If the height of a triangle is decreased by 40% and its base is increased by 40%, what will be the effect on its area?

### Concept & Successive Percentage Change The area of a triangle is dependent on the product of its base and height ($Area = \frac{1}{2} \times b \times h$). A percentage change in both variables requires calculating their combined net effect on the product. ### Step-by-Step Solution * Let the original base be $100$ and the original height be $100$. * The original area index = $100 \times 100 = 10000$ (ignoring the $\frac{1}{2}$ constant as it cancels out in ratio). * The new height after a 40% decrease is $60$. * The new base after a 40% increase is $140$. * The new area index = $60 \times 140 = 8400$. * The decrease in the index is $10000 - 8400 = 1600$. * Percentage decrease = $\frac{1600}{10000} \times 100\% = 16\%$. ### Exam Strategy & Shortcut When a value is increased by $x\%$ and then decreased by $x\%$ (or vice-versa), there is always a net decrease given by the formula: $$\text{Net Decrease \%} = \frac{x^2}{100}$$ Here, $x = 40$. Thus, decrease = $\frac{40^2}{100} = \frac{1600}{100} = 16\%$. ### Common Pitfall Believing that a 40% increase and a 40% decrease cancel each other out, resulting in "No change". Always remember the base value for the second percentage operation has changed. ### Final Answer Therefore, the correct answer is **16 % decrease**.