Discussion
The banker's discount on Rs.1600 at 15% per annum is the same as true discount on Rs.1680 for the same time and at the same rate. The time is?
Answer: B
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.
Let \( k \) be the rate–time product. The banker's discount on Rs. 1600 equals the true discount on Rs. 1680:
\[ 1600\,k = \frac{1680\,k}{1+k} \]
Cancelling \( k \):
\[ 1600(1+k) = 1680 \quad\Longrightarrow\quad 1+k = 1.05 \quad\Longrightarrow\quad k = 0.05 \]
\[ \frac{15\,T}{100} = 0.05 \quad\Longrightarrow\quad T = \frac{1}{3} \text{ year} = 4 \text{ months} \]
The rate enters only at the final step; everything before it depends on the ratio \( 1680/1600 \) alone.
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