UPSC CDS 2 2020 Math Paper

Concept:The net effect after a percentage increase followed by an equal percentage decrease is always a net loss (the final value is less than the original).Explanation:Let the initial number of items be $N$.In the first year, the number increases by $x\%$.The number becomes $N \times \frac{100 + x}{100}$.In the second year, the number decreases by $x\%$ from this new value.So final number = $N \times \frac{100 + x}{100} \times \frac{100 - x}{100}$ = $N \times \frac{(100+x)(100-x)}{100 \times 100}$ = $N \times \frac{10000 - x^{2}}{10000}$ = $N \times \left(1 - \frac{x^{2}}{10000}\right)$.Since $\frac{x^{2}}{10000} > 0$ for any $x > 0$, the factor $\left(1 - \frac{x^{2}}{10000}\right) < 1$.Thus the final number is $N$ multiplied by a number less than 1.Hence the final number is less than $N$.Even when using the successive percentage change formula:Effective change = $x\% + (-x\%) + \frac{x \times (-x)}{100}\% = -\frac{x^{2}}{100}\%$, a negative value.So the net effect is a decrease.Answer:A. Less than N