Find time from true discount: At 5% simple interest, the true discount on a sum of $249 is $9. After how long is the sum due?

Introduction / Context:
True discount TD relates the sum due S, rate r, and time t (years) via TD = S * (r t/100) / (1 + r t/100). This can be inverted to find t given TD, S, and r. It is often quicker to cross-multiply and solve for r t first, then convert to months.

Given Data / Assumptions:

• S = $249.
• TD = $9.
• r = 5% per annum (simple interest).

Concept / Approach:
Start from TD = S * (r t/100)/(1 + r t/100). Multiply both sides to remove the denominator and solve the linear equation in t. Finally, convert years to months.

Step-by-Step Solution:

Verification / Alternative check:
Compute PW = S − TD = 249 − 9 = 240. Check: PW * (1 + r t) = 240 * (1 + 0.05 * 0.75) = 240 * 1.0375 = 249, correct.

Why Other Options Are Wrong:

• 6, 4, 7 months: Do not satisfy the TD equation at 5%.

Common Pitfalls:
Using TD = S * r t/100 (which is banker’s discount), not true discount. Always include the denominator term 1 + r t/100 for TD.

Final Answer:
9 months