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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)=xy−9 xy−5y+3x−15=xy−9 3x−5y−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)=xy+67 ⇒x(y+2)+3(y+2)=xy+67 ⇒xy+2x+3y+6=xy+67 ⇒2x+3y=67−6 ⇒2x+3y=61 ⇒2x+3y−61=0............(ii)  
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 = 1 19  
Therefore,
⇒ y 171 = 1 19 ⇒y= 171 19 ⇒y=9   Now, let us find x.
⇒ x 323 = 1 19 ⇒x= 323 19 ⇒x=17  
Hence, the length of a rectangle is 17 units and the breadth of a rectangle is 9 units.
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