A person invested Rs. 20000 in a bank which is offering 10% per annum simple interest. After two years he withdrew the money from the bank and deposited the total amount in another bank which gives an interest rate of r% p.a. compounded annually. After 2 years he received an amount of Rs. 2460 more than what he had invested in that bank.
If the person had invested Rs. 50,000 instead of 20000 in the bank that offered simple interest, what would have been his net profit after following the same procedure as given above?

Options

A. Rs. 16,800
B. Rs. 16,150
C. Rs. 16,350
D. Rs. 16,000

Answer

To solve the problem step-by-step, we will follow the procedure outlined in the video transcript.

### Step 1: Calculate the Simple Interest for Rs. 20,000
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P = 20000 \) (Principal)
- \( R = 10\% \) (Rate of interest)
- \( T = 2 \) years (Time)

Substituting the values:
\[ \text{SI} = \frac{20000 \times 10 \times 2}{100} = \frac{400000}{100} = 4000 \]

### Step 2: Calculate the Total Amount after 2 Years
The total amount (A) after 2 years is:
\[ A = P + \text{SI} = 20000 + 4000 = 24000 \]

### Step 3: Set Up the Equation for Compound Interest
The person then invests Rs. 24,000 in another bank at an interest rate of \( r\% \) compounded annually for 2 years. According to the problem, he receives Rs. 2460 more than what he had invested in the first bank.

Thus, we have:
\[ \text{Compound Interest (CI)} = A' - A = 2460 \]
Where \( A' \) is the amount received from the second bank.

### Step 4: Express the Amount from Compound Interest
The formula for the amount with Compound Interest is:
\[ A' = P \left(1 + \frac{r}{100}\right)^T \]
Substituting the values:
\[ A' = 24000 \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Set Up the Equation
From the previous steps, we know:
\[ A' - 24000 = 2460 \]
This gives:
\[ 24000 \left(1 + \frac{r}{100}\right)^2 - 24000 = 2460 \]
Simplifying:
\[ 24000 \left(1 + \frac{r}{100}\right)^2 = 26460 \]
Dividing both sides by 24000:
\[ \left(1 + \frac{r}{100}\right)^2 = \frac{26460}{24000} = 1.1025 \]

### Step 6: Solve for \( r \)
Taking the square root of both sides:
\[ 1 + \frac{r}{100} = \sqrt{1.1025} = 1.05 \]
Thus:
\[ \frac{r}{100} = 0.05 \implies r = 5\% \]

### Step 7: Calculate the Simple Interest for Rs. 50,000
Now, if the person had invested Rs. 50,000 instead:
\[ \text{SI} = \frac{50000 \times 10 \times 2}{100} = \frac{1000000}{100} = 10000 \]

### Step 8: Calculate the Total Amount after 2 Years for Rs. 50,000
The total amount after 2 years would be:
\[ A = 50000 + 10000 = 60000 \]

### Step 9: Calculate the Compound Interest for Rs. 60,000
Now, we calculate the amount received from the second bank:
\[ A' = 60000 \left(1 + \frac{5}{100}\right)^2 = 60000 \left(1.05\right)^2 = 60000 \times 1.1025 = 66150 \]

### Step 10: Calculate the Total Profit
The total profit is:
\[ \text{Profit} = A' - \text{Initial Investment} = 66150 - 50000 = 16150 \]

### Final Answer
The net profit after following the same procedure with Rs. 50,000 would be:
\[ \text{Net Profit} = 16150 \]