Area of a rectangle is 150 metre sq. When the breadth of the same rectangle is increased by 2 meter and the length decreased by 5 metre the area of the rectangle decreases by 30 metre square. What is the perimeter of the square whose side are equal to the length of the rectangle? [RBI Officers Gr. 'B' (Phase I) Exam, 2015]

### Concept & Algebraic Formulation This problem requires setting up a system of equations based on the area of a rectangle ($Area = length \times breadth$). By manipulating the dimensions as described, we can solve for the original length and breadth. $$Area = l \times b$$ $$Perimeter_{square} = 4 \times side$$ ### Step-by-Step Solution * Let the original length be $l$ and original breadth be $b$. We are given $l \times b = 150$. * The new breadth is $b + 2$ and the new length is $l - 5$. * The new area decreases by 30, making it $150 - 30 = 120$ m$^2$. * Equation: $(l - 5)(b + 2) = 120$. * Expand the equation: $lb + 2l - 5b - 10 = 120$. * Substitute $lb = 150$: $150 + 2l - 5b - 10 = 120 \Rightarrow 2l - 5b = -20 \Rightarrow 5b - 2l = 20$. * From the first equation, $b = \frac{150}{l}$. Substitute this into the linear equation: $5(\frac{150}{l}) - 2l = 20$. * Multiply through by $l$: $750 - 2l^2 = 20l \Rightarrow 2l^2 + 20l - 750 = 0$. * Divide by 2: $l^2 + 10l - 375 = 0$. * Factor the quadratic equation: $(l + 25)(l - 15) = 0$. Since length cannot be negative, $l = 15$ meters. * The question asks for the perimeter of a square with side equal to this length ($15$ m). * Perimeter = $4 \times 15 = 60$ meters. ### Exam Strategy & Shortcut Instead of fully solving the quadratic, you can test factor pairs of 150 for $l$ and $b$. Pairs of 150: (10, 15), (25, 6), (30, 5). If $l=15$ and $b=10$, new $l=10$ and new $b=12$. New area = $10 \times 12 = 120$. This matches the condition perfectly! Perimeter = $15 \times 4 = 60$. ### Common Pitfall A common error is solving for the length ($15$) and selecting it as the final answer, neglecting the second part of the question which specifically asks for the *perimeter* of a square with that side length. ### Final Answer Therefore, the correct answer is **60 m**.