If the length of a rectangle is increased by 10% and its breadth is decreased by 10%, the change in its area will be (S.S.C. 2007, 2010; P.C.S. 2009; Hotel Managment, 2010)

### Concept & Same Percentage Increase and Decrease When a quantity is successively increased by $R\%$ and then decreased by $R\%$, there is always a net decrease. Formula: $$\text{Net Decrease \%} = \frac{R^2}{100}$$ ### Step-by-Step Solution * **Given:** The length is increased by $10\%$ ($x = +10$). The breadth is decreased by $10\%$ ($y = -10$). * Using the successive percentage change formula: $\text{Net Change} = x + y + \frac{xy}{100}$. * Net Change $= 10 - 10 + \frac{(10)(-10)}{100}$. * Net Change $= 0 - \frac{100}{100} = -1\%$. * The negative sign indicates a decrease. * Therefore, the area decreases by 1%. ### Exam Strategy & Shortcut Memorize the $R^2/100$ rule. For a 10% increase followed by a 10% decrease (or vice-versa), the net change is always a decrease of $10^2/100 = 100/100 = 1\%$. This applies universally to areas and prices. ### Common Pitfall Many students intuitively believe that an equal percentage increase and decrease will cancel each other out, leading to "No change". This is false because the 10% decrease is calculated on the already increased value, making the absolute decrease larger than the absolute increase. ### Final Answer Therefore, the correct answer is **1% decrease**.