The true discount on a bill of Rs 1600 due after a certain time at 5 percent per annum simple interest is Rs 160. After how much time, expressed in months, will the amount fall due?

Introduction / Context:
This problem belongs to the topic of true discount and present worth under simple interest. In commercial mathematics, a bill that is due in the future can be discounted today. The difference between the sum due at maturity and its present worth is called the true discount. Here we are given the true discount, the sum due and the rate of interest, and we are asked to find the time after which the bill is due, expressed in months. This is a classic reverse application of the true discount formula, which is very important for bank discount and bill of exchange questions in aptitude exams.

Given Data / Assumptions:

• Sum due at maturity S = Rs 1600.
• True discount TD = Rs 160.
• Rate of interest r = 5 percent per annum (simple interest).
• Time period t is to be found in years and then converted to months.
• Interest is calculated on simple interest basis.

Concept / Approach:
For simple interest, if a sum S is due after t years at rate r percent per annum, then the present worth PW is:
PW = S / (1 + r * t / 100) The true discount TD is the difference between S and PW:
TD = S - PW Using the above formulas we can express TD directly in terms of S, r and t as:
TD = S * r * t / (100 + r * t) Here S, r and TD are known, so we can solve this equation for t.

Step-by-Step Solution:
Step 1: Write the equation for true discount. 160 = 1600 * 5 * t / (100 + 5 * t) Step 2: Simplify the numerator. 160 = 8000 * t / (100 + 5 * t) Step 3: Multiply both sides by (100 + 5 * t). 160 * (100 + 5 * t) = 8000 * t 16000 + 800 * t = 8000 * t Step 4: Bring like terms together. 16000 = 8000 * t - 800 * t 16000 = 7200 * t t = 16000 / 7200 = 20 / 9 years Step 5: Convert years to months. t in months = (20 / 9) * 12 = 240 / 9 = 80 / 3 months 80 / 3 months is approximately 26.67 months. Rounded to the nearest whole month, the due date is after 27 months.

Verification / Alternative check:
We can quickly verify by converting 27 months into years. That is 27 / 12 = 2.25 years. Substitute t = 2.25 into the formula:
TD approximate = 1600 * 5 * 2.25 / (100 + 5 * 2.25) TD approximate = 1600 * 11.25 / (100 + 11.25) = 18000 / 111.25 This is close to 161.8, slightly higher than 160 because we rounded the exact time. When we use the exact value 20 / 9 years, we get exactly Rs 160 as shown earlier, so the reasoning is sound.

Why Other Options Are Wrong:
Option B (23 months) would give a smaller t and hence a smaller denominator, resulting in a true discount larger than Rs 160. Option C (20 months) gives even smaller time and an even larger discount. Option D (12 months) corresponds to only one year and would make the true discount far too small. Option E (30 months) gives time larger than 27 months and would produce a discount larger than the required value when calculated exactly.

Common Pitfalls:
A common mistake is to confuse true discount with simple interest. True discount is calculated on the sum due, but the formula includes the present worth indirectly through the denominator 100 plus r times t. Another frequent error is to mix months and years without proper conversion. Always convert time to years when applying the formula with annual rate and then convert back to months only at the end. Careful algebraic manipulation avoids errors when solving for t.

Final Answer:
The bill is due after approximately 27 months to produce a true discount of Rs 160 at 5 percent per annum simple interest.