On a certain principal if the simple interest for two years is `Rs. 1400` and compound interest for the two years is `Rs.1449`, what Is the rate of interest?

Options

A. `7 per cent
B. 3.5 per cent
C. 14 per cent
D. 10.5 per cent

Answer

To solve the problem step by step, we will use the information given about Simple Interest (SI) and Compound Interest (CI) to find the rate of interest (R).

### Step 1: Understand the Given Information
- Simple Interest (SI) for 2 years = Rs. 1400
- Compound Interest (CI) for 2 years = Rs. 1449

### Step 2: Set Up the Equations
- Let the Principal amount be \( P \) and the Rate of Interest be \( R \% \).
- The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( T \) is the time in years. Here, \( T = 2 \) years.

### Step 3: Calculate the Principal using Simple Interest
From the SI formula, we can rearrange it:
\[ 1400 = \frac{P \times R \times 2}{100} \]
Multiplying both sides by 100:
\[ 140000 = P \times R \times 2 \]
Dividing by 2:
\[ P \times R = 70000 \quad \text{(Equation 1)} \]

### Step 4: Calculate the Difference between CI and SI
The difference between Compound Interest and Simple Interest for 2 years is given by:
\[ CI - SI = P \times \frac{R^2}{100^2} \times T \]
Substituting the values:
\[ 1449 - 1400 = P \times \frac{R^2}{100^2} \times 2 \]
This simplifies to:
\[ 49 = P \times \frac{R^2}{10000} \]
Multiplying both sides by 10000:
\[ 490000 = P \times R^2 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
From Equation 1, we have \( P = \frac{70000}{R} \). Substitute this into Equation 2:
\[ 490000 = \left(\frac{70000}{R}\right) \times R^2 \]
This simplifies to:
\[ 490000 = 70000R \]
Dividing both sides by 70000:
\[ R = \frac{490000}{70000} = 7 \]

### Step 6: Conclusion
The rate of interest \( R \) is \( 7\% \).