Discussion
The present worth of a bill due some time hence is ₹1,100 and the banker’s gain is ₹11. What is the true discount?
Answer: C
Correct answer: ₹110
The banker’s gain relates to the present worth and the true discount by \[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}}. \] Substituting, \[ 11 = \frac{\text{TD}^{2}}{1100} \;\Rightarrow\; \text{TD}^{2} = 12100 \;\Rightarrow\; \text{TD} = 110. \] The relation is derived by combining two facts: BG is the simple interest on TD, and TD is the simple interest on PW. Writing \( \text{TD} = \text{PW} \cdot \frac{RT}{100} \) and \( \text{BG} = \text{TD} \cdot \frac{RT}{100} \), and eliminating \( \frac{RT}{100} \) between them, gives \( \text{BG} = \text{TD}^{2}/\text{PW} \) directly. Equivalently, PW, TD and BG are in geometric progression, each being the previous one multiplied by \( RT/100 \) - here by \( 1/10 \), since \( 1100, 110, 11 \). That observation often solves such problems by inspection.
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