The length of a rectangle is decreased by $r$%, and the breadth is increased by $(r + 5)$%. Find $r$, if the area of the rectangle is unaltered.

### Concept & Zero Net Change When two percentage changes result in no overall change, their net effect is 0%. We set the successive percentage change formula equal to zero. $$ x + y + \frac{x y}{100} = 0 $$ ### Step-by-Step Solution * Identify the changes: length decreases by $r$\% ($x = -r$), and breadth increases by $(r+5)$\% ($y = r+5$). * Substitute into the formula: $$ -r + (r+5) + \frac{-r(r+5)}{100} = 0 $$ * Simplify the equation: $$ 5 - \frac{r^2 + 5r}{100} = 0 $$ * Multiply through by 100 to clear the fraction: $$ 500 - (r^2 + 5r) = 0 $$ $$ r^2 + 5r - 500 = 0 $$ * Factor the quadratic equation. We need two numbers that multiply to -500 and add to 5. These are 25 and -20. $$ (r + 25)(r - 20) = 0 $$ * Since $r$ must be positive in this context (as an absolute percentage value shown in options), $r = 20$. ### Exam Strategy & Shortcut Instead of solving the quadratic, use **Option Elimination**. Test Option E ($r=20$): Length decreases by 20%, breadth increases by 25%. Net change = $-20 + 25 - \frac{20 \times 25}{100} = 5 - 5 = 0$. The area is unaltered, confirming Option E is correct immediately. ### Common Pitfall Trying to solve the algebra but making a sign error when moving terms across the equals sign, leading to the wrong roots for the quadratic equation. ### Final Answer Therefore, the correct answer is **20**.