Discussion
The banker’s discount and the true discount on a certain sum are ₹42 and ₹40 respectively. What is the sum due?
Answer: D
Correct answer: ₹840
The face value is given by \[ A = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}}. \] Substituting, \[ A = \frac{42 \times 40}{42 - 40} = \frac{1680}{2} = 840. \] The formula comes from the definitions. BD is interest on \( A \) and TD is interest on \( \text{PW} = A - \text{TD} \), so \[ \frac{\text{BD}}{\text{TD}} = \frac{A}{A - \text{TD}}, \] and rearranging gives the result. The denominator \( \text{BD} - \text{TD} \) is the banker’s gain, here 2, so the formula can also be read as \( A = \text{BD} \times \text{TD} / \text{BG} \). As a check, the present worth is \( 840 - 40 = 800 \), and \( 40/800 = 42/840 \), confirming that both discounts are computed at the same rate and time.
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