Difficulty: Medium
Correct Answer: Sum = Rs 400 and Time = 2.5 years
Explanation:
Introduction:
This question gives both the banker's discount (BD) and the true discount (TD) at a known rate and asks us to determine the face value of the sum and the time period. It nicely illustrates how the ratio BD / TD relates to the product r * t in banker's discount theory.
Given Data / Assumptions:
Banker's discount BD = Rs 100. True discount TD = Rs 80. Rate r = 10% per annum. Time t = ? years. Face value (sum) P = ?.
Concept / Approach:
For face value P, rate r and time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). Hence: BD / TD = (100 + r * t) / 100. We can use BD / TD to find r * t, then use BD to find P. Finally we solve for t using r.
Step-by-Step Solution:
Step 1: Use BD / TD. BD / TD = 100 / 80 = 5 / 4. But BD / TD = (100 + r * t) / 100. (100 + r * t) / 100 = 5 / 4. 100 + r * t = 100 * (5 / 4) = 125. r * t = 125 − 100 = 25. Step 2: Since r = 10, we get 10 * t = 25 ⇒ t = 25 / 10 = 2.5 years. Step 3: Use BD to find P. BD = P * r * t / 100 ⇒ 100 = P * 10 * 2.5 / 100. 100 = P * 25 / 100 ⇒ P = 100 * 100 / 25 = Rs 400.
Verification / Alternative check:
Check TD with P = 400, r = 10, t = 2.5: TD = P * r * t / (100 + r * t) = 400 * 25 / (100 + 25) = 10000 / 125 = Rs 80. BD with P = 400: BD = 400 * 25 / 100 = Rs 100. Both match the given values exactly.
Why Other Options Are Wrong:
Sum = Rs 400 and Time = 5 years: Time is too long; r * t would be 50, not 25. Sum = Rs 200 and Time = 2.5 years: BD and TD would be half of the required amounts. Sum = Rs 200 and Time = 5 years: Neither BD nor TD matches the given values. Sum = Rs 500 and Time = 2 years: Gives a different pair of BD and TD entirely.
Common Pitfalls:
Mistakes include adding or subtracting BD and TD in arbitrary ways or trying to guess P and t. The clean route is always via the ratio BD / TD and the formulas that connect them to r * t.
Final Answer:
The correct combination is Sum = Rs 400 and Time = 2.5 years.
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