Find rate from true discount in months: The true discount on Rs. 2575 due 4 months hence is Rs. 75. What is the annual simple-interest rate (percent)?

Introduction / Context:
True discount connects the future amount A, the rate r, and the time t via TD = A * (r * t) / (1 + r * t). With TD, A, and t known, solve for r. Remember to convert months to years and use simple-interest conventions.

Given Data / Assumptions:

• A = 2575.
• TD = 75.
• t = 4 months = 1/3 year.

Concept / Approach:
Let y = r * t (r as percent per annum). In fractional form, use r as a percentage number and divide by 100 when forming y. Solve for y from TD = A * y / (1 + y). Then obtain r from y = (r/100) * t.

Step-by-Step Solution:
75 = 2575 * y / (1 + y).75 (1 + y) = 2575 y ⇒ 75 = (2575 − 75) y = 2500 y ⇒ y = 75 / 2500 = 0.03.But y = (r/100) * (1/3) ⇒ r/300 = 0.03 ⇒ r = 9.

Verification / Alternative check:
Check TD at r = 9%: y = 0.09 * (1/3) = 0.03. TD = 2575 * 0.03 / 1.03 = 77.25 / 1.03 = 75 exactly.

Why Other Options Are Wrong:

• 5%, 6%, 8% yield TD values different from 75 when substituted.

Common Pitfalls:

• Mistreating months as 4/10 of a year instead of 4/12.
• Using banker’s discount formula A * r * t, which omits the 1 + r * t denominator.

Final Answer:
9%