The difference between the compound interest (compounded annually) and simple interest, for two years on the same sum at the amount rate of interest is Rs 370. Find the rate of interest if the simple interest on the amount at the same rate of interest for 1 year is Rs 3700.

Options

A. 0.1
B. 0.12
C. 0.16
D. 0.15

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the given information
We know that:
- The difference between Compound Interest (CI) and Simple Interest (SI) for 2 years is Rs 370.
- The Simple Interest for 1 year is Rs 3700.

### Step 2: Set up the formulas
1. **Simple Interest (SI)** for 1 year is given by the formula:
\[ SI = \frac{P \cdot R \cdot T}{100} \]
where \(P\) is the principal amount, \(R\) is the rate of interest, and \(T\) is the time in years.

2. **Compound Interest (CI)** for 2 years is given by:
\[ CI = P \left(1 + \frac{R}{100}\right)^2 - P \]
This can be simplified to:
\[ CI = P \left(\left(1 + \frac{R}{100}\right)^2 - 1\right) \]

### Step 3: Calculate Simple Interest for 1 year
From the problem, we know:
\[ SI = 3700 \]
Using the formula for SI for 1 year:
\[ 3700 = \frac{P \cdot R \cdot 1}{100} \]
This simplifies to:
\[ P \cdot R = 3700 \cdot 100 = 370000 \]

### Step 4: Calculate Compound Interest for 2 years
Now, we need to express the CI in terms of \(P\) and \(R\):
\[ CI = P \left(\left(1 + \frac{R}{100}\right)^2 - 1\right) \]
Expanding the square:
\[ CI = P \left(\frac{R^2}{10000} + \frac{2R}{100}\right) \]

### Step 5: Find the difference between CI and SI
The difference between CI and SI for 2 years is given as Rs 370:
\[ CI - SI = 370 \]
Substituting the values:
\[ P \left(\frac{R^2}{10000} + \frac{2R}{100}\right) - \frac{2P \cdot R}{100} = 370 \]
This simplifies to:
\[ P \cdot \frac{R^2}{10000} = 370 \]

### Step 6: Substitute \(P\) from Step 3
From Step 3, we know \(P \cdot R = 370000\), so:
\[ P = \frac{370000}{R} \]
Substituting this into the equation:
\[ \frac{370000}{R} \cdot \frac{R^2}{10000} = 370 \]
This simplifies to:
\[ \frac{370000R}{10000} = 370 \]
\[ 37R = 370 \]
\[ R = 10 \]

### Conclusion
The rate of interest \(R\) is **10% per annum**.