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The area of a rectangle gets reduced by $8 m^{2}$ , when its length is reduced by 5m and its breadth is increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by $74 m^{2}$ . Find the length and the breadth of the rectangle.

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Length =19m and Breadth =10m

Length =18m and Breadth =10m

Length =19m and Breadth =11m

Length =19m and Breadth =13m

answer is A.

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Detailed Solution

It is given that when length of rectangle is reduced by 5m and its breadth is increased by 3 m then area of a rectangle gets reduced by

8 m^{2}

.
Let, length of rectangle =

x metre

and breadth of rectangle =

y metre

.
Then, area

= xy m^{2}

Since, length of rectangle is reduced by 5m

\Rightarrow Length = (x - 5)

and breadth is increased by 3 m

\Rightarrow Breadth = (y + 3)

.
Then area of a rectangle gets reduced by

8 m^{2}

.
According to question,

xy - (x - 5) (y + 3) = 8

\Rightarrow xy - xy - 3 x + 5 y + 15 = 8

\Rightarrow 3 x - 5 y = 7

… (1)
Also, when length of rectangle is increased by 3m

\Rightarrow Length = (x + 3)

and its breadth is increased by 2 m

\Rightarrow Breadth = (y + 2)

then area of a rectangle gets increased by

74 m^{2}

.
According to question,

(x + 3) (y + 2) - xy = 74

\Rightarrow xy + 2 x + 3 y + 6 - xy = 74

\Rightarrow 2 x + 3 y = 68

… (2)
On multiplying equation (1) by (2) and equation (2) by 3 we get,

6 x - 10 y = 14

… (3)
and

6 x + 9 y = 204

… (4)
Subtracting equation (3) from equation (4),

y = 10

On putting

y = 10

in equation (1),

3 x - 5 \times 10 = 7 \Rightarrow x = 19

Therefore, length of rectangle =

19 m

and breadth of rectangle =

10 m

.
Hence, option (1) is correct.

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