The true discount on Rs. 1760 due after a certain time at 12% simple interest per annum is Rs. 160. After how much time (in months) is this sum due?

Introduction / Context:
The question gives the true discount on a sum at a known simple interest rate and asks for the time after which the sum is due. It is a direct application of the true discount formula and requires conversion of time in years to months once the result is obtained.

Given Data / Assumptions:

• Sum due, S = Rs. 1760.
• True discount TD = Rs. 160.
• Rate of interest r = 12 percent per annum.
• Time t is unknown, measured in years first, and then converted to months.
• Interest is simple, and the standard true discount relationship holds.

Concept / Approach:
True discount TD for a sum S due after time t years at rate r percent simple interest is: TD = S * r * t / (100 + r * t) Let k = r * t. Then TD = S * k / (100 + k). We know TD, S, and r, so we can solve for k and then find t = k / r. Finally we convert t from years to months by multiplying by 12.

Step-by-Step Solution:
Step 1: Let k = r * t. For r = 12, k = 12 * t. Step 2: Write TD equation: 160 = 1760 * k / (100 + k). Step 3: Multiply both sides: 160 * (100 + k) = 1760 * k. 16000 + 160k = 1760k. Step 4: Move terms: 1760k − 160k = 1600k = 16000. Step 5: Solve: k = 16000 / 1600 = 10. Step 6: Since k = r * t = 12 * t, we have 12 * t = 10. Step 7: t = 10 / 12 year = 5 / 6 year. Step 8: Convert to months: (5 / 6) * 12 = 10 months.

Verification / Alternative check:
We can verify by computing present worth PW and recalculating TD. With S = 1760 and k = 10: PW = S * 100 / (100 + k) = 1760 * 100 / 110 = 176000 / 110 = 1600. TD = S − PW = 1760 − 1600 = 160. This matches the given true discount, confirming that the time is indeed 10 months.

Why Other Options Are Wrong:
Times such as 5, 8, 12, or 15 months do not produce the same effective rate time product when substituted into the true discount formula. They therefore yield different true discounts, not equal to Rs. 160, and do not satisfy the condition in the question.

Common Pitfalls:
Some candidates mistakenly use the simple interest formula SI = S * r * t / 100 instead of the true discount formula and equate SI to Rs. 160, which is incorrect. Others forget to add r * t to 100 in the denominator or misconvert years to months. To avoid errors, always compute k = r * t carefully, solve for k using the given discount, and then use t = k / r followed by time unit conversion.

Final Answer:
The sum is due after 10 months.