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. Find the dimensions of the rectangle.
Given:
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.
To do:
We have to find the dimensions of the rectangle.
Solution:
Let the original length of the rectangle be $l$ and the breadth be $b$.
Area of the original rectangle $=lb$.
In the first case, the length is reduced by 5 units and breadth is increased by 3 units, the area of the rectangle is reduced by 9 square units.
New length $=l-5$
New breadth $=b+3$
The area formed by the new rectangle $=(l-5)(b+3)$ sq. units
According to the question,
$(l-5)(b+3)=lb-9$
$lb-5b+3l-15=lb-9$
$3l-5b=15-9$
$3l-5b=6$.....(i)
In the second case, the length is increased by 3 units and breadth is increased by 2 units, the area is increased by 67 sq. units.
New length $=l+3$
New breadth $=b+2$
The area formed by the new rectangle $=(l+3)(b+2)$ sq. units
According to the question,
$(l+3)(b+2)=lb+67$
$lb+2l+3b+6=lb+67$
$2l+3b=67-6$
$2l+3b=61$.....(ii)
Subtracting $2\times(i)$ from $3\times(ii)$, we get,
$3(2l+3b)-2(3l-5b)=3(61)-2(6)$
$6l-6l+9b+10b=183-12$
$19b=171$
$b=\frac{171}{19}$
$b=9$
$2l+3(9)=61$ (From (ii))
$2l=61-27$
$2l=34$
$l=\frac{34}{2}$
$l=17$
The dimensions of the rectangle are $17$ units (length) and $9$ units (breadth).
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