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

Options

A. `8%`
B. `12%`
C. `14%`
D. `10%`

Answer

To solve the problem, we need to calculate the interest earned from both investments and set up an equation based on the information provided. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Simple Interest Formula
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Calculate Interest for the First Investment
The first investment is Rs. 6,500 at \( x\% \) for 3 years. Thus, the interest earned from this investment is:
\[ SI_1 = \frac{6500 \times x \times 3}{100} = \frac{19500x}{100} = 195x \]

### Step 3: Calculate Interest for the Second Investment
The second investment is Rs. 7,500 at \( (x - 2)\% \) for 3 years. The interest earned from this investment is:
\[ SI_2 = \frac{7500 \times (x - 2) \times 3}{100} = \frac{22500(x - 2)}{100} = 225(x - 2) \]

### Step 4: Set Up the Total Interest Equation
According to the problem, the total interest earned from both investments over 3 years is Rs. 3,750. Therefore, we can write the equation:
\[ SI_1 + SI_2 = 3750 \]
Substituting the values we calculated:
\[ 195x + 225(x - 2) = 3750 \]

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

### 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: Find the Rate of Interest on the Second Investment
The rate of interest on the second investment is \( (x - 2)\% \):
\[ x - 2 = 10 - 2 = 8 \]

### Final Answer
The rate of interest on the second investment is **8%**.

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