The length of a rectangular blackboard is 8 m more than its breadth. If its length is increased by 7m and its breath is decreased by 4 m, its area remains unchanged. The length and breadth of the rectangular blackboard is

### Concept & Equating Areas When dimensions change but the area remains constant, you can set up an algebraic equation by expressing both the original area and the new area in terms of a single variable and setting them equal. ### Step-by-Step Solution * **Given:** The length is 8 m more than the breadth. * Let the breadth be $b$. * Then, the length $l = b + 8$. * Original Area $= l \times b = (b + 8) \times b = b^2 + 8b$. * New dimensions: The length is increased by 7m, so new length $= (b + 8) + 7 = b + 15$. * The breadth is decreased by 4m, so new breadth $= b - 4$. * New Area $= \text{New Length} \times \text{New Breadth} = (b + 15) \times (b - 4)$. * Expand the new area expression: $b^2 - 4b + 15b - 60 = b^2 + 11b - 60$. * Since the area remains unchanged, equate the two areas: * $b^2 + 8b = b^2 + 11b - 60$. * Subtract $b^2$ from both sides: $8b = 11b - 60$. * $3b = 60 \Rightarrow b = 20$ m. * Substitute $b$ back to find length: $l = 20 + 8 = 28$ m. * The dimensions are length = 28 m and breadth = 20 m. ### Exam Strategy & Shortcut Use option elimination to solve this instantly. Check the initial condition: Length is 8m more than breadth. (a) $24 - 16 = 8$ (Possible) (b) $20 - 24 = -4$ (Eliminate) (c) $28 - 16 = 12$ (Eliminate) (d) $28 - 20 = 8$ (Possible) Now test the second condition on the remaining options (a) and (d). Test (d): Original area $= 28 \times 20 = 560$. New length $= 28 + 7 = 35$. New breadth $= 20 - 4 = 16$. New area $= 35 \times 16 = 560$. The area remains unchanged, so (d) is the correct answer. ### Common Pitfall A frequent error is setting up the expressions correctly but making a sign mistake when distributing the terms in $(b + 15)(b - 4)$, leading to an incorrect coefficient for $b$ or the constant term. Always double-check polynomial multiplication. ### Final Answer Therefore, the correct answer is **28 m, 20 m**.