A rectangle becomes a square when its length is reduced by 10 units and its breadth is increased by 5 units. But by this process the area of the rectangle is reduced by 210 sq. units. The area of the rectangle (in square units) is

### Concept & Algebraic Formulation Establish equations for the equality of sides to form a square, and the difference in area between the original rectangle and the new square. $$ \text{Area of Rectangle} = l \times b $$ ### Step-by-Step Solution 1. Let the original length be $l$ and breadth be $b$. 2. The figure becomes a square when $l$ is reduced by $10$ and $b$ is increased by $5$: $l - 10 = b + 5 \Rightarrow l = b + 15$. 3. The new area is $(l - 10)(b + 5)$. The problem states this new area is $210$ less than the original area ($lb$): $lb - (l - 10)(b + 5) = 210$. 4. Expand the equation: $lb - (lb + 5l - 10b - 50) = 210$ $-5l + 10b + 50 = 210$ $-5l + 10b = 160$ 5. Substitute $l = b + 15$ into the equation: $-5(b + 15) + 10b = 160$ $-5b - 75 + 10b = 160$ $5b = 235 \Rightarrow b = 47$. 6. Calculate length: $l = 47 + 15 = 62$. 7. The area of the original rectangle $A = l \times b = 62 \times 47 = 2914$. 8. Evaluate the options based on $A = 2914$: (a) $2950 > 2914 < 2900$ (False) (b) $2900 > 2914 > 2875$ (False) (c) $2925 < 2914 > 2875$ (False) (d) $2925 > 2914 > 2900$ (True, since $2914$ is between $2900$ and $2925$). ### Exam Strategy & Shortcut Write the area equation in terms of the new square's side $S$: Original length = $S + 10$, Original breadth = $S - 5$. Original Area = $(S+10)(S-5) = S^2 + 5S - 50$. New Area = $S^2$. Difference = $(S^2 + 5S - 50) - S^2 = 210$. $5S = 260 \Rightarrow S = 52$. Original Area = $S^2 + 210 = 52^2 + 210 = 2704 + 210 = 2914$. This avoids simultaneous equations. ### Common Pitfall Misinterpreting the confusing inequality notation in the options. Option (d) $2925 > A > 2900$ correctly signifies that $A$ falls within the range $(2900, 2925)$. ### Final Answer Therefore, the correct answer is **2925 > A > 2900**.