A rectangular room is 12 meter long, 6 meter wide and 4 meter high. It has two doors of size 2 meter × 1 meter each and two windows of size 1.5 meter × 1 meter each. If the rate of paint (on the walls) is ₹ 15 per square meter, then total cost of painting the walls is -

Correct — option (4), ₹ 2055.

The paint goes on the four walls only, less the two doors and two windows.

Step 1 — area of the four walls.
Two long walls: 2 × 12 × 4 = 96 m²
Two short walls: 2 × 6 × 4 = 48 m²
Walls = 96 + 48 = 144 m²
Check: 2 × (12 + 6) × 4 = 144 m²

Step 2 — area of the openings.
Doors: 2 × (2 × 1) = 4 m²
Windows: 2 × (1.5 × 1) = 3 m²
Openings = 4 + 3 = 7 m²

Step 3 — area to be painted.
144 − 7 = 137 m²

Step 4 — cost.
137 × ₹ 15 = ₹ 2055

The idea to remember: area of four walls = 2 × (length + breadth) × height; take away every door and window before multiplying by the rate.

Why the others are wrong

• (1)
  ₹ 1955 — ₹ 1955 is ₹ 100 less than the cost of the 137 m² to be painted. At ₹ 15 per m² it would pay for only 1955 ÷ 15 = 130⅓ m².

  The walls come to 144 m² and the openings to 7 m², so 137 m² is painted, costing 137 × 15 = ₹ 2055.

• (2)
  ₹ 2010 — ₹ 2010 = 134 × ₹ 15, the cost of 134 m². That would mean 10 m² of openings taken off the 144 m² of wall.

  The openings are 7 m²: two doors of 2 m × 1 m make 4 m², and two windows of 1.5 m × 1 m make 3 m². 144 − 7 = 137 m², costing ₹ 2055.

• (3)
  ₹ 2155 — ₹ 2155 is ₹ 100 more than the correct cost and ₹ 5 below ₹ 2160, the cost of all 144 m² of wall with nothing taken off.

  The doors and windows are not painted, so their 7 m² comes off first: 137 × 15 = ₹ 2055.

The area of a rectangle is length × breadth. A rectangular room has four walls in two pairs: two of length × height and two of breadth × height.

Their total is 2 × (length + breadth) × height, the perimeter of the floor times the height. This is the lateral surface area of a cuboid.

The total surface area of a cuboid, 2(lb + bh + hl), adds the floor and ceiling. For this room that is 144 + 72 + 72 = 288 m².

For painting, openings such as doors and windows are subtracted, and cost = area × rate per square metre.

Key facts

• Area of the four walls of a rectangular room = 2 × (length + breadth) × height.
• Here the walls measure 2 × (12 + 6) × 4 = 144 m².
• The openings are two 2 m × 1 m doors (4 m²) and two 1.5 m × 1 m windows (3 m²), 7 m² in all.
• Area painted = 144 − 7 = 137 m²; at ₹ 15 per m², the cost is ₹ 2055.
• Total surface area of a cuboid = 2(lb + bh + hl); for this room, 288 m², including a 72 m² floor and a 72 m² ceiling.

Common traps

• Adding the ceiling or floor. The stem prices paint on the walls; the ceiling alone is 12 × 6 = 72 m².
• Forgetting to double. (12 + 6) × 4 = 72 m² covers only one long wall and one short wall.
• Subtracting one door and one window. There are two of each, so the openings are 7 m², not 3.5 m².

• practice — not a real PYQ

  A room is 10 m long, 8 m wide and 3 m high. It has one door of 2 m × 1 m and one window of 1 m × 1 m. What is the cost of painting its walls at ₹ 20 per square metre?

  • (a)₹ 2,160
  • (b)₹ 2,100
  • (c)₹ 1,080
  • (d)₹ 3,700
  Answer(2) — Walls = 2 × (10 + 8) × 3 = 108 m²; openings = 2 + 1 = 3 m²; 105 × ₹ 20 = ₹ 2,100.

  Option (1) takes nothing off for the openings; option (3) prices only two walls, (10 + 8) × 3 = 54 m², with nothing taken off; option (4) adds the 80 m² ceiling.

• practice — not a real PYQ

  The cost of painting the four walls of a room 9 m long and 6 m wide, at ₹ 10 per square metre with no doors or windows, is ₹ 1,500. The height of the room is –

  • (a)5 m
  • (b)10 m
  • (c)2.5 m
  • (d)6 m
  Answer(1) — Wall area = 1,500 ÷ 10 = 150 m², and 2 × (9 + 6) × h = 150 gives 30h = 150, so h = 5 m.

  Option (2) uses (9 + 6) × h, only two walls; option (3) doubles the perimeter twice, 60h = 150; option (4) is the room's width.