The banker's discount and the true discount of a sum at 10% per annum simple interest for the same time are Rs 100 and Rs 80 respectively. What are the sum and the time period?

Introduction / Context:
This question combines all three main quantities in banker's discount problems: banker's discount, true discount and the underlying sum with its time period. We know BD and TD at a given rate and for the same time. From their ratio we can find the product of rate and time, and from BD we can then obtain the face value of the bill. Finally, we deduce the time in years from the product r t.

Given Data / Assumptions:
- Banker's discount BD = Rs 100
- True discount TD = Rs 80
- Rate of simple interest r = 10 percent per annum
- Time t is the same for BD and TD and must be found
- Face value of the bill is the sum to be found

Concept / Approach:
For a bill of face value F, banker's discount is BD = F × r × t / 100 and true discount is TD = F × r × t / (100 + r × t). Their ratio BD / TD simplifies to (100 + r t) / 100. Hence BD / TD allows us to find r t without knowing F. Once r t is known, we can use BD to find F. Then, r and t are related by r t, so we can solve for t using the given rate of 10 percent.

Step-by-Step Solution:
Step 1: Compute BD / TD = 100 ÷ 80 = 5 / 4.Step 2: Use formula BD / TD = (100 + r t) / 100.Step 3: Therefore (100 + r t) / 100 = 5 / 4, so 100 + r t = 125.Step 4: Thus r t = 125 − 100 = 25.Step 5: Since r = 10 percent, 10 × t = 25 and t = 25 ÷ 10 = 2.5 years.Step 6: Now use BD = F × r × t / 100 with BD = 100, r = 10, t = 2.5.Step 7: So 100 = F × 10 × 2.5 / 100 = F × 25 / 100, which gives F = 100 × 100 / 25 = Rs 400.

Verification / Alternative check:
We can confirm by recomputing TD from F and t. With F = 400, r = 10 and t = 2.5, we have r t = 25. True discount TD = F × r t / (100 + r t) = 400 × 25 / 125 = 400 × 1 / 5 = Rs 80, which matches the given TD. Banker's discount BD = F × r t / 100 = 400 × 25 / 100 = Rs 100, which matches the given BD. This consistency across both values confirms that F = 400 and t = 2.5 years is correct.

Why Other Options Are Wrong:
If the sum were Rs 200 or Rs 600 with any of the listed times, substituting into the formulas for BD and TD would not reproduce BD = 100 and TD = 80 at 10 percent. For example, with sum Rs 200 and t = 5 years, r t would be 50, not 25, and both discounts would be different. Only the pair sum Rs 400 and time 2.5 years fits all the given conditions exactly.

Common Pitfalls:
Some learners attempt to guess the time directly or mistakenly use BD − TD instead of BD / TD to find r t. Others forget to use the combined factor r t and try to solve for rate and time independently even though rate is already given. Treating r t as a single variable first simplifies the algebra and reduces errors.

Final Answer:
The sum is Rs. 400 and the time period is 2.5 years.