The true discount on Rs. 2480 due after a certain time at 5% simple interest per annum is Rs. 80. For what time period is the amount due?

Introduction / Context:
This question applies the standard true discount formula to find the time period corresponding to a given true discount, principal, and rate of simple interest. It checks whether the student understands the relation between true discount, sum due, rate, and time, and can convert the resulting time in years into months correctly.

Given Data / Assumptions:

• Face value of the bill, S = Rs. 2480.
• True discount TD = Rs. 80.
• Rate of interest r = 5 percent per annum (simple interest).
• Time is unknown and must be found in months.
• We assume simple interest and standard true discount definitions.

Concept / Approach:
For a sum S due after time t years at simple interest rate r percent per annum, the true discount TD is given by: TD = S * r * t / (100 + r * t) Let k = r * t. We can first find k from the equation using the known true discount, then compute t = k / r. Finally we convert time in years to months by multiplying by 12.

Step-by-Step Solution:
Step 1: Write the true discount equation for S = 2480, TD = 80, r = 5. 80 = 2480 * 5 * t / (100 + 5 * t). Step 2: Set k = 5 * t, so TD = 2480 * k / (100 + k). Step 3: Substitute: 80 = 2480 * k / (100 + k). Step 4: Multiply both sides: 80 * (100 + k) = 2480 * k. 8000 + 80k = 2480k. Step 5: Move terms: 2480k − 80k = 2400k = 8000. Step 6: Solve for k: k = 8000 / 2400 = 80 / 24 = 10 / 3. Step 7: k = r * t = 5 * t so t = (10 / 3) / 5 = 10 / 15 = 2 / 3 year. Step 8: Convert to months: (2 / 3) * 12 = 8 months.

Verification / Alternative check:
We can recompute the true discount using t = 2 / 3 year. First compute r * t: r * t = 5 * 2 / 3 = 10 / 3. Then: TD = 2480 * (10 / 3) / (100 + 10 / 3). Denominator = (300 + 10) / 3 = 310 / 3. TD = 2480 * (10 / 3) * (3 / 310) = 2480 * 10 / 310 = 24800 / 310 = 80. This matches the given true discount, so the time is confirmed as 8 months.

Why Other Options Are Wrong:
Times such as 3 months, 5 months, 6 months, or 10 months do not produce a true discount of Rs. 80 when substituted into the standard formula. They correspond to different effective rate time products and would lead to different discounts, so they do not satisfy the given condition.

Common Pitfalls:
Many students mistakenly use the simple interest formula SI = P * r * t / 100 instead of the true discount formula, which places r * t in the denominator as well. Another common mistake is not converting the final time in years into months properly. Always remember that 1 year equals 12 months, and confirm your result by back-substituting into the original formula to see if you recover the given true discount exactly.

Final Answer:
The amount is due after 8 months.