An equal sum is invested in two different schemes. One scheme gives simple interest and the other gives compound interest (annual compounding). The total interest obtained after 2 years from both the schemes together is Rs 2090. If both the schemes have 18% per annum interest rate, then what is the first year interest (in Rs) of simple interest scheme?

Options

A. 1000
B. 500
C. 1500
D. 1200

Answer

To solve the problem step by step, we will follow the given information about simple interest (SI) and compound interest (CI) schemes.

### Step 1: Define the Variables
Let the principal amount invested in each scheme be \( P \).

### Step 2: Calculate Simple Interest for 2 Years
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R = 18\% \) (rate of interest)
- \( T = 2 \) years

Substituting the values:
\[ SI = \frac{P \times 18 \times 2}{100} = \frac{36P}{100} = 0.36P \]

### Step 3: Calculate Compound Interest for 2 Years
The formula for compound interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values:
\[ CI = P \left(1 + \frac{18}{100}\right)^2 - P = P \left(1.18^2\right) - P \]
Calculating \( 1.18^2 \):
\[ 1.18^2 = 1.3924 \]
Thus,
\[ CI = P \times 1.3924 - P = P(1.3924 - 1) = P \times 0.3924 \]

### Step 4: Total Interest from Both Schemes
The total interest from both schemes after 2 years is given as Rs 2090:
\[ SI + CI = 2090 \]
Substituting the expressions for SI and CI:
\[ 0.36P + 0.3924P = 2090 \]
Combining like terms:
\[ (0.36 + 0.3924)P = 2090 \]
\[ 0.7524P = 2090 \]

### Step 5: Solve for Principal \( P \)
To find \( P \):
\[ P = \frac{2090}{0.7524} \approx 2771.23 \]

### Step 6: Calculate First Year Interest from Simple Interest Scheme
The first year interest from the simple interest scheme can be calculated as:
\[ SI_{1 \text{ year}} = \frac{P \times R \times 1}{100} = \frac{P \times 18}{100} \]
Substituting the value of \( P \):
\[ SI_{1 \text{ year}} = \frac{2771.23 \times 18}{100} \approx 499.00 \]

### Final Answer
The first year interest from the simple interest scheme is approximately Rs 500.

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