Loan Installment Calculation with Compound Interest

The problem asks for the value of each of the two equal annual installments for a loan of Rs. 9,350 at a 20% annual compound interest rate.

Understanding the Annuity Concept

This scenario represents an annuity where the present value (the loan amount) is equal to the sum of the present values of all future installments. Let 'P' be the value of each installment.

Mathematical Formulation

The present value (PV) of an annuity can be calculated using the formula:

$PV = \frac{P}{(1+r)^1} + \frac{P}{(1+r)^2} + ... + \frac{P}{(1+r)^n}$

In this case:

• PV = Rs. 9,350
• r = 20% or 0.20
• n = 2 installments

Substituting the values:

$9350 = \frac{P}{(1+0.20)^1} + \frac{P}{(1+0.20)^2}$

$9350 = \frac{P}{1.20} + \frac{P}{1.44}$

Calculating the Installment Value (P)

Factor out P:

$9350 = P \left( \frac{1}{1.20} + \frac{1}{1.44} \right)$

Find a common denominator for the fractions:

$ \frac{1}{1.20} = \frac{100}{120} = \frac{5}{6}$

$ \frac{1}{1.44} = \frac{100}{144} = \frac{25}{36}$

Add the fractions:

$ \frac{5}{6} + \frac{25}{36} = \frac{5 \times 6}{6 \times 6} + \frac{25}{36} = \frac{30}{36} + \frac{25}{36} = \frac{55}{36}$

Now substitute this back into the equation:

$9350 = P \left( \frac{55}{36} \right)$

Solve for P:

$P = 9350 \times \frac{36}{55}$

$P = \frac{9350}{55} \times 36$

$P = 170 \times 36$

$P = 6120$

Conclusion

Each installment is Rs. 6120.