A trader owes a merchant Rs. 10028 due 1 year hence. The trader wants to settle the account after 3 months. If the rate of interest is 12% per annum simple interest, how much cash should he pay now?

Introduction / Context:
This question asks for the present cash payment required to settle a debt earlier than its due date under simple interest conditions. The trader owes a certain amount 1 year later but wants to close the account after 3 months. Effectively, we need to find the present worth at the earlier date using the concept of true discount with the given annual rate of interest.

Given Data / Assumptions:

• Future debt amount, S = Rs. 10028, due after 1 year.
• Trader settles the debt after 3 months, that is 0.25 year from now.
• This is 9 months, or 0.75 year, earlier than the due date.
• Rate of simple interest r = 12 percent per annum.
• We need the cash amount to be paid at 3 months that is equivalent in value to paying Rs. 10028 at the end of 1 year.

Concept / Approach:
Since the debt is paid earlier than due, the trader is entitled to a true discount for the time saved. Here, the time between settlement and due date is 9 months, or 0.75 year. True discount TD on S for time t at rate r is: TD = S * r * t / (100 + r * t) We first compute r * t and then the true discount. The present worth PW, which is the cash amount to be paid now (at 3 months), is: PW = S − TD.

Step-by-Step Solution:
Step 1: Compute the time between settlement and due date. Original due in 1 year, settlement at 3 months = 0.25 year, so t = 1 − 0.25 = 0.75 year. Step 2: Compute r * t = 12 * 0.75 = 9. Step 3: Use true discount formula. TD = S * r * t / (100 + r * t) = 10028 * 9 / (100 + 9). Step 4: Denominator is 109, so TD = 10028 * 9 / 109. Step 5: Multiply numerator: 10028 * 9 = 90252. Step 6: Divide: 90252 / 109 = 828. Thus the true discount is Rs. 828.
Step 7: Present worth PW = S − TD = 10028 − 828 = 9200. So the trader should pay Rs. 9200 at 3 months to settle the debt.

Verification / Alternative check:
We can verify by growing the present worth forward for 9 months at 12 percent and showing that it equals Rs. 10028. Compute r * t again as 9: PW * (1 + r * t / 100) = 9200 * (1 + 9 / 100) = 9200 * 1.09 = 10028. This exactly matches the future debt amount, so our present worth is correct.

Why Other Options Are Wrong:
A payment of Rs. 9000, Rs. 9002, or Rs. 9120 would grow to less than Rs. 10028 at 12 percent for 9 months, so they would underpay the merchant relative to the agreed rate. A payment of Rs. 8800 is even lower and is clearly insufficient. Only Rs. 9200 grows to exactly Rs. 10028 at the given rate and time.

Common Pitfalls:
Typical mistakes include using the full year as the time period instead of 9 months, or applying the simple interest formula directly to compute interest on the present worth without accounting for the true discount structure. Some learners also confuse the number 10028 with the present worth and subtract interest incorrectly. Using the clear steps of computing r * t, then true discount TD, and finally PW = S − TD is a reliable method.

Final Answer:
The trader should pay Rs. 9200 in cash after 3 months to settle the debt.