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Q.

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. What is the length and breadth of the rectangle?

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a

length = 17 units and breadth = 9 units

b

length = 20 units and breadth = 10 units

c

length = 9 units and breadth = 20 units

d

length = 10 units and breadth = 11 units

answer is A.

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Detailed Solution

Given that the area of a rectangle decreases by 9 square units when the length of the rectangle decreases by 5 units and the breadth of the rectangle increases by 3 units.
Let us assume x as the length of the rectangle and y as the breadth of the rectangle.
Now,
Length =

(x − 5),

Breadth =

(y + 3)

.
So, the area becomes,

(x − 5) (y + 3) = x y − 9 x y − 5 y + 3 x − 15 = x y − 9 3 x − 5 y − 6 = 0..... (i)

Now, when the length of the rectangle is increased by 3 units and the breadth by 2 units, then the area of the rectangle increases by 67 units.
So, length =

(x + 3),

Breadth =

(y + 2)

.
We know that,
Area of rectangle = length × breadth.
Therefore, area of the rectangle is,

(x + 3) (y + 2) = x y + 67 ⇒ x (y + 2) + 3 (y + 2) = x y + 67 ⇒ x y + 2 x + 3 y + 6 = x y + 67 ⇒ 2 x + 3 y = 67 − 6 ⇒ 2 x + 3 y = 61 ⇒ 2 x + 3 y − 61 = 0............ (i i)

Now let us determine the value of x and y.
Compare the eq (i) and eq (ii) with standard pair of linear equations,

a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0

.
Here,

a 1 = 3, b 1 = − 5, c 1 = − 6, a 2 = 2, b 2 = 3, c 2 = − 61.

Use the cross-multiplication method, to solve equations (i) and (ii).

x b 1 c 2 − b 2 c 1 = y c 1 a 2 − c 2 a 1 = 1 a 1 b 2 − a 2 b 1

⇒

x − 5 × − 61 − 3 × − 6 = y − 6 × 2 − − 61 × 3 = 1 3 × 3 − 2 × − 5

⇒

x 305 + 18 = y − 12 + 183 = 1 9 + 10

⇒

x 323 = y 171 = 119

Therefore,

⇒ y 171 = 119 ⇒ y = 17119 ⇒ y = 9

Now, let us find x.

⇒ x 323 = 119 ⇒ x = 32319 ⇒ x = 17

Hence, the length of a rectangle is 17 units and the breadth of a rectangle is 9 units.

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