Discussion
The present worth of a sum due sometime hence is Rs. 576 and the banker's gain is Rs. 16. The true discount is:
Answer: D
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.
Neither the rate nor the time is given, so use the form that needs neither:
\[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}} \]
\[ 16 = \frac{\text{TD}^{2}}{576} \quad\Longrightarrow\quad \text{TD}^{2} = 9216 \]
\[ \text{TD} = \sqrt{9216} = \text{Rs. } 96 \]
This identity follows from \( \text{BG} = \text{TD}\,k \) and \( \text{TD} = \text{PW}\,k \): substituting \( k = \text{TD}/\text{PW} \) into the first gives it directly.
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