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Loan 120,000: number of installments and rate

Short answer

i) At 1% monthly, (P/A,1%,n)=40 → n≈51.34, so about 52 installments (51 of Rs 3,000 plus a smaller last). ii) For Rs 5,000/month over 30 months, (P/A,i,30)=24 gives i≈1.505% per month, nominal annual rate ≈ 18.1%.

Mr. X receives a loan of Rs 120,000 from a bank at an interest rate of 12% per year. i) He wishes to repay the loan in monthly installment with Rs. 3000 per month. How many installments are necessary to complete his payment? ii) What annual interest rate is he paying if the bank asks him to pay Rs 5000 per month for 30 times?

Loan P = Rs 1,20,000 received at t = 0; payments are at the end of each month.

(i) Number of installments at 12% per year

Assumption: 12% nominal, compounded monthly, so i = 1% per month.

P = A (P/A, i, n) so (P/A, 1%, n) = 1,20,000 / 3,000 = 40

[1 − (1.01)^-n] / 0.01 = 40, so (1.01)^-n = 1 − 0.40 = 0.60

n = −ln(0.60) / ln(1.01) = 0.51083 / 0.00995 = 51.34 months

Check: 51 payments have PW = 3,000 × 39.7981 = Rs 119,394, 52 payments have PW = Rs 121,183. So 51 full installments of Rs 3,000 are not enough; a 52nd, smaller payment of about Rs 1,016 clears the loan.

Answer (i): about 52 monthly installments (51 of Rs 3,000 plus a small last one).

(ii) Rate for Rs 5,000 per month for 30 months

(P/A, i, 30) = 1,20,000 / 5,000 = 24

i per month(P/A, i, 30)vs 24
1.0%25.8077too high
2.0%22.3965too low
1.5%24.0158slightly high
1.505%24.0000= 24
  • Nominal annual rate = 12 × 1.505% = 18.06% per year
  • Effective annual rate = (1 + 0.01505)^12 − 1 = 19.63%

Answer (ii): about 18.1% per year nominal (19.6% effective).

✓ Answer checked by Claude Opus 5.5 (high effort)

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