Discussion
The banker’s gain of a certain sum due 3 years hence at 10% per annum is Rs. 27. The present worth is:
Answer: C
Four quantities are related throughout this topic. Writing \( k = \dfrac{RT}{100} \):
\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]
The banker's discount is interest on the face value; the true discount is interest on the present worth; the banker's gain is the interest on the true discount.
With \( k = \dfrac{10 \times 3}{100} = 0.3 \):
\[ \text{TD} = \frac{\text{BG}}{k} = \frac{27}{0.3} = \text{Rs. } 90 \]
\[ \text{PW} = \frac{\text{TD}}{k} = \frac{90}{0.3} = \text{Rs. } 300 \]
Two divisions by the same \( k \), because the banker's gain is interest on the true discount just as the true discount is interest on the present worth.
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