What is the rate of interest if simple interest earned on a certain sum for the 3rd year is ₹ 1,750 and compound interest earned for 2 years is ₹ 3622.52

Options

A. 8
B. 9
C. 10
D. 7

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define Variables
Let the principal amount be \( P \) and the rate of interest be \( R \) (as a decimal).

### Step 2: Set Up the Simple Interest Equation
The simple interest earned in the third year is given as ₹ 1,750. The formula for simple interest for the third year is:
\[ \text{SI} = \frac{P \times R \times 1}{100} \]
Thus, we can write:
\[ \frac{P \times R}{100} = 1750 \]
This gives us our first equation:
\[ PR = 1750 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Compound Interest Equation
The compound interest earned over 2 years is given as ₹ 3,622.52. The formula for compound interest for 2 years is:
\[ \text{CI} = P\left(1 + \frac{R}{100}\right)^2 - P \]
This can be simplified to:
\[ \text{CI} = P\left(\left(1 + \frac{R}{100}\right)^2 - 1\right) \]
We can express this as:
\[ P\left(\frac{R}{100}\right)\left(2 + \frac{R}{100}\right) = 3622.52 \]
This gives us our second equation:
\[ P \times R \times \left(2 + \frac{R}{100}\right) = 3622.52 \quad \text{(Equation 2)} \]

### Step 4: Divide Equation 2 by Equation 1
Now, we will divide Equation 2 by Equation 1 to eliminate \( P \):
\[ \frac{P \times R \times \left(2 + \frac{R}{100}\right)}{PR} = \frac{3622.52}{1750} \]
This simplifies to:
\[ 2 + \frac{R}{100} = \frac{3622.52}{1750} \]

### Step 5: Calculate the Right Side
Calculating the right side:
\[ \frac{3622.52}{1750} \approx 2.07 \]
So we have:
\[ 2 + \frac{R}{100} = 2.07 \]

### Step 6: Solve for \( R \)
Subtract 2 from both sides:
\[ \frac{R}{100} = 0.07 \]
Multiply both sides by 100:
\[ R = 7 \]

### Conclusion
The rate of interest \( R \) is 7%.