If the length of a rectangle is increased by 2 cm and the breadth is decreased by 2 cm, its area decreases by 28 sq. cm. If the length is decreased by 1 cm and the breadth is increased by 2 cm, its area increases by 33 sq. cm. Find the length and bre…

Answer: Length = 23 cm, breadth = 11 cm

Let length \(= x\) cm and breadth \(= y\) cm, so area \(= xy\).

Condition 1: \((x + 2)(y - 2) = xy - 28\)

\(xy - 2x + 2y - 4 = xy - 28 \Rightarrow -2x + 2y = -24 \Rightarrow x - y = 12\) …(I)

Condition 2: \((x - 1)(y + 2) = xy + 33\)

\(xy + 2x - y - 2 = xy + 33 \Rightarrow 2x - y = 35\) …(II)

(II) − (I): \(x = 23\). From (I): \(y = 11\).

Check: original area \(= 253\); \(25 \times 9 = 225 = 253 - 28\) ✓; \(22 \times 13 = 286 = 253 + 33\) ✓.

Thinking point: the \(xy\) terms cancel on both sides, so these equations look quadratic but are really linear.