₹ 20 is the true discount on ₹ 260 due after a certain time. What will be the true discount on the same sum due after half of the former time, the rate of interest being the same? (a) ₹ 15.20 (b) ₹ 10.40 (c) ₹ 10.83 (d) ₹ 13

### Concept & Time Scaling of True Discount True discount is calculated on the Present Worth. When the time frame is halved, the present worth changes, making the scaling non-linear. We extract the Rate $\times$ Time ($RT$) value to solve for the new state. $$ \text{TD}_2 = \frac{\text{Amount} \times (RT/2)}{100 + (RT/2)} $$ ### Step-by-Step Solution - **Initial State:** Amount = ₹ 260, True Discount = ₹ 20. Present Worth ($\text{PW}_1$) = $260 - 20 = 240$. Calculate the $RT$ product from the $\text{TD}_1$ interest equation: $$ 240 \times \frac{R \times T}{100} = 20 \Rightarrow RT = \frac{2000}{240} = \frac{25}{3} $$ - **Halved Time State:** New time factor is $T/2$. Thus the new multiplier is $R \times \frac{T}{2} = \frac{1}{2} \times \frac{25}{3} = \frac{25}{6}$. Substitute into the $\text{TD}_2$ formula: $$ \text{TD}_2 = \frac{260 \times (\frac{25}{6})}{100 + \frac{25}{6}} = \frac{260 \times \frac{25}{6}}{\frac{625}{6}} $$ $$ \text{TD}_2 = \frac{260 \times 25}{625} = \frac{260}{25} = 10.40 $$ ### Exam Strategy & Shortcut Recognize that true discount behaves inversely to present worth. If time is halved, simple interest halves (from 20 to 10), but true discount remains slightly higher than half because the new present worth (base) is larger. $10.40$ is the only logical slight increase over 10. ### Common Pitfall A trap answer is ₹ 10, assuming TD scales linearly like simple interest. ### Final Answer Therefore, the correct answer is **(b) ₹ 10.40**.