Find time from true discount: The true discount on Rs. 1860 at 5% per annum simple interest is Rs. 60. After how much time will it be due?

Introduction / Context:
True discount TD = A * (r * t) / (1 + r * t) relates the due amount A, annual rate r, and time t (in years). With TD, A, r given, solve for t. Finally convert t (in years) to months.

Given Data / Assumptions:

• A = 1860.
• TD = 60.
• r = 5% p.a. simple.

Concept / Approach:
Let x = r * t (with r as a fraction per annum). Then TD = A * x / (1 + x). Solve for x; then find t = x / r.

Step-by-Step Solution:
60 = 1860 * x / (1 + x).60 (1 + x) = 1860 x ⇒ 60 = (1860 − 60) x = 1800 x ⇒ x = 60 / 1800 = 1/30.But x = r * t = 0.05 * t ⇒ 0.05 t = 1/30 ⇒ t = (1/30) / 0.05 = (1/30) / (1/20) = 20/30 = 2/3 year.t = 2/3 year = 8 months.

Verification / Alternative check:
Compute TD at t = 8 months: r * t = 0.05 * 2/3 = 1/30. TD = 1860 * (1/30) / (1 + 1/30) = 62 / (31/30) = 62 * 30 / 31 = 60, confirming.

Why Other Options Are Wrong:

• 6, 9, 10 months produce TD values different from 60 at 5%.

Common Pitfalls:

• Treating TD as A * r * t (banker’s discount) and missing the denominator term.
• Forgetting to convert years to months properly.

Final Answer:
8 months