Calculating Borrowed Sum with Compound Interest Installments
This problem involves calculating the original principal amount borrowed when the repayment is made in equal installments over a period with compound interest.
Understanding the Concepts
When a loan is repaid in equal installments over time, the principal amount borrowed is equivalent to the sum of the present values of all the future installments. The present value is calculated by discounting each future installment back to the time the loan was taken, using the given compound interest rate.
Key Information:
- Installment Amount (I): Rs. 882
- Number of Installments (n): 2 (paid annually)
- Interest Rate (r): 5% per annum = 0.05
- Compounding Frequency: Annual
- Loan Duration: 2 years
Setting up the Calculation
Let the principal sum borrowed be P. The borrower pays back the loan in two equal annual installments. This means the first installment is paid at the end of the first year, and the second installment is paid at the end of the second year.
The present value of the first installment (paid after 1 year) is calculated as:
$$ \text{PV}_1 = \frac{I}{(1 + r)^1} $$
The present value of the second installment (paid after 2 years) is calculated as:
$$ \text{PV}_2 = \frac{I}{(1 + r)^2} $$
The total principal amount borrowed (P) is the sum of the present values of these two installments:
$$ P = \text{PV}_1 + \text{PV}_2 $$
$$ P = \frac{I}{(1 + r)^1} + \frac{I}{(1 + r)^2} $$
Step-by-Step Calculation
- Substitute the values: Plug the known values into the formula. $$ P = \frac{882}{(1 + 0.05)^1} + \frac{882}{(1 + 0.05)^2} $$
- Simplify the denominators: $$ P = \frac{882}{1.05} + \frac{882}{(1.05)^2} $$
- Calculate the values of the fractions: First installment's present value: $$ \frac{882}{1.05} = 840 $$ Second installment's present value: $$ (1.05)^2 = 1.1025 $$ $$ \frac{882}{1.1025} = 800 $$
- Sum the present values: Add the present values of both installments to find the total principal amount borrowed. $$ P = 840 + 800 $$ $$ P = 1640 $$
Conclusion
Therefore, the sum of money the man borrowed was Rs. 1,640.
| Installment Number | Time Period (Years) | Calculation of Present Value | Present Value (Rs.) |
|---|---|---|---|
| 1st | 1 | $$ \frac{882}{(1.05)^1} $$ | 840 |
| 2nd | 2 | $$ \frac{882}{(1.05)^2} $$ | 800 |
| Total Principal (P) | - | Sum of Present Values | 1640 |
Important Questions from Installments
A sum of Rs. 4,620 is to be paid back in 2 equal annual instalments. How much is each instalment (in Rs.) if the interest is compounded annually at 10% per annum?
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:
A loan is to be returned in two equal yearly instalments. If the rate of interest is 10% p.a., compounded annually, and each instalment is Rs. 5,808, then the total interest charged in this scheme is:
What annual instalment will discharge a debit of ₹5,664 in 4 years at 12% simple interest?
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% compounded annually, then the value of P is:
