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

Introduction / Context:
This question gives the discount on a sum at a known rate of simple interest and asks for the time until the sum is due. The term discount here is interpreted as true discount, which is the difference between the face value and its present worth. Using the true discount formula, we can connect the discount amount, the face value, the rate and the time, and solve for the time period.

Given Data / Assumptions:
- Face value F = Rs 498
- True discount TD = Rs 18
- Rate of simple interest r = 5 percent per annum
- Time t in years is unknown and must be found
- We assume simple interest and standard definitions of true discount.

Concept / Approach:
The true discount at simple interest is given by TD = F × r × t / (100 + r × t). Here F, r and TD are known, and t is the unknown variable. We can let x = r × t to simplify the algebra, convert the equation to one in x and solve. Once we know x, we can obtain t by dividing x by r, and finally express the time in months by multiplying the time in years by 12.

Step-by-Step Solution:
Step 1: Let x = r × t so that TD = F × x / (100 + x).Step 2: Substitute known values: 18 = 498 × x / (100 + x).Step 3: Cross multiply to get 18 × (100 + x) = 498x.Step 4: Expand the left side: 1800 + 18x = 498x.Step 5: Rearrange to group x terms: 1800 = 498x − 18x = 480x.Step 6: Solve for x: x = 1800 ÷ 480 = 15 ÷ 4 = 3.75.Step 7: Since x = r × t and r = 5, we have 5t = 3.75 and t = 3.75 ÷ 5 = 0.75 year.Step 8: Convert 0.75 year to months: 0.75 × 12 = 9 months.

Verification / Alternative check:
We can verify by recomputing TD using t = 0.75 year. Then r × t = 5 × 0.75 = 3.75. True discount TD should be F × r t / (100 + r t) = 498 × 3.75 / 103.75. This simplifies numerically to approximately 1867.5 / 103.75 = Rs 18, matching the given discount. This confirms that the time until the sum is due is 0.75 year, or 9 months.

Why Other Options Are Wrong:
If the time were 8 months, t would be 2 / 3 year, giving r t = 10 / 3 and a different true discount. Similarly, 10 or 11 months would yield other values for r t and hence different discounts than Rs 18. A full year or 12 months would be longer than needed and would produce a larger discount. Only 9 months matches the given discount exactly at 5 percent simple interest.

Common Pitfalls:
Sometimes learners confuse true discount with simple interest or mistakenly use SI = P r t / 100 directly with TD as if it were the interest on the face value. Another error is to forget to include 100 + r t in the denominator of the true discount formula. Expressing the problem in terms of x = r t and solving step by step helps prevent algebraic mistakes.

Final Answer:
The sum is due after 9 months.