Step 1: Concept Identification: This problem necessitates the computation of an Equated Monthly Installment (EMI) for a loan, applying the reducing balance method where interest is calculated monthly on the outstanding principal.
Step 2: Formula/Methodology: The EMI is calculated using the formula:
\[ EMI = P \times r \times \frac{(1+r)^n}{(1+r)^n - 1} \]
where:
- \(P\) represents the principal loan amount.
- \(r\) denotes the monthly interest rate.
- \(n\) signifies the total number of monthly installments.
Step 3: Calculation Breakdown:
1. Principal Loan Amount (P) Calculation:
\[ P = \text{Total House Cost} - \text{Down Payment} \]
\[ P = 39,65,000 - 5,00,000 = 34,65,000 \]
2. Monthly Interest Rate (r) Calculation:
Given an annual rate of 6%, compounded monthly:
\[ r = \frac{6\%}{12} = 0.5\% = 0.005 \]
3. Number of Installments (n) Calculation:
For a loan term of 25 years:
\[ n = 25 \text{ years} \times 12 \text{ months/year} = 300 \text{ months} \]
4. EMI Calculation:
With \((1+r)^n = (1.005)^{300} = 4.465\):
Substitute values into the EMI formula:
\[ EMI = 34,65,000 \times 0.005 \times \frac{(1.005)^{300}}{(1.005)^{300} - 1} \]
\[ EMI = 17,325 \times \frac{4.465}{4.465 - 1} \]
\[ EMI = 17,325 \times \frac{4.465}{3.465} \]
\[ EMI \approx 17,325 \times 1.2886002886 \]
\[ EMI \approx 22324.59 \]
Step 4: Conclusion: The calculated EMI, rounded to the nearest rupee, is Rupees 22,325.