The true discount on a bill of $5450 due in 9 months is $450. Find the annual rate of interest (per annum, simple interest).

Introduction / Context: We determine the rate from a known true discount over a given period. The time is in months and must be converted to years.

Given Data / Assumptions:

• F = 5450
• TD = 450
• t = 9 months = 0.75 years

Concept / Approach: With x = r * t, TD/F = x/(1 + x) → x = TD/(F − TD). Then r = x/t.

Step-by-Step Solution: x = TD/(F − TD) = 450/(5450 − 450) = 450/5000 = 0.09. t = 0.75 → r = x/t = 0.09/0.75 = 0.12 = 12% p.a.

Verification / Alternative check: Present worth P = 5450/(1 + 0.09) = 5000; TD = 450, consistent; and r * t = 0.12 * 0.75 = 0.09.

Why Other Options Are Wrong: 12.5%, 11.5%, 13.1%, 10% do not satisfy the exact relation between TD and F for the given time.

Common Pitfalls: Forgetting to convert months into years; mixing up TD with SI on face value; or solving for r without first finding x.

Final Answer: 12%