Product-Method Interest on a PACS Crop Loan
On 1 April a Primary Agricultural Credit Society disburses a crop loan of ₹1,50,000 to a member at 12% per annum simple interest, with the interest year ending 31 March. The member repays ₹50,000 on 1 August and ₹60,000 on 1 December, and pays no interest in cash during the year. The society charges interest on the product method: each repayment takes effect from the first day of the month in which it is received, and every intervening calendar month is counted as one unit of one-twelfth of a year, regardless of the number of days in it. The secretary wants two things before the annual closing: the exact interest income to be booked for the year, and confirmation of the balance the member must pay to close the account. A director has separately proposed an alternative - a single repayment of ₹1,10,000 on 1 October - and claims it would be costlier because “one lump payment is always heavier than two smaller ones”.
1. Compute the interest income for the year under the actual repayment pattern, and the amount the member must pay on 31 March to close the account.
Read the worked answer
Build the rupee-month products.
| Period | Principal outstanding | Months | Product |
|---|---|---|---|
| 1 Apr – 31 Jul | 1,50,000 | 4 | 6,00,000 |
| 1 Aug – 30 Nov | 1,00,000 | 4 | 4,00,000 |
| 1 Dec – 31 Mar | 40,000 | 4 | 1,60,000 |
Total product = 6,00,000 + 4,00,000 + 1,60,000 = 11,60,000 rupee-months.
Interest = product × rate ÷ (100 × 12) = 11,60,000 × 12 ÷ 1200 = ₹11,600.
Sanity check: an untouched ₹1,50,000 for a full year at 12% would earn ₹18,000, so ₹11,600 is plausible for a balance that falls to ₹40,000 by December.
Closing balance on 31 March = outstanding principal ₹40,000 + unpaid interest ₹11,600 = ₹51,600. Interest is not itself charged interest under simple interest, so it is added once, at the end.
2. Now compute the director's alternative: a single repayment of ₹1,10,000 on 1 October. What interest is earned and what is the closing balance?
Read the worked answer
With one repayment on 1 October the account carries ₹1,50,000 from 1 April to 30 September (6 months) and ₹40,000 from 1 October to 31 March (6 months).
Products:
- 1,50,000 × 6 = 9,00,000
- 40,000 × 6 = 2,40,000
- Total = 11,40,000 rupee-months
Interest = 11,40,000 × 12 ÷ 1200 = ₹11,400.
Closing balance = ₹40,000 principal + ₹11,400 interest = ₹51,400.
The principal outstanding at year end is identical (₹40,000) in both patterns, because the member repays ₹1,10,000 either way. Only the interest differs, and it differs only because of the timing of the outstanding balance, not the number of receipts.
3. Is the director right that the single lump repayment is costlier? Settle the point with numbers and state the principle the branch should apply.
Read the worked answer
The director is wrong on both direction and reason. The lump repayment costs ₹11,400 against ₹11,600 for the two-instalment plan - it is ₹200 cheaper, not costlier.
Why: compare the products. In the two-instalment plan the member still owed ₹60,000 during August, September, October and November, adding 60,000 × 4 = 2,40,000 rupee-months that the single-repayment plan does not have. The extra interest is 2,40,000 × 12 ÷ 1200 = ₹200, exactly the gap.
Principle: interest is a charge on rupee-months of outstanding balance, not on the number of receipts. A clerk who assumes “more instalments = cheaper” or “one big payment = dearer” will misquote every settlement figure. Timing dominates size: if the same ₹1,10,000 had been paid on 1 August instead of 1 October, the product would fall to (1,50,000 × 4) + (40,000 × 8) = 9,20,000, giving interest of only ₹9,200 - a saving of ₹2,400 over the 1 October plan for the same amount of money.