A person borrows  ₹ 2,00,000. He pays  ₹ 75,000 at the end of 3 years and  ₹ 50,000 at the end of 5 years. What final payment should be made at the end of 6 years to settle the debt if interest rate 6% compounded continuously.


Select one:
 ₹ 1,43,782.75
 ₹ 1,42,287.57
 ₹ 1,42,827.57
 ₹ 1,43,287.75

To solve this problem, we need to calculate the final payment required to settle the debt, considering the continuously compounded interest.

Given:
- Principal amount (P) = ₹ 2,00,000
- Interest rate (r) = 6% per annum (compounded continuously)
- Payments made: ₹ 75,000 at the end of 3 years, ₹ 50,000 at the end of 5 years
- Time for the final payment: 6 years

### Step 1: Calculate the future value of the payments made using continuous compounding.

The formula for the future value with continuous compounding is:

\[
FV = PV \times e^{rt}
\]

Where:
- \( PV \) = Present value (initial amount)
- \( r \) = Interest rate (as a decimal)
- \( t \) = Time in years
- \( e \) = Euler's number (approximately 2.71828)

### Step 2: Calculate the future value of the first payment

For the payment of ₹ 75,000 made at the end of 3 years:

\[
FV_1 = 75,000 \times e^{0.06 \times (6-3)}
\]

This is the value at the end of 6 years.

### Step 3: Calculate the future value of the second payment

For the payment of ₹ 50,000 made at the end of 5 years:

\[
FV_2 = 50,000 \times e^{0.06 \times (6-5)}
\]

### Step 4: Calculate the total debt at the end of 6 years

The original principal grows with continuous compounding over 6 years:

\[
FV_{\text{debt}} = 2,00,000 \times e^{0.06 \times 6}
\]

### Step 5: Subtract the future values of the payments from the total debt

The final payment required will be:

\[
\text{Final Payment} = FV_{\text{debt}} - (FV_1 + FV_2)
\]

Let's calculate this step by step.

### Calculation:

1. \( FV_{\text{debt}} = 2,00,000 \times e^{0.36} \)
2. \( FV_1 = 75,000 \times e^{0.18} \)
3. \( FV_2 = 50,000 \times e^{0.06} \)

Finally,

\[
\text{Final Payment} = FV_{\text{debt}} - (FV_1 + FV_2)
\]

I'll now compute the final payment amount.The final payment required to settle the debt at the end of 6 years is approximately **₹ 1,43,782.75**.

Therefore, the correct answer is:

**₹ 1,43,782.75**

Would you like more details or have any further questions?

Here are some related questions you might find useful:

1. How does continuous compounding differ from regular compounding?
2. What would the final payment be if the interest rate were different?
3. How is the formula for future value derived for continuous compounding?
4. How do different payment schedules affect the final amount to be paid?
5. What would happen if there were more payments made during the loan period?

**Tip:** Continuous compounding can be more costly than other compounding methods as it assumes interest is being added continuously, leading to a higher future value.

Learn how to calculate the final payment required to settle a debt using continuous compounding at an interest rate of 6% per annum. This problem involves determining the future value of payments made at different times and subtracting them from the total debt amount. Suitable for advanced mathematics enthusiasts and finance professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation