Simple interest equals true discount: The simple interest on Rs. 750 for 2 years is equal to the true discount on Rs. 810 for the same time period and at the same rate. What is the annual rate of interest (percent per annum)?

Introduction / Context:
This problem compares two time-value measures at the same rate and period: simple interest (on a present principal) and true discount (the rebate that converts a future amount to present worth). When they are set equal, the common rate can be solved directly. Such questions are common in true-discount chapters and test formula fluency and algebraic manipulation.

Given Data / Assumptions:

• Principal for simple interest = 750.
• Future amount for true discount = 810.
• Common time = 2 years; common annual rate = r%.
• Assume simple-interest conventions; present worth PW = A / (1 + r * t), true discount TD = A − PW = A * (r * t) / (1 + r * t).

Concept / Approach:
Use SI = P * r * t / 100 and TD = A * (r * t / 100) / (1 + r * t / 100). Equate these two because the problem states they are equal for the same r and t. Solve for r as a percentage.

Step-by-Step Solution:
Simple interest on 750 for 2 years: SI = 750 * r * 2 / 100 = 15r.True discount on 810 for 2 years: TD = 810 * (2r/100) / (1 + 2r/100).Equate: 15r = [810 * (2r/100)] / (1 + 2r/100).Cancel r (r ≠ 0): 15 = 16.2 / (1 + 0.02r).Invert: 1 + 0.02r = 16.2 / 15 = 1.08 ⇒ 0.02r = 0.08 ⇒ r = 4.

Verification / Alternative check:
At r = 4%: SI = 15 * 4 = 60. Also TD = 810 * (2 * 4 / 100) / (1 + 0.08) = 810 * 0.08 / 1.08 = 64.8 / 1.08 = 60. Both match.

Why Other Options Are Wrong:

• 41/3% and 51/6% do not satisfy the equality when substituted.
• 5% makes TD ≠ SI; TD becomes 75/1.10 = 68.18…, not equal to SI = 75.

Common Pitfalls:

• Using banker's discount A * r * t instead of true discount A * r * t / (1 + r * t).
• Forgetting to express r as a percent (divide by 100) consistently.

Final Answer:
4%