A rectangular room is “half as long again as it is broad”, meaning the length is 1.5 times the breadth. The cost of carpeting the floor at ₹5 per sq m is ₹270. The cost of papering the four walls at ₹10 per sq m is ₹1720. A door and 2 windows together occupy 8 sq m of wall area (so they are not papered). Find the dimensions of the room: breadth b, length l, and height H (in metres).

Introduction / Context:
This is a combined mensuration problem using both floor area (for carpeting) and four-walls area (for papering). You must translate costs into areas using the given rates. Then, use the relationship between length and breadth to solve for floor dimensions. Finally, use wall papering area (after adding back openings) to solve for the height using 2 * (l + b) * H.

Given Data / Assumptions:

• Length l = 1.5 * breadth b
• Carpet rate = ₹5 per sq m, carpet cost = ₹270
• Paper rate = ₹10 per sq m, paper cost = ₹1720
• Door + 2 windows area (unpapered) = 8 sq m
• Floor area = l * b
• Four walls area = 2 * (l + b) * H

Concept / Approach:
Convert cost to area. Use l = 1.5b in l*b to find b and l. Convert papering cost to papered area, add openings to get total wall area, then solve for height.

Step-by-Step Solution:

Verification / Alternative check:
Check carpeting: 9 * 6 = 54 sq m, cost = 54 * 5 = ₹270. Check walls: 2*(9+6)*6 = 180 sq m total, minus openings 8 gives 172 sq m papered, cost = 172 * 10 = ₹1720. Everything matches perfectly.

Why Other Options Are Wrong:

Common Pitfalls:
Common mistakes include interpreting “half as long again” incorrectly (it means 1.5 times, not 0.5 times). Another mistake is forgetting to add back the 8 sq m openings to compute total wall area for the 2(l+b)H formula. Some also confuse floor area with four walls area.

Final Answer:
b = 6 m, l = 9 m, H = 6 m