Rekha borrowed a sum of Rs. 24,000 at 12% p.a. compound interest. interest compounded yearly, for 3 years. At the end of first year, she repays a sum of Rs. 6,880 and at the end of second year, she repays Rs. 5,400. What amount should she pay at the end of third year to discharge the loan?
Rs. 19,040
This problem involves calculating the outstanding loan amount at the end of three years for a sum borrowed under compound interest with periodic repayments. We will track the principal amount year by year, considering the interest accrued and the repayments made.
Compound Interest Calculation Breakdown
Let's break down the calculation for each year to determine the final amount Rekha needs to pay.
Year 1: Principal and First Repayment
- Initial Principal (P): Rs. 24,000
- Interest Rate (R): 12% per annum, compounded yearly
At the end of the first year, the interest accrued on the initial principal is calculated as follows:
\[ \text{Interest for Year 1} = \text{Principal} \times \text{Rate} \]
\[ I_1 = 24000 \times 0.12 = \text{Rs. } 2,880 \]
The total amount at the end of the first year, before any repayment, is:
\[ \text{Amount at end of Year 1} = \text{Initial Principal} + \text{Interest for Year 1} \]
\[ A_1 = 24000 + 2880 = \text{Rs. } 26,880 \]
Rekha repays a sum of Rs. 6,880 at the end of the first year. So, the principal outstanding for the second year will be:
\[ \text{Principal for Year 2} = A_1 - \text{Repayment at end of Year 1} \]
\[ P_2 = 26880 - 6880 = \text{Rs. } 20,000 \]
Year 2: Interest and Second Repayment
Now, we calculate the interest for the second year on the new principal amount (Rs. 20,000).
\[ \text{Interest for Year 2} = \text{Principal for Year 2} \times \text{Rate} \]
\[ I_2 = 20000 \times 0.12 = \text{Rs. } 2,400 \]
The total amount at the end of the second year, before the second repayment, is:
\[ \text{Amount at end of Year 2} = \text{Principal for Year 2} + \text{Interest for Year 2} \]
\[ A_2 = 20000 + 2400 = \text{Rs. } 22,400 \]
Rekha repays Rs. 5,400 at the end of the second year. The principal outstanding for the third year will be:
\[ \text{Principal for Year 3} = A_2 - \text{Repayment at end of Year 2} \]
\[ P_3 = 22400 - 5400 = \text{Rs. } 17,000 \]
Year 3: Final Payment Calculation
Finally, we calculate the interest for the third year on the remaining principal amount (Rs. 17,000).
\[ \text{Interest for Year 3} = \text{Principal for Year 3} \times \text{Rate} \]
\[ I_3 = 17000 \times 0.12 = \text{Rs. } 2,040 \]
The total amount Rekha should pay at the end of the third year to discharge the loan is the principal for Year 3 plus the interest for Year 3:
\[ \text{Amount to pay at end of Year 3} = \text{Principal for Year 3} + \text{Interest for Year 3} \]
\[ \text{Final Payment} = 17000 + 2040 = \text{Rs. } 19,040 \]
Summary of Loan Transactions
The following table summarizes the step-by-step calculations for Rekha's loan:
| Year | Opening Principal (Rs.) | Interest (12%) (Rs.) | Amount before Repayment (Rs.) | Repayment (Rs.) | Closing Principal (Rs.) |
|---|---|---|---|---|---|
| 1 | 24,000 | \(24000 \times 0.12 = 2880\) | \(24000 + 2880 = 26880\) | 6,880 | \(26880 - 6880 = 20000\) |
| 2 | 20,000 | \(20000 \times 0.12 = 2400\) | \(20000 + 2400 = 22400\) | 5,400 | \(22400 - 5400 = 17000\) |
| 3 | 17,000 | \(17000 \times 0.12 = 2040\) | \(17000 + 2040 = 19040\) | (Final Payment) | 0 |
Therefore, Rekha should pay Rs. 19,040 at the end of the third year to completely discharge her loan.
Important Questions from Installments
A loan of Rs. 1,50,000 is availed with compound interest rate of 10% per annum for two years compounded annually. It is to be paid in equal yearly installments, and the installment is to be paid at the end of each year. The value of the equal yearly installment is : (Rounded off to two places of decimal)
To buy a car, Sahil borrowed some amount at 10% simple interest rate. He has paid some loan amount but Rs. 1,81,500 is due which he has to pay in 3 years. What will be the annual installment (in Rs.)?
What annual installment will discharge a debt of Rs. 35,455 due in three years at 12% simple interest per annum (Installments will be paid at the end of the years)(Round off to nearest Rupee)?
Ravi borrowed ₹1,380 from a bank which he repaid in 6 years at the rate of 6% per annum simple interest. If payment was made in six equal instalment, then each instalment was:
A loan of Rs 15000 is to be repaid in 4 equal annual installments. If the compound interest rate is 10% per annum, what is the approximate amount of each installment?
