = 6 + 50/165 = 6 + 10/33 days.

Verification / Alternative check:
Check that in 6 10/33 days they cover 1040 m²: 165 * (6 10/33) = 165 * ( (6*33 + 10) / 33 ) = 165 * (208 / 33) = (165 / 33) * 208 = 5 * 208 = 1040 m², confirming the calculation.

Why Other Options Are Wrong:
If the time were 6, 7 10/33, or 8 days, multiplying by 165 m² per day would not yield 1040 m². For example, 6 days gives only 990 m². Therefore those options correspond to either underestimating or overestimating the work done.

Common Pitfalls:
A common mistake is to forget the ceiling area or to include the floor by accident. Others miscompute individual rates or incorrectly add them. Always handle area and rate calculations separately and carefully.

Final Answer:
Working together, the painters will complete the job in 6 10/33 days.