In simple interest and true discount problems: The true discount (TD) on a bill of $161 due after 2 years and 6 months (i.e., 2.5 years) is $21. Calculate the annual rate of interest (per annum) at which this true discount is computed.

Introduction / Context:
This problem tests the connection between True Discount (TD), present worth, and the simple interest rate for a given time. We use the standard TD relation with simple interest to find the annual rate.

Given Data / Assumptions:

• Face value F = 161
• True discount TD = 21
• Time t = 2.5 years
• Simple interest framework; TD is computed at the same rate r per annum.

Concept / Approach:
For simple interest, TD = F * (r * t) / (1 + r * t). Let x = r * t. Then TD/F = x/(1 + x). Solve for x, then r = x/t.

Step-by-Step Solution:
Let x = r * t. TD/F = x/(1 + x) = 21/161. Cross-multiply: 21(1 + x) = 161x. 21 + 21x = 161x → 21 = 140x → x = 21/140 = 0.15. Since x = r * t and t = 2.5, r = x / t = 0.15 / 2.5 = 0.06 = 6%.

Verification / Alternative check:
Compute present worth P = F/(1 + r*t) = 161/(1 + 0.06*2.5) = 161/1.15 = 140. TD = F - P = 161 - 140 = 21, which matches the given TD.

Why Other Options Are Wrong:
4% and 5% give smaller x and thus smaller TD than 21; 8% and 10% give larger x, producing TD greater than 21. Only 6% satisfies the relation exactly.

Common Pitfalls:
Confusing TD with simple interest on face value; forgetting the denominator (1 + r*t); or mixing years and months without converting to years.

Final Answer:
6%