A person invested a sum of Rs 18600 at x% p.a. and another sum that is twice the former at (x + 2)% p.a., both at simple interest. If the total interest earned on both investments for 3% years is Rs 23110.50, then the rate of interest p.a. on the second investment is:

Options

A. 0.11
B. 0.105
C. 0.13
D. 0.125

Answer

To solve the problem step by step, we will use the formula for simple interest, which is given by:

\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]

where:
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum,
- \( T \) is the time in years.

### Step 1: Identify the given values
- Principal for the first investment, \( P_1 = 18600 \) Rs
- Rate for the first investment, \( R_1 = x \% \)
- Time for both investments, \( T = 3.5 \) years (which is \( \frac{7}{2} \) years)
- Principal for the second investment, \( P_2 = 2 \times P_1 = 2 \times 18600 = 37200 \) Rs
- Rate for the second investment, \( R_2 = (x + 2) \% \)
- Total interest earned from both investments = Rs 23110.50

### Step 2: Write the equation for the first investment
Using the simple interest formula for the first investment:
\[ \text{SI}_1 = \frac{P_1 \times R_1 \times T}{100} = \frac{18600 \times x \times \frac{7}{2}}{100} \]

### Step 3: Write the equation for the second investment
Using the simple interest formula for the second investment:
\[ \text{SI}_2 = \frac{P_2 \times R_2 \times T}{100} = \frac{37200 \times (x + 2) \times \frac{7}{2}}{100} \]

### Step 4: Set up the total interest equation
The total interest from both investments is given as:
\[ \text{SI}_1 + \text{SI}_2 = 23110.50 \]
Substituting the expressions from Steps 2 and 3:
\[ \frac{18600 \times x \times \frac{7}{2}}{100} + \frac{37200 \times (x + 2) \times \frac{7}{2}}{100} = 23110.50 \]

### Step 5: Simplify the equation
Multiply through by 100 to eliminate the denominator:
\[ 18600 \times x \times \frac{7}{2} + 37200 \times (x + 2) \times \frac{7}{2} = 2311050 \]
Factoring out \( \frac{7}{2} \):
\[ \frac{7}{2} \left(18600x + 37200(x + 2)\right) = 2311050 \]
Now, distribute the \( 37200 \):
\[ \frac{7}{2} \left(18600x + 37200x + 74400\right) = 2311050 \]
Combine like terms:
\[ \frac{7}{2} \left(55800x + 74400\right) = 2311050 \]

### Step 6: Solve for \( x \)
Multiply both sides by \( \frac{2}{7} \):
\[ 55800x + 74400 = \frac{2 \times 2311050}{7} \]
Calculating the right side:
\[ 55800x + 74400 = 660300 \]
Now, isolate \( x \):
\[ 55800x = 660300 - 74400 \]
\[ 55800x = 585900 \]
\[ x = \frac{585900}{55800} = 10.5 \]

### Step 7: Find the rate of interest for the second investment
The rate of interest for the second investment is:
\[ R_2 = x + 2 = 10.5 + 2 = 12.5\% \]

### Final Answer
The rate of interest per annum on the second investment is **12.5%**.