Discussion
The certain worth of a certain sum due sometime hence is Rs. 1600 and the true discount is Rs. 160. The banker's gain 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.
\[ k = \frac{\text{TD}}{\text{PW}} = \frac{160}{1600} = 0.1 \]
\[ \text{BG} = \text{TD} \times k = 160 \times 0.1 = \text{Rs. } 16 \]
Equivalently \( \text{BG} = \dfrac{\text{TD}^{2}}{\text{PW}} = \dfrac{160^{2}}{1600} = 16 \) - a form worth remembering, since it needs neither the rate nor the time.
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