Ratio and Proportion

Concept:The problem uses total budget fixed, so daily expense changes inversely with number of days.Explanation:Let the original tour duration be $x$ days.Original daily expense = $\frac{9600}{x}$.After extending by 16 days, new duration = $x+16$ days.New daily expense = $\frac{9600}{x+16}$.The expense is reduced by ₹20 per day:$\frac{9600}{x} - \frac{9600}{x+16} = 20$.Factor out $9600$: $9600\left(\frac{1}{x} - \frac{1}{x+16}\right) = 20$.Combine fractions: $9600 \cdot \frac{16}{x(x+16)} = 20$.Multiply both sides by $x(x+16)$: $9600 \times 16 = 20x(x+16)$.Divide by 20: $480 \times 16 = x(x+16)$ ⟹ $7680 = x^2 + 16x$.Rearrange: $x^2 + 16x - 7680 = 0$.Solve quadratic: discriminant $= 16^2 + 4(7680) = 256 + 30720 = 30976$.$\sqrt{30976} = 176$.$x = \frac{-16 \pm 176}{2}$.Positive root: $\frac{160}{2} = 80$.Thus original duration is 80 days.Answer:Original duration = 80 days, which matches option C.