Surekha borrowed a sum of money and returned it in two equal annual installments of Rs. 5,547 each. If the rate of interest was \(7 \frac{1}{2}\%\) p.pa compounded yearly, then the total interest paid by her was:
Rs. 1,134
Understanding the Compound Interest Installment Problem
This question asks us to calculate the total interest paid on a sum of money borrowed, which is returned in two equal annual installments. The key concepts here are compound interest and loan repayment through installments. When a loan is repaid through installments, each installment includes a part of the principal and the interest accrued on the outstanding principal for that period.
To find the total interest paid, we first need to determine the original sum of money borrowed (the principal). We can do this by calculating the present value of each installment. The present value of an amount is the value today of money to be received or paid in the future, discounted at a specific interest rate.
Calculating the Present Value of Installments
The formula for the present value (PV) of an amount \(I\) received or paid \(n\) years from now, with an interest rate \(r\) compounded annually, is:
\[PV = \frac{I}{(1+r)^n}\]
In this problem:
- Each installment amount (\(I\)) is Rs. 5,547.
- The interest rate (\(r\)) is \(7 \frac{1}{2}\%\) p.a., which is \(7.5\%\) or 0.075 in decimal form.
- There are two annual installments. The first is paid after 1 year (\(n=1\)), and the second is paid after 2 years (\(n=2\)).
Present Value of the First Installment
The first installment of Rs. 5,547 is paid at the end of the first year. Its present value is:
\[PV_1 = \frac{5547}{(1+0.075)^1} = \frac{5547}{1.075}\]
Calculating the value:
\(PV_1 = 5160\)
The present value of the first installment is Rs. 5,160.
Present Value of the Second Installment
The second installment of Rs. 5,547 is paid at the end of the second year. Its present value is:
\[PV_2 = \frac{5547}{(1+0.075)^2} = \frac{5547}{(1.075)^2}\]
First, calculate \((1.075)^2\):
\((1.075)^2 = 1.075 \times 1.075 = 1.155625\)
Now, calculate \(PV_2\):
\[PV_2 = \frac{5547}{1.155625}\]
Calculating the value:
\(PV_2 = 4800\)
The present value of the second installment is Rs. 4,800.
Determining the Principal Borrowed
The total principal borrowed is the sum of the present values of all the installments.
Principal (\(P\)) = \(PV_1 + PV_2\)
\(P = 5160 + 4800\)
\(P = 9960\)
So, the sum of money borrowed was Rs. 9,960.
Calculating the Total Amount Paid
The total amount paid back is the sum of the two annual installments.
Total Paid = Number of installments \(\times\) Installment amount
Total Paid = \(2 \times 5547\)
Total Paid = \(11094\)
The total amount paid back by Surekha was Rs. 11,094.
Calculating the Total Interest Paid
The total interest paid is the difference between the total amount paid back and the principal borrowed.
Total Interest = Total Paid - Principal Borrowed
Total Interest = \(11094 - 9960\)
Total Interest = \(1134\)
The total interest paid by Surekha was Rs. 1,134.
Summary of Calculations
| Installment Amount | Rs. 5,547 |
| Number of Installments | 2 |
| Rate of Interest | \(7.5\%\) p.a. |
| Present Value of 1st Installment | Rs. 5,160 |
| Present Value of 2nd Installment | Rs. 4,800 |
| Principal Borrowed | Rs. 9,960 |
| Total Amount Paid | Rs. 11,094 |
| Total Interest Paid | Rs. 1,134 |
Revision Table: Compound Interest Installments
| Concept | Description | Formula Example (Annual Compounding) |
|---|---|---|
| Compound Interest | Interest calculated on the initial principal and also on the accumulated interest of previous periods. | Amount \(A = P(1+r)^n\) |
| Installment Repayment | Loan repaid in periodic equal payments over a set time. Each payment covers interest and reduces principal. | Installment = Principal / (Sum of Present Value Factors) |
| Present Value (PV) | The current value of a future sum of money given a specified rate of return. | \(PV = \frac{FV}{(1+r)^n}\) |
Additional Information: Loan Amortization
The process of paying off a debt over time through regular payments is called amortization. In loan amortization with equal installments, the amount of interest decreases with each payment because the principal outstanding decreases. Conversely, the portion of the payment applied towards reducing the principal increases over time. While we calculated the total interest here by finding the principal first, a detailed amortization schedule would show how each installment is split between interest and principal reduction.
Understanding the present value concept is crucial in financial calculations, including valuing investments, calculating loan payments, and determining the true cost of borrowing when payments are spread over time. The interest rate acts as the discount rate when calculating present values. A higher interest rate leads to a lower present value of a future sum.
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