What annual installment will discharge a debt of ₹5,460 due in 5 years at 10% simple interest per annum?
To determine the annual installment required to discharge a debt of ₹5,460 due in 5 years at a simple interest rate of 10% per annum, we need to calculate the Equal Annual Installments (EAI).
The total amount paid in installments will be the sum of the principal and the interest accrued over 5 years.
Let's calculate the simple interest first:
Simple Interest (SI) for 5 years = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}
= \frac{5460 \times 10 \times 5}{100}= ₹2,730
The total amount including interest to be discharged is:
Total Amount = Principal + Simple Interest = ₹5,460 + ₹2,730 = ₹8,190
The Equal Annual Installment (EAI) should be such that the present value of all these installments equals the total amount, ₹8,190.
Let the annual installment be ₹X. The installment for each year with the present value at the end of the 5th year can be expressed using:
- Installment at the end of Year 1 = X (1 + 0.1)^4
- Installment at the end of Year 2 = X (1 + 0.1)^3
- Installment at the end of Year 3 = X (1 + 0.1)^2
- Installment at the end of Year 4 = X (1 + 0.1)^1
- Installment at the end of Year 5 = X
Using the formula for the future value of an installment, we can calculate the required installment:
Total future value of all installments = X \left[ (1.1)^4 + (1.1)^3 + (1.1)^2 + (1.1)^1 + 1 \right]
Since this sum should equal ₹8,190, solve for X:
X \left[ 1.4641 + 1.331 + 1.21 + 1.1 + 1 \right] = 8190X \times 6.1051 = 8190
X = \frac{8190}{6.1051} \approx 910
Therefore, the correct answer is ₹910. Thus, the annual installment that will discharge the debt in 5 years at 10% simple interest per annum is ₹910.
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