If the compound interest on a certain sum for 2 years is Rs. 60.60 and the simple interest is Rs. 60, then the rate of interest per annum is

Answer

To solve the problem step by step, we will follow the information provided in the question regarding compound interest (CI) and simple interest (SI).

### Step 1: Understand the given information
- Compound Interest for 2 years (CI) = Rs. 60.60
- Simple Interest for 2 years (SI) = Rs. 60
- Let the principal amount be \( P \).
- Let the rate of interest per annum be \( R \).
- Time \( T = 2 \) years.

### Step 2: Write the formula for Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 60 = \frac{P \times R \times 2}{100} \]
This simplifies to:
\[ P \times R = 30 \quad \text{(Equation 1)} \]

### Step 3: Write the formula for Compound Interest
The formula for Compound Interest after 2 years is:
\[ CI = P \left(1 + \frac{R}{100}\right)^2 - P \]
This can be rewritten as:
\[ CI = P \left[\left(1 + \frac{R}{100}\right)^2 - 1\right] \]
Substituting the value of CI:
\[ 60.60 = P \left[\left(1 + \frac{R}{100}\right)^2 - 1\right] \]

### Step 4: Expand the Compound Interest formula
Using the binomial expansion:
\[ \left(1 + \frac{R}{100}\right)^2 = 1 + 2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Thus,
\[ \left(1 + \frac{R}{100}\right)^2 - 1 = 2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Substituting this back into the CI equation:
\[ 60.60 = P \left[2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2\right] \]

### Step 5: Substitute \( P \) from Equation 1
From Equation 1, we have \( P = \frac{30 \times 100}{R} = \frac{3000}{R} \). Substitute this into the CI equation:
\[ 60.60 = \frac{3000}{R} \left[2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2\right] \]
This simplifies to:
\[ 60.60 = \frac{3000}{R} \left[\frac{2R}{100} + \frac{R^2}{10000}\right] \]

### Step 6: Simplify the equation
Multiply both sides by \( R \):
\[ 60.60R = 3000 \left[\frac{2R}{100} + \frac{R^2}{10000}\right] \]
This simplifies to:
\[ 60.60R = 60R + \frac{3000R^2}{10000} \]
\[ 60.60R = 60R + 0.3R^2 \]
Rearranging gives:
\[ 0.3R^2 - 0.60R = 0 \]

### Step 7: Factor the equation
Factoring out \( R \):
\[ R(0.3R - 0.60) = 0 \]
This gives us:
\[ R = 0 \quad \text{or} \quad 0.3R = 0.60 \]
Solving for \( R \):
\[ R = \frac{0.60}{0.3} = 2 \]

### Step 8: Conclusion
Thus, the rate of interest per annum is:
\[ \text{Rate of Interest} = 2\% \]