Discussion
The banker’s gain on a bill due 1 year hence at 12% per annum is ₹6. What is the true discount?
Answer: C
Correct answer: ₹50
The banker’s gain is the simple interest on the true discount, so \[ \text{BG} = \frac{\text{TD} \times R \times T}{100}. \] Substituting, \[ 6 = \frac{\text{TD} \times 12 \times 1}{100} = 0.12\,\text{TD} \;\Rightarrow\; \text{TD} = \frac{600}{12} = 50. \] That identity is the most useful single fact in this topic. It follows from BD being interest on the face value and TD being interest on the present worth: the difference in principal is exactly TD, so the difference in interest is the interest on TD. From it the rest follows here. The present worth is \( \text{TD}^{2}/\text{BG} = 2500/6 \approx 416.67 \), the face value is about 466.67, and the banker’s discount is \( 50 + 6 = 56 \). Note that BG is always the smallest of the three quantities.
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