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:
65,600
Understanding Loan Instalment Calculation with Compound Interest
This problem involves a loan that is paid back in equal yearly instalments. The key here is to understand how compound interest affects the value of future payments when calculating the original principal amount borrowed. The principal amount (P) is equal to the sum of the present values of each future instalment.
The formula for the present value (PV) of a single amount received in the future is:
\(PV = \frac{Future\ Value}{(1 + \frac{r}{100})^n}\)
Where:
- \(Future\ Value\) is the amount received in the future (the instalment).
- \(r\) is the annual interest rate.
- \(n\) is the number of years from the present.
In this specific problem:
- Each yearly instalment is Rs. 35,280.
- The interest rate is 5% per annum, compounded annually.
- There are two equal yearly instalments.
Step-by-Step Calculation of Principal Amount (P)
To find the principal amount (P), we need to calculate the present value of each instalment and sum them up.
Step 1: Calculate the present value of the first instalment.
The first instalment of Rs. 35,280 is paid at the end of the first year (n=1). The interest rate is 5%.
\(PV_{1st\ instalment} = \frac{35280}{(1 + \frac{5}{100})^1} = \frac{35280}{(1 + 0.05)^1} = \frac{35280}{1.05}\)
Calculating the value:
\(PV_{1st\ instalment} = \frac{35280}{1.05} = 33600\)
So, the present value of the first instalment is Rs. 33,600.
Step 2: Calculate the present value of the second instalment.
The second instalment of Rs. 35,280 is paid at the end of the second year (n=2). The interest rate is still 5%.
\(PV_{2nd\ instalment} = \frac{35280}{(1 + \frac{5}{100})^2} = \frac{35280}{(1 + 0.05)^2} = \frac{35280}{(1.05)^2}\)
First, calculate \((1.05)^2\):
\((1.05)^2 = 1.05 \times 1.05 = 1.1025\)
Now, calculate the present value:
\(PV_{2nd\ instalment} = \frac{35280}{1.1025} = 32000\)
So, the present value of the second instalment is Rs. 32,000.
Step 3: Calculate the total principal amount (P).
The total principal amount borrowed is the sum of the present values of all the instalments.
\(P = PV_{1st\ instalment} + PV_{2nd\ instalment}\)
\(P = 33600 + 32000\)
\(P = 65600\)
Therefore, the value of P, the principal amount borrowed, is Rs. 65,600.
Comparing this result with the given options, we find that it matches one of the options.
| Instalment | Year (n) | Instalment Amount | Calculation of Present Value | Present Value |
|---|---|---|---|---|
| 1st | 1 | Rs. 35,280 | \(\frac{35280}{(1.05)^1}\) | Rs. 33,600 |
| 2nd | 2 | Rs. 35,280 | \(\frac{35280}{(1.05)^2}\) | Rs. 32,000 |
| Total Principal (P) | Rs. 65,600 | |||
Revision Table: Key Concepts in Loan Instalments
| Term | Definition | Relevance to Problem |
|---|---|---|
| Principal Amount (P) | The initial amount of money borrowed. | This is the value we needed to find. |
| Instalment | A fixed amount paid periodically to repay a loan, including both principal and interest. | Given as Rs. 35,280 per year. |
| Interest Rate | The cost of borrowing money, expressed as a percentage of the principal. | Given as 5% compounded annually. |
| Compound Interest | Interest calculated on the initial principal and also on the accumulated interest of previous periods. | Used for calculating the future value and subsequently the present value of instalments. |
| Present Value (PV) | The current value of a future sum of money, given a specified rate of return. | Used to discount future instalments back to the present to find the original principal. |
Additional Information on Loan Repayments and Present Value
When a loan is repaid with equal instalments over time, each instalment includes a portion of the principal repayment and the interest accumulated since the last payment. Earlier instalments primarily cover interest, while later instalments pay down more principal. However, from the lender's perspective (or when calculating the original principal), the loan amount is the sum of the present values of all the future payments they expect to receive.
The concept of present value is fundamental in finance. It tells us how much a future amount is worth today, considering the time value of money and the earning potential (interest rate). A higher interest rate means a lower present value for a future sum because the discounting effect is stronger.
For a loan with equal instalments (an annuity), there is a general formula for the present value, but breaking it down by calculating the present value of each instalment separately, as done here, is also a valid and often clearer approach, especially for a small number of periods like two years.
In summary, calculating the principal amount of a loan repaid by equal instalments involves discounting each future instalment back to the present using the given interest rate compounded annually and summing these present values.
Similar Questions
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. 6534, then the total interest charged (in Rs.) is:
A loan has to be returned in two equal yearly instalments each of Rs. 44,100. If the rate of interest is 5% p.a compounded annually, the total interest paid is:
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:
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?
A certain loan was returned in two equal half - yearly instalments each of Rs. 6,760. If the rate of interest was 8% p.a., compounded yearly, how much was the interest paid on the loan?
What annual instalment will discharge a debit of ₹5,664 in 4 years at 12% simple interest?
What annual installment will discharge a debt of Rs. 9,600 due in 5 years at 10% simple interest?
A person borrowed ₹2,000 at 5% annual simple interest repayable in 3 equal annual installments. What will be the annual installment?
Rajnish borrowed ₹1,500 from a bank and repaid the entire amount with interest in two equal annual instalments, the first instalment being paid a year after Rajnish borrowed from the bank. If the rate of interest was 40% per annum, compounded annually, then what was the value (in ₹) of each instalment paid by Rajnish?
Important Questions from Installments
A computer is available for Rs. 39,000 on cash payment or Rs. 19,000 as cash payment followed by five monthly instalments of Rs. 4,200 each. What is the rate of interest per annum under the instalment plan?
What is the amount (in Rs.) of debt that will be discharged in 6 equal instalments of Rs. 800 each, if the debt is due in 6 years at 5% per annum?
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 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 computer is available for ₹75,300 cash or for ₹25,740 cash down payment and two equal half-yearly instalments. If the dealer charges interest at 20% p.a., compounded half-yearly, then the total interest to be paid by a customer who buys it in instalment scheme is:
