Discussion
What is the true discount if a banker’s discount on a bill due 4 months hence is 420 at 15%?
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 \( T = \dfrac{4}{12} = \dfrac{1}{3} \) year at 15%:
\[ k = \frac{15 \times 1/3}{100} = 0.05 \]
Since \( \text{BD} = A\,k \) and \( \text{TD} = \dfrac{A\,k}{1+k} \), the two differ only by the factor \( 1+k \):
\[ \text{TD} = \frac{\text{BD}}{1+k} = \frac{420}{1.05} = \text{Rs. } 400 \]
The face value is \( 420/0.05 = \text{Rs. } 8400 \), and the banker's gain is Rs. 20 - but neither is needed once the \( 1+k \) relation is used.
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