Question

A person wishes to purchase a house for Rupess 39,65,000 with a down payment of Rupees 5,00,000 and balance in equal monthly installments (EMI) for 25 years. If bank charges 6% per annum compounded monthly, then EMI on reducing balance payment method is:
[Given \((1.005)^{300} = 4.465\)]

• A. Rupees 22325
• B. Rupees 36542
• C. Rupees 21652
• D. Rupees 34500

Hint:

When an exam question provides a calculated value like \((1.005)^{300} = 4.465\), it's a strong hint that you are on the right track and should use this value directly in your formula. This saves you from performing complex exponentiation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires the calculation of an Equated Monthly Installment (EMI) for a loan. The reducing balance method means that interest is calculated each month on the outstanding principal.
Step 2: Key Formula or Approach:
The formula to calculate EMI is: \[ EMI = P \times r \times \frac{(1+r)^n}{(1+r)^n - 1} \] where: - \(P\) is the principal loan amount. - \(r\) is the monthly interest rate. - \(n\) is the number of monthly installments.
Step 3: Detailed Explanation:
1. Calculate the Principal Loan Amount (P): \[ P = \text{Total House Cost} - \text{Down Payment} \] \[ P = 39,65,000 - 5,00,000 = 34,65,000 \] 2. Calculate the Monthly Interest Rate (r): The annual rate is 6%, compounded monthly. \[ r = \frac{6%}{12} = 0.5% = 0.005 \] 3. Calculate the Number of Installments (n): The loan term is 25 years. \[ n = 25 \text{ years} \times 12 \text{ months/year} = 300 \text{ months} \] 4. Calculate the EMI: We are given that \((1+r)^n = (1.005)^{300} = 4.465\). Now, substitute the 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: Final Answer:
Rounding to the nearest rupee, the EMI is Rupees 22,325.