Find the rate percent from mixed BD and TD data: The banker’s discount on a sum for 1.5 years is ₹60, and the true discount on the same sum for 2 years is ₹75. Determine the annual simple interest rate.
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A5%
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B6%
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C62 / 3%
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D31 / 3%
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ENone of these
Answer
Correct Answer: 31 / 3%
Explanation
Introduction / Context:This problem combines banker’s discount (computed on the face value) and true discount (computed via present worth) for different times on the same sum to identify the simple interest rate r. It showcases how BD and TD connect through r and t.
Given Data / Assumptions:
- BD for t1 = 1.5 years is ₹60
- TD for t2 = 2 years is ₹75
- Same face value A and same annual rate r
Concept / Approach:Formulas:
BD = A * r * t TD = A * (r * t) / (1 + r * t)Step-by-Step Solution:
From BD: 60 = A * r * 1.5 ⇒ A r = 40 From TD: 75 = A * (2r) / (1 + 2r) Substitute A(2r) = 80 ⇒ 75 = 80 / (1 + 2r) 1 + 2r = 80/75 = 16/15 ⇒ 2r = 1/15 ⇒ r = 1/30 = 3 1/3 %Verification / Alternative check:Check BD at 1.5 y: A r = 40, BD = 40*1.5 = 60 (ok). TD at 2 y: numerator 80, denominator 1 + 2r = 16/15, TD = 80 / (16/15) = 75 (ok).
Why Other Options Are Wrong:Other rates do not satisfy both BD and TD equations simultaneously with the same A.
Common Pitfalls:Forgetting the (1 + r t) divisor in TD or mixing BD and TD definitions. Keep r as a per-year fraction throughout.
Final Answer:31 / 3%