QUESTION: The length of a rectangle exceeds its breadth by 6cms. If the length were halved and the breadth were increased by 7cms, the area would be decreased by 33cm². What is the length of the rectangle?
Let the breadth of first rectangle = B —————–(a)
The length of a rectangle exceeds its breadth by 6cm, can be written mathematically as
Length = B + 6 ——————-(b)
So area of first rectangle = Length * Breadth = (B + 6) * B ——————-(1)
now, there is another rectangle whose length, breadth , area is related to the first one.
If the length were halved,
Length of second rectangle = (Length of first)/2 = (B + 6) /2 Using equation (b)
the breadth were increased by 7cms,
Breadth of second rectangle = Breadth of first + 7 = B + 7 Using equation (a)
Area of this second rectangle = Length of second rectangle * Breadth of second rectangle
= {(B + 6 ) / 2 } * ( B + 7) ———————(2)
the area would be decreased by 33cm², means area of second rectangle is 33cm² less than the first one.
Area of second rectangle = Area of first rectangle – 33
Put values using equations (1) and (2)
{(B + 6 ) / 2 } * ( B + 7) = {(B + 6) * B} – 33
Multiply both sides by 2
(B + 6 ) * ( B + 7) = {2(B + 6) * B} – (2)33
B² + 7B + 6B + 42 = 2B² + 12 B – 66
2B² – B² – 13B + 12 B – 66 – 44 = 0
B² – B – 108 = 0
Find roots of this quadratic equation
B = 
B = 
B = 
B= 10.9 or -9.9
As breadth could not be a negative value so
B = 10.9cm ( B = Breadth of first rectangle from equation (a) above)
Now find the desired length of first rectangle.
From equation (b)
length = B + 6
Put value of B=10.9
So length = 10.9 + 6 = 16.9 cm
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