Discussion
The banker’s discount on ₹1,800 at 12% per annum is equal to the true discount on ₹1,872 at the same rate and for the same time. What is the time?
Answer: A
Correct answer: 4 months
Write both in terms of the time \( T \) in years. The banker’s discount is interest on the face value: \[ \text{BD} = \frac{1800 \times 12 \times T}{100} = 216T. \] The true discount is \[ \text{TD} = \frac{1872 \times 12 \times T}{100 + 12T} = \frac{22464T}{100 + 12T}. \] Equating and cancelling \( T \), \[ 216(100 + 12T) = 22464 \;\Rightarrow\; 21600 + 2592T = 22464 \;\Rightarrow\; T = \frac{864}{2592} = \frac{1}{3}\text{ year}, \] which is 4 months. Cancelling \( T \) from both sides is the step that makes the algebra manageable; it is legitimate because \( T \) cannot be zero. The pattern - BD on one sum equalling TD on a larger one - always reduces to a linear equation in this way.
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