From PW and true discount, find banker’s discount and banker’s gain: The present worth (PW) of a bill due sometime hence is Rs 1600, and its true discount (TD) is Rs 160. Find the banker’s discount (BD) and the extra gain (banker’s gain, BG).
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ARs 176, Rs 18
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BRs 186, Rs 16
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CRs 176, Rs 16
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DNone of these
Answer
Correct Answer: Rs 176, Rs 16
Explanation
Introduction / Context:Relations among present worth (PW), true discount (TD), banker’s discount (BD), and banker’s gain (BG) under simple interest are standard results used in banker’s discount problems.
Given Data / Assumptions:
- PW = Rs 1600.
- TD = Rs 160.
- Face value S = PW + TD = 1760.
Concept / Approach:Let x = r * t (rate × time). Then TD = S * x / (1 + x) and BD = S * x. Also BG = BD − TD = S * x^2 / (1 + x). We can find x from PW and TD without knowing r or t individually.
Step-by-Step Solution:
Since PW = S / (1 + x), we have 1600 = 1760 / (1 + x) → 1 + x = 1760 / 1600 = 1.1 → x = 0.1.BD = S * x = 1760 * 0.1 = Rs 176.BG = BD − TD = 176 − 160 = Rs 16.Verification / Alternative check:TD = S * x / (1 + x) = 1760 * 0.1 / 1.1 = 160 ✔️; BG = S * x^2 / (1 + x) = 1760 * 0.01 / 1.1 = 16 ✔️.
Why Other Options Are Wrong:Rs 176, Rs 18 gives a BG inconsistent with the identity BG = BD − TD; Rs 186, Rs 16 implies x > 0.1 and contradicts PW.
Common Pitfalls:Using BD = TD or mixing up PW with face value; forgetting that x = r * t and cancels in ratios.
Final Answer:Banker’s Discount = Rs 176; Banker’s Gain = Rs 16