A person invested a sum of Rs. 6,500 x% per annum at simple interest and a sum of Rs. 7,500 at (x-2)% per annum at simple interest. If the total interest earned on both the investment for 3 years is Rs. 3,750, then the rate of interest on the first investment is

Options

A. 0.14
B. 0.1
C. 0.08
D. 0.12

Answer

To solve the problem, we need to find the rate of interest on the first investment. Let's break it down step by step.

### Step 1: Understand the given information
We have two investments:
1. Rs. 6,500 at x% per annum
2. Rs. 7,500 at (x - 2)% per annum

The total interest earned from both investments over 3 years is Rs. 3,750.

### Step 2: Write the formula for Simple Interest
The formula for simple interest is:
\[ \text{Simple Interest} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \]

### Step 3: Calculate the interest for both investments
For the first investment (Rs. 6,500 at x%):
\[ \text{Interest}_1 = \frac{6500 \times x \times 3}{100} = \frac{19500x}{100} = 195x \]

For the second investment (Rs. 7,500 at (x - 2)%):
\[ \text{Interest}_2 = \frac{7500 \times (x - 2) \times 3}{100} = \frac{22500(x - 2)}{100} = 225(x - 2) = 225x - 450 \]

### Step 4: Set up the equation for total interest
The total interest from both investments is given as Rs. 3,750:
\[ 195x + (225x - 450) = 3750 \]

### Step 5: Simplify the equation
Combine like terms:
\[ 195x + 225x - 450 = 3750 \]
\[ 420x - 450 = 3750 \]

### Step 6: Solve for x
Add 450 to both sides:
\[ 420x = 3750 + 450 \]
\[ 420x = 4200 \]
Now, divide both sides by 420:
\[ x = \frac{4200}{420} = 10 \]

### Step 7: Conclusion
The rate of interest on the first investment is:
\[ \text{Rate} = x = 10\% \]