The banker's discount on a certain sum due 2 years hence is eleven-tenths of the true discount. If the rate of simple interest is the same for both discounts, what is the rate percent per annum?

Introduction / Context:
This problem deals with the relationship between banker's discount and true discount at the same rate and time. We are told that the banker's discount is eleven-tenths of the true discount for a bill due 2 years hence, and we are asked to determine the rate of interest. This is a formula based question but requires understanding of how the ratio BD over TD depends on rate and time.

Given Data / Assumptions:
- Time until the bill is due t = 2 years
- Banker's discount BD is equal to 11 divided by 10 times the true discount TD, that is BD = (11 / 10) × TD
- Rate of simple interest per annum is r percent, same for both BD and TD
- We assume simple interest throughout.

Concept / Approach:
For a bill of face value F due after t years at rate r percent, true discount TD is given by TD = F × r × t / (100 + r × t), while banker's discount BD is BD = F × r × t / 100. The ratio BD / TD simplifies nicely to (100 + r × t) / 100. Here we know BD / TD as 11 / 10 and time t as 2 years, so we can set up an equation in terms of r × t and solve for r.

Step-by-Step Solution:
Step 1: Use the formula BD / TD = (100 + r × t) / 100.Step 2: Given BD = (11 / 10) TD, so BD / TD = 11 / 10.Step 3: Therefore (100 + r × t) / 100 = 11 / 10.Step 4: Cross multiply: 10 × (100 + r × t) = 11 × 100.Step 5: This gives 1000 + 10 r t = 1100, so 10 r t = 100 and r t = 10.Step 6: Time t = 2 years, so r × 2 = 10 and r = 10 ÷ 2 = 5 percent per annum.

Verification / Alternative check:
To confirm, take an assumed face value, for example F = 1000. With r = 5 percent and t = 2 years, we have r t = 10. Then BD = 1000 × 10 / 100 = 100. True discount TD = 1000 × 10 / (100 + 10) = 1000 × 10 / 110 = 10000 / 110 ≈ 90.91. The ratio BD / TD equals 100 / 90.91 which is approximately 1.1, that is 11 / 10. This matches the condition in the question, confirming that 5 percent is correct.

Why Other Options Are Wrong:
If r were 10 percent, then r t would be 20 and BD / TD would equal (100 + 20) / 100 = 1.2 which is 6 / 5, not 11 / 10. If r were 7 or 8 percent, the product r t would differ and the ratio would no longer be 11 / 10. Only r = 5 percent gives r t = 10 and exactly satisfies the required ratio.

Common Pitfalls:
Many learners try to apply direct percentage difference between BD and TD without recalling the correct formula for their ratio. Another mistake is to confuse the time as 10 years instead of interpreting r t = 10 correctly. Always express BD / TD in terms of r and t first, then substitute the given ratio and time to solve cleanly for the rate.

Final Answer:
The required rate of simple interest is 5% per annum.