The sums of ₹ 1,00,000,  ₹ 2,00,000 and  ₹. 4,00,000 are due at the end of 1, 3 and 5 years respectively. It is proposed to replace this series of payments by a single sum  ₹ 5,72,500 payable at the end of n years. If the rate of interest is 12% compounded semiannually, find the value of n.


Select one:
2.15
2.1
2
2.2

To solve this problem, we'll use the concept of the present value of money, considering that the interest rate is compounded semiannually. The present value of the original series of payments should equal the present value of the proposed single sum payment.

### Step 1: Calculate the Present Value of the Original Payments
Given that the interest rate is 12% compounded semiannually, the effective rate per period (6 months) is:

\[
\text{Effective interest rate per period} = \frac{12\%}{2} = 6\% = 0.06
\]

We now calculate the present value of each payment at time \( t = 0 \).

1. **₹ 1,00,000 due at the end of 1 year (2 periods):**
   \[
   PV_1 = \frac{1,00,000}{(1.06)^2} = \frac{1,00,000}{1.1236} = ₹ 89,015.15
   \]

2. **₹ 2,00,000 due at the end of 3 years (6 periods):**
   \[
   PV_2 = \frac{2,00,000}{(1.06)^6} = \frac{2,00,000}{1.4185} = ₹ 1,41,019.87
   \]

3. **₹ 4,00,000 due at the end of 5 years (10 periods):**
   \[
   PV_3 = \frac{4,00,000}{(1.06)^{10}} = \frac{4,00,000}{1.7908} = ₹ 2,23,442.97
   \]

### Step 2: Calculate the Total Present Value of the Original Payments
\[
PV_{\text{original}} = PV_1 + PV_2 + PV_3 = ₹ 89,015.15 + ₹ 1,41,019.87 + ₹ 2,23,442.97 = ₹ 4,53,477.99
\]

### Step 3: Calculate the Present Value of the Proposed Payment
The proposed payment is ₹ 5,72,500 at the end of \( n \) years (or \( 2n \) periods since the interest is compounded semiannually). We equate the present value of this payment to the total present value of the original payments:

\[
\text{Present Value of Proposed Payment} = \frac{5,72,500}{(1.06)^{2n}}
\]

We set this equal to the total present value:

\[
\frac{5,72,500}{(1.06)^{2n}} = 4,53,477.99
\]

### Step 4: Solve for \( n \)

First, solve for \( (1.06)^{2n} \):

\[
(1.06)^{2n} = \frac{5,72,500}{4,53,477.99} \approx 1.262
\]

Now, take the natural logarithm of both sides:

\[
2n \times \ln(1.06) = \ln(1.262)
\]

\[
2n = \frac{\ln(1.262)}{\ln(1.06)} \approx \frac{0.2338}{0.05827} \approx 4.01
\]

\[
n = \frac{4.01}{2} = 2.005
\]

### Final Answer
The value of \( n \) is closest to 2, matching one of the provided options.

**Correct answer: 2**

### Would you like more details or have any questions?

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Here are 5 related questions to explore:

1. How does the present value change if the interest rate is compounded quarterly instead of semiannually?
2. What if the payments were spread over a different time horizon, how would that affect the value of \( n \)?
3. How does the effective interest rate affect the computation of \( n \)?
4. How would the solution change if the single sum payment were proposed at a different amount?
5. What is the impact of changing the frequency of compounding on the value of \( n \)?

**Tip:** Always ensure to align the periods in interest rate calculations, especially when dealing with different compounding frequencies.

Learn how to calculate present value using compound interest and solve for n in a series of payments versus a single sum scenario. This problem involves understanding the time value of money and compound interest at a 12% semiannual rate.

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