The Banker's discount on a certain sum due after 2 years is $\frac{11}{10}$ of the actual discount. The rate percentage is:

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to understand the relationship between the banker's discount and the actual discount, and how this relates to the rate of interest.

### Explanation:

1. **Definitions:**
   - **Banker's Discount (BD):** The interest on the face value of a bill for the period from the date of discounting to the due date.
   - **Actual Discount (AD):** The difference between the face value and the present value of the bill.

2. **Given Condition:**
   - The banker's discount is $\frac{11}{10}$ of the actual discount.

3. **Formulae:**
   - Let $ F $ be the face value, $ P $ be the present value, $ r $ be the rate of interest, and $ t $ be the time period in years.
   - Banker's Discount, $ BD = F \times r \times t $.
   - Actual Discount, $ AD = (F - P) = F - \frac{F}{1 + rt} $.

4. **Relationship:**
   - Given $ BD = \frac{11}{10} \times AD $.

5. **Derivation:**
   - Substitute the expressions for BD and AD:
     $$
     F \times r \times t = \frac{11}{10} \times \left( F - \frac{F}{1 + rt} \right)
     $$
   - Simplify the equation:
     $$
     F \times r \times t = \frac{11}{10} \times \left( \frac{F \times rt}{1 + rt} \right)
     $$
   - Cancel $ F $ from both sides (assuming $ F \neq 0 $):
     $$
     r \times t = \frac{11}{10} \times \frac{rt}{1 + rt}
     $$
   - Simplify further:
     $$
     1 + rt = \frac{11}{10}
     $$
   - Solve for $ rt $:
     $$
     rt = \frac{11}{10} - 1 = \frac{1}{10}
     $$
   - Since $ t = 2 $ years, we have:
     $$
     r \times 2 = \frac{1}{10}
     $$
   - Therefore, $ r = \frac{1}{20} = 0.05 $.

6. **Conclusion:**
   - The rate percentage is $ 0.05 $, which corresponds to Option 3.

### Explanation of Incorrect Options:

- **Option 1 (0.11):** This would imply a much higher rate than derived from the given condition. The calculations show that the rate is $ 0.05 $, not $ 0.11 $.

- **Option 2 (0.1):** This rate would suggest that the banker's discount is equal to the actual discount, which contradicts the given condition that the banker's discount is $\frac{11}{10}$ of the actual discount.

- **Option 4 ($5\frac{1}{3}\%$):** This is equivalent to $0.0533$, which is not consistent with the derived rate of $0.05$.

Thus, based on the calculations and logical reasoning, Option 3 is indeed the correct answer.