Understanding the Loan Repayment Problem
This question involves calculating the size of equal annual installments needed to repay a loan taken under compound interest. We are given:
- Principal Loan Amount (PV): ₹10,000
- Annual Interest Rate (R): 10% per annum, which is 0.10 in decimal form.
- Loan Duration (n): 3 years
- Interest Compounding: Annually
- Repayment Structure: 3 equal annual installments
We need to find the amount of each equal installment, let's call it 'x'. The core idea is that the original loan amount must be equal to the sum of the present values of all the installments paid back.
Formula for Equal Installment Calculation
When a loan is repaid in equal installments, it can be treated as an annuity. The formula connecting the Present Value (PV) of the loan to the equal installment amount (x), interest rate per period (r), and the number of periods (n) is derived from the present value of an ordinary annuity formula:
\(PV = x \times \frac{1 - (1 + r)^{-n}}{r}\)
Where:
- PV is the initial loan amount (Principal).
- x is the amount of each equal installment.
- r is the interest rate per installment period (as a decimal).
- n is the total number of installments.
Applying the Formula to the Problem
Let's substitute the given values into the formula:
- PV = 10,000
- r = 10% = 0.10
- n = 3
So, the equation becomes:
\(10,000 = x \times \frac{1 - (1 + 0.10)^{-3}}{0.10}\)
Step-by-Step Calculation
First, calculate the value of \((1 + r)^{-n}\):
\((1 + 0.10)^{-3} = (1.1)^{-3}\)
\((1.1)^3 = 1.331\)
\((1.1)^{-3} = \frac{1}{1.331} \approx 0.751315\)
Next, calculate the term in the fraction (the annuity factor):
\(\frac{1 - (1.1)^{-3}}{0.10} = \frac{1 - 0.751315}{0.10}\)
\(= \frac{0.248685}{0.10}\)
\(= 2.48685\)
Now, we can find the installment amount 'x' by rearranging the formula:
\(x = \frac{PV}{\text{Annuity Factor}}\)
\(x = \frac{10,000}{2.48685}\)
Performing the final division:
\(x \approx 4021.14\)
Conclusion
The calculation shows that the amount of each installment is approximately ₹4,021.14. Looking at the options provided, 4,021 is the closest approximation.
Similar Questions
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Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
The difference between the compound interest (compounding annually) and the simple interest on a sum of money at the rate of 40 per cent per annum for 2 years is Rs. 2400. What is the amount?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
