Anil invests Rs 22000 for 6 years in a scheme with 4% interest per annum, compounded half-yearly. Separately, Sunil invests a certain amount in the same scheme for 5 years, and then reinvests the entire amount he receives at the end of 5 years, for one year at 10% simple interest. If the amounts received by both at the end of 6 years are equal, then the initial investment, in rupees, made by Sunil is

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Accepted Answer

Let the amount invested by Sunil initially be $$ P $$.

#### Step 1: Calculate Anil's final amount  
Anil invests Rs. 22,000 for 6 years at 4% per annum, compounded half-yearly. The formula for compound interest is:  
$$
A = P \left(1 + \frac{r}{100n}ight)^{n \cdot t}
$$  
Here:  
- $$ P = 22000 $$ (principal amount)  
- $$ r = 4\% $$ (annual interest rate)  
- $$ n = 2 $$ (compounding frequency per year)  
- $$ t = 6 $$ (time in years)

Substituting the values:  
$$
A = 22000 \left(1 + \frac{4}{2 \cdot 100}ight)^{2 \cdot 6} = 22000 \left(1 + 0.02ight)^{12} = 22000 (1.02)^{12}
$$

#### Step 2: Calculate Sunil's final amount  
Sunil invests $$ P $$ for 5 years in the same scheme. At the end of 5 years, the amount he receives is:  
$$
A_5 = P (1.02)^{10}
$$

In the 6th year, Sunil reinvests this amount at a simple interest rate of 10%. The interest earned in one year is:  
$$
	ext{Interest} = A_5 \cdot 0.1 = P (1.02)^{10} \cdot 0.1
$$

The total amount Sunil has at the end of the 6th year is:  
$$
A_{	ext{Sunil}} = A_5 + 	ext{Interest} = P (1.02)^{10} + P (1.02)^{10} \cdot 0.1 = P (1.02)^{10} (1 + 0.1)
$$  
$$
A_{	ext{Sunil}} = P (1.02)^{10} \cdot 1.1
$$

#### Step 3: Equate Anil's and Sunil's final amounts  
Since both amounts are equal at the end of 6 years:  
$$
22000 (1.02)^{12} = P (1.02)^{10} \cdot 1.1
$$

Simplify to solve for $$ P $$:  
$$
P = \frac{22000 (1.02)^2}{1.1}
$$

#### Step 4: Calculate $$ P $$:  
First, calculate $$ (1.02)^2 $$:  
$$
(1.02)^2 = 1.0404
$$

Substitute into the equation:  
$$
P = \frac{22000 \cdot 1.0404}{1.1} = \frac{22888.8}{1.1} = 20808
$$

Thus, Sunil's initial investment is **Rs. 20,808**.

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