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Mensuration MCQ 1: (Carpet length and width ratio, area)

Mensuration MCQ 1: An order was placed for the supply of a carper whose length and breadth were in the ratio of 3 : 2. Subsequently, the dimensions of the carpet were altered such that its length and breadth were in the ratio 7 : 3 but were was no change in its perimeter. Find the ratio of the areas of the carpets in both the cases.

A. 4 : 3                      b) 8 : 7                  c)4 : 1                      d) 6 : 5

Solution:

We consider the shape of the carpet in two cases separately.

Case 1:

In the first case let the length and breadth of the carpet be in the ratio;

3x:2x

i-e if length = 3x then breadth = 2x

So by applying formula of perimeter of a rectangle which is;

\[ Perimeter Of Rectangle= 2(Length+Breadth) \]

So,

\[ Perimeter Of Carpet= 2(3x+2x)=10x \]

Case 2

In the second case we suppose that length and breadth are in the ratio;

7y:3y

i-e If length = 7y then breadth =3y

Again using perimeter formula;

\[ Carpet Perimeter = 2(7y+3y)=20y \]

But according to the question the perimeter does not change after amendments in length and breadth, so perimeters of both casese should be equal;

i-e

10x=20y

or

x=2y

Now consider areas of the carpet in both cases;

We know area of a rectangle ;

\[ Rectangle Area = Length\times Breadth \]

So, area in first case ;

\[ Area1=3x\times2x=6x^2 \]

Area in second case;

\[ Area2=7y\times 3y=21y^2 \]

Ratio of both cases area;

\[ 21y^2\::\:6x^2 \]

But x= 2y

so,

\[ 21y^2\:;\;6(2y)^2 \]

\[ 21y^2\:;\;24y^2 \]

\[ 7\::\;8 \]

So the ratio is 7:8 or 8:7 –   ———– option b is correct!

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