Calculating Equal Annual Installments with Compound Interest

This problem asks us to find the amount of each equal annual installment required to pay back a sum of Rs. 4,620 over two years, with interest compounded annually at a rate of 10%.

When a loan is paid back in equal installments under compound interest, each installment covers both the interest due for that period and a portion of the principal amount. The present value of all future installments must be equal to the original principal amount borrowed.

The formula for calculating the present value of a series of equal annual installments (x) for a principal amount (P) at an annual interest rate (r) for n years is:

\(P = \frac{x}{(1+r)^1} + \frac{x}{(1+r)^2} + \dots + \frac{x}{(1+r)^n}\)

In this specific problem:

• Principal amount (P) = Rs. 4,620
• Annual interest rate (r) = 10% or 0.10
• Number of installments (n) = 2
• Amount of each installment = x (what we need to find)

Using the formula for n=2:

\(P = \frac{x}{(1+r)^1} + \frac{x}{(1+r)^2}\)

Substitute the given values:

\(4620 = \frac{x}{(1+0.10)^1} + \frac{x}{(1+0.10)^2}\)

\(4620 = \frac{x}{(1.1)} + \frac{x}{(1.1)^2}\)

\(4620 = \frac{x}{1.1} + \frac{x}{1.21}\)

Now, we can factor out x:

\(4620 = x \left( \frac{1}{1.1} + \frac{1}{1.21} \right)\)

To add the fractions, find a common denominator, which is 1.21:

\(\frac{1}{1.1} = \frac{1 \times 1.1}{1.1 \times 1.1} = \frac{1.1}{1.21}\)

So, the equation becomes:

\(4620 = x \left( \frac{1.1}{1.21} + \frac{1}{1.21} \right)\)

\(4620 = x \left( \frac{1.1 + 1}{1.21} \right)\)

\(4620 = x \left( \frac{2.1}{1.21} \right)\)

Alternatively, using fractions:

\(1.1 = \frac{11}{10}\)

\(1.21 = (1.1)^2 = \left(\frac{11}{10}\right)^2 = \frac{121}{100}\)

So, the equation is:

\(4620 = \frac{x}{11/10} + \frac{x}{121/100}\)

\(4620 = \frac{10x}{11} + \frac{100x}{121}\)

Find a common denominator, which is 121:

\(4620 = \frac{10x \times 11}{11 \times 11} + \frac{100x}{121}\)

\(4620 = \frac{110x}{121} + \frac{100x}{121}\)

\(4620 = \frac{110x + 100x}{121}\)

\(4620 = \frac{210x}{121}\)

Now, solve for x:

\(x = 4620 \times \frac{121}{210}\)

We can simplify the calculation:

\(x = \frac{4620}{210} \times 121\)

\(\frac{4620}{210} = \frac{462}{21}\)

Dividing 462 by 21:

\(462 \div 21 = 22\)

So,

\(x = 22 \times 121\)

\(x = 2662\)

Thus, each equal annual installment is Rs. 2,662.

Revision Table: Key Terms and Concepts

Additional Information on Loan Repayment Installments

When a loan is repaid through installments, the payment includes both interest and a portion of the principal. With each subsequent payment, the amount of interest decreases (as the remaining principal balance reduces), and consequently, the amount applied towards reducing the principal increases.

For a loan repaid in equal installments under compound interest, the calculation essentially finds the future value of the principal amount and equates it to the future value of an annuity (the series of equal installments). However, using the present value approach (sum of present values of installments equals principal) is often more straightforward for calculating the installment amount itself.

Understanding the concept of present value is crucial in such calculations, as it discounts future payments back to their worth today at the given interest rate. This ensures that the total value received by the lender through installments is equivalent to the original amount lent, considering the time value of money.