Discussion
The banker’s gain of a certain sum due 2 years hence at 5% per annum is Rs. 80. The present worth is ?
Answer: A
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 \( k = \dfrac{5 \times 2}{100} = 0.1 \):
\[ \text{TD} = \frac{\text{BG}}{k} = \frac{80}{0.1} = \text{Rs. } 800 \]
\[ \text{PW} = \frac{\text{TD}}{k} = \frac{800}{0.1} = \text{Rs. } 8000 \]
Rs. 800 is only the true discount, one step short of the answer. As a check, the face value is \( 8000 + 800 = 8800 \) and the banker's discount is \( 8800 \times 0.1 = 880 \), exactly Rs. 80 above the true discount.
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