The simple interest and the true discount on a certain sum for a given time and at a given rate are ₹ 85 and ₹ 80 respectively. The sum is

### Concept & SI-TD Relationship A very specific formula links the Amount (Sum due), Simple Interest (SI), and True Discount (TD) when they correspond to the same time and rate. $$ \text{Sum (Amount)} = \frac{\text{SI} \times \text{TD}}{\text{SI} - \text{TD}} $$ ### Step-by-Step Solution - Given Simple Interest (SI) = ₹ 85 - Given True Discount (TD) = ₹ 80 - Substitute these values directly into the derived relationship formula: $$ \text{Sum} = \frac{85 \times 80}{85 - 80} $$ $$ \text{Sum} = \frac{6800}{5} $$ $$ \text{Sum} = 1360 $$ ### Exam Strategy & Shortcut Memorize the formula $\text{Amount} = \frac{\text{SI} \times \text{TD}}{\text{SI} - \text{TD}}$. Without it, the derivation requires establishing $SI$ as interest on Amount and $TD$ as interest on Present Worth, creating a lengthy system of simultaneous equations that will easily burn 3-4 minutes on an exam. With the formula, it resolves in 15 seconds. ### Common Pitfall Applying the formula backward or dividing by the sum of SI and TD instead of the difference. Always remember the denominator is the difference ($\text{SI} - \text{TD}$). ### Final Answer Therefore, the correct answer is **₹ 1360**.