Example 33: Y bought a TV costing 13,000 by making a down payment of ₹3000 and agreeing to make equal annual payment for four years. How much would be each payment if the interest on unpaid amount be 14% compounded annually 

To solve this problem, we can use the formula for calculating the equated annual installment (EAI) when the interest is compounded annually. This involves calculating the present value of an annuity.

### Steps to Solve:

1. **Determine the Loan Amount:**
   - The TV costs ₹13,000.
   - The down payment is ₹3,000.
   - Therefore, the loan amount (principal) is:
     \[
     P = 13,000 - 3,000 = 10,000
     \]

2. **Identify the Variables:**
   - \( P = ₹10,000 \) (Loan amount)
   - \( r = 14\% = 0.14 \) (Annual interest rate)
   - \( n = 4 \) years (Number of payments)

3. **Use the Present Value of Annuity Formula:**

   The formula to find the equal annual payment \( A \) is:
   \[
   P = A \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right]
   \]
   Rearrange to solve for \( A \):
   \[
   A = \frac{P \times r}{1 - (1 + r)^{-n}}
   \]

4. **Substitute the Values:**
   \[
   A = \frac{10,000 \times 0.14}{1 - (1 + 0.14)^{-4}}
   \]
   First, calculate \( (1 + 0.14)^{-4} \):
   \[
   (1.14)^{-4} \approx 0.59208
   \]
   Then, calculate \( 1 - 0.59208 = 0.40792 \).

   Now, substitute back into the equation:
   \[
   A = \frac{10,000 \times 0.14}{0.40792} \approx \frac{1,400}{0.40792} \approx 3,430.61
   \]

So, the equal annual payment \( A \) is approximately **₹3,430.61**.

Would you like further details or have any questions?

Here are 5 related questions you might consider:
1. How would the annual payment change if the interest rate were 10% instead of 14%?
2. What if the loan were to be paid off over 5 years instead of 4 years?
3. How does the concept of amortization work in such loan repayment scenarios?
4. What would be the total interest paid over the four years?
5. How would you calculate the remaining loan balance after two years of payments?

**Tip:** Understanding how to calculate the equated annual installment helps in planning finances, especially for large purchases or loans.

Learn how to calculate the equal annual payment for buying a TV with a detailed step-by-step solution. This problem involves finance concepts such as annuities and compound interest, specifically addressing a purchase scenario with a down payment and annual payments over four years.

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