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# The length, breadth, and height of a room are in the ratio 3:2:1. If the breadth and height are halved, while the length is doubled, then the total area of the four walls of the room will:

# (1) Remain the same (2) Decrease by 30% (3) Decrease by 15% (4) Decrease by 18.75%

The area of the room is calculated by the concept of the lateral surface area of the cuboid.

## Answer: If the breadth and height are halved, while the length is doubled, then the total area of the four walls of the room will decrease by 30%.

Let's understand the surface area of the cuboid in detail.

**Explanation:**

Let the length, breadth, and height of the cuboid as 3x, 2x, and x respectively.

Then, the area of four walls is given by 2(bh + lh) = 2h(l + b) = 2(x)(3x + 2x) = 2(x)(5x) = 10x^{2 }sq.m

Given that,

- length is doubled: (3x) × 2 = 6x
- breadth is halved: 2x / 2 = x
- height is halved: x / 2 = x/2

Now, the new area of four walls

\text { Area }_{\text {new }} = 2(x/2)(6x + x) = 2 × (x/2) × (7x) = 7x^{2}

\text { Area }_{\text {original }}_{ }= 10x^{2}

The percentage increase = \left(\text { Are } a_{\text {new }} \text { - Area }_{\text {original }}\right) / \text { Area }_{\text {original }}

The percentage difference in the area of four walls = [ (7x^{2} - 10x^{2}) / 10x^{2 }] × 100 = [ - 3x^{2} / 10x^{2 }] × 100 = [ -3/10 ] × 100 = - 30 %

The percentage difference in the area of four walls is - 30% (Negative sign shows decrease in area)

### Thus, if the breadth and height are halved while the length is doubled, then the total area of the four walls of the room will decrease by 30%.

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