If the discount (true discount) on Rs 498 at 5% simple interest is Rs 18, after how long (in months) is the sum due?

Introduction:
This question gives the true discount on a sum at a known rate and asks for the time until the sum is due. Recognizing that the term “discount” here represents true discount, we can use the true discount formula to solve for the product r * t and thus find t in months.

Given Data / Assumptions:
Sum (face value) P = Rs 498. True discount TD = Rs 18. Rate r = 5% per annum (simple interest). Time t = ? years (we will later convert to months).

Concept / Approach:
For true discount: TD = P * r * t / (100 + r * t). Let x = r * t. Then: TD = P * x / (100 + x). We know TD, P, and r, so we can solve for x, then find t = x / r, and finally convert t to months.

Step-by-Step Solution:
Step 1: Set up the equation with x = r * t. 18 = 498 * x / (100 + x). 18(100 + x) = 498x. 1800 + 18x = 498x. 1800 = 498x − 18x = 480x. x = 1800 / 480 = 15 / 4 = 3.75. So r * t = x = 3.75. Step 2: Find t using r = 5. r * t = 3.75 ⇒ 5 * t = 3.75. t = 3.75 / 5 = 0.75 years. Step 3: Convert years to months. 0.75 years = 0.75 * 12 = 9 months.

Verification / Alternative check:
Recompute TD using t = 0.75 years: r * t = 5 * 0.75 = 3.75. TD = 498 * 3.75 / (100 + 3.75) = 1867.5 / 103.75 = Rs 18 (approximately), which matches the given discount.

Why Other Options Are Wrong:
8, 10, 11, or 6 months: Each corresponds to a different t in years, and none give TD exactly equal to Rs 18 when substituted in the true discount formula with P = 498 and r = 5%.

Common Pitfalls:
A major pitfall is to treat the discount as simple interest on the present worth or to apply the simple interest formula P * r * t / 100 directly to the face value, which gives the banker's discount, not the true discount. Using the correct TD formula is essential for the right answer.

Final Answer:
The sum is due after 9 months.