Which is the better financial offer: (i) an immediate cash payment of Rs. 8100 now or (ii) a credit payment of Rs. 8250 after 6 months at 6.5% simple interest per annum?

Introduction / Context:
This question compares two alternative payment options using the concept of present worth and true discount under simple interest. Students must decide whether it is better to receive a smaller amount immediately or a slightly larger amount after 6 months when the money carries simple interest at 6.5 percent per annum.

Given Data / Assumptions:

• Offer (i): Immediate cash payment of Rs. 8100.
• Offer (ii): Rs. 8250 received after 6 months.
• Rate of simple interest r = 6.5 percent per annum.
• Time for second offer t = 6 months = 0.5 year.
• Simple interest and true discount concepts apply.

Concept / Approach:
We compare both offers on the same date by converting the future amount into its present worth. Under simple interest, if amount due after time t is A, present worth PW is given by: PW = A * 100 / (100 + r * t) where r is the annual rate in percent and t is the time in years. The offer with higher present worth is financially better.

Step-by-Step Solution:
Step 1: For offer (i), present worth PW1 = Rs. 8100 (already paid now). Step 2: For offer (ii), amount A = Rs. 8250, r = 6.5, t = 0.5 year. Step 3: Compute r * t = 6.5 * 0.5 = 3.25. Step 4: Present worth of offer (ii): PW2 = 8250 * 100 / (100 + 3.25). Step 5: PW2 = 825000 / 103.25 which is approximately equal to Rs. 7990.31. Step 6: Compare PW1 and PW2: Rs. 8100 is greater than about Rs. 7990.31. Thus the immediate cash payment has a higher present value.

Verification / Alternative check:
We can also find the true discount on Rs. 8250 for 6 months and subtract it from the nominal amount. True discount TD is: TD = A * r * t / (100 + r * t) = 8250 * 3.25 / 103.25 ≈ Rs. 259.69. Present worth = 8250 − 259.69 ≈ Rs. 7990.31. Again this is less than Rs. 8100, so our earlier conclusion is consistent.

Why Other Options Are Wrong:
Option (ii) is worse because Rs. 8250 after 6 months is worth only about Rs. 7990.31 today, which is lower than Rs. 8100. The two offers are not equivalent, so saying both are same is wrong. It is also not true that neither offer is acceptable; one is clearly better under the given rate, and the data is fully sufficient for a decision.

Common Pitfalls:
A common mistake is to compare only the nominal amounts 8250 and 8100 without adjusting for time value of money. Another error is to compute interest on the present worth instead of on the future value when applying formulas. Some students forget to convert 6 months to 0.5 year or miscalculate r * t. Always ensure that the comparison is done at the same point in time, usually by converting future sums to present worth using the correct simple interest expression.

Final Answer:
The better offer is (i): an immediate cash payment of Rs. 8100 now.