Discussion
The gain on abill due 1 year hence at 12% per annum is Rs.6. 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.
With \( k = \dfrac{12 \times 1}{100} = 0.12 \):
\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{6}{0.12} = \text{Rs. } 50 \]
Checking the whole chain: PW = \( 50/0.12 = 416.67 \), face value \( 466.67 \), and BD = \( 466.67 \times 0.12 = 56 \), which is indeed Rs. 6 more than the true discount.
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