Discussion
The difference between Simple Interest and True Discount on a certain sum of money for 6 months at 12 1/2% per annum is Rs 25. Find the sum.
Answer: A
Simple interest is charged on the amount and true discount on the present worth, so with \( k = \dfrac{RT}{100} \):
\[ \text{SI} - \text{TD} = A\,k - \frac{A\,k}{1+k} = \frac{A\,k^{2}}{1+k} \]
Here \( T = 0.5 \) year at \( 12\tfrac{1}{2}\% \):
\[ k = \frac{12.5 \times 0.5}{100} = 0.0625 \]
\[ 25 = \frac{A \times 0.0625^{2}}{1.0625} = \frac{A \times 0.00390625}{1.0625} \]
\[ A = \frac{25 \times 1.0625}{0.00390625} = \text{Rs. } 6800 \]
The gap between simple interest and true discount is itself the banker's gain, which is why it depends on \( k^{2} \) and stays small for short periods.
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