Banker’s gain to banker’s discount relation: On a sum due in 3 years at 10% per annum (simple discounting convention), the banker’s gain (BG) is $180. Find the banker’s discount (BD).
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A$ 680
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B$ 780
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C$ 580
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D$ 480
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ENone of these
Answer
Correct Answer: $ 780
Explanation
Introduction / Context:For a face value S, time t, and discount rate d% (per annum), banker’s discount (BD) is S * d * t / 100. True discount (TD) equals S − S/(1 + d t/100). Banker’s gain (BG) is BD − TD. Given BG, we can solve for S and then find BD straightforwardly.
Given Data / Assumptions:
- t = 3 years, d = 10% per annum.
- BG = $180.
Concept / Approach:Compute TD in terms of S: TD = S − S/(1 + 0.10 * 3) = S − S/1.3 = S * (0.3/1.3). BD = 0.3S. Then BG = BD − TD = 0.3S − S*(0.3/1.3). Solve for S, then find BD = 0.3S.
Step-by-Step Solution:
TD = S*(0.3/1.3) ≈ 0.230769 S.BD = 0.3 S.BG = BD − TD = (0.3 − 0.230769) S = 0.0692308 S.Set BG = 180 ⇒ S = 180 / 0.0692308 ≈ 2600.BD = 0.3 S = 0.3 * 2600 = $780.Verification / Alternative check:TD ≈ 2600*(0.3/1.3) ≈ $600. BG = 780 − 600 = $180, consistent.
Why Other Options Are Wrong:
- $680, $580, $480: Do not satisfy BG = BD − TD with d = 10% and t = 3 years.
Common Pitfalls:Confusing TD formula with BD. TD is based on present-worth relation; BD is simple-interest on face value.
Final Answer:$ 780