BG as a fraction of BD (time = 2.5 years) → Rate %: The banker's gain on a certain sum due 2 1/2 years hence is 2/23 of the banker's discount on it for the same time and rate. Find the rate of interest (per annum).

Introduction / Context:This problem ties BG and BD via a proportion. Using the simple-interest identities BD = F * x and BG = F * x^2 / (1 + x) with x = r * t, we can solve for x from the ratio, then obtain r by dividing x by t.

Given Data / Assumptions:

• BG = (2/23) * BD.
• t = 2.5 years; r unknown; x = r * t.

Concept / Approach:Set F * x^2 / (1 + x) = (2/23) * F * x ⇒ cancel F * x (x ≠ 0) and solve for x. Then r = x / t.

Step-by-Step Solution:x / (1 + x) = 2 / 23 ⇒ 23x = 2 + 2x ⇒ 21x = 2 ⇒ x = 2/21.r = x / t = (2/21) / 2.5 = (2/21) * (2/5) = 4 / 105 ≈ 0.038095.Rate % ≈ 3.8095% ≈ 4%.

Verification / Alternative check:Plug r back: x = r * t ≈ 0.038095 * 2.5 ≈ 0.095238. Then BG/BD = x/(1 + x) ≈ 0.095238 / 1.095238 ≈ 0.086956 ≈ 2/23, consistent.

Why Other Options Are Wrong:5%, 6%, 8%, 3% give x values that do not satisfy BG = (2/23) * BD.

Common Pitfalls:Using BD/BG instead of BG/BD; forgetting to divide by t to get r.

Final Answer:4%