A sum of Rs. 1100 was taken as a loan. This is to be paid in two equal installments. If the rate of interest is 20% per annum, compounded annually, find the amount payable in each installment.
Rs. 720
Calculating Equal Loan Installments with Compound Interest
This problem requires us to find the amount of two equal annual installments needed to repay a loan of Rs. 1100 at a 20% annual compound interest rate.
When a loan is repaid in equal installments, each installment includes both principal repayment and interest. The concept used here is the present value of an annuity. The sum of the present values of all future installments must equal the original loan amount (principal).
Let:
- Principal loan amount \(P = 1100\)
- Annual interest rate \(r = 20\% = 0.20\)
- Number of equal installments \(n = 2\)
- Amount of each installment \(x\) (which we need to find)
The present value of the first installment, payable after 1 year, is given by \( \frac{x}{(1+r)^1} \). The present value of the second installment, payable after 2 years, is given by \( \frac{x}{(1+r)^2} \).
The total principal amount is the sum of the present values of the installments:
\[P = \frac{x}{(1+r)} + \frac{x}{(1+r)^2}\]Now, let's substitute the given values into the formula:
\(1+r = 1+0.20 = 1.20\)
\((1+r)^2 = (1.20)^2 = 1.44\)
So the equation becomes:
\[1100 = \frac{x}{1.20} + \frac{x}{1.44}\]To solve for \(x\), we can find a common denominator for the fractions on the right side, which is 1.44:
\[1100 = \frac{x \times 1.20}{1.20 \times 1.20} + \frac{x}{1.44}\] \[1100 = \frac{1.20x}{1.44} + \frac{x}{1.44}\] \[1100 = \frac{1.20x + x}{1.44}\] \[1100 = \frac{2.20x}{1.44}\]Now, multiply both sides by 1.44:
\[1100 \times 1.44 = 2.20x\] \[1584 = 2.20x\]Finally, divide by 2.20 to find the value of \(x\):
\[x = \frac{1584}{2.20}\] \[x = \frac{15840}{22}\] \[x = 720\]Therefore, the amount payable in each equal installment is Rs. 720.
| Parameter | Value |
|---|---|
| Principal Loan Amount (P) | Rs. 1100 |
| Annual Interest Rate (r) | 20% (0.20) |
| Number of Installments (n) | 2 |
| Amount of each Installment (x) | Rs. 720 |
Revision Table: Loan Installment Calculation
Let's quickly summarize the key figures from our loan calculation:
| Item | Detail |
|---|---|
| Original Loan | Rs. 1100 |
| Interest Rate | 20% compounded annually |
| Number of Installments | 2 (equal annual) |
| Calculated Installment Amount | Rs. 720 |
Additional Information: Loan Amortization and Present Value
Understanding loan installments involves concepts like compound interest and present value.
- Compound Interest: Interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. This means the amount owed grows faster over time compared to simple interest.
- Present Value: This is the current value of a future sum of money or stream of cash flows, given a specified rate of return. In loan calculations, the loan amount is the present value of all future installment payments.
- Annuity: A series of equal payments made at regular intervals. Loan installment payments often form an annuity. The formula used above for calculating the installment amount is derived from the formula for the present value of an ordinary annuity (where payments are made at the end of each period), adapted for a specific number of periods.
- Loan Amortization: The process of paying off a debt over time through regular payments. Each payment typically covers the interest accrued since the last payment, with the remainder going towards reducing the principal balance. Over time, the portion of the payment going towards principal increases, and the portion going towards interest decreases.
For a loan with 20% annual compound interest, each Rs. 720 installment helps reduce the loan balance, considering the interest accrued on the remaining balance. The sum of the present values of these two Rs. 720 payments at a 20% discount rate equals the initial Rs. 1100 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)
A sum of Rs. P was borrowed and paid back in two equal yearly instalments, each of Rs. 35,280. If the rate of interest was 5% per annum and interest is compounding annually, then the value of P is ________.
A sum of Rs. 16400 is borrowed to be paid back in 2 years by equal payments allowing 5% compound interest. Find the annual payment.
The formula for finding the annual installment, when A is the amount taken on loan, where r% is the rate of interest, n is the number of installments, is:
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?
