Difficulty: Medium
Correct Answer: 4 months
Explanation:
Introduction:
This problem compares a banker's discount on one sum with the true discount on another sum, at the same rate and for the same time. Using the formulas for BD and TD, we can form an equation and solve for the time period t.
Given Data / Assumptions:
Sum 1 (for BD) P1 = Rs 1800. Sum 2 (for TD) P2 = Rs 1872. Rate r = 12% per annum. Time t = ? years (same for both). Banker's discount on P1 equals true discount on P2.
Concept / Approach:
For the first sum P1: BD = P1 * r * t / 100. For the second sum P2: TD = P2 * r * t / (100 + r * t). Given that BD = TD, we set these expressions equal and solve for t. Once t (in years) is found, we convert it into months.
Step-by-Step Solution:
Step 1: Write the equality. P1 * r * t / 100 = P2 * r * t / (100 + r * t). For r = 12%, P1 = 1800, P2 = 1872. 1800 * 12 * t / 100 = 1872 * 12 * t / (100 + 12 * t). Simplify left side: 1800 * 12 / 100 = 216. So 216 * t = 1872 * 12 * t / (100 + 12 * t). Cancel t (t ≠ 0): 216 = 1872 * 12 / (100 + 12 * t). Compute numerator: 1872 * 12 = 22464. So 216 * (100 + 12 * t) = 22464. 100 + 12 * t = 22464 / 216 = 104. 12 * t = 104 − 100 = 4 ⇒ t = 4 / 12 = 1/3 year. Convert to months: (1/3) year = 4 months.
Verification / Alternative check:
Substitute t = 1/3 into both expressions: BD = 1800 * 12 * (1/3) / 100 = 1800 * 4 / 100 = Rs 72. TD = 1872 * 12 * (1/3) / (100 + 12 * (1/3)) = 1872 * 4 / (100 + 4) = 7488 / 104 = Rs 72. Since BD = TD = Rs 72, the time t = 4 months is validated.
Why Other Options Are Wrong:
7 months, 6 months, 3 months, 5 months: None of these values, when converted to years and substituted into the formulas, make BD equal TD.
Common Pitfalls:
Students often forget to cancel t correctly or mis-handle the algebra when solving for t. Another mistake is to assume that the time must be a whole year, which is not necessary; here it is a fraction of a year, corresponding to 4 months.
Final Answer:
The required time period is 4 months.
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