Find the annual rate from a BG/BD ratio: For a sum due in 5 years, the banker’s gain (BG) is 3/23 of the banker’s discount (BD). Find the annual rate of interest (simple discounting convention).

Introduction / Context:
There is a standard relation between BG and BD for banker’s discount at rate d% per annum for time t years. Using BG/BD in terms of the single variable x = d t / 100 allows us to solve for x and then recover d since t is given. This avoids explicitly computing the face value or present worth.

Given Data / Assumptions:

• t = 5 years.
• BG = (3/23) * BD.
• Banker’s discount framework: BD = S * d * t / 100; TD = S * (d t/100)/(1 + d t/100); BG = BD − TD.

Concept / Approach:
We have BG/BD = (x/(1 + x)), where x = d t / 100. Set x/(1 + x) = 3/23 and solve for x, then back out d = 100x/t.

Step-by-Step Solution:

Verification / Alternative check:
Plug x = 0.15 into BG/BD: 0.15/1.15 = 3/23 (since 0.15/1.15 = 15/115 = 3/23). Checks out.

Why Other Options Are Wrong:

• 6%, 5%, 4%, 2%: Do not satisfy the specified BG-to-BD ratio at t = 5 years.

Common Pitfalls:
Using BD/BG instead of BG/BD or forgetting to divide by t when converting x back to the annual rate.

Final Answer:
3%